C O N FO R M A L M E T H O D S I N G E N E R A L R E L AT I V I T YThis book is a systematic exposition of conformal methods and how they canbe used to study the global properties of solutions to the equations of Einstein’s
theory of gravity. It shows that combining these ideas with techniques of thetheory of partial diﬀerential equations can elucidate the stability of the basic
solutions of the theory. Introducing the diﬀerential geometric, spinorial and PDE
background required to gain a deep understanding of conformal methods, thistext provides an accessible account of key results in mathematical relativity overthe last 30 years, including the stability of de Sitter and Minkowski spacetimes.
For graduate students and researchers, this self-contained account includes
useful visual models to help the reader grasp abstract concepts and a list offurther reading, making this the perfect reference companion on the topic.
This title, first published in 2017, has been reissued as an Open Accesspublication on Cambridge Core.
J u a n A . Va l i e n t e K ro o n is a Reader in Applied Mathematics at Queen
Mary University of London. He was a Lise Meitner fellow of the Austrian
Science Fund (FWF), an Engineering and Physical Sciences (EPSRC) Advanced
Research fellow and he specialises in various aspects of mathematical generalrelativity.

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Conformal Methods in General
Relativity

J U A N A . VA L I E N T E K RO O N
Queen Mary University of London

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(Let no one untrained in geometry enter)– Epigram at the Academy of Plato

Contents

PrefaceAcknowledgements
List of Symbols
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7

Introduction
On the Einstein ﬁeld equations
Exact solutions
The Cauchy problem in general relativity
Conformal geometry and general relativity
Existence of asymptotically simple spacetimes
Perspectives
Structure of this bookPart I
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Geometric tools

page xv
xviii
xx
1
1
3
4
8
15
21
21
25

Diﬀerential geometry
Manifolds
Vectors and tensors on a manifold
Maps between manifolds
Connections, torsion and curvature
Metric tensors
Frame formalisms
Congruences and submanifolds
Further reading

27
27
30
36
38
44
51
53
63

3
Spacetime spinors
3.1 Algebra of 2-spinors
3.2 Calculus of spacetime spinors
3.3 Global considerations
3.4 Further reading
Appendix: the Newman-Penrose formalism

64
65
81
90
91
91

4
4.1
4.2
4.3
4.4

94
94
97
105
111

Space spinors
Hermitian inner products and 2-spinors
The space spinor formalism
Calculus of space spinors
Further reading

x
5
5.1
5.2
5.3
5.4
5.5
5.6

Contents
Conformal geometry
Basic concepts of conformal geometry
Conformal transformation formulae
Weyl connections
Spinorial expressions
Conformal geodesics
Further reading

Part II

General relativity and conformal geometry

112
112
114
119
122
126
137
139

6
6.1
6.2
6.3
6.4
6.5
6.6

Conformal extensions of exact solutions
Preliminaries
The Minkowski spacetime
The de Sitter spacetime
The anti-de Sitter spacetime
Conformal extensions of static and stationary black hole spacetimes
Further reading

141
141
145
155
159
163
177

7
7.1
7.2
7.3
7.4

Asymptotic simplicity
Basic deﬁnitions
Other related deﬁnitions
Penrose’s proposal
Further reading

178
178
181
182
183

8
8.1
8.2
8.3
8.4
8.5

The conformal Einstein ﬁeld equations
A singular equation for the conformal metric
The metric regular conformal ﬁeld equations
Frame and spinorial formulation of the conformal ﬁeld equations
The extended conformal Einstein ﬁeld equations
Further reading

184
184
185
194
200
209

9
9.1
9.2
9.3
9.4
9.5

Matter models
General properties of the conformal treatment of matter models
The Maxwell ﬁeld
The scalar ﬁeld
Perfect ﬂuids
Further reading

211
211
213
216
219
221

10
Asymptotics
10.1 Basic set up: general structure of the conformal boundary
10.2 Peeling properties
10.3 The Newman-Penrose gauge
10.4 Other aspects of asymptotics
10.5 Further reading
Appendix: spin-weighted functions

222
223
226
231
238
241
241

Contents

xi

Part III Methods of the theory of partial diﬀerential
equations245

11
The conformal constraint equations
11.1 General setting and basic formulae
11.2 Basic notions of elliptic equations
11.3 The Hamiltonian and momentum constraints
11.4 The conformal constraint equations
11.5 The constraints on compact manifolds
11.6 Asymptotically Euclidean manifolds
11.7 Hyperboloidal manifolds
11.8 Other methods for solving the constraint equations
11.9 Further reading
Appendix: some results of analysis

247
247
252
253
259
268
270
284
286
291
291

12
Methods of the theory of hyperbolic diﬀerential equations
12.1 Basic notions
12.2 Uniqueness and domains of dependence
12.3 Local existence results for symmetric hyperbolic systems
12.4 Local existence for boundary value problems
12.5 Local existence for characteristic initial value problems
12.6 Concluding remarks
12.7 Further reading
Appendix

294
294
301
306
313
319
328
329
329

13
Hyperbolic reductions
13.1 A model problem: the Maxwell equations on a ﬁxed
background13.2 Hyperbolic reductions using gauge source functions
13.3 The subsidiary equations for the standard conformal
ﬁeld equations
13.4 Hyperbolic reductions using conformal Gaussian systems
13.5 Other hyperbolic reduction strategies
13.6 Further reading
Appendix A.1: the reduced Einstein ﬁeld equations
Appendix A.2: diﬀerential forms

331

354
366
383
385
386
389

14
14.1
14.2
14.3
14.4
14.5
14.6

390
390
395
398
401
402
404

Causality and the Cauchy problem in general relativity
Basic elements of Lorentzian causality
PDE causality versus Lorentzian causality
Cauchy developments and maximal Cauchy developments
Stability of solutions
Causality and conformal geometry
Further reading

332
336

xii

Contents

Part IV

Applications

405

15
De Sitter-like spacetimes
15.1 The de Sitter spacetime as a solution to the conformal
ﬁeld equations
15.2 Perturbations of initial data for the de Sitter spacetime
15.3 Global existence and stability using gauge source functions
15.4 Global existence and stability using conformal Gaussian systems
15.5 Extensions
15.6 Further reading

408
416
420
426
434
436

16
16.1
16.2
16.3
16.4
16.5

437
438
442
444
450
452

Minkowski-like spacetimes
The Minkowski spacetime and the conformal ﬁeld equations
Perturbations of hyperboloidal data for the Minkowski spacetime
A priori structure of the conformal boundary
The proof of the main existence and stability result
Extensions and further reading

17
17.1
17.2
17.3
17.4

407

Anti-de Sitter-like spacetimes
General properties of anti-de Sitter-like spacetimes
The formulation of an initial boundary value problem
Covariant formulation of the boundary conditions
Other approaches to the construction of anti-deSitter-like spacetimes
17.5 Further reading

454
455
460
468

18
Characteristic problems for the conformal ﬁeld equations
18.1 Geometric and gauge aspects of the standard characteristic
initial value problem
18.2 The conformal evolution equations in the standard characteristic
initial value problem
18.3 A local existence result for characteristic problems
18.4 The asymptotic characteristic problem on a cone
18.5 Further reading

477

485
491
496
502

19
Static solutions
19.1 The static ﬁeld equations
19.2 Analyticity at inﬁnity
19.3 A regularity condition
19.4 Multipole moments
19.5 Uniqueness of the conformal structure of static metrics
19.6 Characterisation of static initial data
19.7 Further reading
Appendix 1: Hölder conditions
Appendix 2: the Cauchy-Kowalewskaya theorem

504
504
512
517
518
522
524
525
526
526

475
475

478

Contents

xiii

20
Spatial inﬁnity
20.1 Cauchy data for the conformal ﬁeld equations near spatial inﬁnity
20.2 Massless and purely radiative spacetimes
20.3 A regular initial value problem at spatial inﬁnity
20.4 Spatial inﬁnity and peeling20.5 Existence of asymptotically simple spacetimes
20.6 Obstructions to the smoothness of null inﬁnity
20.7 Further reading
Appendix: properties of functions on the complex null cone

527
527
531
538
554
555
556
557
558

21
21.1
21.2
21.3
21.4
21.5

560
560
562
564
567
568

Perspectives
Stability of cosmological models
Stability of black hole spacetimes
Conformal methods and numerics
Computer algebra
Concluding remarks

References
Index

569
587

PrefaceThis book discusses an approach to the study of global properties of solutions
to the equations of general relativity, the Einstein ﬁeld equations, in which thenotion of conformal transformation plays a central role. The use of conformaltransformations in diﬀerential geometry dates back, at least, to the work ofHermann Weyl in the 1920s.1 Their application to global questions in generalrelativity, as presented in this book, stems from the seminal work of Roger
Penrose in the 1960s in which the close connection between the global causal
structure of the solutions to the equations of general relativity and conformal
geometry was established.2 Penrose’s key insights are that the close relation
between the propagation of the gravitational ﬁeld and the structure of light
cones which holds locally in a spacetime is also preserved in the case of large
scales and that the asymptotic behaviour of the gravitational ﬁeld can beconveniently analysed in terms of conformal extensions of the spacetime. In
the following decade Penrose’s ideas were polished, extended and absorbed
into the mainstream research of general relativity by a considerable number of
researchers3 – ﬁnally leading to the inﬂuential notion of asymptotic simplicity.
The subject reached its maturity when this formal theory was combinedwith the methods of the theory of partial diﬀerential equations (PDEs). This
breakthrough is mainly due to the work of Helmut Friedrich in the early 1980s,
who – through the conformal Einstein ﬁeld equations 4 – showed that ideas ofconformal geometry can be used to establish the existence of large classes ofsolutions to the Einstein ﬁeld equations satisfying Penrose’s notion of asymptotic
simplicity. As a result of this work it is now clear that Penrose’s original insights
hold for large classes of spacetimes and not only for special explicitly known
solutions.
This book develops the theory of the conformal Einstein ﬁeld equations fromthe ground up and discusses their applications to the study of asymptoticallysimple spacetimes. Special attention is paid to results concerning the existence
and stability of de Sitter-like spacetimes, the semiglobal existence and stabilityof Minkowski-like spacetimes using hyperboloidal Cauchy problems and the
1 See Weyl (1968).
2 See Penrose (1963, 1964).
3 See e.g. Hawking and Ellis (1973); Geroch (1976).
4 See Friedrich (1981a,b, 1983).

xvi

Prefaceconstruction of anti-de Sitter-like spacetimes from initial boundary value problems. These results belong to the canon of modern mathematical relativity. In
addition to their mathematical interest, they are of great physical relevance as
they express, among other things, the internal consistency of general relativity
and provide an approach for the global evaluation of spacetimes by means of
numerical methods.
Why a book on the subject? The applications of conformal methods in generalrelativity constitute a mature subject with a number of core results which willwithstand the pass of time. Still, it provides a number of challenging open
questions whose resolution will strengthen its connections with other researchstrands in general relativity. This book aims at making the subject accessible tophysicists and mathematicians alike who want to make use of conformal methods
to analyse the global structure and properties of spacetimes. Hopefully, this bookwill provide an alternative to the use of original references while learning the
subject or doing research.
Anyone who wants to engage with the subject of this book faces a numberof challenges. To begin with, one has a vast literature spreading over morethan 50 years. As it is to be expected from a living subject, the perspectives
change through time, the importance of certain problems rise and wane andit is sometimes hard to diﬀerentiate the fundamental from the subsidiary. The
combination of results from various references is often hindered by changing
notation and conﬂicting conventions. Moreover, to appreciate and understand theresults of the theory one requires a considerable amount of background material:conformal geometry, spinors, PDE theory, causal theory, etc. These methods arean essential part of the toolkit of a modern mathematical relativist. This bookendeavours to bring together in a single volume all the required backgroundmaterial in a concise and coherent manner.
As a cautionary note, it should be mentioned what this book is not intended
to be. This book is not an introductory book to general relativity. A certain
familiarity with the subject is assumed from the outset – ideally at the level ofPart I of R. Wald’s book General Relativity.5 This is also not a book on theapplications of the theory of PDEs in general relativity. For this, there are otherbooks available.6 Also, although the Cauchy problem in general relativity is aleading theme, this book should not be viewed as a monograph on the topic –for this, I refer the interested reader to H. Ringström’s monograph.7
I have endeavoured to write a book which not only serves as an introduction to
the subject but also is a tool for research. With this idea in mind, I have striven toprovide as much detail as possible of the arguments and calculations. However,at some stages supplying further details is neither possible nor desirable. Indeed,
quoting the preface of J. L. Synge’s classical book on general relativity: “There
5 See Wald (1984).
6 See e.g. Choquet-Bruhat (2008); Rendall (2008).
7 See Ringström (2009).

Prefacexvii

are heavy calculations in the book, but there are places where the reader willﬁnd me sitting on the fence, whistling, instead of rushing into the fray”; see
Synge (1960).8 In an attempt to keep the readability and the length of the text
under control, I have not endeavoured to provide completely general or optimal
theorems – the attentive reader will realise this and is referred to the literaturefor further details, if required. As a picture is better than a thousand words, I
have complemented the text with a considerable number of ﬁgures and diagrams
which, I hope, will help to explain the content of the main text and provide
useful visual models.
In writing this book, I have assumed the reader to have a certain mathematicalmaturity. Some basic knowledge of topology is needed – Appendix A in Wald’sbook contains the required background – as well as familiarity with basic tensorial
calculus. I have, however, not assumed any prior knowledge of 2-spinors. The
necessary toolkit is developed in the course of two chapters. Readers looking
for a supplementary source on the topic are referred to J. Stewart’s book.9
The applications of conformal methods discussed in this book require certain
knowledge of the theory of PDEs. I provide all the required material in a chapter
of its own – nevertheless, some previous exposure to the basic ideas of thetheory of PDEs is an advantage. Some arguments in the book make use of very
concrete results of analysis. In these cases, I have included the necessary ideas
in appendices to the various chapters.

8 I am thankful to R. Beig for bringing my attention to this quote.
9 See Stewart (1991).

Acknowledgements

First and foremost, I would like to acknowledge the inﬂuence that Helmut
Friedrich had in the development of my understanding of the theory of
the conformal Einstein ﬁeld equations. He has been an inexhaustible sourceof knowledge and insights. I also would like to acknowledge the role that
Robert Beig, Sergio Dain and Malcolm MacCallum have played in shaping
complementary perspectives on the subject. I am also grateful for the extensive
discussions with my collaborators Christian Lübbe and Jose Luis Jaramillo whichhave allowed to form, deﬁne and reﬁne some of the points of view and concernspresented in this book.
I want to thank my students (Diego Carranza, Michael Cole, Edgar Gasperı́n,
Adem Hursit and Jarrod Williams), collaborators and colleagues (Artur Alho,
Robert Beig, Alberto Carrasco, Daniela Pugliese, Alfonso Garcı́a-Parrado, David
Hilditch, Jose Luis Jaramillo, Malcolm MacCallum, Belgin Seymenoglu andClaire Sommé) who have carefully read various parts of this book, thus providing
valuable feedback and a seemingly never-ending list of typos. Of course, I am
fully responsible for any mistakes or omissions left in the text. Finally, I would
also like to thank Jean-Phillipe Nicolas for useful conversations on some of the
topics covered in this book.
Although all the calculations presented in this book have been carried out byhand, the system xAct for Mathematica has been a valuable resource in caseof trouble. I am thankful to Thomas Bäckdahl and Alfonso Garcı́a-Parrado for
their help at various stages with this system. I also thank Peter Hübner and Anil
Zenginoglu for allowing me to use ﬁgures and diagrams from their publications.
I would like to thank my home department, the School of Mathematical
Sciences of Queen Mary University of London, for the support granted during
the course of the project and for the sabbatical term in spring 2014 when aconsiderable portion of this text took form. The primal notes which eventuallyturned into this book stem from a minicourse I held at the Centre of Mathematicsof the University of Minho in Portugal in April 2011; further notes come from aminicourse held at the Institute of Mathematics and System Sciences of theChinese Academy of Sciences in June 2013. I thank these institutions andparticularly my hosts (Filipe Mena and Xiaoning Wu, respectively) for theinvitation and hospitality. I also thank the Gravitational Physics Group of theAcknowledgements

xix

Faculty of Physics of the University of Vienna for their hospitality at various
times during the last years.
Finally, I would like to express my deepest gratitude to my wife, Christiane
Maria Losert-Valiente Kroon, for her constant support during the course of thisproject and for a careful and critical reading of the whole manuscript and for
spotting an incredible amount of typos and mismatching indices in the equations.
The conclusion of this book would have taken much longer without her help!
Finally I would like to thank my father, Antonio Valiente, for instilling in me
the desire of writing a book and my in-laws, Linde and Peter Losert, for theinterest they have taken in this project.
Juan A. Valiente Kroon

Symbolsd∗ij alternative description of the components of the magnetic part of therescaled Weyl tensor, page 264(ea , Γa b c , Ξ, s, Lab , da bcd , Tab ) unknowns in the frame version of the standard
conformal ﬁeld equations, page 196(ea , Γ̂a b c , L̂ab , da bcd , Tab , Ξ, da ) unknowns in the frame version of theextended conformal ﬁeld equations, page 205(eAA , ΓAA BC , Ξ, s, LAA BB  , φABCD , TAA BB  ) unknowns in the spinorial
version of the standard conformal ﬁeld equations, page 199(eAA , Γ̂AA BC , L̂AA BB  , φABCD , TAA BB  , Ξ, dAA ) unknowns in the
spinorial version of the extended conformal ﬁeld equations, page 208(o, ι) spin basis in index-free notation, page 66(Σ̂AA BB  , Ξ̂C DAA BB  , Δ̂CC  DD BB  , Λ̂BB  CD , δAA , ςAA BB  , γAA BB  )zero quantities in the spinorial extended conformal ﬁeld equations,page 208(Σ̂ab , Ξ̂c dab , Δ̂cdb , Λbcd , δa , γab , ςab ) zero quantities in the frame extended
conformal ﬁeld equations, page 205(M, g) generic spacetime, page 45(U , ϕ) coordinate chart, page 28(ΣAA BB  , ΞC DAA BB  , ZAA BB  , ZAA , ΔCDBB  , ΛBB  CD , Z, MAA ) zero
quantities in the spinorial version of the standard conformal equations,page 199(Σab , Ξc dab , Zab , Za , Δcdb , Λbcd , Z, Ma ) zero quantities in the frame version of
the standard conformal ﬁeld equations, page 196(gab , Ξ, s, Lab , da bcd , Tab ) unknowns in the metric standard conformal ﬁeld
equations, page 191(hij , sij , ζ, ς) unknowns in the conformal static equations, page 511(oA , ιA ) spin basis in abstract index notation, page 71(u, r, θA ) Bondi coordinates, page 236(x(τ ), β(τ )) conformal geodesic with parameter τ , page 127(xμ ) local coordinates in a four-dimensional manifold, page 28[∇a , ∇b ] commutator of covariant derivatives, page 39[[ξ, η]] antisymmetric product of ξ, η ∈ S, page 65[g] conformal class of the metric g, page 113[u, v] commutator of the vector ﬁelds u and v, page 34

List of Symbols

xxi

αa , βa , ωa , . . . components of the covectors α, β, ω, . . . with respect to theframe {ea }, page 51
αa , βa , ωa , . . . generic covectors in abstract index notation≈ diﬀeomorphism between sets, page 27ξ¯A , η̄A , . . . complex conjugates of the spinors ξ A , ηB , . . ., page 72
β 2 norm of the covector β̃, page 134
βa covector associated to a conformal geodesic in abstract index notation,
page 127α, β, ω, . . . generic covectors in index-free notation
β covector associated to a conformal geodesic in index-free notation, page 127
χ Weingarten map, page 56
D generic three-dimensional connection in index-free notationd rescaled conformal geodesic covector, page 201
δ Euclidean metric on R3 , page 143e, eAB space spinor irreducible components of the frame vector eAA , page 104
three-dimensional Lorentzian metric on the conformal boundary of an anti-deSitter-like spacetime, page 456
f covector deﬁning a Weyl connection in index-free notation, page 119
f unphysical conformal geodesic covector, page 201
g generic four-dimensional Lorentzian metric tensor in index-free notationg  generic contravariant four-dimensional Lorentzian metric tensor inindex-free notationg E standard metric on the Einstein cylinder, page 144
γ metric in the quotient manifold, page 141
h generic (negative deﬁnite) Riemannian three-dimensional metrich̄ standard metric on the unit 3-sphere, page 142
K extrinsic curvature tensor of a hypersurface in index-free notation, page 61
k intrinsic metric of compact two-dimensional surfaces
M , N , . . . generic higher rank tensors in index-free notation
N tangent to the generators of null inﬁnity¯ generic linear connections in index-free notation, page 38∇, ∇ν unit normal to a hypersurface S, page 54ω, ω AB space spinor irreducible components of the frame covector ω AA ,page 104∂ μ coordinate basis vector
Q transition tensor between connections in index-free notation, page 42
q intrinsic metric of null inﬁnity
Σ torsion tensor of a connection ∇ in index-free notation, page 39
σ standard metric on the unit 2-sphere
σ L (ξ) symbol of a diﬀerential operator L, page 252
t vector ﬁeld generating a timelike congruenceτ vector counterpart of the spinor τAA , page 102v, u, w, . . . generic vectors in index-free notation

xxii

List of Symbols

ς shear tensor, page 226z, ζ deviation vector and covector, respectively, page 135
Lh conformal Killing operator of the metric h, page 257ŭ perturbation quantity in an evolution system
χ(AB)CD spinorial counterpart of the Weingarten tensor
χAB spinorial counterpart of the acceleration vector◦ composition of functions, page 36
disjoint union of sets
δ(i) Dirac’s delta, page 279
Δh Laplacian operator of the Riemannian metric hδμ ν , δa b , δi j , δα β , δA B , δA B , δa b , δi j Kronecker’s deltaδαβ components of the three-dimensional Euclidean metric in Cartesian
coordinates, page 47
δAB Sen connection on a timelike conformal boundary, page 471
δij components of a three-dimensional Riemannian metric with respect to an
orthonormal basis, page 45
γ̇(s) tangent vector to a curve, page 30ẋ(s) alternative notation for the tangent vector to a curve, page 30
 = ±1 encodes the causal character of a hypersurface, page 54
abcd components of the volume form with respect to an orthonormal basis
AB , AB components of the spinors AB , AB with respect to a spin basis,
page 71 
A B  , A B complex conjugates of the spinors AB , AB
AA BB  CC  DD spinorial counterpart of the volume form, page 78
ABCDEF spinorial counterpart of the three-dimensional volume form, page 105
abcd volume form of a metric gab , page 49
AB , AB antisymmetric spinors, page 67≡ deﬁnition
ηABCD components of the electric part of the Weyl spinor, page 373
ηab components of a four-dimensional Lorentzian metric with respect to an
orthonormal basis, page 45
ημν components of the Minkowski metric tensor in Cartesian coordinates,
page 47ð, ð̄ eth and eth-bar operators, page 241
exp exponential map, page 275
Γ geodesic distance, page 276
γ(s) curve in a manifold with parameter s, page 30
Γa c b connection coeﬃcients of ∇ with respect to {ea }γi j k connection coeﬃcients of the three-dimensional connection D withrespect to the frame {ei }, page 59νΓμ λ Christoﬀel symbols of the metric g in the coordinates (xμ )ΓA A BB CC  spinorial counterpart of the connection coeﬃcients Γa b c , page 82
ΓAA B C reduced spin connection coeﬃcients, page 82

List of Symbols

xxiii

ΓABCD space spinor counterpart of the reduced spin connection coeﬃcientsΓAA CD , page 107
C
γAB D reduced spatial spin connection coeﬃcients, page 109γAB CD EF spinorial counterpart of the three-dimensional connection
coeﬃcients γi j k , page 109Ὼ, Ω̌ massless and, respectively, massive part of the conformal factor
associated to Euclidean initial data sets, page 529
ˆ generic Weyl connection in index-free notation, page 119∇
ˆ page 119Γ̂a b c connection coeﬃcients of a Weyl connection ∇,BΓ̂AA C reduced Weyl connection spin coeﬃcients, page 206
ˆ a generic Weyl connection in abstract index notation, page 119∇ρ̂c dab Weyl connection algebraic curvature, page 203
ρ̂ABCC  DD Weyl connection reduced spinorial algebraic curvature, page 207
P̂ c dab Weyl connection geometric curvature, page 203
P̂ABCC  DD Weyl connection reduced spinorial geometric curvature, page 207
κ conformal factor associated to the construction of the cylinder at spatial
inﬁnity, page 541
Λ Newman-Penrose Ricci scalar, page 87
λ cosmological constant, page 2Λ(ABCD) , ΛAB irreducible components of the spinorial Bianchi equation,
page 351
ω, v action of the covector ω on the vector v
t⊥ |p subspace orthogonal to t, page 55
t one-dimensional subspace spanned by t, page 55ξ, η Hermitian product of ξ, η ∈ S, page 94
∇a , ∇b  modiﬁed commutator of covariant derivatives, page 40
Hn n-dimensional half Euclidean space, page 29
R+ non-negative real numbers
R2 Euclidean plane
Rn n-dimensional Euclidean space
S2 2-sphere
S3 three-dimensional unit sphere, page 142A∗ transpose of the complex conjugate of the matrix A
A3 normal matrix in an initial boundary value problem, page 314
Aμ symmetric matrices in a symmetric hyperbolic system, page 294
d exterior derivative (diﬀerential), page 31
dxμ coordinate basis covector
L generic diﬀerential operatorL∗ formal adjoint of the diﬀerential operator L
Lh Yamabe operator, page 256
T map associated to the prescription of boundary conditions in an initialboundary value problem, page 314
u, v, w, . . . CN -valued functions

xxiv

List of Symbols

Ba (p) ball of radius a > 0 centred at the point p
Cp null cone at a point p ∈ M, page 45
Cp+ , Cp− future and, respectively, past null cone at a point p ∈ M, page 45
D a generic derivation, page 30
DAB Sen connection of ∇AA induced by τAA , page 105
E corner in an initial boundary value problem, page 314
G generic lens-shaped domain, page 301
Hk standard hyperboloids, page 154
I cylinder at spatial inﬁnity, page 542
I 0 intersection of the cylinder at spatial inﬁnity with a Cauchy initial
hypersurface, page 542
I ± critical sets where null inﬁnity touches spatial inﬁnity, page 542
M, N generic (unphysical) spacetime manifolds
N , N  initial null hypersurfaces in a characteristic problem, page 320
Ni complex null cone at i, page 522
NC (i) complexiﬁcation of the null cone through i, page 532
P covariant derivative in the direction of τAA , page 105
Q generic quotient manifold, page 141
R generic subset of a hypersurface S
S generic hypersurface on a manifold M
T timelike boundary, page 314
U , V generic open subsets of a manifold or Rn
UC complexiﬁcation of a neighbourhood U of the point at inﬁnity, page 532
Z intersection of initial null hypersurfaces in a characteristic problem, page 320
S complex vector space, page 65
S(M) spin structure (spin bundle) over M, page 81
S(S) space spinor structure over a three-dimensional manifold S, page 101
S∗ dual of the complex vector space S, page 65
S• (M), SA (M), SA (M), SAA B (M), . . . various spin bundles over M
S• spin algebra, page 66
SA , SA , . . . alternative notation for the vector spaces S, S∗ , . . ., page 66
SA , SA B  , . . . complex conjugates of the spaces SA , SAB , . . ., page 72
T• (M) tensor bundle over M, page 34
Ta (M) alternative notation for the tangent bundle over M, page 36
Ta1 ···ak b1 ···bl (M) alternative notation for the tensor bundle over M, page 36
Ta (M) alternative notation for the cotangent bundle over M, page 36
X(M) set of of scalar ﬁelds over M, page 30
ů background quantity in an evolution system
C generic cut of null inﬁnity
C ﬁduciary cut of null inﬁnity
E extension operator of functions between Sobolev spaces, page 308
I part of the conformal boundary that is a hypersurface, page 178
I ± future and, respectively, past null inﬁnity

List of Symbols

xxv

Ni + , Ni − null cones generated by the null geodesics through i, page 531
Nu outgoing null hypersurface associated to the retarded time u
Rh linearised Ricci operator, page 289
Z generic intersection of null inﬁnity with a null hypersurface
int A topological interior of the set A, page 397
i square root of −1
μABCD components of the magnetic part of the Weyl spinor, page 373∇a covariant directional derivative in the direction of ea , page 51∇u v covariant derivative of v with respect to u, page 38¯ a generic linear connections in abstract index notation, page 38∇a , ∇∇AA directional spinorial covariant derivative, page 82
˜ AA , . . . spinor covariant derivatives, page 81∇AA , ∇∇AB space spinor counterpart of ∇AA , page 105
Ω generic three-dimensional conformal factor⊕ direct sum⊗ tensor product between tensors or tensor spaces
A topological closure of the set A, page 394
u S,m Sobolev norm of order m of a function over S, page 306∂Hn boundary of the n-dimensional half Euclidean space, page 29∂M boundary of M
φ unphysical conformally coupled scalar ﬁeld, page 216
φ0 radiation ﬁeld in the asymptotic characteristic problem on a cone, page 500
ΦABA B  spinorial counterpart of the trace-free Ricci tensor, page 89
Φab trace-free Ricci tensor of a connection ∇a in abstract index notation,
page 48
φAB unphysical Maxwell spinor, page 215
Π generic distribution, page 55
Π |p hyperplane induced by a distribution at a point p ∈ M, page 55£v Lie derivative in the direction of v, page 37
ΨABCD Weyl spinor, page 87
ρ boundary-deﬁning function, page 285
ρ polar radial coordinate, page 514
ρα three-dimensional unit position vector, page 514
ρC DAA BB  reduced spinorial algebraic curvature, page 198
ρc dab components of the algebraic curvature, page 195ρAA spatial spinor used to introduce a 1 + 1 + 2 spinor formalism, page 464
Ric, Ric[g] Ricci tensor of a connection ∇ in index-free notation, page 48
Riem Riemann curvature tensor of a connection ∇ in index-free notation,
page 40
Schouten, Schouten[g] Schouten tensor of a connection ∇ in index-freenotation, page 48
σ Newman-Penrose spin connection coeﬃcient corresponding to Γ01 00
σ a AA , σa AA spacetime Infeld-van der Waerden symbols, page 74

xxvi

List of Symbols

Σa c b components of the torsion tensor with respect to an orthonormal frame,
page 53
σi k j , Πk lij , πklij components of the three-dimensional torsion, geometric andalgebraic curvatures, page 264
σi AB , σ i AB spatial Infeld-van der Waerden symbols, page 99
Σa c b torsion tensor of a connection ∇a in abstract index notation, page 39
equality at the conformal boundary
 D’Alembertian operator, page 89
AB box commutator, page 89
equality at a ﬁduciary cut of null inﬁnity
τAA privileged timelike spinor inducing a space spinor formalism, page 102
Θ conformal factor associated to a conformal geodesic, page 132
θ = (θA ) local coordinates on S2
ΘABCD space spinor counterpart of the components of the Schouten tensor ofa Weyl connection, page 373
η̃ Minkowski metricg̃ E metric of the anti-de Sitter spacetime, page 159
g̃ S metric of the Schwarzschild spacetime, page 163
g̃ dS metric of the de Sitter spacetime, page 155
Ẽk asymptotic ends of asymptotically Euclidean manifold S̃, page 272
F̃ab self-dual Faraday tensor, page 213
M̃ generic (physical) spacetime manifoldφ̃ physical conformally coupled scalar ﬁeld, page 216
φ̃AB physical Maxwell spinor, page 215
˜ density of a perfect ﬂuid, page 219
˜ energy density, page 254
F̃ab physical Faraday tensor, page 213j̃k energy ﬂux vector, page 254p̃ pressure of a perfect ﬂuid, page 219
T̃ab physical energy-momentum tensorũa physical 4-velocity of a perfect ﬂuid, page 219
x spatial coordinates (x1 , x2 , x3 )Υa logarithmic gradient of a conformal factor, page 116
ΥAA spinorial counterpart of the logarithmic gradient of a conformal factor,
page 123ϕ∗ pull-back, page 36
ϕ∗ push-forward, page 36
AA components of AA with respect to a spin basis, page 95
AA Hermitian spinor assocated to a Hermitian inner product, page 95
 conformally rescaled density of a perfect ﬂuid, page 220
 unphysical energy density, page 255
W eyl, W eyl[g] Weyl tensor of a connection ∇ in index-free notation, page 48
ξ A , ηA , . . . components of the spinors ξ A , ηA , . . . with respect to a spin basis

List of Symbols

xxvii

ξ A , ηA , . . . generic spinors in abstract-index notation
ξABCC , χABCD real and imaginary parts of ΓABCD , page 107
Ξij , Si , Sij , Hkij zero quantities associated to the conformal static ﬁeld
equations, page 511
ζ0 , . . . ζ4 components of the spin-2 zero-rest mass ﬁeld ζABCD , page 551
ζABCD spin-2 zero-rest mass ﬁeld, page 551{ci } global orthonormal frame on S3 , page 142{ea } vector basis in index-free notation, page 31{ω a } covector basis in index-free notation, page 31{St }t∈R foliation of M, page 54{ei } three-dimensional vector basis in index-free notation, page 59{eAA } alternative index-free notation for the Newman-Penrose null tetrad,
page 79{eAB }, {ω AB } three-dimensional basis and cobasis with spin frame indices,page 109{l, n, m, m̄} Newman-Penrose null tetrad in index-free notation, page 77{ω i } three-dimensional covector basis in index-free notation, page 59
{ω AA } soldering form, page 79{A A }, {A A } alternative abstract index notation for a spin basis and its dual,
page 71{ω a a } covector basis in abstract index notation, page 36{ω i i } three-dimensional covector basis in index-free notation, page 59{ea a } vector basis in abstract index notation, page 36{ei i } three-dimensional vector basis in abstract index notation, page 59{la , na , ma , m̄a } Newman-Penrose null tetrad in abstract index notation,
page 77{m, mα , mα1 α2 , . . .} sequence of multipole moments of a static spacetime,
page 519
bABCD Cotton spinor, page 512
Cp∗ characteristic set of a symmetric hyperbolic system at the point p, page 297
C ∞ class of inﬁnitely diﬀerentiable (smooth) functions
C ∞ (R3 , CN ) space of smooth functions from R3 to CN , page 306
C c dab Weyl tensor of a connection ∇a in abstract index notation, page 48
C k class of k-times diﬀerentiable functions
C k (R3 , CN ) set of C k functions from R3 to CN , page 307

C k [0, T ]; H m (R3 , CN ) set of C k functions from [0, T ] to H m (R3 , CN ),page 307
D bounded open subset of H m (R3 , CN ) such that for w ∈ D the matrix
A0 (0, x, w) is positive deﬁnite bounded away from zero by δ > 0 for allp ∈ R3 , page 309
D(R) domain of dependence of R, page 304
D, Δ, δ, δ̄ Newman-Penrose directional covariant derivatives, page 92
D± (A), D(A) future/past and total domain of dependence of a set A, page 392

xxviii

List of Symbols

da bcd rescaled Weyl tensor, page 188
da components of the rescaled physical conformal geodesics covector, page 203
Di three-dimensional directional covariant derivative in the direction of ei ,page 59
Di generic three-dimensional connection in abstract index notation
DAB three-dimensional covariant directional derivative, page 109
dij , dijk components of the electric and magnetic parts of the rescaled Weyltensor, page 261
DAB spinorial counterpart of a three-dimensional Levi-Civita connection D,
page 106
F a (x), F μ (x) coordinate gauge source functions, page 339
fa components of the unphysical conformal geodesics covector, page 203
fa covector deﬁning a Weyl connection in abstract index notation, page 119
FAB (x) frame gauge source functions, page 345
Fab unphysical Faraday tensor, page 214
g ab generic contravariant four-dimensional Lorentzian metric tensor in abstract
index notation
Gab Einstein tensor of a metric gab
gab generic four-dimensional Lorentzian metric tensor in abstract index
notation
H ± (A), H(A) future/past and total Cauchy horizons of the set A, page 394
H m (R3 , CN ) Sobolev space of order m of functions from R3 to CN , page 307
ha b projector associated to a distribution Π, page 55
hABCD components of hABCD with respect to a spin frame {A A }, page 99

hAA BB spinorial counterpart of the projector ha b , page 98
hABCD space spinor counterpart of hAA BB and of a three-dimensional
Riemannian metric, page 98
I generic interval in R
i0 spatial inﬁnity
i± future and, respectively, past timelike inﬁnity
I ± (U ) chronological future and, respectively, past of a set U , page 391
J + (o, M ) set consisting of o and all points of M which can be joined to o bya causal curve in M , page 497±
J (U ) causal future and, respectively, past of a set U , page 391
jk unphysical ﬂux vector, page 255
Jjk , Jj normal components of the rescaled Cotton tensor, page 262
Kij extrinsic curvature tensor of a hypersurface in abstract index notation,
page 61
Lab Schouten tensor of a connection ∇a in abstract index notation, page 48
lij three-dimensional Schouten tensor, page 60
p ≺≺ q timelike related points, page 391
p conformally rescaled pressure of a perfect ﬂuid, page 220p ≺ q strictly causally related points, page 391

List of Symbols

xxixp  q causally related points, page 391
P C DAA BB  reduced spinorial geometric curvature, page 198
P c dab components of the geometric curvature, page 194

P CC DD AA BB  spinorial geometric curvature, page 197(α,β)
Pn(τ ) Jacobi polynomial of degree n with parameters (α, β), page 553
Qa b c transition tensor between connections in abstract index notation, page 42r three-dimensional Ricci scalar, page 60
R(x) conformal gauge source function, page 348
R, R[g] Ricci scalar of a connection ∇a , page 48
Rc dab components of the Riemann tensor with respect to an orthonormal
frame, page 53dR cab Riemann curvature tensor of a connection ∇a in abstract index notation,
page 40rk lij three-dimensional Riemann curvature tensor in abstract index notation,
page 60
rABCDEF GH spinorial counterpart of the three-dimensional Riemann curvaturetensor, page 110
Rab Ricci tensor of a connection ∇a in abstract index notation, page 48
rACEF GH , rABCE reduced three-dimensional curvature spinors, page 110
RCC  DD AA BB  spinorial counterpart of the Riemann curvature tensor,
page 86
RCDAA BB  reduced Riemann curvature spinor, page 86
rij three-dimensional Ricci tensor in abstract index notation, page 60s the Friedrich scalar, page 186
sABCD spinorial counterpart of the three-dimensional trace-free Ricci tensor,
page 110
sij three-dimensional trace-free Ricci tensor, page 60
SO(3) three-dimensional special orthogonal group
T (M) tangent bundle over M, page 34
T |p (M) tangent space at a point p ∈ M, page 31
T ∗ (M) cotangent bundle over M, page 34
T ∗ |p (M) cotangent space at a point p ∈ M, page 31
T • |p (M) tensor algebra at p ∈ M, page 33
Tlk |p (M) space of (k, l)-tensors at the point p ∈ M, page 33
T a1 ···ak b1 ···bl arbitrary (k, l)-tensor in abstract index notation
Tab unphysical energy-momentum tensor
Tcdb rescaled Cotton tensor, page 189
u retarded time
U, X A , ω, ξ A components of an adapted frame in the asymptotic
characteristic problem, page 482
u, v retarded and, respectively, advanced time coordinates
ua , v a , wa , . . . components of the vectors u, v, w with respect to the coframe{ω a }, page 51

xxx

List of Symbols

ua unphysical 4-velocity of a perfect ﬂuid, page 220
ua , v a , wa , . . . generic vectors in abstract index notationv norm of a static Killing vector, page 504
x(s) alternative notation for a curve with parameter s, page 30
XCDAB , YCDA B  curvature spinors, page 86
Y [h] Yamabe invariant, page 280
Yabc four-dimensional Cotton tensor, page 116
yijk three-dimensional Cotton tensor, page 118
yij three-dimensional Cotton-York tensor, page 118
zAA , z, z(AB) spacetime and space spinor components of the spinorial
counterpart of the deviation vector of a congruence of conformal
geodesics, page 383∗
Fab Hodge dual of an antisymmetric tensor Fab , page 50∗∗
Rabcd , Rabcd
left and, respectively, right duals of the tensor Rabcd , page 50+
Hermitian conjugation, page 96† ‡, generalised dualisation operations, page 50
 , musical operators, page 44
α , β , γ , . . . spatial coordinate indicesA , B , C , . . . spinor frame indices, page 74
a , b , . . . spacetime frame indices ranging 0, . . . , 3
i , j , k , . . . frame indices ranging either 0, 1, 2 or 1, 2, 3
⊥ perpendicular component
μ , ν , λ , . . . spacetime coordinate indicesA , B , C , . . . abstract spinor indices, page 66
a , b , c . . . abstract spacetime indicesi , j , k , . . . abstract spatial indicess Ylm spin-weighted spherical harmonics(a1 ···al ) symmetrisation over the indices a1 ···al , page 36[a1 ···al ] antisymmetrisation over the indices a1 ···al , page 36
A , B ,. . . arbitrary string of indices{a1 ···al } symmetric trace-free part over the indices a1 ···al , page 47

1
Introduction

This book discusses an approach to the analysis of asymptotic and global
properties of solutions to the equations of Einstein’s theory of general relativity(the Einstein ﬁeld equations) based on ideas arising in conformal geometry. Thisapproach allows a geometric and rigorous formulation of problems and notionsof great physical relevance in the context of general relativity. At the same time,
it provides valuable insights into the properties of the Einstein ﬁeld equations
under optimal regularity conditions.
Before entering into the subject, it is useful to discuss the motivation behind
this type of endeavour. Accordingly, a brief account of certain aspects of what
can be called mathematical general relativity is necessary.
1.1 On the Einstein ﬁeld equations
Einstein’s theory of general relativity is the best theory of gravity we have. It
is a relativistic theory of gravity which considers four-dimensional diﬀerentiable,orientable manifolds M̃ endowed with a Lorentzian metric g̃; a discussion of these
diﬀerential geometric notions is provided in Chapter 2. The pair (M̃, g̃) is calleda spacetime. Here, and in the rest of this book, quantities associated to thespacetime (M̃, g̃) will be distinguished by a tilde (˜); the motivation behind thisnotation will become clear in the following. The gravitational ﬁeld is describedin general relativity as a manifestation of the curvature of spacetime.
The fundamental equations of general relativity, the Einstein ﬁeld equations, describe how matter produces the curvature of spacetime. They are given,in the abstract index notation discussed in Section 2.2.6, by
1R̃ab − R̃g̃ab + λg̃ab = T̃ab ,
2

(1.1)

where g̃ab is the abstract index version of g̃, and where R̃ab and R̃ denote,
respectively, the Ricci tensor and Ricci scalar of the metric g̃. Moreover, λ isthe so-called cosmological constant and T̃ab denotes the energy–momentum

2

Introduction

tensor of the matter in the spacetime. Precise deﬁnitions and conventions for thecurvature tensors are provided in Chapter 2, while a discussion of the energy–
momentum tensors for a range of matter models is provided in Chapter 9. Theenergy–momentum tensor satisﬁes the conservation equation
˜ a T̃ab = 0,∇
˜ a denotes the covariant derivative of the metric g̃. The Bianchi identity
where ∇
satisﬁed by the Riemann curvature tensor R̃a bcd of the metric g̃ ensures theconsistency between the conservation equation and the Einstein ﬁeld equations.
A solution to the Einstein ﬁeld equations is a pair (M̃, g̃), together witha g̃-divergence-free tensor T̃ab such that Equation (1.1) holds. In suitable open
subsets of M̃ the metric g̃ is expressed, using some local coordinates (xμ ), in
terms of its components (g̃μν ); here and in what follows, Greek indices areused as coordinate indices. In general, several coordinate charts will be needed
to cover the spacetime manifold M̃. Two metrics g̃ and ḡ over M̃ are said
to be isometric if they are related, everywhere on M̃, by some coordinatetransformation.
In the cases where T̃ab = 0, a direct computation shows that Equation (1.1)
impliesR̃ab = λg̃ab .

(1.2)

In what follows, the latter will be known as the vacuum Einstein ﬁeld
equations and a solution thereof as an Einstein spacetime. The full curvatureof a four-dimensional manifold is described by the tensor R̃a bcd . This tensor
has 20 independent components. By contrast, the Ricci tensor appearing in the
Einstein ﬁeld Equations (1.1) and (1.2) has only 10 independent components.
Hence, even in the absence of a cosmological constant, where the vacuum ﬁeld
Equations (1.2) reduce toR̃ab = 0,(1.3)

it is possible to have solutions with a non-vanishing Riemann tensor. As aconsequence, solutions to the vacuum ﬁeld equations play a special role in generalrelativity, as they describe pure gravitational conﬁgurations. Vacuum spacetimes
are often deemed more fundamental, as they exclude potential pathologies which
may arise from the choice of a particular matter model.
General relativity has two main domains of applicability: cosmology and isolated systems. To make use of the Einstein ﬁeld Equations (1.1) within these twodomains, one requires a number of idealisations. On the one hand, in cosmology it
is usually assumed that the matter content of the universe can be described by aperfect ﬂuid with an equation of state which depends on a particular cosmologicalera. It is a convention in mathematical relativity to refer to spacetimes withcompact spacelike sections as cosmological spacetimes. On the other hand,isolated systems are convenient idealisations of astrophysical objects for which1.2 Exact solutions

3

it is assumed that the cosmological expansion has no inﬂuence. The transition
between the regime of isolated systems and the cosmological one is a topic
of fundamental relevance for the understanding of the physical content of theEinstein ﬁeld equations; see, for example, Ellis (1984, 2002).
The validity of general relativity has been veriﬁed in a number of experiments
covering a wide range of scenarios ranging from the dynamics of the solar system
to cosmological scales; see, for example, Will (2014) for a discussion of thesubject. Surveys of the physical content of general relativity and its various
domains of applicability can be found, for example, in Poisson and Will (2015)
and Shapiro (1999).
Note. In the remainder of this chapter, in order to simplify the presentation, thediscussion will be restricted to Einstein spaces, that is, solutions to the vacuum
Equations (1.2). The inclusion of matter very often requires a case-by-case analysis.
1.2 Exact solutions
A natural ﬁrst step to developing an understanding of the properties of solutions
to the Einstein ﬁeld equations is the construction of exact solutions, that is,
explicit solutions written in terms of elementary functions of some coordinates.
The ﬁrst non-trivial exact solution to the Einstein ﬁeld equations ever obtainedis the Schwarzschild solution. It describes a static spherically symmetric vacuum
conﬁguration; see Schwarzschild (1916), an English translation of which canbe found in Schwarzschild (2003). Remarkably, despite the complexity of theﬁeld equations, the literature contains a vast number of exact solutions to
the equations of general relativity; see, for example, Stephani et al. (2003)
for a monograph on the subject. The number of solutions with a physical orgeometric signiﬁcance is, arguably, much smaller; see, for example, Bičák (2000)
and Griﬃths and Podolský (2009).
1.2.1 Construction of exact solutions
The construction of exact solutions to the Einstein ﬁeld equations requires anumber of assumptions concerning the nature of the solutions. The most natural
assumptions involve the presence of continuous symmetries (Killing vectors) of
some type in the solution, for example, spherical symmetry, axial symmetry,stationarity (including staticity) and homogeneity. Other types of assumptions
involve the algebraic structure of the curvature tensors of the spacetime (e.g.
the Petrov type of the Weyl tensor). These types of assumptions are harder tojustify on a physical basis.
Exact solutions are usually constructed in a coordinate system adapted tothe assumptions being made. Very often, these natural coordinates cover only aportion of the whole spacetime manifold. Thus, one needs to ﬁnd new coordinate
systems (charts) for the exact solution which allow one to uncover a full maximal

4

Introduction

analytic extension of the spacetime. This maximal extension usually paves theway to the interpretation of the exact solution and gives access to its global
properties.
1.2.2 The limitations of exact solutions
Several of the well-known consequences of general relativity have been developed
through the analysis of exact solutions, for example, the notion of a black
hole. Thus, the study of exact solutions to the Einstein ﬁeld equations helps todevelop a physical and geometric intuition which, in turn, can lead to questions
concerning more generic solutions. However, despite the valuable insights they
provide, the construction of exact solutions is not a systematic approach to
explore the space of solutions of the theory. In particular, this approach leaves
open the question of whether certain properties of a solution are generic, thatis, satisﬁed by a broader class of spacetimes. Moreover, exact solutions do notlend themselves to the analysis of dynamic situations such as, for example, thedescription of the gravitational radiation produced by an isolated system. Thus,
it is not possible to address issues involving stability just by means of exact
solutions. In order to analyse the above issues one has to consider whether it ispossible to formulate an initial value problem for the Einstein ﬁeld equations bymeans of which large classes of solutions can be constructed.
1.3 The Cauchy problem in general relativity
As in the case of many other physical theories, general relativity admits theformulation of an initial value problem (Cauchy problem). This aspect of thetheory is obscured by both the tensorial character of the Einstein ﬁeld equations
and the absence of a background geometry in the theory; it is a priori not clearthat the ﬁeld equations give rise to a system of partial diﬀerential equations(PDEs) of a recognisable type.
Classical physical theories are expected to satisfy a causality principle:
the future of an event in spacetime cannot inﬂuence its past, and, moreover,signals must propagate at ﬁnite speed . Among the three main types of PDEs
(elliptic, hyperbolic and parabolic), hyperbolic diﬀerential equations are the only
ones compatible with the causality principle. This observation suggests it should
be possible to extract from the Einstein ﬁeld equations a system of evolution
equations with hyperbolic properties.
1.3.1 Hyperbolic reductions
The seminal work of Fourès-Bruhat (1952) has shown that the hyperbolic
properties of the Einstein ﬁeld equations can be made manifest by means ofa suitable choice of coordinates. Following modern terminology, a choice of
coordinates is a particular example of gauge choice. Indeed, by choosing the1.3 The Cauchy problem in general relativity

5

spacetime coordinates (xμ ) in such a way that they satisfy the wave equation
associated with the metric g̃, the Einstein ﬁeld equations can be shown to
imply a system of quasilinear wave equations for the components (g̃μν ) of the(a priori unknown) metric g̃ with respect to the wave coordinates. For quasilinear
wave diﬀerential equations there exists a developed theory which allows theformulation of a well-posed Cauchy problem. The use of wave coordinates is not
the only way of bringing to the fore the hyperbolic aspects of the Einstein ﬁeld
equations. In this book, it will be shown that the Einstein ﬁeld equations canbe reformulated in such a way that after a suitable gauge choice they imply aso-called (ﬁrst order) symmetric hyperbolic evolution system – a class of PDEs
with properties similar to those of wave equations and for which a comparable
theory is available. The procedure of extracting suitable hyperbolic evolution
equations through a particular reformulation of the Einstein ﬁeld equations
and a suitable gauge choice is known as a hyperbolic reduction; hyperbolic
reductions are further discussed in Chapter 13. Besides its natural relevancein mathematical relativity, the construction of hyperbolic reductions for the
Einstein ﬁeld equations is of fundamental importance for numerical relativity;see, for example, Alcubierre (2008) and Baumgarte and Shapiro (2010).
In the same way that the Einstein ﬁeld equations are geometric in nature,a proper formulation of the Cauchy problem in general relativity must also bedone in a geometric way; see, for example, Choquet-Bruhat (2007). This ideais, in principle, in conﬂict with the discussion of hyperbolicity properties of theEinstein ﬁeld equations, as the associated procedure of gauge ﬁxing breaks thespacetime covariance of the ﬁeld equations. As will be seen in the following, thistension can be resolved in a satisfactory manner.
1.3.2 Initial data and the constraint equations
The formulation of an initial value problem for the Einstein ﬁeld equations
requires the prescription of suitable initial data for the evolution equations ona three-dimensional manifold S̃. This manifold will be later interpreted as a
hypersurface of the spacetime (M̃, g̃). An important feature of general relativity
is that the initial data for the evolution equations implied by the Einstein ﬁeld
equations are constrained. The constraint equations of general relativity(Einstein constraints) can be formulated as a set of equations intrinsic to theinitial hypersurface S̃ for a pair of symmetric tensors h̃ and K̃ describing,respectively, the intrinsic geometry of the hypersurface (intrinsic metric or
ﬁrst fundamental form) and the way the initial hypersurface is curved
within the spacetime (M̃, g̃) – the so-called extrinsic curvature or second
fundamental form. A priori, it is not clear what the freely speciﬁable data forthese constraint equations consist of, or whether, given a particular choice of
free data, the equations can be solved. The systematic analysis of the constraint
equations has shown that under suitable assumptions, they can be recast as a setof elliptic partial diﬀerential equations; see, for example, Bartnik and Isenberg

6

Introduction(2004). For this type of equation a theory is available to discuss the existence
and uniqueness of solutions.
The constraint equations play a fundamental role in the theory and ensure thatthe solution of the evolution equations is, in fact, a solution to the Einstein ﬁeld
equations; this type of analysis is often called the propagation of the constraints.
The constraint equations of general relativity will be discussed in Chapter 11.
1.3.3 The well-posedness of the Cauchy problem in general relativity
The formulation of the Cauchy problem in general relativity ensures, at least
locally, the existence of a solution to the Einstein ﬁeld equations which isconsistent with the prescribed initial data. More precisely, one has the following
result ﬁrst proven in Fourès-Bruhat (1952).
Theorem 1.1 (local existence of solutions to the initial value problem)
Given a solution (h̃, K̃) to the Einstein constraint equations on a threedimensional manifold S̃ there exists a vacuum spacetime (M̃, g̃) such that S̃
is a spacelike hypersurface of M̃, h̃ is the intrinsic metric induced by g̃ on S̃
and K̃ is the associated extrinsic curvature.
The spacetime (M̃, g̃) obtained as a result of Theorem 1.1 is called adevelopment of the initial data set (S̃, h̃, K̃). Not every spacetime canbe globally constructed from an initial value problem. Those which can beconstructed in this way are said to be globally hyperbolic. There are important
examples of spacetimes which do not possess this property – most noticeably,
the anti-de Sitter spacetime. A general result concerning globally hyperbolic
spacetimes states that their topology is that of R× S̃ with each slice S̃t ≡ {t}× S̃being intersected only once by each timelike curve in the spacetime. The slicesS̃t are known as Cauchy surfaces. The above points will be further discussed
in Chapter 14.
The Cauchy problem for the Einstein ﬁeld equations provides an appropriate
setting for the discussion of dynamics. In particular, it allows one to investigate
whether a given solution of the Einstein ﬁeld equations is stable, that is, whether
its essential features are retained if the initial data set is perturbed. Moreover,it also allows one to analyse whether a given property of a solution is generic,that is, whether the property holds for all solutions in an open set in the spaceof initial data.
1.3.4 Geometric uniqueness and the maximal globally hyperbolic
development
An important observation concerning Theorem 1.1 is that it does not ensure
the uniqueness of the development (M̃, g̃) of the initial data set (S̃, h̃, K̃):a diﬀerent hyperbolic reduction procedure will, in general, give rise to analternative development (M̃ , g̃  ). From the point of view of the Cauchy problem

1.3 The Cauchy problem in general relativity

7of general relativity, the solution manifold is not known a priori. Instead, it isobtained as a part of the evolution process.
Given that an initial data set for the Einstein ﬁeld equations gives rise to
an inﬁnite number of developments (one for each reasonable gauge choice), it isnatural to ask whether it is possible to combine these various developments toobtain a maximal development. This question is answered in the positive by thefollowing fundamental result; see Choquet-Bruhat and Geroch (1969).
Theorem 1.2 (existence of a maximal development) Given an initial data
set for the Einstein ﬁeld equations (S̃, h̃, K̃), there exists a unique maximal
development (M̃, g̃), that is, a development such that if (M̃ , g̃  ) is anotherdevelopment, then M̃ ⊆ M̃ and on M̃ the metrics g̃ and g̃  are isometric.
The maximal development (M̃, g̃) is also known as the maximal globally
hyperbolic development of the data (S̃, h̃, K̃). Theorem 1.2 clariﬁes the sense inwhich one can expect uniqueness from the Cauchy problem in general relativity;
this idea is known as geometric uniqueness.
One can think of the maximal development of an initial data set as the largest
spacetime that can be uniquely constructed out of an initial value problem. The
boundary of this maximal development, if any at all, sets the limits of predictability of the data – accordingly, one has a close link with the notion of classicaldeterminism. In certain spacetimes, it is possible to extend the maximal development of a hypersurface to obtain a maximal extension. Accordingly, in general,maximal developments and maximal extensions do not coincide. A further
discussion of the Cauchy problem in general relativity is provided in Chapter 14.
1.3.5 Construction of maximal developments and global existenceof solutions
Given some initial data set (S̃, h̃, K̃), it is natural to ask, How can one
construct its maximal development (M̃, g̃)? In general, this is a very diﬃcult
task, as it requires controlling the evolution dictated by the Einstein ﬁeld
equations under very general circumstances – something for which the requiredmathematical technology is not yet available. There are, nevertheless, someconjectures concerning the global behaviour of maximal developments. The originof these conjectures goes back to Penrose (1969) – see Penrose (2002) for a
reprint – and are usually known by the name cosmic censorship. In particular,
the so-called strong cosmic censorship states that the maximal development ofgeneric initial data for the Einstein ﬁeld equations cannot be extended as a
Lorentzian manifold.
Given an exact solution to the Einstein equations, if one knows its maximal
extension, one can determine the maximal development (M̃, g̃) of one of its(Cauchy) hypersurfaces, say, S̃. In what follows, let (h̃, K̃) denote the initial data
implied on S̃ by the spacetime metric g̃. The explicit knowledge of the maximal
development allows one to provide a physical interpretation of the solution and8Introduction

to analyse its global structure in some detail. One can now ask whether certain
aspects of (M̃, g̃) – say, its basic global structure – are shared by a wider classof solutions to the Einstein ﬁeld equations. A strategy to address this question
within the framework of the Cauchy problem in general relativity is to consider
initial data sets (S̃, h̄, K̄) which are, in some sense, close to the initial data for theexact solution. One can then try to show that the associated maximal globally
hyperbolic development (M̄, ḡ) has the desired global properties. If this is thecase, one has obtained a statement about the stability of the solution and thegenericity of the property one is interested in. The standard convention, to be
used in this book, is to call (M̃, g̃) and (S̃, h̃, K̃), respectively, the backgroundspacetime and the background initial data set and (M̄, ḡ) and (S̃, h̄, K̄) the
perturbed spacetime and perturbed initial data set, respectively. In practice, the notion of closeness between initial data sets is dictated by the requirements of the PDE theory used to prove the existence of solutions to the evolution
equations. In the previous discussion it has been assumed that the 3-manifolds
on which the background and perturbed initial data are prescribed are the same.
The stability analysis allows one to conclude that the spacetime manifolds M̃
and M̄ are the same – they are, however, endowed with diﬀerent metrics.
In analysing the stability of the background solution (M̃, g̃) one needs to showthat the solutions to the evolution equations with perturbed initial data exist
as long as the background solution. The expectation is that the assumption of
having initial data close to data for an exact solution whose global structure iswell understood will ease this task. In the following sections a strategy to exploit
this assumption will be discussed.
1.4 Conformal geometry and general relativity
Special relativity provides a framework for the discussion of the notion of
causality – that is, the relation between cause and eﬀect – which is consistent
with the principle of relativity. The causal structure of special relativity isdetermined by the light cones associated with the Minkowski metric η̃. It allows
the determination of whether a signal travelling not faster than the speed of light
can be sent between two events – if this is the case, then the two events are said
to be causally related. More generally, one can talk of Lorentzian causality:
any Lorentzian metric g̃ gives rise to a causal structure determined by the light
cones associated to g̃. Thus, general relativity provides a natural generalisationof the notions of causality of special relativity – one in which the light cones
vary from event to event in spacetime. Crucially, however, in general relativity
the causal structure is a basic unknown of the theory.
The theory of hyperbolic diﬀerential equations provides notions of causality
which, in principle, are independent from the notions of Lorentzian causality.
It is, nevertheless, a remarkable feature of general relativity that locally, thepropagation of ﬁelds dictated by the Einstein ﬁeld equations is governed by
the structure of the light cones of the solutions – the so-called characteristic

1.4 Conformal geometry and general relativity

9

surfaces of the evolution equations. Thus, the notions of Lorentzian and PDE
causality coincide. This aspect of the Einstein ﬁeld equations is further discussed
in Chapter 14.
1.4.1 Conformal transformations and conformal geometry
Locally, a light cone can be described (away from its vertex) in terms of acondition of the form φ(xμ ) = constant where φ : M̃ → R is such thatg̃ μν ∂μ φ∂ν φ = 0.

(1.4)

The structure of the light cones of a spacetime (M̃, g̃) is preserved by conformal
rescalings, that is, transformations of the spacetime metric of the formg̃ → g ≡ Ξ2 g̃,

Ξ>0

(1.5)

where Ξ is a smooth function on M̃ – the so-called conformal factor.
Throughout this book, the metrics g̃ and g will be called the physical metricand the unphysical metric, respectively. The rescaling (1.5) gives rise toa conformal transformation of (M̃, g̃) to (M̃, g). Precise deﬁnitions andfurther discussion of these notions are provided in Chapter 5. In elementarygeometry, conformal transformations are usually described as transformations
preserving the angle between vectors. In Lorentzian geometry, they preserve thelight cones; from (1.4) it follows that g μν ∂μ φ∂ν φ = 0, so that the condition
φ(xμ ) = constant also describes the light cones of the metric g.
One key aspect of conformal rescalings is that they allow one to introduce
conformal extensions of the spacetime (M̃, g̃); see Figure 1.1. In a Riemanniansetting, the most basic example of conformal extensions of manifolds is the socalled conformal completion of the Euclidean plane R2 into the 2-sphere S2 by

g = Ξ2g̃
˜ g)
˜
(M,

(M,g)

∂M

Figure 1.1 Schematic representation of the conformal extension of a manifold.
The physical manifold (M̃, g̃) has inﬁnite extension, while the unphysical(extended) manifold (M, g) is compact with boundary ∂M. The boundary∂M corresponds to the points for which Ξ = 0. Further details can be foundin Chapter 5. Adapted from Penrose (1964).

10Introduction
(a)

I+

(b)

(c)

I+
I
I−

I−

Figure 1.2 Penrose diagrams of the three spacetimes of constant curvature: (a)the de Sitter spacetime; (b) the anti-de Sitter spacetime; (c) the Minkowski
spacetime. Details of these constructions can be found in Chapter 6.

means of stereographic coordinates. By suitably choosing the conformal factor Ξ,
the metric g given by the rescaling (1.5) may be well deﬁned even at the points
where Ξ = 0. If this is the case, it can be veriﬁed that the set of points ∂M forwhich Ξ = 0 corresponds to ideal points at inﬁnity for the spacetime (M̃, g̃) andis called the conformal boundary . The pair (M, g) where M is the extended
manifold obtained from attaching to M̃ its conformal boundary is usually known
as the unphysical spacetime. Of particular interest are the portions of theconformal boundary which are hypersurfaces of the manifold M – these setsare characterised by the additional requirement of dΞ = 0, so that they have awell-deﬁned normal. This part of the conformal boundary is denoted by I .
Explicit calculations show that the three spacetimes of constant curvature –the Minkowski, de Sitter and anti-de Sitter spacetimes – can be conformallyextended. The details of these constructions are described in Chapter 6. These
conformal extensions are conveniently represented in terms of Penrose diagrams;
see Figure 1.2. A discussion of the construction of Penrose diagrams can also
be found in Chapter 6. The insights provided by the conformal extensions ofthese solutions are, in great measure, the fundamental justiﬁcation for the useof conformal methods in general relativity.
1.4.2 Conformal geometry
The study of properties which are invariant under conformal transformations ofa manifold is known as conformal geometry . Associated to the metric g of
the unphysical spacetime (M, g) one has its covariant derivative (connection) ∇aand its curvature tensors, say, Ra bcd , Rab , R. These objects can be related to the˜ a , R̃a bcd , R̃ab and R̃)corresponding objects associated to the physical metric g̃ (∇and the conformal factor Ξ and its derivatives. Their transformation laws show,in particular, that the Riemann tensor, the Ricci tensor and the Ricci scalar are
not conformal invariants. There is, however, another part of the curvature which1.4 Conformal geometry and general relativity

11

is conformally invariant. It is described by the Weyl tensor, for which it holds thaton M̃.

C̃ a bcd = C a bcd ,In view of the above, one can regard the Weyl tensor as a property of thecollection of metrics conformally related to g̃ – the conformal class [g̃]. If thevacuum Einstein ﬁeld Equations (1.3) hold, the Bianchi identities imply that˜ a C̃ a bcd = 0
∇

(1.6)

irrespectively of the value of the cosmological constant.
1.4.3 Conformal invariance of equations of physics
A number of equations in physics have nice conformal properties. The prototypical example is given by the source-free Maxwell equations
˜ a F̃ab = 0,∇

˜ [a F̃bc] = 0,
∇

(1.7)

where F̃ab denotes the Faraday tensor. One can introduce an unphysical
Faraday tensor Fab by requiring it to coincide with F̃ab on M̃. Using the
˜ a and ∇a , it follows
transformation properties relating the covariant derivatives ∇that the Maxwell equations are conformally invariant; that is, one has that∇a Fab = 0,∇[a Fbc] = 0.

The above equations are well deﬁned everywhere on the unphysical spacetime
manifold M, in particular at the conformal boundary. These equations allow
the extension of the deﬁnition of the unphysical ﬁeld Fab to the conformalboundary ∂M.
In contrast to the Maxwell equations, the vacuum Einstein ﬁeld Equations(1.2) are not conformally invariant. The transformation law for the Ricci tensor
under the rescaling (1.5) implies the equation


1
2
3
cd
∇c ∇d Ξ − 2 ∇c Ξ∇d Ξ .
(1.8)
Rab = − ∇a ∇b Ξ − gab gΞΞΞThe above equation is, at least formally, singular at the points where Ξ = 0.
Thus, it does not provide a good equation for the analysis of the evolution of theunphysical metric g on M. Nevertheless, as pointed out by Penrose (1963) the
Bianchi identity (1.6) has a nice conformal covariance property. More precisely,
one has that
˜ a Ξ−1 C̃ a bcd = 0.
∇
The above equation suggests deﬁning the rescaled Weyl tensor da bcd ≡Ξ−1 C̃ a bcd . Under certain assumptions, the Weyl tensor can be shown to vanish at
I so that the rescaled Weyl tensor is well deﬁned at this portion of the conformalboundary – this important result is analysed in detail in Chapter 10. The rescaled12Introduction

Weyl tensor is not a conformal invariant; it transforms in a homogeneous fashion
under the rescaling (1.5). The above discussion leads to the equation
˜ a da bcd = 0,
∇

(1.9)

the so-called Bianchi equation. In addition, in view of the symmetries of the
Weyl tensor it can be shown that˜ [e da |b|cd] = 0.
∇(1.10)

Note the similarity between Equations (1.9) and (1.10) and the Maxwell
Equations (1.7). In particular, the equations are regular even at the conformalboundary. These equations are full of physical signiﬁcance, as the Weyl tensor canbe thought of as describing the free gravitational ﬁeld, that is, a gravitational
analogue of the Faraday tensor. Chapter 8 provides a detailed derivation anddiscussion of the equations presented in this section.
1.4.4 Asymptotics of the gravitational ﬁeld and asymptotic
simplicity
One of the basic predictions of general relativity is the existence of gravitational
waves propagating at the speed of light across the fabric of spacetime. Asa dynamical process governed by the Einstein ﬁeld equations, gravitationalradiation is closely related to the structure of the light cones of spacetime –thus, if one wants to analyse gravitational radiation one has to examine thepropagation of the gravitational ﬁeld along null directions. This analysis iscomplicated by the absence of a background geometry so that, a priori, it is notclear what the asymptotic behaviour of the gravitational ﬁeld should be. This
concern lies at the heart of the subject of the asymptotics of spacetime – thatis, the study of the limit behaviour of ﬁelds at large distances and large times
and the characterisation of spacetimes by data obtained by taking such limits.
In theories which describe ﬁelds on a given background, one can discuss limits
at inﬁnity in a meaningful way in terms of the background geometry. The
situation is radically diﬀerent in general relativity, where the spacetime (M̃, g̃)– with respect to which the limits of ﬁelds derived from g̃ are to be formulated –is the central objects of study. Accordingly, making sense of limiting proceduresin general relativity is a delicate process and requires a careful analysis of thegeometry and the way it is determined by the Einstein ﬁeld equations. An
approach to this analysis is provided by Penrose’s suggestion that the close
relation between the propagation of the gravitational ﬁeld and the structureof null cones which holds locally is also preserved at large scales and that theasymptotic behaviour of the gravitational ﬁeld can be conveniently analysed interms of conformal extensions of the spacetime; see Penrose (1963, 1964) andPenrose (2011) for a reprint of the latter reference. With this idea in mind,
Penrose introduced the notion of asymptotically simple spacetimes, namely,
spacetimes admitting a smooth conformal extension which is similar to that1.4 Conformal geometry and general relativity

13

of one of the three constant curvature spacetimes. Proceeding in this manner,
one attempts to single out a class of suﬃciently well-behaved spacetimes forwhich it is possible to relate the structure of the light cones in spacetime to thestructure of the ﬁeld equations and the large-scale behaviour of their solutions.
For asymptotically simple spacetimes the causal character of I is determinedby the sign of the cosmological constant; moreover, as already seen, the Weyltensor vanishes on the conformal boundary – the latter is the basic observation
in a collection of results known generically as peeling.
Minkowski-like spacetimes, that is, those asymptotically simple spacetimes
for which λ = 0, are of particular relevance in the study of asymptotics withregard to their connection to the notion of isolated systems in general relativity;
compare the discussion at the end of Section 1.1. For this type of spacetime I isa null hypersurface describing idealised observers at inﬁnity. Penrose’s original
insight was to use the notion of asymptotic simplicity as a way of characterising
isolated systems in general relativity – this idea has been called Penrose’s
proposal by Friedrich (2002). One of the appealing features of this approach tothe study of isolated systems is that it provides a general framework in whichnotions of physical interest such as gravitational radiation and the associatedmass/momentum-loss can be rigorously formulated and analysed. A substantial
amount of work has been invested in pursuing these ideas, as attested by the
sprawling literature on the subject. An exposition of the notion of asymptotic
simplicity, some of its basic consequences and Penrose’s proposal is given in
Chapters 7 and 10.
1.4.5 The conformal Einstein ﬁeld equations
In view of Penrose’s ideas on the relation between general relativity andconformal geometry one can ask: to what extent is it possible to draw conclusions
about the global structure of spacetimes from an analysis of the behaviour, underconformal rescalings, of the Einstein ﬁeld equations? As will be seen in this book,by considering this question one is led to analyse the behaviour of solutions to
the Einstein ﬁeld equations under optimal regularity conditions. To address theabove question one needs a suitable set of equations to work with. As alreadyobserved, the direct transcription of the Einstein ﬁeld equations as an equation
for the unphysical metric g does not provide a set of equations which are adequate
from the point of view of PDE theory.
An alternative set of ﬁeld equations, the so-called conformal Einstein ﬁeld
equations, has been constructed in the seminal work by Friedrich (1981a,b,
1983). The construction of this conformal representation of the equations begins
with a revised reading of the singular Equation (1.8) not as an equation for theunphysical metric (or alternatively, its Ricci tensor) but for the derivatives of theconformal factor Ξ. To complete this alternative point of view one upgrades thecurvature tensors to the level of unknowns and, accordingly, provides equations
for them. The required equations are supplied by the Bianchi identities in a way

14

Introduction

which is consistent with the Einstein ﬁeld equations satisﬁed by the physicalmetric g̃. The resulting system consists of equations for the conformal factor
and its ﬁrst- and second-order derivatives, the unphysical metric g (through thedeﬁnition of its Ricci tensor), the unphysical Ricci tensor Rab and the rescaled
Weyl tensor da bcd – the equation for the latter ﬁeld is Equation (1.9). The
equations derived by Friedrich have two key properties: (i) they are formallyregular even at the points where Ξ = 0 and (ii) whenever Ξ = 0, they imply
a solution to the Einstein ﬁeld equations. The considerations leading to theconformal Einstein ﬁeld equations will be discussed in Chapter 8.
The equations described in the previous paragraph are usually known as
the metric conformal ﬁeld equations. One can extend the basic construction toincorporate more gauge freedom so as to obtain a more ﬂexible set of equations.
A natural ﬁrst step in this direction consists of rewriting the ﬁeld equations ina frame formalism. This leads, in turn, in an almost direct way to the spinorial
version of the equations; see below. A more extreme generalisation consists of a
ˆ a whichreformulation of the ﬁeld equations in terms of a covariant derivative ∇is not the Levi-Civita connection of a metric, but which nevertheless respects the
structure of the conformal class [g̃], a so-called Weyl connection. The resulting
equations are known as the extended conformal Einstein ﬁeld equations. As will
be seen below, this particular formulation of the equations allows the use ofgauges with conformally privileged properties.
Friedrich’s conformal Einstein equations are not the only possible type ofconformal representation of the Einstein ﬁeld equations; see, for example, Mason(1995) and Anderson (2005a). In any case, they are the ones which have beenstudied in a more systematic manner in the literature.
1.4.6 Gauge conditions and conformal geodesics
As already mentioned, the procedure of hyperbolic reduction requires the
speciﬁcation of a gauge in terms of which the evolution equations are to be
expressed. Earlier in this chapter, the notion of a gauge choice had been restricted
to a speciﬁcation of coordinates. For the conformal ﬁeld equations, the gauge
speciﬁcation involves three aspects: a coordinate, a frame and a conformal aspect.
The precise choice of these three aspects of the gauge depends on the particularsof the problem at hand. A discussion of the gauge freedom contained in theconformal ﬁeld equations is given in Chapter 13.
The presence of a conformal gauge freedom – that is, the freedom to specify
the representative in the conformal class one wants to work with – is one of themost attractive aspects of the conformal ﬁeld equations. Given the bewildering
freedom one has in this respect, the use of conformal gauges related to conformal
invariants is a natural choice. Conformal geodesics are a good example of the typeof invariants one can consider. These curves are deﬁned through a set of equations
which are invariant under conformal rescalings. In general, the conformal class[g̃] does not contain a metric for which the conformal geodesics can be recast

1.5 Existence of asymptotically simple spacetimes

15

as standard (metric) geodesics. However, there is always a Weyl connection forwhich they are aﬃne geodesics. Conformal geodesics can be used to constructconformal Gaussian gauge systems for which coordinates and an adapted frame
are propagated oﬀ an initial hypersurface. Conformal geodesics allow one to
specify a privileged unphysical metric g = Θ2 g̃ where Θ is a conformal factor
determined through the conformal geodesic equations. Crucially, for solutions
to the vacuum ﬁeld equations (1.2), the conformal factor Θ can be determined
explicitly from the initial data for a congruence of these curves – it turns out to bea quadratic polynomial of a suitable parameter of the curves in the congruence.
To fully exploit the advantages provided by conformal Gaussian systems, it isnecessary to express the conformal ﬁeld equations in terms of Weyl connections –these considerations lead to the already mentioned extended conformal ﬁeld
equations. Conformal geodesics and their properties are analysed in Chapter 5.
1.4.7 Spinors
This book adopts an approach to the extraction of information from theconformal Einstein ﬁeld equations which makes systematic use of a formalism
based on the so-called 2-spinors. The use of spinors to carry out this analysis is
not essential to the purposes of the book, but it has the advantage of simplifying
certain algebraic aspects of the discussion.
Spinors are the most basic objects subject to Lorentz transformations. To
every tensor and tensorial operation there exists a spinorial counterpart. Moreprecisely, to every tensor of rank k there corresponds a spinor of rank 2k. In someparticular cases – for example, null vectors or the Weyl tensor – by exploiting
symmetries one can associate to the tensor a spinor of the same rank k.
Spinors are well adapted to the discussion of the geometry of null hypersurfaces. Thus, it is not surprising that they are a valuable tool in the discussion of
the Einstein ﬁeld equations. In this book, spinorial representations of the conformal ﬁeld equations are systematically used as a part of the hyperbolic reduction
procedure. In particular, a 2-spinor formalism usually known as the space spinor
formalism, which can be regarded as a spinorial analogue of the 1+3 formalism
for tensors, provides an almost completely algorithmic approach to the decomposition of the ﬁeld equations into (symmetric hyperbolic) evolution equations andconstraint equations. The basic spinorial formalism used in this book is describedin Chapter 3, while the space spinor formalism is dealt with in Chapter 4.
1.5 Existence of asymptotically simple spacetimes
The conformal ﬁeld equations provide a powerful tool for the analysis andconstruction of asymptotically simple spacetimes. In broad terms, they allow
the reformulation of problems involving unbounded domains in the physicalspacetime (M̃, g̃) as problems on bounded domains of the unphysical spacetime(M, g). From the point of view of PDE theory, problems involving a ﬁnite

16

Introduction

existence time are simpler to analyse than global existence questions. Under
the appropriate conditions, the existence of solutions to hyperbolic diﬀerential
equations on a ﬁxed ﬁnite time interval can be shown by invoking the propertyof Cauchy stability; this and other basic notions of PDE theory are discussed
in Chapter 12 where a brief account of basic existence results for symmetric
hyperbolic systems is given. Prior to its use with the conformal Einstein ﬁeld
equations, the technique for the analysis of evolution equations based on a
combination of conformal techniques and Cauchy stability had been used to show
the existence of global solutions of the Yang-Mills equations on the Minkowski
and de Sitter spacetimes; see Choquet-Bruhat and Christodoulou (1981).
The remainder of this section provides a brief survey of some of the existence
results for asymptotically simple spacetimes which have been obtained using theconformal Einstein equations. These results will be elaborated in Part IV of thisbook.
1.5.1 Characteristic initial value problems
Characteristic problems are a particular type of initial value problem wheredata are prescribed on null initial hypersurfaces. Typically, these data areprescribed on two intersecting null hypersurfaces N1 and N2 . The relevant PDE
theory then allows one to conclude the existence and uniqueness of solutions onneighbourhoods of N1 ∩ N2 which are either to the future or to the past of theirintersection. In a diﬀerent type of characteristic problem one prescribes initial
data on a null cone N , including its vertex, and one endeavours to obtain asolution inside the cone – at least in a neighbourhood of the vertex. Conformal
methods allow the formulation of characteristic problems for which initial data
are prescribed on a null conformal boundary – in this case one talks of an
asymptotic characteristic initial value problem; see Friedrich (1981a,b,
1982, 1986c). An attractive feature of characteristic initial value problems is thatthe ﬁeld equations, expressed in an adapted gauge, have structural properties
which simplify their analysis. In particular, the constraint equations on the initialnull hypersurfaces reduce to ordinary diﬀerential equations.
Asymptotic characteristic problems allow the aspects of the theory of theasymptotics of isolated systems to be set on a rigorous footing. The basic
theory of characteristic problems for hyperbolic equations is discussed in Chapter
12. Applications of this theory to the conformal ﬁeld equations are given in
Chapter 18.
1.5.2 De Sitter-like spacetimes
The simplest type of standard (i.e. non-characteristic) initial value problem forthe conformal Einstein ﬁeld equations involves the construction of de Sitter-likespacetimes. In this case one considers compact initial hypersurfaces S which are
diﬀeomorphic to the 3-sphere S3 . One has the following concise statement ﬁrst
proved in Friedrich (1986b).

1.5 Existence of asymptotically simple spacetimes

17

Theorem 1.3 (global existence and stability of de Sitter-like spacetimes) Solutions to the Einstein ﬁeld Equations (1.2) with a de Sitter-like valueof the cosmological constant arising from Cauchy initial data close to data forthe de Sitter spacetime are asymptotically simple.
The proof of this result relies on the fact that a conformal representation of theexact de Sitter spacetime can be recast as a solution of the conformal Einsteinﬁeld equations which extends beyond the conformal boundary. It follows fromthe general theory of hyperbolic equations that the solution of the evolution
equations for an initial data set which is close to initial data for the backgroundsolution will give rise, in its development, to a spacelike hypersurface on which
the conformal factor vanishes. This hypersurface can then be interpreted as theconformal boundary of the perturbed spacetime. Thus, the resulting perturbed
spacetime has the same global structure as the de Sitter spacetime, and one cansay that, in this case, the notion of asymptotic simplicity is stable. Remarkably,a variation of Theorem 1.3 allows for the possibility of prescribing initial dataon the conformal boundary.
Theorem 1.3 can be extended to include the coupling of the gravitational ﬁeld
with various types of trace-free matter. A detailed discussion of the proof of
Theorem 1.3 is given in Chapter 15.
1.5.3 Anti-de Sitter-like spacetimes
As already mentioned, the anti-de Sitter spacetime provides one of the basic
examples of non-globally hyperbolic spacetimes. This peculiarity of the spacetime
can be attributed to the timelike nature of its conformal boundary; this is further
discussed in Chapter 14. As a consequence of the above, spacetimes with a global
structure which is similar to that of the anti-de Sitter spacetime cannot be
constructed using a standard initial value problem, and the initial data haveto be supplemented by suitable boundary data on the hypothetic conformalboundary. This type of setting was ﬁrst analysed in Friedrich (1995) and requires
the identiﬁcation of initial data which can be described as anti-de Sitter-likeand appropriate boundary data for the conformal Einstein ﬁeld equations on
a timelike hypersurface representing the conformal boundary. It turns out that
initial data sets (S̃, h̃, K̃) for anti-de Sitter-like spacetimes are characterised by
the fact that they admit a conformal extension (S, h, K) such that S has aboundary ∂S with the topology of the 2-sphere S2 . Based on the example of theexact anti-de Sitter spacetime one expects the conformal boundary to intersect S
on ∂S and be of the form Ic = (−c, c)×∂S for some c > 0. A detailed analysis of
the conformal evolution equations on Ic reveals that suitable boundary data forthe conformal ﬁeld equations consists of a three-dimensional Lorentzian metric. In order to ensure the smoothness of solutions, the underlying PDE theory
requires certain compatibility conditions (corner conditions) between the initial
and the boundary data which are implied by the conformal ﬁeld equations.
Taking into account the above observations one has the following.

18

Introduction

Theorem 1.4 (local existence of anti-de Sitter-like spacetimes) Consider
an anti-de Sitter-like initial data set (S̃, h̃, K̃) for the Einstein ﬁeld equations anda Lorentzian three-dimensional metric on Ic . Assume that the above data satisfy suitable corner conditions. Then, there exists a solution to the Einstein ﬁeld
equations (M̃, g̃) with anti-de Sitter-like cosmological constant and an associatedconformal extension (M, g) such that S̃ is a spacelike hypersurface of (M̃, g̃) andso that (h̃, K̃) coincides with the intrinsic metric and extrinsic curvature impliedby (M̃, g̃) on S̃. Furthermore, Ic is the conformal boundary of (M, g) and the
intrinsic metric of Ic implied by g belongs to the conformal class of .
The proof of the above theorem is described in Chapter 17. The above theorem
ensures only local existence of anti-de Sitter-like spacetimes, that is, the existenceof a solution close to S̃. It says nothing about the global existence or stability ofsolutions. Accordingly, it does not require assumptions on the smallness of thedata. At the time of writing, the question of the stability (or lack thereof) is an
open problem.
1.5.4 Minkowski-like spacetimes
The analysis of Minkowski-like spacetimes gives rise to some of the mostchallenging open problems in the application of conformal methods in generalrelativity.
In principle, one would like to construct Minkowski-like spacetimes byprescribing suitable asymptotically Euclidean initial data on a three-dimensional
manifold S̃ which is a Cauchy hypersurface of the hypothetic spacetime. However,it turns out that a simpler problem consists of the speciﬁcation of initial dataon a 3-manifold H̃ describing a hypersurface of M̃ which in the conformal
extension intersects I —a so-called hyperboloid . Hyperboloidal initial data
sets (H̃, h̃, K̃) admit conformal extensions (H, h, K) for which H is a manifoldwith boundary ∂H which has the topology of the 2-sphere S2 – this boundary
corresponds to the intersection of the hyperboloid with I . Hyperboloidal initial
data sets are similar in structure to anti-de Sitter-like initial data. There is, infact, a correspondence between the two; this relation is explored in Chapter 11.
An important feature of hyperboloids is that they are not Cauchy hypersurfaces;that is, they do not allow the reconstruction of a whole Minkowski-like spacetime.
Despite this shortcoming, one has the following semi-global existence andstability result ﬁrst proved in Friedrich (1986b).
Theorem 1.5 (semi-global existence and stability of the hyperboloidal
initial value problem) Solutions to the hyperboloidal initial value problem
for the Einstein Equation (1.3) with initial data (H̃, h̃, K̃) which are suitable
perturbations of Minkowski hyperboloidal data are asymptotically simple to the
future of H̃ and have a conformal boundary with the same global structure as theconformal boundary of Minkowski spacetime.

1.5 Existence of asymptotically simple spacetimes

19

A detailed account of this result is given in Chapter 16. Aside from sometechnical details, the key ideas of the proof of this result are similar to those of
Theorem 1.3 for de Sitter-like spacetimes. Again, a conformal point of view allows
one to provide a global existence result for the Einstein ﬁeld equations in termsof a problem involving a ﬁnite existence time. A proof of the non-linear stabilityof the Minkowski spacetime making use of initial data prescribed on a Cauchy
initial hypersurface has been given in the work by Christodoulou and Klainerman(1993). This proof relies on a detailed analysis of the decay of the gravitational
ﬁeld using carefully constructed estimates. Remarkably, the main result of thiswork does not provide enough regularity at inﬁnity for us to conclude that the
spacetime obtained is asymptotically simple.
Time-independent solutions
An important source of intuition on the behaviour of general Minkowskilike spacetimes is provided by the analysis of time-independent spacetimes,that is, spacetimes possessing a continuous symmetry which (at least) in theasymptotic region is timelike. If the Killing vector of a time-independent solution
is hypersurface orthogonal, then one speaks of a static spacetime. Otherwise,
one has a stationary solution. In the vacuum case, static and stationary
solutions can be thought of as describing the exterior gravitational ﬁeld ofsome compact matter conﬁguration. In addition, the Schwarzschild and Kerr
spacetimes describe time-independent black holes. From the point of view ofconformal geometry, their relevance lies in that they allow a detailed analysis ofspatial inﬁnity, that is, the portion of the conformal boundary intersecting theconformal extension S of a Cauchy hypersurface S̃. Vacuum time-independent
spacetimes can be shown to admit conformal extensions which are as smooth as
one can expect.
Time-independent spacetimes are described by equations which, in a suitable
gauge, are elliptic. This feature of this class of solutions explains many of theirrigidity and uniqueness properties – in particular, they are characterised througha sequence of multipole moments. The analysis of these expansions and otherasymptotic properties of static and stationary solutions can be performed ina very convenient manner through conformal methods. In addition, and quite
remarkably, static spacetimes can be shown to have a close relation to spacetimes
constructed from an asymptotic characteristic initial value problem on a lightcone. These and further aspects of static solutions are discussed in Chapter 19.
Spatial inﬁnity
The asymptotic region of Cauchy hypersurfaces of Minkowski-like spacetimes
can be conformally extended to include a further point – the point at inﬁnity. In
these conformal extensions, domains in the asymptotic region are transformed
into suitable neighbourhoods of the point at inﬁnity. This point compactiﬁcation
procedure is a generalisation of the compactiﬁcation of R2 into S2 . From a20

Introduction

spacetime perspective, the point at inﬁnity gives rise to spatial inﬁnity i0 .
In this picture, i0 can be thought of as the vertex of the light cone of I ,and the Minkowski-like spacetime corresponds to the exterior of the cone; thisconstruction is analysed in Chapters 19 and 20.
The construction of Minkowski-like asymptotically simple spacetimes from
Cauchy initial data requires a precise understanding of the behaviour of thegravitational ﬁeld in a neighbourhood of spatial inﬁnity. It was ﬁrst observed
by Penrose (1965) that for spacetimes with non-vanishing mass the conformal
structure becomes singular at spatial inﬁnity. As a consequence, the initial data
implied by the Bianchi Equation (1.9) – which, as already discussed, is one of the
key constituents of the conformal ﬁeld equations – blows up at spatial inﬁnity.
The resulting singularity makes the analysis of solutions to the conformal ﬁeld
equations in this region of spacetime particularly challenging. This observation
explains, to some extent, why the ﬁrst results on the existence of Minkowskilike spacetimes were restricted to the developments of hyperboloidal initial data.
Early attempts to analyse this situation – see, for example, Beig and Schmidt(1982), Beig (1984) and Friedrich (1988) – reached an impasse due to the lackof a suitable representation of spatial inﬁnity. A breakthrough in this direction
was given in Friedrich (1998c) where a representation of spatial inﬁnity based on
the properties of conformal geodesics, the so-called cylinder at spatial inﬁnity,
allows one to formulate a regular ﬁnite initial value problem for the conformal
ﬁeld equations at spatial inﬁnity. In recent years, a considerable amount ofwork has been devoted to exploring the implications of this construction.
The picture that has progressively emerged is that the conditions required toensure the existence of asymptotically simple developments out of asymptoticallyEuclidean initial data are much more restrictive than what one would ﬁrst
expect.
The analysis of the structure of spatial inﬁnity has been informed bydevelopments in the construction of solutions to the constraint equations ofgeneral relativity. The exterior asymptotic gluing constructions introduced inCorvino (2000) and Corvino and Schoen (2006) allow one to glue static andstationary asymptotic regions to otherwise completely general asymptoticallyEuclidean initial data sets, the basic ideas of the exterior asymptotic gluing
construction are brieﬂy discussed in Chapter 11. As already observed, timeindependent solutions to the Einstein ﬁeld equations are well behaved in aneighbourhood of spatial inﬁnity. Chruściel and Delay (2002) have shown that
it is possible to combine this observation with Theorem 1.5 to obtain complete
Minkowski-like asymptotically simple spacetimes. The spacetimes obtained inthis manner are very special, as they are exactly static, or, more generally,
stationary in a neighbourhood of spatial inﬁnity – nevertheless, radiation isregistered at null inﬁnity. It is natural to ask whether it is possible to relax
this rigid behaviour so as to obtain more general types of asymptotically simple
spacetimes. The analysis of the problem of spatial inﬁnity remains a challenging

1.7 Structure of this book21

open area of research; an introductory discussion to the problem of spatial inﬁnity
is provided in Chapter 20.
1.6 Perspectives
At the time of writing, the use of conformal methods to analyse the global
existence and stability of solutions to the Einstein ﬁeld equations has been
mainly restricted to asymptotically simple spacetimes. One of the motivations
behind this book is to encourage researchers interested in the open problems
of mathematical relativity to further extend the available conformal methods
so as to make them suitable for the analysis of more complicated spacetimes– for example, black holes. From the author’s point of view, the realisationof this vision requires the development of not only analytic tools, but also acomputational framework which allows one to perform numerical relativity using
the conformal ﬁeld equations. Some ideas in this direction are put forward in the
concluding Chapter 21.
1.7 Structure of this book
This book is divided in four parts. Throughout, a combination of abstract index
notation and index-free notation has been used. An index-free notation has
been preferred whenever it simpliﬁes the presentation and emphasises structural
aspects of an equation, while abstract indices are used, mostly, in detailed
calculations. The spinorial conventions follow those in the monograph of Penrose
and Rindler (1984). In view of the systematic use of spinors, this book adopts a(+−−−) convention for the signature of Lorentzian metrics. As a consequence of
this convention the sign of the cosmological constant in the de Sitter spacetime
is negative, while for the anti-de Sitter spacetime it is positive. In order toavoid confusion – inasmuch as it is possible – with other sources, a negative
cosmological constant will be described as being de Sitter-like and a positive
one as being anti-de Sitter-like. Further details on conventions can be found in
Chapters 2, 3 and 4.
Throughout this book bold italics are systematically used to denote that agiven concept is being deﬁned, while italics are used to highlight an idea; theattentive reader will realise that sometimes the distinction between these two isblurry.
The content of the four parts of this book can be brieﬂy described as follows.
Part I (Geometric tools) provides a self-contained discussion of the diﬀerential
geometric and spinorial notions that will be used throughout the book. The
presentation and selection of material is tailored to the needs of the discussionin Parts II and III and the applications in Part IV. Chapter 2 gives a brief
account of the required notions of diﬀerential geometry. The purpose of thechapter is not only to serve as a quick reference in later parts of the book
but also to elaborate certain ideas which are not readily available elsewhere22

Introductionin the literature. Chapter 3 provides an account of 2-spinors, while Chapter 4
develops the so-called space spinor formalism. Chapter 5 provides an introduction
to conformal geometry which covers not only the transformation formulae forthe connection and curvature but also not so well-known topics such as Weyl
connections and conformal geodesics – two key notions which will be further
developed in Parts II and III.
Part II (General relativity and conformal geometry) provides an introduction to
the use of conformal methods in general relativity. It also develops a toolkit ofother mathematical methods which will be used to extract information from theEinstein ﬁeld equations. Chapter 6 provides a brief survey of the constructionof conformal extensions of basic solutions to the Einstein ﬁeld equations – theMinkowski, de Sitter, anti-de Sitter and Schwarzschild spacetimes – as well asa general framework for the construction of Penrose diagrams of sphericallysymmetric static spacetimes. Chapter 7 provides a discussion of one of the leading
themes of this book, the concept of asymptotically simple spacetimes and aformulation of the so-called Penrose’s proposal. Chapter 8 gives a derivationand detailed discussion of the main tool of this book, the conformal Einsteinﬁeld equations. Several versions of the equations are considered – metric, frame,spinorial and in terms of Weyl connections. Chapter 9 complements Chapter
8 and describes matter models amenable to treatment by means of conformal
methods. Several of the main results of this book for the vacuum case canbe generalised by including these matter models. Chapter 10 provides a brief
discussion of the formal theory of the asymptotics of spacetime – sometimes also
called asymptopia. This is a vast topic with a sprawling literature. It is thus
impossible to do full justice to the subject in a concise chapter. Accordingly, thedecision has been made to restrict the material to aspects of the subject which
motivate the later parts of the book.
Part III (Methods of PDE theory) provides an account of PDE and spinorial
methods that will be used systematically in Part IV to obtain statements about
the existence of various types of solutions to the Einstein ﬁeld equations. Chapter
11 provides a discussion of the constraint equations implied by the conformal
Einstein ﬁeld equations on spacelike and timelike hypersurfaces – the so-called
conformal constraint equations. The proper discussion of this material requires
the introduction of certain notions of elliptic PDE theory. This is done at various
places in the chapter. Chapter 12 provides a discussion of the methods of thetheory of hyperbolic PDEs which will be used in the latter parts of the book. This
chapter has been written with the applications in Part IV in mind and covers
basic local existence and uniqueness results for initial value, boundary value
and characteristic initial value problems. Chapter 13 discusses in detail various
hyperbolic reduction procedures for the conformal Einstein ﬁeld equations bymeans of spinorial methods. The analysis is not restricted to the evolution
systems, but also considers the subsidiary evolution equations required to prove
the propagation of the constraints. Part III of the book concludes with Chapter

1.7 Structure of this book23

14 where a brief discussion of Lorentzian causality and key aspects of the Cauchy
problem in general relativity are given.
Part IV (Applications) is concerned with applications of the conformal Einstein ﬁeld equations to the analysis of the existence of asymptotically simple
spacetimes. Chapter 15 analyses the global existence and stability of de Sitterlike spacetimes; see Theorem 1.3. Two diﬀerent proofs are provided: the ﬁrst
one makes use of the standard conformal ﬁeld equations and gauge source
functions, and the second one relies on the extended conformal ﬁeld equations
and conformal Gaussian systems. Chapter 16 provides a proof of the semiglobal
existence and stability result for hyperboloidal initial data for the Minkowski
spacetime and a detailed analysis of the structure of the conformal boundary of
the resulting spacetimes; see Theorem 1.5. Chapter 17 provides a discussion of
the construction of anti-de Sitter-like spacetimes by means of an initial boundary
value problem; see Theorem 1.5. Chapter 18 discusses a diﬀerent setting for theconstruction of solutions to the conformal ﬁeld equations, that of asymptotic
characteristic initial value problems either on intersecting null hypersurfaces (oneof them representing null inﬁnity) or on a cone (representing past null inﬁnity).
Chapter 19 analyses the properties of static solutions by means of conformal
methods. The main purpose of this chapter is to pave the way for the discussionof the problem of spatial inﬁnity, which is analysed in Chapter 20. In particular,a discussion of the construction of the so-called cylinder at spatial inﬁnity isprovided.
The book concludes with Chapter 21, which provides a subjective selection ofopen problems in mathematical general relativity where it is felt that the use ofconformal methods can provide fresh insights.
Further reading sections. Each chapter provides a brief literature survey. The
purpose of this is to provide the interested reader a convenient point of entry into
the literature in case more details or an alternative perspective on the subject
are required.

Part I
Geometric tools

2
Diﬀerential geometry

The language of general relativity is diﬀerential geometry. The present chapter
provides a brief review of the ideas and notions of diﬀerential geometry thatwill be used in this book. It also serves the purpose of setting the notation andconventions. The chapter assumes a prior knowledge of the subject at the level,
say, of the ﬁrst chapter of Choquet-Bruhat (2008) or Stewart (1991), or chapters
2 and 3 of Wald (1984). In view of the applications in later parts of this book,some topics which may not be regarded as belonging to the standard baggageof a relativist are discussed in some detail – for example, general (i.e. non-LeviCivita) connections, the so-called 1 + 3 split of tensors – that is, a split based ona congruence of timelike curves, rather than on a foliation, as in the usual 3 + 1– and the analysis of the geometry of submanifolds using a frame formalism.
2.1 Manifolds
The basic objects of study in diﬀerential geometry are diﬀerentiable manifolds.
Intuitively, a manifold is a space that, locally, looks like Rn for some n ∈ N.
Despite this simplicity at a small scale, the global structure of a manifold can bemuch more complicated and leads to considerations of diﬀerential topology.
2.1.1 On the deﬁnition of a manifold
A diﬀerentiable function f between open sets U , V ⊂ Rn , f : U → V, iscalled a diﬀeomorphism if it is bijective and if its inverse f −1 : V →U is diﬀerentiable. If f and f −1 are C k functions, then one has a C k diﬀeomorphism. Furthermore, if f and f −1 are C ∞ functions, one speaks ofa smooth diﬀeomorphism and one writes U ≈ V. Throughout this book, theword smooth will be used as a synonym for C ∞ . The words function, map andmapping will be used as synonyms of each other.
A topological space is a set with a well-deﬁned notion of open and closed
sets. Given some topological space M, a chart on M is a pair (U , ϕ), with

28

Diﬀerential geometry

U ⊂ M and ϕ a bijection from U to an open set ϕ(U ) ⊂ Rn such that givenp∈Uϕ(p) ≡ (x1 , . . . , xn ).
The entries x1 , . . . , xn are called local coordinates of the point p ∈ U . The set
U is called the domain of the chart. Two charts (U1 , ϕ1 ) and (U2 , ϕ2 ) are said
to be C k -related if the map
ϕ2 ◦ ϕ−1
1 : ϕ1 (U1 ∩ U2 ) → ϕ2 (U1 ∩ U2 )deﬁnes changes of local coordinates
and its inverse are C k . The map ϕ1 ◦ ϕ−12(xμ ) = (x1 , . . . , xn ) → (y μ ) = (y 1 , . . . , y n ) in the intersection U1 ∩ U2 ; see
Figure 2.1. Thus, one can regard the coordinates (y μ ) as functions of thecoordinates (xμ ). All throughout this book the Greek letters μ, ν, . . . will be used
to denote coordinate indices. The functions y μ (x1 , . . . , xn ) are C k and, moreover,
the Jacobian det(∂y μ /∂xν ) is diﬀerent from zero.
A C k -atlas on M is a collection of charts whose domains cover the set M.
The collection of all C k -related charts is called a maximal atlas. The pairconsisting of the space M together with its maximal C k -atlas is called a C k diﬀerentiable manifold. If the charts are C ∞ -related, one speaks of a smooth
diﬀerentiable manifold. If for each ϕ in the atlas, the map ϕ : U → Rn has
the same n, then the manifold is said to have dimension n. In what follows,
the discussion will be restricted to manifolds of dimension 3 and 4.
Remark. In introductory discussions of diﬀerential geometry one generally
considers smooth structures. However, as will be seen in later chapters, when
one looks at general relativity from the perspective of conformal geometry, the

U1

U2

ϕ1

ϕ2

ϕ2 ◦ ϕ1−1RnRnFigure 2.1 Schematic representation of the change of coordinates between
charts – see the main text for further details. The ﬁgure is adapted from
Stewart (1991).

2.1 Manifolds

29

smoothness (or lack thereof) encodes important physical content. Accordingly,
one is led to consider the more general class of C k -diﬀerentiable manifolds.
The diﬀerentiable manifolds used in general relativity are generally assumed
to be Hausdorﬀ and paracompact. A diﬀerentiable manifold is Hausdorﬀ if
every two points in it admit non-intersecting open neighbourhoods. The reason
for requiring the Hausdorﬀ condition is to ensure that a convergent sequence ofpoints cannot have more than one limit point. If M is paracompact, then there
exists a countable basis of open sets. Paracompactness is used in several basic
constructions in diﬀerential geometry. In particular, it is required to show thatevery Riemannian manifold admits a metric. In what follows, all diﬀerentiable
manifolds to be considered will be assumed to be Hausdorﬀ and paracompact.
Accordingly, in the rest of the book Hausdorﬀ, paracompact diﬀerentiable
manifolds will be simply called manifolds.
Orientability
An open set of Rn is naturally oriented by the order of the coordinates (xμ ) =(x1 , . . . , xn ). Hence, a chart (U , ϕ) inherits an orientation from its image in Rn . In
an orientable manifold the orientation of these charts matches together properly.
More precisely, a manifold is said to be orientable if its maximal atlas is suchthat the Jacobian of the coordinate transformation for each pair of overlapping
charts is positive.
An alternative description of the notion of orientability in terms of orthonormal
frames will be given in Section 2.5.3. Orientability is a necessary and suﬃcientcondition for the existence of a spinorial structure on M; see, for example,
Chapter 3.
2.1.2 Manifolds with boundary
Manifolds with boundary arise naturally when discussing general relativity fromthe perspective of conformal geometry. In order to introduce this concept one
requires the following subsets of Rn :Hn ≡ {(x1 , · · · , xn ) ∈ Rn | xn ≥ 0},∂Hn ≡ {(x1 , · · · , xn ) ∈ Rn | xn = 0}.
One says that M is a manifold with boundary if it can be covered with charts
mapping open subsets of M either to open sets of Rn or to open subsets of Hn .
The boundary of M, ∂M, is the set of points p ∈ M for which there is a chart(U , ϕ) with p ∈ U such that ϕ(U ) ⊂ Hn and ϕ(p) ∈ ∂Hn . The boundary ∂M
is an (n − 1)-dimensional diﬀerentiable manifold in its own right. Hence, it is asubmanifold of M – see Section 2.7.1.

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Diﬀerential geometry
2.2 Vectors and tensors on a manifold

In order to probe the geometric properties of a manifold one needs vectors
and, more generally, tensors. This section provides a brief discussion of thesefundamental notions.
2.2.1 Some ancillary notions
Derivations
Denote by X(M) the set of scalar ﬁelds (i.e. functions) over M; that is,smooth functions f : M → R.
Deﬁnition 2.1 (derivations) A derivation is a map D : X(M) → X(M)such that:(i) Action on constants. For all constant ﬁelds c, D(c) = 0.
(ii) Linearity. For all f, g ∈ X(M), D(f + g) = D(f ) + D(g).
(iii) Leibnitz rule. For all f, g ∈ X(M), D(f g) = D(f )g + f D(g).
The connection between derivations and covariant derivatives is discussed in
Section 2.4.1.
CurvesThe notion of a vector is intimately related to that of a curve. Given an open
interval I = (a, b) ⊂ R where either or both of a, b can be inﬁnite, a smooth
curve on M is a map γ : I → M such that for any chart (U , ϕ), the compositionϕ◦γ : I → Rn is a smooth map. One often speaks of the curve γ(s) with s ∈ (a, b);s is called the parameter of the curve. If the domain (a, b) of a curve can beextended to, say, [a, b] while keeping γ(s) smooth, one has an extendible curve.
A curve which is not extendible is called inextendible.
A tangent vector to a curve γ(s) at a point p ∈ M, to be denoted as γ̇(p),is the map deﬁned byγ̇(p) : f →d(f ◦ γ) p = γ̇(f ) p ,dsf ∈ X(M).

Given a chart (U , ϕ) with local coordinates (xμ ), the components of γ̇(p) withrespect to the chart are given byẋμ (p) ≡d μx (γ(s)) p .
ds

In a slight abuse of notation the points of the curve γ will often be denoted byx(s) ∈ M and its tangent vector by ẋ(s).

2.2 Vectors and tensors on a manifold

31

2.2.2 Tangent vectors and covectors
To each point p ∈ M, one can associate a vector space T |p (M), the tangent
space at p consisting of all the tangent vectors at p. In what follows, the elementsof this space will be simply known as vectors. All throughout, vectors will
mostly be denoted with lowercase bold Latin letters: v, u, w, . . . Abstract index
notation will also be used to denote vectors; see Section 2.2.6. The tangent space
T |p (M) can be characterised either as the set of derivations at p of smoothfunctions on M or as the set of equivalence classes of curves through p undera suitable equivalence relation. With the ﬁrst characterisation one considers thevectors as directional derivatives, while with the second one they are considered
as velocities. If the dimension of the manifold M is n, then T |p (M) is avector space of dimension n. Local coordinates (xμ ) in a neighbourhood of thepoint p give a basis of T |p (M) consisting of the partial derivative operators{∂/∂xμ }; where no confusion arises about which coordinates are meant, one
simply writes {∂ μ }. In particular, for the vector tangent to a curve one hasthat ẋ(s) = ẋμ (s)∂ μ . In this last expression and in what follows, Einstein’ssummation convention has been adopted – that is, repeated up and down
coordinate indices indicate summation for all values of the range of the index.
That is,nẋμ (s)∂ μ ≡ẋμ (s)∂ μ .
μ=1Covectors
The dual space T ∗ |p (M), the cotangent space at p, is the vector space of
linear maps ω : T |p (M) → R. Generic elements of T ∗ |p (M) will be denotedby lowercase bold Greek letters: α, β, ω, . . . . Being dual to T |p (M), the space
T ∗ |p (M) has also dimension n, and its elements are called covectors. If ω acts
on v ∈ T |p (M), then one writes ω, v ∈ R.
Given f ∈ X(M), for each v ∈ T |p (M), one has that v(f ) is a scalar. Hence,
f deﬁnes a map, the diﬀerential of f , df : T |p (M) → R via
df (v) = v(f ).
As a consequence of the linearity of v one has that df is linear, and thus df ∈T ∗ |p (M). Given a chart (U , ϕ) with coordinates (xμ ), the coordinate diﬀerentials
dxμ form a basis for T ∗ |p (M), the so-called dual basis. The dual basis satisﬁes
dxμ , ∂ ν  = δν μ , where δν μ is the so-called Kronecker’s delta. It follows thatevery covector ω at p ∈ M can be written as ω = ω, ∂ μ dxμ .
Bases
The previous discussion is extended in a natural way to more general bases. Given
any basis {ea } of T |p (M), its dual basis {ω b } of T ∗ |p (M) is deﬁned by the

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Diﬀerential geometry

condition ω b , ea  = δa b . In the rest of the book, lowercase bold indices such asa, b, . . . denote spacetime frame indices ranging 0, . . . , 3. These will be used
when working with four-dimensional manifolds. The lowercase bold Latin lettersi, j, k, . . . will range, depending on the context, over either 0, 1, 2 or 1, 2, 3.
For simplicity of presentation, and unless explicitly stated, a four-dimensional
manifold will be assumed in the subsequent discussion.
Given another pair of bases {ẽa } and {ω̃ b } of T |p (M) and T ∗ |p (M),respectively, these are related to the bases {ea } and {ω b } by non-singularmatrices (Aa b ) and (Aa b ) such thatẽa = Aa b eb ,ω̃ a = Aa b ω b ,(2.1)

satisfying Aa b Ab c = δc a so that (Aa b ) and (Aa b ) are inverses of each other. In
these last expressions and in what follows, Einstein’s summation convention
for repeated contravariant and covariant frame indices has been adopted so thata sum from b = 0 to b = 3 is implied.
Condition (2.1) ensures that the new bases {ẽa } and {ω̃ b } are dual to eachother; that is, ω̃ b , ẽa  = δa b . Given v ∈ T |p (M), α ∈ T ∗ |p (M), the abovetransformation rules for the bases implyv = v a ea = ṽ a ẽa = (ṽ a Aa b )eb ,α = αa ω a = α̃a ω̃ a = (α̃a Aa b )ω b .
The two bases are said to have the same orientation if det(Aa b ) > 0.
2.2.3 Higher rank tensors
Higher rank tensors can be constructed using elements of T |p (M) andT ∗ |p (M) as basic building blocks. A contravariant tensor of rank k at the
point p is a multilinear map
M : T ∗ |p (M) × · · · × T ∗ |p (M) −→ R,k termsthat is, a function taking k covectors as arguments. Similarly, a covariant
tensor of rank l at the point p is a multilinear map
N : T |p (M) × · · · × T |p (M) −→ R,l termsthat is, a function taking l vectors as arguments. More generally, one can also
have tensors of mixed type: a (k, l) tensor at p is a multilinear map
T : T ∗ |p (M) × · · · × T ∗ |p (M) × T |p (M) × · · · × T |p (M) −→ R,k terms

l termsso that T takes k covectors and l vectors as arguments. In particular, a (k, 0)tensor corresponds to a contravariant tensor of rank k, while a (0, l)-tensor is2.2 Vectors and tensors on a manifold

33

a covariant tensor of rank l. The space of (k, l)-tensors at the point p will
be denoted by Tlk |p (M). In particular, one has the identiﬁcations T 1 |p (M) =T |p (M) and T1 |p (M) = T ∗ |p (M). Formally, the space Tlk |p (M) is obtained as
the tensor product of k copies of T ∗ |p (M) and l copies of T |p (M). That is, one
has that
Tlk |p (M) = T |p (M) ⊗ · · · ⊗ T |p (M) ⊗ T ∗ |p (M) ⊗ · · · ⊗ T ∗ |p (M) .
k terms

l terms

The ordering given in the previous expression is known as the standard order.
Notice, however, that an arbitrary tensor does not need not to have its arguments
in standard order.
As an example of the previous discussion consider v ∈ T |p (M) and α ∈∗T |p (M). Their tensor product v ⊗ α is then deﬁned by(v ⊗ α)(u, β) = β, vα, u,u ∈ T |p (M),β ∈ T ∗ |p (M).

(2.2)

One readily sees that v ⊗ α is a bilinear map and thus a (1, 1)-tensor at p ∈ M.
The action of the tensor product given in Equation (2.2) can be extended directly
to an arbitrary (ﬁnite) number of tensors and covectors. If {ea } and {ω b } denote,
respectively, bases of T |p (M) and T ∗ |p (M), then a basis of Tlk |p (M) is given by{eb1 ⊗ · · · ⊗ ebk ⊗ ω a1 ⊗ · · · ⊗ ω al }.
The collection of all the tensor spaces of the form Tlk |p (M) is called the tensor
algebra at p and will be denoted by T • |p (M). The tensor algebra is deﬁned bymeans of a direct sum.
Symmetries of tensors
A covariant tensor of rank l, say, S, is said to be symmetric with respect to its
ith and jth arguments if
S(v 1 , . . . , v i , . . . , v j , . . . , v l ) = S(v 1 , . . . , v j , . . . , v i , . . . , v l ).

(2.3)

Similarly, A it is said to be antisymmetric if
A(v 1 , . . . , v i , . . . , v j , . . . , v l ) = −A(v 1 , . . . , v j , . . . , v i , . . . , v l ).

(2.4)

If the properties (2.3) and (2.4) hold under interchange of any arbitrary pairof indices, one says that S is totally symmetric and A is totally antisymmetric, respectively. The above deﬁnitions can be extended to contravariant
tensors of arbitrary rank. A totally antisymmetric covariant tensor of rank l isalso called an l-form. Symmetry properties of tensors are best expressed interms of abstract index notation.

34

Diﬀerential geometry
2.2.4 Tensor ﬁelds

The discussion in the previous subsections concerned the notion of a tensor ata point p ∈ M. The tensor bundle over M, T• (M), is the disjoint union of
the tensor algebras T • |p (M) for all p ∈ M:
T• (M) ≡T • |p (M).
p∈M

The disjoint union emphasises that although for p, q ∈ M, p = q, the spaces
T • |p (M) and T • |q (M) are isomorphic; they are regarded as diﬀerent sets.
Important subsets of the tensor bundles are the tangent bundle and thecotangent bundle given, respectively, byT |p (M),T ∗ (M) ≡T ∗ |p (M).
T (M) ≡p∈M

p∈M

A smooth tensor ﬁeld over M is a prescription of a tensor T ∈ T • |p (M) at
each p ∈ M such that when T is represented locally in a system of coordinatesaround p, the corresponding components are smooth functions on the local
chart and, more generally, across the atlas. This idea can be naturally extended
to consider tensor ﬁelds which are not smooth but just C k for some positive
integer k. An important property of tensor ﬁelds is that they are multilinear
over X(M). This property is often referred to as X-linearity. It can be used tocharacterise tensors. More precisely, one has the following lemma which will be
used repeatedly (see Penrose and Rindler (1984) for a proof):
Lemma 2.1 (characterisation of tensors) A map
T : T ∗ (M) × · · · × T ∗ (M) × T (M) × · · · × T (M) → X(M)is induced by a (k, l)-tensor ﬁeld if and only if it is multilinear over X(M).
The discussion of tensor ﬁelds and the tensor bundle is naturally carried out
using the language of ﬁbre bundles; see, for example, Kobayashi and Nomizu(2009). This point of view will, however, not be used in this book.
2.2.5 The commutator of vector ﬁelds
Given u, v ∈ T (M), their commutator [u, v] ∈ T (M) is the vector ﬁeld
deﬁned by[u, v]f ≡ u(v(f )) − v(u(f )),for f ∈ X(M). Given a basis {ea } one has that the components of thecommutator with respect to this basis are given by[u, v]a = u(v a ) − v(ua ),ua ≡ ω a , u,v a ≡ ω a , v.

2.2 Vectors and tensors on a manifold

35

One can readily verify that[u, v] = −[v, u],[u + v, w] = [u, w] + [v, w],[[u, v], w] + [[v, w], u] + [[w, u], v] = 0.
The last identity is known as the Jacobi identity – not to be confused with
the Jacobi identity for spinors, to be discussed in Chapter 3.
2.2.6 Abstract index notation for tensors
The presentation of tensors in this section has so far used an index-freenotation. In the sequel, the so-called abstract index notation will also be
used where convenient; see Penrose and Rindler (1984). To this end, lowercase
Latin indices will be employed. Accordingly, a vector ﬁeld v ∈ T (M) will also be
written as v a . Similarly, for α ∈ T ∗ (M) one writes αa . More generally, a (k, l)tensor T will be denoted by T a1 ···ak b1 ···bl . It is important to stress that the indicesin these expressions do not represent components with respect to some coordinatesor frame. These components are denoted, respectively, by Greek indices and boldlowercase Latin indices such as in v μ and v a . The role of the abstract indices isto specify in a simple way the nature of the object under consideration and todescribe in a convenient fashion operations between tensors. In particular, theaction α, v of a 1-form on a vector is denoted in abstract index notation by
αa v a , while its tensor product α ⊗ v is written as αa v b . Similarly, the operation
deﬁned in Equation (2.2) is expressed as αa ua βb v b .
The idea behind the use of abstract indices is to have a notation for tensorial
expressions that mirrors the expressions for their basis components (had abasis been introduced). Using the index notation one can write only tensorial
expressions since no basis has been speciﬁed; see, for example, Wald (1984) fora further discussion on this subject.
Each type of notation has its own advantages. In particular, the index-freenotation is better to describe conceptual and structural aspects, while theabstract index notation is useful in explicit computations. In particular, theabstract index notation allows the expression, in a convenient way, of tensors
whose arguments are not given in standard order as in Fab c d .
An operation which has a particularly convenient description in terms ofabstract indices is the contraction between a contravariant and a covariant
index. For example, given Fab c d , the contraction between the contravariant
index c and, say, the covariant index d is denoted by Fab c c . Following theconvention that repeated indices are dummy indices one has, for example, that
Fab c c = Fab d d . Given a basis {ea } and a cobasis {ω a }, their elements aredenoted, using abstract index notation, as ea a and ω a a , respectively. If Fab c d ≡Fab c d ea a eb b ω c c ed d denotes the components of Fab c d with respect to a basis {ea }and its associated cobasis {ω a }, then the components of the contraction Fab c c

36

Diﬀerential geometry

are given by Fab c c . Following Einstein’s summation convention, a sum on theindex c is understood. Although this deﬁnition is given in terms of components
with respect to a basis, the contraction is a geometric (i.e. coordinate- and baseindependent) operation transforming a tensor of rank (k, l) into a tensor of rank(k − 1, l − 1).
Symmetries of tensors are expressed in a convenient fashion using abstract
index notation. For example, if Sab and Aab denote, respectively, symmetric andantisymmetric covariant tensors of rank 2, then Sab = Sba and Aab = −Aba . Moregenerally, given Mab , its symmetric and antisymmetric parts are deﬁned,
respectively, by the expressions
1
1(Mab + Mba ),M[ab] ≡ (Mab − Mba ).
2
2
The operations of symmetrisation and antisymmetrisation can be extended tohigher rank tensors. In particular, it is noticed that for a rank 3 covariant tensor
Tabc one has
M(ab) ≡

1
(Tabc + Tbca + Tcab − Tacb − Tcba − Tbac ).
3!
If a tensor Sa1 ···al is symmetric with respect to the indices a1 , . . . , al , then one
writes Sa1 ···al = S(a1 ···al ) . Similarly, if Aa1 ···al is antisymmetric with respect to
a1 , . . . , al , one writes Aa1 ···al = A[a1 ···al ] and Aa1 ···al is said to be an l-form.
T[abc] ≡Consistent with the abstract index notation for tensors, it is convenient tointroduce a similar convention to denote the various tensor spaces. Accordingly
the bundle Tkl (M) will, in the following, be denoted by Ta1 ···ak b1 ···bl (M). In
particular, in this notation the tangent bundle T (M) is denoted by Ta (M),while the cotangent bundle T ∗ (M) is given by Ta (M).
A further discussion of the abstract index notation with speciﬁc remarks inthe treatment of spinors is given in Section 3.1.4.
2.3 Maps between manifolds
This section discusses maps between manifolds. In what follows let M and N
denote two manifolds. These manifolds could be the same.
2.3.1 Push-forwards and pull-backs
A map ϕ : N → M is said to be smooth (C ∞ ) if for every smooth functionf ∈ X(M), the composition ϕ∗ f ≡ f ◦ ϕ : N → R is also smooth. Given p ∈ N ,let T |p (N ), T |ϕ(p) (M) denote, respectively, the tangent spaces at p ∈ N andϕ(p) ∈ M. The map ϕ : N → M induces a map ϕ∗ : T |p (N ) → T |ϕ(p) (M), thepush-forward, through the formula(ϕ∗ v)f (p) ≡ v(f ◦ ϕ)(p),v ∈ T |p (N ).

It can be readily veriﬁed that ϕ∗ so deﬁned is a X-linear map; that is, givenv, u ∈ T |p (N ) and a function f ∈ X(M) one has ϕ∗ (f v + u) = f ϕ∗ v + ϕ∗ u.

2.3 Maps between manifolds

37

Note that the above deﬁnition is made in a point-wise manner. Smooth vectorﬁelds do not, in general, push forward to smooth vector ﬁelds, except in the caseof diﬀeomorphisms. For example, if ϕ is not surjective, then there is no wayof deciding which vector to assign to a point not on the image of ϕ. If ϕ is
not injective, then for some points of M, there may be several diﬀerent vectors
obtained as push-forwards of a vector on N . However, given ϕ : N → M a
diﬀeomorphism, for every v ∈ T (N ) there exists a unique vector ﬁeld on T (M)obtained as the pull-back of v; see Lee (2002).
The push-forward ϕ∗ : T (N ) → T (M) can be used, in turn, to deﬁne a map∗
ϕ : T ∗ (M) → T ∗ (N ), the pull-back, as
ϕ∗ ω, v ≡ ω, ϕ∗ v,ω ∈ T ∗ (M),v ∈ T (N ).

Again, it can be readily veriﬁed that ϕ∗ so deﬁned is X-linear: ϕ∗ (f ω + ζ) =f ∗ ϕ∗ ω + ϕ∗ ζ for ω, ζ ∈ T ∗ (M). The pull-back commutes with the diﬀerential
d; that is, ϕ∗ (df ) = d(ϕ∗ f ). Contrary to the case of push-forwards, pull-backs ofsmooth covector ﬁelds always lead to smooth covector ﬁelds. There is no ambiguityin the construction. In the case that ϕ : N → M is a diﬀeomorphism, then the
inverse pull-back (ϕ∗ )−1 is well deﬁned so that covectors can be pulled back fromT ∗ (N ) to T ∗ (M).
The operations of push-forward and pull-back can be extended in a naturalway, respectively, to arbitrary contravariant and covariant tensors. The case of
most relevance for the subsequent discussion is that of a covariant tensor of rank
2, g ∈ T2 (M). Its pull-back ϕ∗ g ∈ T2 (N ) satisﬁes(ϕ∗ g)(u, v) = g(ϕ∗ u, ϕ∗ v),

u, v ∈ T (N ).

2.3.2 Lie derivatives
Smooth maps of the manifold into itself, ϕ : M → M, lead to the notion of theLie derivative. Given a vector v, the Lie derivative £v measures the change
of a tensor ﬁeld along the integral curves of v.
In what follows, let f ∈ X(M) denote a smooth function and u, v ∈ T (M),α ∈ T ∗ (M). The action of £v on functions and vectors is given by£v f ≡ v(f ),£v u ≡ [v, u].

The Lie derivative can be extended to act on covectors by requiring the Leibnitz
rule£v α, u = £v α, u + α, £v u.
A coordinate expression can be obtained from the latter. The action of £v canbe extended to arbitrary tensor ﬁelds by means of the Leibnitz rule£v (S ⊗ T ) = £v S ⊗ T + S ⊗ £v T .

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Diﬀerential geometry

The reader interested in the derivation of the above expressions and their precise
relation to the notions of push-forward and pull-back of tensor ﬁelds is referred
to, for example, Stewart (1991) where a list of coordinate expressions for thecomputation of the derivatives is also provided.
2.4 Connections, torsion and curvatureThis section discusses the further structure required on a manifold to describe
the geometric notion of curvature – a key ingredient of the equations of generalrelativity.
2.4.1 Covariant derivatives and connections
The notion of linear connection allows one to relate tensors at diﬀerent pointsof the manifold M.
Deﬁnition 2.2 (linear connection) A linear connection (connection for
short) is a map ∇ : T1 (M) × T1 (M) → T1 (M) sending the pair of vector ﬁelds(u, v) to a vector ﬁeld ∇v u satisfying:(i) ∇u+v w = ∇u w + ∇v w(ii) ∇u (v + w) = ∇u v + ∇u w(iii) ∇f u v = f ∇u v(iv) ∇u (f v) = u(f )v + f ∇u v
for f ∈ X(M). The vector ∇u v is called the covariant derivative of v withrespect to u.
Any manifold admits a connection. In four dimensions this can be shown
through the speciﬁcation of 43 functions on the spacetime manifold M; see,for example, Willmore (1993). The reason behind this result becomes moretransparent once the so-called connection coeﬃcients have been introduced; see
Section 2.6.
As a consequence of the requirement (iv) ∇u v is not X-linear in v; however, it
is X-linear in u. Thus, using Lemma 2.1 for a ﬁxed second argument it deﬁnes amixed (1, 1)-tensor. Using abstract index notation the latter is denoted by ∇a v b ,so that ∇a v b ∈ Ta b (M).
From the discussion in the previous paragraph it follows that one can regard
the connection ∇ as a map ∇a : Tb (M) → Ta b (M). Moreover, a connection ∇induces a map ∇a : Tb (M) → Tab (M) via(∇a ωb )v b = ∇a (ωb v b ) − ωb (∇a v b ).
This map is ﬁxed if one requires the Leibnitz rule to hold between the product ofa vector and a covector. To extend the covariant derivative to arbitrary tensors
one uses again the Leibnitz rule. For example, from

2.4 Connections, torsion and curvature39∇e (ωa T a bcd ub v c wd ) = (∇e ωa )T a bcd ub v c wd + ωa (∇e T a bcd )ub v c wd
+ ωa T a bcd (∇e ub )v c wd + ωa T a bcd ub (∇e v c )wd+ ωa T a bcd ub v c (∇e wd ),it follows that(∇e T a bcd )ωa ub v c wd = ∇e (ωa T a bcd ub v c wd ) − (∇e ωa )T a bcd ub v c wd
− ωa T a bcd (∇e ub )v c wd − ωa T a bcd ub (∇e v c )wd− ωa T a bcd ub v c (∇e wd ),so that one obtains a X-linear map Ta bcd (M) → Te a bcd (M).
The subsequent discussion will make use of the commutator of covariant
derivatives. This is deﬁned as[∇a , ∇b ] ≡ 2∇[a ∇b] .
One has that[∇a , ∇b ](TA + SA ) = [∇a , ∇b ]TA + [∇a , ∇b ]SA ,[∇a , ∇b ](TA RB ) = ([∇a , ∇b ]TA )RB + TA ([∇a , ∇b ]RB ),where A and B denote an arbitrary string of (covariant and contravariant) indices.
Covariant derivatives and derivations on a manifold are related in a natural
way: given a derivation D and a connection ∇ on M there exists a uniquev ∈ T (M) such that Df = v a ∇a f for any f ∈ X(M) ; see, for example, O’Neill(1983).
2.4.2 Torsion of a connection
The notion of torsion arises from the analysis of the action of the commutator ofcovariant derivatives on scalar ﬁelds. For convenience the abstract index notation
is used. Consider xab ∈ Tab (M) and f, g ∈ X(M). One readily has thatxab [∇a , ∇b ](f + g) = xab [∇a , ∇b ]f + xab [∇a , ∇b ]g,
xab [∇a , ∇b ](f g) = (xab [∇a , ∇b ]f )g + f (xab [∇a , ∇b ]g).
It follows from the latter that the operator xab [∇a , ∇b ] must be a derivation; seeDeﬁnition 2.1. Thus, there exists ua ∈ Ta (M) such thatxab [∇a , ∇b ] = ua ∇a .

(2.5)

The map xab → ua ∇a deﬁned by Equation (2.5) is X-linear. It deﬁnes a tensor
Σ, the torsion tensor of the connection ∇, via uc = xab Σa c b . Hence,∇a ∇b f − ∇b ∇a f = Σa c b ∇c f,

f ∈ X(M).

(2.6)

40

Diﬀerential geometry

One readily sees thatΣa c b = −Σb c a .
That is, the torsion is an antisymmetric tensor. If a connection ∇ is such thatΣa c b = 0, then it is said to be torsion-free.
Remark. Alternatively, one could have deﬁned the torsion via the relation
Σ(u, v) = ∇u v − ∇v u − [u, v],u, v ∈ T (M).

(2.7)

2.4.3 Curvature of a connection
In order to discuss the notion of curvature of a connection it is convenient todeﬁne the modiﬁed commutator of covariant derivatives
∇a , ∇b  ≡ [∇a , ∇b ] − Σa c b ∇c .
Clearly, one has that ∇a , ∇b f = 0 for f ∈ X(M) so that∇a , ∇b (f TA ) = f ∇a , ∇b TA ,for A denoting an arbitrary string of covariant or contravariant indices. In
particular, one has that∇a , ∇b (f uc ) = f ∇a , ∇b uc ,∇a , ∇b (uc + v c ) = ∇a , ∇b uc + ∇a , ∇b v c .
From the previous expressions one concludes that the map ud → ∇a , ∇b ud is
X-linear. Thus, using Lemma 2.1 it deﬁnes a tensor ﬁeld Rd cab , the Riemanncurvature tensor of the connection ∇. One writes
∇a , ∇b ud = ([∇a , ∇b ] − Σa c b ∇c ) ud = Rd cab uc .

(2.8)

Alternatively, one has that(∇a ∇b − ∇b ∇a ) ud = Rd cab uc + Σa c b ∇c ud .
The antisymmetry of ∇a , ∇b  on the indices a and b is inherited by the Riemanncurvature tensor, so thatRd cab = −Rd cba .
The action of the commutator of covariant derivatives can be extended toother tensors using the Leibnitz rule. For example, from∇a , ∇b (ωd v d ) = (∇a , ∇b ωd ) v d + ωd ∇a , ∇b v d ,one can conclude that(∇a ∇b − ∇b ∇a ) ωd = −Rc dab ωc + Σa c b ∇c ωd .

2.4 Connections, torsion and curvature

41

Similarly, evaluating ∇a , ∇b (S d ef ωd ue v f ), one concludes that(∇a ∇b − ∇b ∇a ) S d ef = Rd cab S c ef − Rc eab S d cf − Rc f ab S d ec + Σa c b ∇c S d ef .
Remark. The curvature can be deﬁned in an alternative way via the relation
Riem(u, v)w = ∇u ∇v w − ∇v ∇u w − ∇[u,v] w,

u, v, w ∈ T (M), (2.9)

where the expression Riem(u, v)w corresponds to Rd cab wc ua v b in abstract
index notation.
Bianchi identities
In order to investigate further symmetries of the curvature tensor, consider the
triple derivative ∇[a ∇b ∇c] f of f ∈ X(M). A computation shows, on the one
hand, that2∇[a ∇b ∇c] f = 2∇[[a ∇b] ∇c] f = [∇[a , ∇b ]∇c] f= Σ[a d b ∇|d| ∇c] f − Rd [cab] ∇d f,
and on the other hand that2∇[a ∇b ∇c] f = 2∇[a ∇[b ∇c]] f= ∇[a [∇b , ∇c] ]f = ∇[a Σb d c] ∇d f

= ∇[a Σb d c] ∇d f + Σ[b d c ∇a] ∇d f.
Putting these two computations together and using the deﬁnition of the torsion
tensor, Equation (2.6), one concludes that∇[a Σb d c] ∇d f + Rd [cab] ∇d f + Σ[a d b Σc] e d ∇e f = 0.
As the scalar ﬁeld f is arbitrary, one concludes thatRd [cab] + ∇[a Σb d c] + Σ[a e b Σc] d e = 0.

(2.10)

This is the so-called ﬁrst Bianchi identity . In the case of a torsion-freeconnection it takes the familiar form
Rd [cab] = 0.
As a consequence of the antisymmetry in the last two indices, the latter can be
written as
Rd cab + Rd abc + Rd bca = 0.

42

Diﬀerential geometry

Next, consider the action of ∇[a ∇b ∇c] on a vector ﬁeld v d . As in the case of
the ﬁrst Bianchi identity, one can compute this object in two diﬀerent ways. On
the one hand, one has that2∇[a ∇b ∇c] v d = 2∇[[a ∇b] ∇c] v d= [∇[a , ∇b ]∇c] v d= ∇[a , ∇b ∇c] v d + Σ[a e b ∇|e| ∇c] v d= − Re [cab] ∇e v d + Rd e[ab ∇c] v e + Σ[a e b ∇|e| ∇c] v d ,and on the other hand that2∇[a ∇b ∇c] v d = 2∇[a ∇[b ∇c]] v d= 2∇[a ∇b , ∇c] v d + ∇[a Σb e c] ∇e v d= ∇[a Rd |e|bc] v e + Rd e[bc ∇a] v e + ∇[a Σb e c] ∇e v d + Σ[b e c ∇a] ∇e v d .

Equating the two expressions for 2∇[a ∇b ∇c] v d and using the ﬁrst Bianchi
identity, Equation (2.10), to eliminate covariant derivatives of the torsion tensor
one concludes that∇[a Rd |e|bc] + Σ[a f b Rd |e|c]f = 0.

(2.11)

This is the so-called second Bianchi identity . For a torsion-free connection
one obtains the well-known expression∇[a Rd |e|bc] = 0.

(2.12)

2.4.4 Change of connection¯ on the manifold M. A natural question toConsider two connections ∇ and ∇be asked is whether there is any relation between these connections and theirassociated torsion and curvature tensors. By deﬁnition one has that¯ a − ∇a )f = 0,(∇

f ∈ X(M).

Moreover, one also has that¯ a − ∇a )v a .
¯ a − ∇a )(f v a ) = f (∇(∇¯ a − ∇a )v a is X-linear, so that, invoking Lemma
It follows that the map v b → (∇2.1, there exists a tensor ﬁeld, the transition tensor Qa b c , such that¯ a − ∇a )v b = Qa b c v c .
(∇(2.13)

Now, from¯ a − ∇a )(ωb v b ) = 0,(∇
one readily concludes that¯ a − ∇a )ωb = −Qa c b ωc .
(∇(2.14)

2.4 Connections, torsion and curvature

43

A diﬀerent choice of covariant derivatives gives rise to a diﬀerent choice oftransition tensor. The set of connections over a manifold M is an aﬃne space:given a connection ∇ on the manifold, any other connection can be obtained bya suitable choice of transition tensor. If Q denotes the index-free version of the¯ will be denoted,
tensor Qa b c , then the relation between the connection ∇ and ∇in a schematic way, as¯ − ∇ = Q.
∇
In Chapter 5 speciﬁc forms for the transition tensor will be investigated.
Transformation of the torsion and the curvature
A direct computation using Equations (2.6) and (2.13) renders the following¯ and ∇:
relation between the torsion tensors of the connections ∇Σ̄a c b − Σa c b = −2Q[a c b] .

(2.15)

In particular, it follows that if Qa c b = 12 Σa c b , then Σ̄a c b = 0. That is, it is always
possible to construct a connection which is torsion-free.
An analogous, albeit lengthier computation using Equations (2.6) and (2.8)
renders the following relation between the respective curvature tensors:R̄c dab − Rc dab = 2∇[a Qb] c d − Σa e b Qe c d + 2Q[a c |e| Qb] e d .

(2.16)

2.4.5 The geodesic and geodesic deviation equations
Given a covariant derivative ∇, one can introduce the notion of parallel
propagation. Given u, v ∈ T (M), then u is said to be parallely propagated in
the direction of v if it satisﬁes the equation ∇v u = 0.
A geodesic γ ⊂ M is a curve whose tangent vector is parallely propagatedalong itself. Following the convention of Section 2.2.1, let ẋ denote the tangent
vector to γ. One has that∇ẋ ẋ = 0.

(2.17)A congruence of geodesics is the set of integral curves of a vector ﬁeldẋ satisfying Equation (2.17). Any vector z such that [ẋ, z] = 0 is called adeviation vector of the congruence of geodesics. Assuming that the connection∇ is torsion-free so that ∇ẋ z = ∇z ẋ, a computation shows that z satisﬁes thegeodesic deviation equation∇ẋ ∇ẋ z = Riem(ẋ, z)ẋ.
Remark. The set of geodesics emanating from a point p ∈ M allows one todeﬁne a diﬀeomorphism between a neighbourhood of the origin of T |p (M) and asuitably small neighbourhood U of p, the so-called exponential map. A precise

44

Diﬀerential geometry

deﬁnition of the exponential map is given in Section 11.6.2. Further properties
and applications are given in Sections 14.2 and 18.4.1.
2.5 Metric tensors
A metric on the manifold M is a symmetric rank 2 covariant tensor ﬁeld g– to be denoted by gab in abstract index notation. The metric tensor g is said
to be non-degenerate if g(u, v) = 0 for all u if and only if v = 0. In thesequel, and unless otherwise explicitly stated, it is assumed that all the metrics
under consideration are non-degenerate. If g(u, v) = 0, then the vectors u andv are said to be orthogonal . Pointwise, the components gab ≡ g(ea , eb ) withrespect to a basis {ea } deﬁne a symmetric (n × n)-matrix (gab ). As this matrix
is symmetric, it has n real eigenvalues. The signature of g is the diﬀerence
between the number of positive and negative eigenvalues. If the signature is n or
−n, then g is said to be a Riemannian metric. If the signature is ±(n − 2),then g is a Lorentzian metric.
From the non-degeneracy of g it follows that there exists a unique contravariant
rank 2 tensor to be denoted by either g  or g ab such thatgab g bc = δa c .
In terms of components with respect to a basis this means that the matrices (gab )and (g ab ) are inverses of each other. Accordingly, g  is also non-degenerate andone obtains an isomorphism between the vector spaces T |p (M) and T ∗ |p (M).
More precisely, given v ∈ T |p (M), then v  ≡ g(v, ·) ∈ T ∗ |p (M) as g(u, v) ∈R for any u ∈ T |p (M). Similarly, given ω ∈ T ∗ |p (M), one has that ω  ≡g  (ω, ·) ∈ T |p (M). In terms of abstract indices, the operations  (ﬂat) and (sharp) correspond to the operations of lowering and raising of indices bymeans of gab and g ab :va ≡ gab v b ,ω a ≡ g ab ωb .

The operations  and  are inverses of each other. They can be extended in anatural way to tensors of arbitrary rank.
Given two manifolds M and M̄ with metrics g and ḡ, respectively, adiﬀeomorphism ϕ : M → M̄ is called an isometry if ϕ∗ ḡ = g. If an isometry
exists, then the pairs (M, g) and (M̄, ḡ) are said to be isometric. If M = M̄and g = ḡ, one speaks of an isometry of M.
Remark. Most of the Lorentzian metrics to be considered in this book will
be associated to four-dimensional manifolds. These Lorentzian metrics will
be assumed to have signature −2. This convention leads one to consider
three-dimensional negative-deﬁnite Riemannian metrics, that is, metrics withsignature −3. In this book, only three-dimensional Riemannian manifolds will beconsidered. In the sequel, the symbol g will be used to denote a generic Lorentzian
metric, while h will be used for a generic negative-deﬁnite Riemannian metric.

2.5 Metric tensors

45

Speciﬁcs for Lorentzian metrics
Following the standard terminology of general relativity, a pair (M, g) consistingof a four-dimensional manifold and a Lorentzian metric will be called aspacetime. The metric g can be used to classify vectors in a pointwise manner
as timelike, null or spacelike depending on whether g(v, v) > 0, g(v, v) = 0
or g(v, v) < 0, respectively. A basis {ea } is said to be orthonormal ifg(ea , eb ) = ηab ,ηab ≡ diag(1, −1, −1, −1).

It follows that g can be written asg = ηab ω a ⊗ ω b ,(2.18)

where {ω a } denotes the coframe dual to {ea }. A change of basis, as given by
Equation (2.1), preserving Equation (2.18), is called a Lorentz transformation. A calculation readily shows that for a Lorentz transformation one hasthatηab Aa c Ab d = ηcd .
Further aspects of Lorentz transformations are discussed in Sections 3.1.9, 3.1.12
and 5.1.1.
The set of null vectors at a point p ∈ M is called the null cone at p and will
be denoted by Cp . By deﬁnition timelike vectors lie inside the null cone, while
spacelike ones lie outside it. The null cone is made of two half cones. If one ofthese half cones can be singled out and called the future half cone Cp+ andthe other the past half cone Cp− , then T |p (M) is said to be time oriented.
A timelike vector inside Cp+ is said to be future directed ; similarly a timelike
vector inside Cp− is called past directed . If T (M) can be time oriented in acontinuous manner for all p ∈ M, then (M, g) is said to be a time-oriented
spacetime. A curve γ ⊂ M with a timelike, future-oriented tangent vector ẋ issaid to be parametrised by its proper time if g(ẋ, ẋ) = 1.
Speciﬁcs for Riemannian metrics
A Riemannian metric h endows the tangent spaces of the manifold with an innerproduct. Because of the signature conventions, this inner product is negative
deﬁnite. A basic result of Riemannian geometry is that every diﬀerential manifold
admits a Riemannian metric. The proof of this argument relies heavily on theparacompactness of the manifold; see, for example, Choquet-Bruhat et al. (1982).
In the case of a Riemannian metric h, a basis {ei } is said to be orthonormal ifh(ei , ej ) = −δij ,δij ≡ diag(1, 1, 1).

Thus, using the associated coframe basis {ω i } one can writeh = −δij ω i ⊗ ω j .

46

Diﬀerential geometry
2.5.1 Metric connections and Levi-Civita connections

Two further conditions which are usually required from a connection are metriccompatibility and torsion-freeness. In this section the consequences of theseassumptions are brieﬂy reviewed.
Metric connections
A connection ∇ on M is said to be metric with respect to g if ∇g = 0 (i.e.
∇a gbc = 0). The Riemann curvature tensor of the connection ∇ acquires, byvirtue of the metricity condition, a further symmetry. This can be better seen
by applying the modiﬁed commutator ∇a , ∇b  to the metric gab . On the one
hand, by the assumption of metricity one has ∇a , ∇b gcd = 0, while on the otherhand∇a , ∇b gcd = −Re cab ged − Re dab gce = −Rdcab − Rcdab ,where Rdcab ≡ gde Re cab . Hence, one concludes that
Rcdab = −Rdcab .

(2.19)The Levi-Civita connection
A connection ∇ is said to be the Levi-Civita connection of the metric g
if ∇ is torsion-free and metric with respect to g. The Fundamental Theorem ofRiemannian Geometry (also valid in the Lorentzian case) ensures that the LeviCivita connection of a metric g is unique. The proof of this result is well known
and readily available in most books on Riemannian geometry; see, for example,
Choquet-Bruhat et al. (1982). The Levi-Civita connection ∇ of the metric g ischaracterised by the so-called Koszul formula2g(∇v u, w) = v(g(u, w)) + u(g(w, v)) − w(g(v, u))− g(v, [u, w]) + g(u, [w, v]) + g(w, [v, u]).

(2.20)

Of particular interest are the further symmetries that the Riemann tensor
of a Levi-Civita connection possesses. First of all, because of the metricity, thecurvature tensor has the symmetry given in Equation (2.19). Furthermore, as
the connection is torsion-free, the ﬁrst Bianchi identity implies Rc[dab] = 0. Fromthe latter one readily has that
2Rcdab = Rcdab + Rdcba= −Rcabd − Rcbda − Rdbac − Rdacb= −Racdb − Rbcad − Rbdca − Radbc= Rabcd + Rbadc .

2.5 Metric tensors

47

Hence, one recovers the well-known symmetry of interchange of pairs
Rcdab = Rabcd .
Characterisation of ﬂatness
An open subset U ⊂ M of a spacetime (M, g) is said to be ﬂat if the metric g
on U is isometric to the Minkowski metricη ≡ ημν dxμ ⊗ dxν ,(ημν ) ≡ diag(1, −1, −1, −1).

In the case of a three-dimensional Riemannian manifold (S, h), ﬂatness implies
a local isometry with the three-dimensional Euclidean metricδ ≡ −δαβ dxα ⊗ dxβ ,(δαβ ) ≡ diag(1, 1, 1).

The Riemann tensor of a Levi-Civita connection provides a local characterisationof the ﬂatness of a manifold. More precisely, a metric is ﬂat on U if and only
if its Riemann tensor vanishes on U . The if part of the result follows by direct
evaluation of the Riemann tensor. The only if part is more complicated; see, forexample, Choquet-Bruhat et al. (1982), page 310 for a proof.
Traces
A metric g on a manifold M allows one to introduce a further operation on
tensors which reduces their rank by 2 – the trace with respect to g. Given
T ∈ T2 (M), its trace, trg T , is the scalar described in abstract index notation by
g ab Tab . Observing that g ab Tab = T a a , one sees that taking the trace of a tensor isa generalisation of the operation of contraction. The operation of taking the trace
can be generalised to any pair of indices of the same type in an arbitrary tensor– for example, g ac Mabcd and g bc Mabcd denote the traces of Mabcd with respect
to the ﬁrst and third arguments and the second and third ones, respectively.
Given a symmetric tensor on a four-dimensional manifold M, Tab = T(ab) ∈Tab (M), its trace-free part T{ab} is given by
1
T{ab} ≡ Tab − gab g cd Tcd .
4
In the case of a three-dimensional manifold S with metric h, the above deﬁnition
has to be modiﬁed to1
T{ij} ≡ Tij − hij hkl Tkl ,3
for a symmetric tensor Tij ∈ Tij (S). The operation of taking the trace-free partof a tensor can be extended to tensors of arbitrary rank. Unfortunately, theexpressions to compute them become increasingly cumbersome. A more eﬃcientapproach to describe this operation is in terms of spinors; see Chapters 3 and 4.
A tensor Ma1 ···ak is said to be trace-free if Ma1 ···ak = M{a1 ···ak } .

48

Diﬀerential geometry
2.5.2 Decomposition of the Riemann tensor¯ on M – not
In what follows, consider a spacetime (M, g) and a connection ∇necessarily the Levi-Civita connection of the metric g. Let R̄a bcd denote the¯ A concomitant of R̄a bcd is any
Riemann curvature tensor of the connection ∇.
tensorial object which can be constructed from the curvature tensor by means of
the operations of covariant diﬀerentiation and contraction with gab and g ab . The
basic concomitant of R̄a bcd is the Ricci tensor R̄cd deﬁned by the contraction
R̄bd ≡ R̄a bad .
When working in index-free notation the Ricci tensor will be denoted by Ric.
Using the contravariant metric g ab one can deﬁne a further concomitant, theRicci scalar relative to the metric g, R̄, asR̄ ≡ g bd R̄bd .
A concomitant of R̄a bcd which will appear recurrently in this book is the
Schouten tensor relative to g, L̄ab . In four dimensions it is deﬁned as
1
1R̄ab − R̄gab .
2
12The deﬁnition of the Schouten tensor is dimension dependent. The deﬁnition forthree dimensions will be discussed in Section 2.7. When working in index-freenotation the Schouten tensor will be denoted by Schouten. In the discussion of
spinors in Chapter 3 a further concomitant arises in a natural way: the trace-free
Ricci tensor Φ̄ab . In four dimensions one has that


1
1
1
Φ̄ab ≡ R̄{ab} =R̄(ab) − R̄gab ,22
4L̄ab =

where the overall factor of 12 is conventional. It is important to observe that the¯ is a Levi-Civita connection.
tensors R̄ab and L̄ab are not symmetric unless ∇¯ relative to g, C̄ a bcd , as the fully
Finally, one can deﬁne the Weyl tensor of ∇trace-free part of R̄a bcd . When working in index-free notation the Weyl tensor
will be denoted by W eyl.
The case of a Levi-Civita connection¯ is the Levi-Civita connection of the metric g, so that ∇¯ = ∇, it can be
If ∇shown that
Rc dab = C c dab + 2(δ c [a Lb]d − gd[a Lb] c ),cce

= C dab + 2Sd[a Lb]e ,(2.21a)(2.21b)where
Sab cd = δa c δb d + δa d δb c − gab g cd .
This tensor will play a special role in the context of conformal geometry; see
Chapter 5. A spinorial derivation of this decomposition is provided in Chapter 3.

2.5 Metric tensors

49

Remark. The decomposition given by Equations (2.21a) and (2.21b) is unique;that is, the Rieman tensor cannot be reconstructed from any other combinationof the Schouten and Weyl tensors. Moreover, if C c dab = 0 and Lab = 0, thennecessarily Rc dab = 0. These remarks also hold for the generalisations of thedecomposition to Weyl connections; see Section 5.3 and, in particular, Equation(5.28a).
The Einstein tensor
An important concomitant of the Riemann tensor of a Levi-Civita connection ∇is the Einstein tensor G deﬁned in four dimensions by1
Gab ≡ Rab − Rgab .
2Starting from the second Bianchi identity, Equation (2.12), contracting theindices d and b and then contracting the resulting expression with g ae yields∇a Rab =

1
∇b R,
2

∇a Gab = 0.

that is,That is, the Einstein tensor is divergence-free.
2.5.3 Volume forms and Hodge duals
The spacetime volume form of the metric g, abcd , is deﬁned by the conditions
abcd = [abcd] ,abcd abcd = −24,and
abcd e0 a e1 b e2 c e3 d = 1,
where {ea } is a g-orthonormal frame. A spacetime (M, g) has a non-vanishingvolume element if and only if M is orientable; see, for example, O’Neill (1983);
Willmore (1993). The following properties can be directly veriﬁed:abcd pqrs = −24δa [p δb q δc r δd s] ,(2.22a)abcd pqrd

= −6δa δb δc ,(2.22b)abcd pqcd

= −4δa δb ,(2.22c)abcd pbcd

= −6δa ;(2.22d)

[p

[p

p

q

q]

r]

see, for example, Penrose and Rindler (1984). If ∇ denotes the Levi-Civita
covariant derivative of the metric g, one can then readily verify that ∇a bcde = 0.
That is, the volume form is compatible with the Levi-Civita connection of themetric g.

50

Diﬀerential geometry
The Hodge duals

Given an antisymmetric tensor Fab = F[ab] , one can use the volume form todeﬁne its Hodge dual ∗ Fab as∗1
Fab ≡ − ab cd Fcd .
2This deﬁnition can be naturally extended to any tensor with a pair ofantisymmetric indices. Using the identity (2.22c) one readily ﬁnds that∗∗

Fab = −Fab .

Of special relevance are the Hodge duals of the Riemann and Weyl tensors. If
Rabcd denotes the Riemann curvature of the Levi-Civita connection ∇, then one
can deﬁne a left dual and a right dual , respectively, by∗1
Rabcd ≡ − ab pq Rpqcd ,
2

1
∗
Rabcd
≡ − cd pq Rabpq .
2

The Hodge dual can be used to recast the Bianchi identities in an alternative
way. More precisely, one has that
1
1
Ra[bcd] = δ[b p δc q δd] r Rapqr = − sbcd (spqr Rapqr ) = sbcd R∗ ap sp .
6
3Thus, the ﬁrst Bianchi identity Ra[bcd] = 0 is equivalent to
∗ cb
Rab
= 0.

Furthermore,
1 abc
f ∇[a Rd |e|bc] = ∇a2(2.23)

1 abc df R ebc2= −∇a R∗d ef a .

Thus, one has that∇a R∗ abcd = 0.
Finally, it is noticed that the duals of the Weyl tensor satisfy∗∗
Cabcd = Cabcd.

Sometimes it is convenient to make use of operations of dualisation on
one or three indices. Given an arbitrary tensor Ja and another tensor Kabc
antisymmetric in abc one deﬁnes†

Jabc ≡ abc d Jd ,

‡

Ka ≡

1 bcd
a Kbcd .
6(2.24)

Using the properties of contractions of the volume form, it can be shown that‡†

Ja = Ja ,†‡Kabc = Kabc .

Further details on the calculations required to obtain all of the properties
discussed in this section can be found in Penrose and Rindler (1984).

2.6 Frame formalisms

51

2.6 Frame formalisms
Frame formalisms have been used in many areas of relativity to analyse theproperties of the Einstein ﬁeld equations and their solutions; see, for example,
Ellis and van Elst (1998); Ellis et al. (2012); Wald (1984). One of the advantagesof frame formalisms is that they lead to consider scalar objects and equations,which are, in general, simpler to manipulate than their tensorial counterparts. A
further advantage of frames is that they lead to a straight forward transcriptionof tensorial expressions into spinors; see Chapter 3.
The purpose of this section is to develop and ﬁx the conventions of a frame
formalism used in Friedrich (2004).
2.6.1 Basic deﬁnitions and conventions
Given a spacetime (M, g), let {ea } denote a frame and let {ω b } denote its dual
coframe basis. For the time being, this frame is not assumed to be g-orthogonal.
By deﬁnition one has thatω b , ea  = δa b .

(2.25)In what follows, it will be assumed one has a connection ∇ which, for the time
being, is assumed to be general ; that is, it is not necessarily metric or torsionfree. The connection coeﬃcients of ∇ with respect to the frame {ea }, to be
denoted by Γa b c , are deﬁned via∇a eb = Γa c b ec ,(2.26)

where ∇a ≡ ea a ∇a denotes the covariant directional derivative in the
direction of ea . As ∇a eb is a vector, it follows thatω c , ∇a eb  = ω c , Γa d b ed  = Γa d b ω c , ed  = Γa c b .
This expression could have been used, alternatively, as a deﬁnition of theconnection coeﬃcients. In order to carry out computations one also needs an
expression for ∇a ω b . By analogy with Equation (2.26) one can write ∇a ω b =0a b c ω c . The coeﬃcients 0a b c can be expressed in terms of the connection
coeﬃcients Γa c b by diﬀerentiating Equation (2.25) with respect to ∇d . Notingthat δa b is a constant scalar one has, on the one hand, that∇d (ω b , ea ) = ed (ω b , ea ) = ed (δa b ) = 0,
while, on the other hand, one has∇d (ω b , ea ) = ∇d ω b , ea  + ω b , ∇d ea  = 0d b c + Γd b c ω c , ea ,
so that 0d b c = −Γd b c . Consequently, one has∇a ω b = −Γa b c ω c .

(2.27)

52

Diﬀerential geometry

It is observed that the speciﬁcation of the 43 connection coeﬃcients Γa b c fully
determines the connection ∇; a generalisation of this argument shows that every
manifold admits a connection; see, for example, Willmore (1993).
Consider now v ∈ T (M) and α ∈ T ∗ (M). Writing the above in terms of theframe and coframe, respectively, one hasv = v a ea ,v a ≡ ω a , v,α = αa ω a ,αa ≡ α, ea .

In order to further develop the frame formalism it will be convenient to deﬁne∇a v b ≡ ω b , ∇a v,∇a αb ≡ ∇a α, eb .

It follows from Equations (2.26) and (2.27) that∇a v b = ea (v b ) + Γa b c v c ,∇a αb = ea (αb ) − Γa c b αc .

(2.28)

The above expressions extend in the obvious way to higher rank components.
Notice, in particular, that∇a δb c = −Γa d b δd c − Γa c d δb d = −Γa c b + Γa c b = 0.
Metric connections
Now assume that the connection ∇ is g-compatible (i.e. ∇g = 0) and that theframe {ea } is g-orthogonal; that is, g(ea , eb ) = ηab . It follows then that∇a (g(eb , ec )) = ea (ηbc ) = 0
and that∇a g(eb , ec ) = g(∇a eb , ec ) + g(eb , ∇a ec ).
Thus, using Equation (2.26) one concludes thatΓa d b ηdc + Γa d c ηbd = 0.

(2.29)

Finally, in the case of a Levi-Civita connection and with the choice of a coordinate basis {∂ μ }, the Koszul formula, Equation (2.20), shows that the connection
coeﬃcients reduce to the classical expression for the Christoﬀel symbols:Γμ ν λ =1 νρg (∂μ gρλ + ∂λ gμρ − ∂ρ gμλ ).
22.7 Congruences and submanifolds

53

2.6.2 Frame description of the torsion and curvature
Following the spirit of the previous subsections, letΣa c b ≡ ea a eb b ω c c Σa c b
denote the components of the torsion tensor Σa c b with respect to {ea } and{ω a }. Given f ∈ X(M), a short computation shows thatΣa c b ec (f ) = ∇a eb (f ) − ∇b ea (f )= (ea eb (f ) − Γa c b ec (f )) − (eb ea (f ) − Γb c a ec (f ))= [ea , eb ](f ) − (Γa c b − Γb c a ) ec (f ),where it has been used that ∇a f = ea (f ). Thus, one obtains thatΣa c b ec = [ea , eb ] − (Γa c b − Γb c a ) ec .

(2.30)

To obtain a frame description of the Riemann curvature tensor one makes useof Equation (2.8) with uc = ed c , and contracts with ea a eb b ω c d . One then has that
Rc dab ≡ ea a eb b ed d ω c c Rc dab .
Furthermore, one can computeea a eb b ω c c ∇a ∇b ed c = ω c c ∇a (∇b ed c ) − ω c c (∇a eb b )(∇b ed c ),= ω c c ∇a (Γb f d ef c ) − ω c c Γa f b ∇f ed c
= ω c c ea (Γb f d )ef c + ω c c Γb f d ∇a ef c − Γa f b Γf c d= ea (Γb c d ) + Γb f d Γa c f − Γa f b Γf c d .
A similar computation can be carried out for ea a eb b ω c c ∇b ∇a ed c so that one
obtains
Rc dab = ea (Γb c d ) − eb (Γa c d ) + Γf c d (Γb f a − Γa f b )+ Γb f d Γa c f − Γa f d Γb c f − Σa f b Γf c d .

(2.31)

Remark. Equations (2.30) and (2.31) are sometimes known as the (Cartan)
structure equations. They can be conveniently expressed in the language ofdiﬀerential forms; see, for example, Frankel (2003); Wald (1984).
2.7 Congruences and submanifolds
The formulation of an initial value problem in general relativity requires thedecomposition of tensorial objects in terms of temporal and spatial components.
This decomposition requires, in turn, an understanding of the way geometricstructures of the spacetime are inherited by suitable subsets thereof. For
concreteness, in what follows a spacetime (M, g) is assumed. Hence M is afour-dimensional manifold and g denotes a Lorentzian metric.

54

Diﬀerential geometry
2.7.1 Basic notions
Submanifolds

Intuitively, a submanifold of M is a set N ⊂ M which inherits a manifold
structure from M. A more precise deﬁnition of submanifolds requires the conceptof embedding. Given two smooth manifolds M and N , an embedding is a map
ϕ : N → M such that:(a) The push-forward ϕ∗ : T |p (N ) → T |ϕ(p) (M) is injective for every pointp ∈ N.
(b) The manifold N is diﬀeomorphic to the image ϕ(N ).
In terms of the above, one deﬁnes a submanifold N of M as the image,ϕ(S) ⊂ M, of a k-dimensional manifold S (k < 4) by an embedding ϕ : S →M. Often it is convenient to identify N with ϕ(S) and denote, in an abuse ofnotation, both manifolds by N . A three-dimensional submanifold of M is calleda hypersurface. In what follows, a generic hypersurface will be denoted by
S. As a consequence of its manifold structure, one can associate to S tangent
and cotangent bundles, T (S) and T ∗ (S) and, more generally, a tensor bundle
T• (S).
A vector u (ui ) on S can be associated to a vector of M by the push-forwardϕ∗ u. A vector on v ∈ T (M) is said to be normal to S if g(v, ϕ∗ u) = 0 for
all u ∈ T (S). If  ≡ g(v, v) = ±1, one speaks of a unit normal vector – inthis case the surface is said to be timelike if  = −1 and spacelike if  = 1.
A hypersurface S of a Lorentzian manifold M is orientable if and only if there
exists a unique smooth normal vector ﬁeld on S; see, for example, O’Neill (1983).
A natural way of specifying a hypersurface is as the level surface of somefunction f ∈ X(M). In this case one has that the gradient df ∈ T ∗ (M) gives
rise to a normal vector (df ) ∈ T (M). The unit normal of S, ν (νa ), is thendeﬁned as a unit 1-form in the direction of df ; that is, g  (ν, ν) = . The normalof S is deﬁned in the restriction to S of the cotangent bundle T ∗ (M). In thecase of a spacelike hypersurface, the normal constructed in this way is taken,
conventionally, to be future pointing.
Foliations
A foliation of a spacetime (M, g) is a family, {St }t∈R , of spacelike hypersurfaces
St , such that
St = M,
St1 ∩ St2 = Ø for t1 = t2 .
t∈R

The hypersurfaces St are called the leaves or slices of the foliation. The foliation{St }t∈R can be deﬁned in terms of a scalar ﬁeld f ∈ X(M) such that the leavesof the foliation are level surfaces of f . That is, given p ∈ St , then f (p) = t. The
scalar ﬁeld f is said to be a time function. In what follows, it will be convenient2.7 Congruences and submanifolds

55

to identify f and t. The normal of a foliation is a normalised vector ﬁeld νorthogonal to each leaf of a foliation. The gradient dt provides a further 1-form
normal to the leaves. In general, one has that
ν = N dt.
The proportionality factor N is called the lapse of the foliation.
Distributions
A distribution Π is an assignment at each p ∈ M of a k-dimensional subspaceΠ|p of the tangent space T |p (M). The vector spaces Π|p are called hyperplanes
if their dimension is one less than that of M. A submanifold N of M such that
Π|p = T |p (N ) for all p ∈ N is said to be an integrable manifold of Π. If forevery p ∈ M there is an integrable manifold, then Π is said to be integrable.
One has the following result (see e.g. Choquet-Bruhat et al. (1982) for details):
Theorem 2.1 (Frobenius theorem) A distribution Π on M is integrable if
and only if for u, v ∈ Π, one has [u, v] ∈ Π.
The projector associated to the distribution Π is a tensor ﬁeld ha bsatisfying ha b hb c = δa c such that for v a ∈ T(M) one has that ha b v a ∈ Π.

2.7.2 Geometry of congruences
Integral curves
A curve γ : I → M is the integral curve of a vector v if the tangent vectorof the curve γ coincides with v. Standard theorems of the theory of ordinary
diﬀerential equations – see, for example, Hartman (1987) – ensure that, givenv ∈ T (M), for all p ∈ M there exists an interval I  0 and a unique integral
curve γ : I → M of v such that γ(0) = p. If the domain of an integral curve isR, then the integral curve is said to be complete.
Congruences
The notion of a congruence of geodesics has been discussed in Section 2.4.5. Moregenerally, a congruence of curves is the set of integral curves of a (nowhere
vanishing) vector ﬁeld v on M. In the remaining part of this section it will beassumed that the curves of a congruence are non-intersecting and timelike. This
will be the case of most relevance in this book. In what follows, t will denotethe vector ﬁeld generating a timelike congruence. Without loss of generality it isassumed that g(t, t) = 1.
As in previous sections let {ea } denote a g-orthonormal frame. The orthonormal frame can be adapted to the congruence deﬁned by the vector ﬁeld t by

56

Diﬀerential geometry

setting e0 = t. Given a point p ∈ M, the tangent space T |p (M) is naturally
split in a part tangential to t, to be denoted by t|p (the one-dimensional
subspace spanned by t), and a part orthogonal to it which will be denotedby t⊥ |p = ei |p (the three-dimensional subspace generated by {ei } withi = 1, 2, 3). The space t⊥ |p is an example of a hyperplane. One writes then
T |p (M) = t|p ⊕ t⊥ |p ,(2.32)

where ⊕ denotes the direct sum of vectorial spaces – that is, any vector in
T |p (M) can be written in a unique way as the sum of an element in t|p andan element in t⊥ |p . Hence, one sees that the congruence generated by t gives
rise to a three-dimensional distribution Π. At every point p ∈ M, the subspaceΠp ⊂ T |p (M) corresponds to ei |p ; that is, {ei } is a basis of Πp . In the sequel,t and t⊥ will denote, respectively, the disjoint union of all the spaces t|p⊥and t|⊥p , p ∈ M, and one has that Π = t . The Frobenius theorem, Theorem
2.1, gives the necessary and suﬃcient conditions for the distribution deﬁned byt|⊥
p to be integrable; that is, for the vector t to be the unit normal of a foliation{St } of the spacetime.
Making use of g  one obtains an analogous decomposition for the cotangent
space. Namely, one has thatT ∗ |p (M) = t |p ⊕ t ⊥ |p ,(2.33)

with t ⊥ |p = ω i |p . The decompositions (2.32) and (2.33) can be extended
in a natural way to higher rank tensors by considering tensor products. Givena tensor Tab with components with respect to the frame {ea } given by Tab ,one has that Tij ≡ ei a ej b Tab and T00 ≡ ta tb Tab correspond, respectively, to thecomponents of Tab transversal and longitudinal to t; ﬁnally, T0i ≡ ta ei b Tab
and Ti0 ≡ ei a tb Tab are mixed transversal-longitudinal components.
The covariant derivative of t
To further discuss the geometry of the congruence generated by the timelike
vector t it is convenient to introduce the Weingarten map χ : t⊥ → t⊥deﬁned byχ(u) ≡ ∇u t,

u ∈ t⊥ .

One can readily verify thatg(t, χ(u)) = g(t, ∇u t) =1
∇u (g(t, t)) = 0,2(2.34)

so that indeed χ(u) ∈ t⊥ . Hence, it is enough to consider the Weingarten map
evaluated on a basis {ei } of t⊥ . Accordingly, one deﬁnes
χi ≡ χ(ei ) = χi j ej ,χi j ≡ ω j , χi .

2.7 Congruences and submanifolds

57

In the following, it will be more convenient to work with χij ≡ ηjk χi k . The
scalars χij can be considered as the components of a rank 2 covariant tensor onχ ∈ t⊥ ⊗ t⊥ – the Weingarten (or shape) tensor of the congruence. The
symmetric part θij ≡ χ(ij) and the antisymmetric part ωij ≡ χ[ij] are called the
expansion and the twist of the congruence, respectively. From g(t, ei ) = 0 it
follows that g(∇j t, ei ) = −g(t, ∇j ei ). Hence, one can computeχij = g(ei , χj ) = g(ei , ∇j t) = −g(t, ∇j ei )= −g(t, ∇i ej − [ei , ej ]) = g(∇i t, ej ) + g(t, [ei , ej ])= g(χi , ej ) + g(t, [ei , ej ])= χji + g(t, [ei , ej ]),where in the third line it has been used that ∇i ei − ∇j ei = [ei , ej ] as ∇is torsion-free. Hence, by the Frobenius theorem, Theorem 2.1, the symmetry
relation χij = χji holds if and only if the distribution t⊥ is integrable. The
components χij are related to the connection coeﬃcients of ∇ as can be seenfrom
χi j = ω j , χi  = ω j , ∇i e0  = ω j , Γi b 0 eb  = Γi j 0 .
Alternatively, one has thatχij = Γi c 0 ηcj = −Γi c j ηc0 = −Γi 0 j ,where the last two equalities follow from the metricity of the connection; see
Equation (2.29). Now, from g(t, t) = 1, it readily follows that g(∇a t, t) = 0.
Consequently, one has that the acceleration of the congruence, a ≡ ∇0 t =∇0 e0 , if non-vanishing, must be spatial; that is, g(a, t) = 0 so that a ∈ t⊥ .
Using the deﬁnition of connection coeﬃcients of the connection ∇ it follows thatai ≡ ω i , a = Γ0 i 0 .
2.7.3 Geometry of hypersurfaces
Given a spacetime (M, g) and a hypersurface thereof, S, the embedding ϕ :S → M induces on S a rank 2 covariant tensor h, the intrinsic metric or
ﬁrst fundamental form of S via the pull-back of g to S:h ≡ ϕ∗ g.
As a consequence of the deﬁnition of an embedding, the intrinsic metric h will
be non-degenerate if the hypersurface S is timelike or spacelike. Its signature
will be (+, −, −) in the former case and (−, −, −) in the latter. The (unique)Levi-Civita connection of h will be denoted by D. Alternatively, one can deﬁne
the pull-back connectionϕ∗ ∇ : T (S) × T (S) → T (S)58

Diﬀerential geometryviaϕ∗ (ϕ∗ ∇)v u ≡ ∇ϕ∗ v (ϕ∗ u),u, v ∈ T (S).

(2.35)

It can be veriﬁed that ϕ∗ ∇ as deﬁned above is indeed a linear connection. Givena function f ∈ X(M), the action of ϕ∗ ∇ on the pull-back ϕ∗ f is deﬁned by(ϕ∗ ∇)v (ϕ∗ f ) ≡ ϕ∗ (∇ϕ∗ v f ) ∈ T (M).

(2.36)In order to deﬁne the action of ϕ∗ ∇ on covectors, one requires the Leibnitz rule(ϕ∗ ∇)v ϕ∗ ω, u = (ϕ∗ ∇)v (ϕ∗ ω), u + ϕ∗ ω, (ϕ∗ ∇)v u,for ω ∈ T ∗ (M) and u, v ∈ T (S). A calculation using this expression with thedeﬁnitions (2.35) and (2.36) shows that for ω ∈ T ∗ (M) one has(ϕ∗ ∇)v ϕ∗ ω ≡ ϕ∗ (∇ϕ∗ v ω).
In a natural way, the embedding ϕ : S → M takes the connection ∇ to theconnection D. More precisely, one has the following result:Lemma 2.2 Given u, w ∈ T (S)ϕ∗ (Dw u) = ∇ϕ∗ w (ϕ∗ u).
Proof(2.37)

Given a function f ∈ X(M) one has that(ϕ∗ ∇)u ((ϕ∗ ∇)v (ϕ∗ f )) = (ϕ∗ ∇)u (ϕ∗ (∇ϕ∗ v f ))= ϕ∗ (∇ϕ∗ u ∇ϕ∗ v f )= ϕ∗ (∇ϕ∗ v ∇ϕ∗ u f )= (ϕ∗ ∇)v ((ϕ∗ ∇)u (ϕ∗ f )) ,where to pass from the second to the third line it has been used that theconnection ∇ is torsion-free. One thus concludes that the connection ϕ∗ ∇ isindeed torsion free. Finally, it can be readily veriﬁed that one has compatibility
with the metric h. Indeed,(ϕ∗ ∇)v h = (ϕ∗ ∇)v (ϕ∗ g) = ϕ∗ (∇ϕ∗ v g) = 0,
where the last equality follows from the g-compatibility of the connection ∇. Asϕ∗ ∇ is torsion-free and h-compatible, it follows from the fundamental theoremof Riemannian geometry that it must coincide with the connection D. In otherwords, one has that ϕ∗ ∇ = D, as given in Equation (2.37).

2.7 Congruences and submanifolds

59

A frame formalism on hypersurfaces
The present discussion of the geometry of hypersurfaces is valid for boththe spacelike and timelike case. To accommodate these two possibilities, allthroughout, the following conventions concerning frame indices will be used:
if the hypersurface is timelike so that  = 1, the frame indices i, j, k, . . . takethe values 1, 2, 3; if the hypersurface is spacelike so that  = −1, the indicesi, j, k, . . . take the values 0, 1, 2.
Following the conventions given in the previous paragraph, let {ei } ⊂ T (S)denote a triad of h-orthogonal vectors. If S is spacelike one has that h(ei , ej ) =−δij , while in the timelike case h(ei , ej ) = diag(1, −1, −1). Using the pushforward ϕ∗ : T (S) → T (M) one obtains the vectors ϕ∗ ei deﬁned on the
restriction of T (M) to S. The triad {ei } can be naturally extended to a tetrad{ea } on the restriction of T (M) to S by setting e0 = ν  in the spacelike case and
e3 = ν  in the timelike case. In order to discuss these two cases simultaneously,
the notation e⊥ will be used. Similarly, the notation ω ⊥ will be used to denotethe normal element of the coframe, that is, ω 0 or ω 3 . Given v and α on the
restriction of T (M) and T ∗ (M) to S, their components along the normal will
be denoted by v ⊥ and α⊥ , respectively.
To simplify the presentation, the notation ei will often be used to denote boththe vectors of T (S) and their push-forward to T (M). The appropriate point ofview should be clear from the context. In the cases where confusion may arise,
it is convenient to make use of abstract index notation: given ei ∈ T (S), weshall write ei i ; its push-forward ϕ∗ ei ∈ T (M) will be denoted by ei a . Similarly,ω i ∈ T ∗ (M) will be written as ω i a , while the pull-back ϕ∗ ω i ∈ T ∗ (S) will bedenoted by ω i i . Given u ∈ T (S), one has that ui ≡ ϕ∗ ω i , u = ω i , ϕ∗ u.
Written in index notation ua ω i a = ui ω i i ; that is, the (spatial) components of u
and its push-forward ϕ∗ u coincide.
As a consequence of the existence of two covariant derivatives, one also has
two sets of directional covariant derivatives. Firstly, acting on spacetime objects,∇a = ea a ∇a , so that in particular ∇i = ei a ∇a . Secondly, acting on hypersurfacedeﬁned objects, one has Di = ei i Di . The connection coeﬃcients of D withrespect to {ei } are given by γi j k ≡ ω j , Di ek . Now, given u ∈ T (S) andα ∈ T ∗ (S) and deﬁning
Di uj ≡ ω j , Di u,Di αj ≡ Di α, ej ,

one has, by analogy to Equation (2.28), thatDi uj = ei (uj ) + γi j k uk ,Di αj = ei (αj ) − γi k j αk .

To investigate relations between the directional covariant derivatives ∇i and
Di one makes use of the formula (2.37) with w = ei , u = ej so that ϕ∗ (Di ej ) =∇i (ϕ∗ ej ) = ∇i ej – the last equality given in a slight abuse of notation as,
strictly speaking, ∇i acts on spacetime objects. From the deﬁnition of connection
coeﬃcients one has that

60

Diﬀerential geometry
Γi j k = ω j , ∇i ek  = ω j , ϕ∗ (Di ek )
= ϕ∗ ω j , Di ek  = γi j k .

(2.38)

Given a spatial vector u ∈ T (M) (i.e. u⊥ ≡ na ua = 0) and recalling that∇a ub ≡ ea a ω b b ∇a ub , using Equation (2.28) one has that∇a ub = ea (ub ) + Γa b k uk .

(2.39)

Restricting the free frame indices in the above expression and using (2.38) one
ﬁnds∇i uj = ei (uj ) + Γi j k uk
= ei (uj ) + γi j k uk = Di uj .
The intrinsic curvature tensors on the hypersurface
In order to describe the intrinsic curvature of the submanifold S, one considers
the three-dimensional Riemann curvature tensor rk lij of the Levi-Civita
connection D of the intrinsic metric h. Given v ∈ T (S), and recalling that D is
torsion-free, one has by analogy to Equation (2.8) thatDi Dj v k − Dj Di v k = rk lij v l .
As rk lij is the Riemann tensor of a Levi-Civita connection one has the symmetries
rklij = r[kl]ij = rkl[ij] = r[kl][ij] ,rklij = rijkl ,rk[lij] = 0.

In what follows, let rlj ≡ rk lkj and r ≡ hlj rlj denote, respectively, the Ricci
tensors and scalars of D. It is convenient to also consider the trace-free partof the three-dimensional Ricci tensor sij and the three-dimensional
Schouten tensor lij given by1
sij ≡ r{ij} = rij − rhij ,3

lij ≡ sij +1
rhij .
12

The three-dimensionality of the submanifold S leads to the decomposition
rklij = 2hk[i lj]l + 2hl[j li]k .

(2.40)

A computation using the above expressions shows that the second Bianchi
identity D[i rjk]lm = 0 takes, in this case, the form
Di sij =

1
Dj r.
6

2.7 Congruences and submanifolds

61

Given the h-orthogonal triad {ei } and its associated coframe basis {ω i } one
deﬁnes the components rk lij ≡ ei i ej j ω k k el l rk lij . A computation similar to thatleading to Equation (2.31) yields
rk lij = ei (γj k l ) − ej (γi k l ) + γm k l (γj m i − γi m j )+ γj m l γi k m − γi m l γj k m .

(2.41)

Moreover, the deﬁnition of the torsion tensor implies:[ei , ej ] = γi k j − γj k i ek .

(2.42)

Remark. Equations (2.41) and (2.42) are the three-dimensional analogue of the(Cartan) structure Equations (2.30) and (2.31).
Extrinsic curvature
The discussion in Section 2.7.2 concerning the Weingarten map can be specialised
to the case of the tangent space of a hypersurface. This leads to the notion of
extrinsic curvature or second fundamental form of the hypersurface S.
The latter is deﬁned via the map K : T (S) × T (S) → R given by
K(u, v) ≡ ∇u ν, v = g(∇u ν  , v).

(2.43)

From the discussion of the Weingarten map it follows that K as deﬁned above isa symmetric three-dimensional tensor. In abstract index notation the latter will
be written as Kij .
Now, given an orthonormal frame {ei } on S and choosing v = ei and u = ej
in formula (2.43) one ﬁnds that the components Kij are given by
Kij = ∇i νj ≡ ∇j ν, ei  = ∇j ω ⊥ , ei 

(2.44)

so that, comparing Equation (2.44) with the deﬁnition of the connection
coeﬃcients one ﬁnds that
Kij = Γi a ⊥ ηaj= −Γi a j ηa⊥ = −Γi ⊥ j .

(2.45)

Now, looking again at Equation (2.39) and setting a → i, b → ⊥ one obtains∇i v ⊥ = ei (v ⊥ ) + Γi ⊥ k v k= Γi ⊥ k v k = −Kik v k ,as v ∈ T (S) so that v ⊥ = 0.

(2.46)

62

Diﬀerential geometry
The Gauss-Codazzi and Codazzi-Mainardi equations

The curvature tensors of the connections ∇ and D are related to each other bymeans of the Gauss-Codazzi equation
Rijkl = rijkl + Kik Kjl − Kil Kjk ,(2.47)and the Codazzi-Mainardi equationRi⊥jk = Dj Kki − Dk Kji .

(2.48)The proof of the Gauss-Codazzi equation follows by considering the commutator of ∇, Equation (2.8), acting on the frame vectors el :∇a ∇b el c − ∇b ∇a el c = Rc dab el d ≡ Rc lab .
Contracting the previous equation with ei a ej b ω k c , and using∇b el c = ω b b Γb a l ea c ,together with formulae (2.38) and (2.46) and the expression for the componentsof the three-dimensional Riemann tensor in terms of the connection coeﬃcients,
Equation (2.41), yields (2.47). The proof of the Codazzi-Mainardi Equation(2.48) involves less computation. In this case one evaluates the commutator ofcovariant derivatives on the covector ν. Contracting with ei a ej b ek c one readilyﬁnds that∇i ∇j νk − ∇j ∇i νk = −R⊥ kij ,where R⊥ kij ≡ Rd cab νd ek c ei a ej b . Now, using Equation (2.44) one ﬁnds that∇i Kjk − ∇j Kik = −R⊥ kij .
Formula (2.48) follows from the above expression by noticing that ∇i Kjk =Di Kjk as Kjk corresponds to the spatial components of a spatial tensor.
A remark concerning foliations
The discussion in the previous subsections was restricted to a single hypersurface
S. However, it can be readily extended to a foliation {St }. In this case thecontravariant version of the normal ν  and the unit vector t generating thecongruence coincide. Moreover, one
a distribution
which is integrable so

 has
⊥
⊥
that the Weingarten tensor χ ∈ t ⊗ t |p , for p ∈ M can be identiﬁed
with the second fundamental form K ∈ T |p (St(p) ) ⊗ T |p (St(p) ) where t(p) ∈ R
is the only value of the time function such that p ∈ St(p) . In particular one hasthat χij = χ(ij) .

2.8 Further reading

63

2.8 Further reading
There is a vast choice of books on diﬀerential geometry ranging from introductory
texts to comprehensive monographs. An introductory discussion geared towards
applications in general relativity can be found in the ﬁrst chapter of Stewart(1991) or the second and third chapters of Wald (1984). A more extensive
introduction with broader applications in physics is Frankel (2003). A moreadvanced discussion, again aimed at applications in physics, is the classicaltextbook by Choquet-Bruhat et al. (1982). A systematic and coherent discussionof the theory from a modern mathematical point of view covering topologicalmanifolds, smooth manifolds and diﬀerential geometry can be found in Lee(1997, 2000, 2002). A more concise alternative to the latter three books is given
in Willmore (1993). A monograph on Lorentzian geometry with applications togeneral relativity is O’Neill (1983). Readers who like the style of this reference will
also ﬁnd the brief summary of diﬀerential geometry given in the ﬁrst chapter of
O’Neill (1995) useful. The present discussion of diﬀerential geometry has avoided
the use of the language of ﬁbre bundles. Readers interested in the latter arereferred to Taubes (2011).
Books on numerical relativity like Baumgarte and Shapiro (2010) andAlcubierre (2008) also provide introductions to the 3+1 decomposition of generalrelativity. In these references, the reader will encounter an approach to this topic
based on the so-called projection formalism. A more detailed discussion, also
aimed at numerical relativity, can be found in Gourgoulhon (2012).

3Spacetime spinors

The notion of spinors arises naturally in the construction of a relativistic ﬁrstorder equation for a quantum wave function – the so-called Dirac equation.
Spinors are the most basic objects to which one can apply a Lorentz transformation. The seminal work in Penrose (1960) has shown that spinors constitutea powerful tool to analyse the structure of the Einstein ﬁeld equations and theirsolutions. Most applications of spinors in general relativity make use not of the
Dirac spinors but of the so-called 2-spinors. The latter are more elementary
objects, and indeed, the whole theory of the Dirac equation can be reformulated
in terms of 2-spinors. In the sequel, 2-spinors will be very often simply calledspinors.
The purpose of this chapter is to develop the basic formalism of spinors in aspacetime. Accordingly, one speaks of spacetime spinors, sometimes also called
SL(2, C) spinors; see, for example, Ashtekar (1991). A discussion of spinors in thepresence of a singled-out timelike direction, the so-called space spinor formalism,is given in Chapter 4. One of the motivations for the use of spinors in generalrelativity is that they provide a simple representation of null vectors and ofseveral tensorial operations. Although spinors will be used systematically in thisbook, they are not essential for the analysis. All the key arguments could be
carried out in a tensorial way at the expense of lengthier and less transparent
computations.
The presentation in this chapter diﬀers sligthly in focus and content fromthat given in other texts; see, for example, Penrose and Rindler (1984); Stewart(1991); O’Donnell (2003). For reasons to be discussed in the main text, asystematic use of the so-called Newman-Penrose formalism will be avoided –although the basic notational conventions of Penrose and Rindler (1984), theauthoritative work on the subject, are retained.

3.1 Algebra of 2-spinors

65

3.1 Algebra of 2-spinors
In what follows let (M, g) be a spacetime. The present discussion begins byanalysing spinorial structures at a given point p of the spacetime manifold M.
The concept of a spinor is closely related to the representation theory of thegroup SL(2, C). This group has two inequivalent representations in terms of
two-dimensional complex vector spaces which are complex conjugates of eachother; for a discussion of this aspect of the theory, see, for example, Carmeli(1977); Sexl and Urbantke (2000). Thus, the discussion of this chapter starts
with a brief discussion of complex vector spaces.
3.1.1 Complex vector spaces
By a complex vector space it will be understood a vector space over the ﬁeldof the complex numbers, C. In what follows let S denote a complex vector space,and let S∗ denote its dual, that is, the complex vector space of all linear maps
from S to C. As in the case of real vector spaces, given ς ∈ S and ζ ∈ S∗ , theapplication of ζ on ς will be denoted by ζ, ς. Notice, however, that in this caseζ, ς ∈ C.
Given S, it is natural to deﬁne an operation of complex conjugation over S:
given ς ∈ S, its complex conjugate ς̄ is deﬁned via
ζ, ς̄ ≡ ζ, ς,ζ ∈ S∗ .

The operation of complex conjugation from S to S∗ can be deﬁned in an
analogous way: given ζ ∈ S∗ , its complex conjugate ζ̄ satisﬁesζ̄, ς ≡ ζ, ς,ς ∈ S.

Given ξ, ζ ∈ S and z ∈ C, the complex conjugate of the linear combination
ξ+zζ is ξ̄+z̄ ζ̄. Thus, the operation of complex conjugation is not an isomorphism
between S and itself, but an anti-isomorphism between S and the vector spaceS̄, the complex conjugate of S. Similarly, the complex conjugation deﬁnes ananti-isomorphism between S∗ and the space, S∗ , the complex conjugate of S∗ .
If one considers the complex conjugate of the spaces S and S∗ , one recovers
the spaces S and S∗ , respectively. Moreover, because of the way the complex
conjugate operation has been deﬁned, one has that S∗ = S̄∗ , so that S and S∗
are duals of each other.
The vector spaces S, S∗ , S and S∗ will be regarded as the elementary building
blocks in the construction of a spinorial formalism. As in the case of real vector
spaces one can construct higher rank objects by considering arbitrary tensor
products of these vector spaces. This will be discussed later in the chapter once
further structure and an abstract index notation for spinors has been introduced.
3.1.2 Simplectic vector spaces
Key to the notion of spinors is the deﬁnition of a symplectic vector space.

66

Spacetime spinors

Deﬁnition 3.1 (simplectic vector space) A simplectic vector spaceconsists of an even-dimensional vector space S endowed with a function [[·, ·]] :S × S → C which is:(i) antisymmetric (skew); that is, given ξ, η ∈ S[[ξ, η]] = −[[η, ξ]](ii) bilinear; that is,[[ξ + zζ, η]] = [[ξ, η]] + z[[ζ, η]],[[ξ, η + zζ]] = [[ξ, η]] + z[[ξ, ζ]](iii) non-degenerate; that is, if [[ξ, η]] = 0 for all η then ξ = 0.
The antisymmetric product [[·, ·]] deﬁnes in a canonical way an isomorphism
between S and S∗ : to ξ ∈ S one associates ξ  ≡ [[ξ, ·]] ∈ S∗ . A transformation Q : S → S satisfying [[Qξ, Qη]] = [[ξ, η]] is called a symplectictransformation.
Remark. The rest of this book will be concerned only with the case where thedimension of S is 2.
3.1.3 Spin bases
From the deﬁnition of a symplectic vector space it follows directly that givennon-zero ξ, η ∈ S such that [[ξ, η]] = 0, there exists z ∈ C, z = 0 such that
ξ = zη. Alternatively, given ξ, η ∈ S, they are linearly independent if and only
if [[ξ, η]] = 0. This observation leads to the idea of a spin basis.
Deﬁnition 3.2 (spin basis) Given non-zero o, ι ∈ S, the pair {o, ι} is said
to be a spin basis for S if [[o, ι]] = 1.
Now, given ξ ∈ S, the components of ξ with respect to the basis {o, ι} aredeﬁned by the equationξ = ξ 0 o + ξ 1 ι,
where
ξ 0 ≡ [[ξ, ι]],ξ 1 ≡ −[[ξ, o]].

3.1.4 Abstract index notation for spinors
The discussion of spinors in this book makes use of a combination of index-free
and abstract index notations. Following the general discussion on abstract index
notation given in Penrose and Rindler (1984), an element ξ ∈ S will also bedenoted by ξ A , where the abstract superindex A provides information about thevector space to which the object belongs – in this case S. Similarly given η ∈ S∗ ,

3.1 Algebra of 2-spinors

67

it will often be written as ηA . This notation of abstract sub- and superindices will
also be extended to the vector spaces themselves; thus, the symbols SA and SA
will be used, respectively, instead of S and S∗ . Furthermore, given ξ A ∈ SA ,then ξA will denote ξ  , the dual of ξ under the antisymmetric product in S.
Following this notation, the product [[η, ξ]] = η  , ξ will be written as ηA ξ A .
In order to extend the formalism, one introduces an inﬁnite number of copies(realisations) of the spaces S and S∗ : SA , SB , . . . and SA , SB , . . . . The
diﬀerent realisations are connected to each other by a sameness map such that
ξ A and ξ B correspond to two diﬀerent copies of the same object ξ belonging to
diﬀerent realisations of S, that is, SA and SB . A peculiarity of the abstract
index notation is that although ξ A and ξ B describe the same object, expressions
like ξ A = ξ B are not allowed – the indices in an equation must be balanced.
Objects like ξ A and ηB are called valence 1 spinors. Following theterminology used for tensors, ξ A is said to be contravariant, while ηA is
said to be covariant. Higher valence spinors can be introduced using thetensorial product ⊗ of the basic vector spaces S and S∗ . The use of the abstract
index notation simpliﬁes the underlying discussion of these tensorial products.
For example, a valence 3 spinor χAB C is deﬁned through a multilinear map
χ : SA × SB × SC → C. As a consequence of the S-linearity of this mapping,
there exists a spinor χAB C ∈ SAB C . The space SAB C is a vector space. This
procedure extends in a natural way to higher valence spinors with arbitrary
combinations of covariant and contravariant indices. The collection of all the
spaces of the form SA···C D···F is called the spin algebra and is denoted by S• .
The spin algebra ensures that the multiplication of spinors renders a spinor. The
operation of addition in S• is deﬁned only between spinors of the same type, thatis, the same rank and same combination of covariant and contravariant indices.
3.1.5 The spinor AB
As the antisymmetric 2-form [[·, ·]] is a function from S ⊗ S to C, it follows that
there exists a valence 2 spinor AB ∈ SAB such that[[ξ, η]] = AB ξ A η B .
The spinor AB is called the -spinor . Now, as [[ξ, η]] = −[[η, ξ]], it follows
that AB = −BA ; that is, AB is antisymmetric. It has already been shown that[[ξ, η]] can be written as ξA η A ; thus, it follows that
ξB = AB ξ A = ξ A AB .

(3.1)

That is, AB can be regarded as an index lowering object. In other words, thespinor AB provides a convenient way to express the duality between the spaces
S and S∗ . This duality is a bijection, so that it follows that there must exist afurther spinor, (−1 )AB ∈ SAB , by means of which one can raise back the index

68Spacetime spinorsof the spinor ξA ; that is, ξ A = (−1 )CA ξC . In order to simplify the appearanceof the above expressions it is convenient to deﬁne a further spinor AB ∈ SAB
via
AB ≡ −(−1 )AB ,(3.2)

ξ A = −CA ξC .

(3.3)

so that one obtains

Combining Equations (3.1) and (3.3) one obtains ξB = −AB CA ξC , which
together with the requirement that AB and (−1 )AB represent inverse operations, implies
δB C = −AB CA ,with δB C the two-dimensional Kronecker’s delta. The spinor AB is also
antisymmetric. This can be seen from[[ξ, η]] = ξB η B = ξB δC B η C = −ξB (DC BD )η C
= BD ξB (CD η C ) = BD ξB ηD .
A similar computation shows that [[η, ξ]] = DB ηD ξB . Finally, as [[ξ, η]] =−[[η, ξ]] one concludes that AB = −BA as claimed.
If A C and A C denote the spinors in S• obtained by raising the ﬁrst andsecond index of AB , respectively, it follows from the above calculations thatC A = −A C = δC A ,AB AB = A A = 2.

The above formulae lead to the so-called see-saw rule. Given a spinor χP ···QA
one has thatχP ···QA = AB χP ···Q B = −χP ···Q B BA = χP ···QB B A ,χP ···QA = −AB χ

P ···QB=χP ···QBBA = −χP ···QB

B  A.

(3.4a)(3.4b)

Comparing the above expressions one concludes thatχP ···Q A A = −χP ···QA A .
3.1.6 The Jacobi identity and decompositionsin irreducible components
As S is a vector space of dimension 2, it follows that any antisymmetrisation overa set of three or more spinorial indices must vanish. In particular, one obtains
what is known as the Jacobi identity :A[B CD] = AB CD + AC DB + AD BC = 0.
A direct consequence of the Jacobi identity is the following lemma:(3.5)

3.1 Algebra of 2-spinors

69

Lemma 3.1 (irreducible decomposition of a pair of indices) Consider
the spinor ζ···AB··· . Then1
ζ···AB··· = ζ···(AB)··· + AB ζ···C C ··· .
2
ProofConsider the Jacobi identity rewritten in the form
A C B D − B C A D = AB CD ,and multiply it by ζ···CD··· . One readily obtains
2ζ···[AB]··· = AB ζ···C C ··· .
Finally, combining the latter with the identity
ζ···AB··· = ζ···(AB)··· + ζ···[AB]··· ,one obtains the required result.
The previous result can be used to interchange the order of two spinorial
indices. In this case Lemma 3.1 directly yields
ζ···BA··· = ζ···AB··· − AB ζ···P P ··· .

(3.6)

The above lemma leads to the following result:
Proposition 3.1 (irreducible decomposition of spinors) Any spinor ζA···F
can be decomposed as the sum of the spinor ζ(A···F ) and products of -spinors withsymmetrised contractions of ζA···F .
Proof Assume ζABC···F to have valence n. In the following argument, thesymbol ∼ between two spinors indicates that their diﬀerence is a linear
combination of the outer product of -spinors and spinors of lower valence. The
key idea of the decomposition is to show that
ζABC···EF ∼ ζ(ABC···EF ) .
To this end, one ﬁrst notices thatnζ(ABC···EF ) = ζA(BC···EF ) + ζB(AC···EF ) + ζC(AB···EF ) + · · · + ζF (AB···E) . (3.7)
Now, one looks at the terms in the right-hand side of the above equation andconsiders the diﬀerence between the ﬁrst and the second term, the ﬁrst and thethird term and so on. Using Lemma 3.1, these diﬀerences can be rewritten as
ζA(BC···EF ) − ζB(AC···EF ) = −ζ X (XC···EF ) AB ,ζA(BC···EF ) − ζC(AB···EF ) = −ζ X (XB···EF ) AC ,..
.
ζA(BC···EF ) − ζF (ABC···E) = −ζ X (XBC···E) AF .

70

Spacetime spinors

The above expressions can be used in Equation (3.7) to eliminate the terms
ζB(AC···EF ) ,ζB(AC···EF ) ,···ζF (ABC···E) .

One obtains
ζ(ABC···EF ) = ζA(BC···EF ) +1 X
1ζ (XC···EF ) AB + · · · + ζ X (XBC···E) AF .
nnThat is,ζ(ABC···EF ) ∼ ζA(BC···EF ) .
The procedure described above can be repeated for each of the terms
ζ X (XC···EF ) ,···ζ X (XB···E) ,to obtain
ζ(ABC···EF ) ∼ ζA(BC···EF ) ∼ ζAB(C···EF ) ∼ · · · ∼ ζABC···(EF ) ∼ ζABC···EF .

Remark. If one has a spinor with a set of contravariant indices, these can belowered so that Proposition 3.1 applies.
The type of decompositions of spinors provided by Proposition 3.1 will be usedsystematically in the rest of the book. A particularly useful example is given by1
1
1
χABCD = χ(ABCD) + χ(AB)P P CD + χP P (CD) AB + χP P Q Q AB CD
2
2
4
1
1
1
+ A(C χD)B + B(C χD)A − A(C D)B χ,
(3.8)
2
2
3
with
χAB ≡ χQ(AB) Q ,χ ≡ χP Q P Q .

A decomposition like the one given in Equation (3.8) will be called a decomposition in irreducible components. The spinors χ(ABCD) , χ(AB)P P , . . . , χ
are independent in the sense that χABCD = 0 if and only if
χ(ABCD) = 0,

χ(AB)P P = 0,···

χ = 0.

The latter fact will be used repeatedly in the following. Finally, it is observedthat the number of independent components an arbitrary symmetric spinor canhave is given by the following proposition; see Penrose and Rindler (1984).
Proposition 3.2 (number of independent components) If ζA···C = ζ(A···C)
is of valence p, then it has (p + 1) independent components.
In conjunction with Proposition 3.1 the latter result can be used to count the
total number of independent components of an arbitrary spinor.

3.1 Algebra of 2-spinors

71

3.1.7 Components with respect to a basis
As in the case of tensors, it is often convenient to discuss spinors in terms ofa speciﬁc basis. To express this idea, it is convenient to introduce bold indicesAand ηA represent the components of
A , B , . . . ranging over 0 and 1. Thus, ξAξ and ηB with respect to a speciﬁc basis. This idea extends in a natural way
to higher valence spinors.
Given a spin basis {o, ι}, one often requires a notation to describe the basis
in a more systematic manner. This will be done by means of the symbol A A
where
0 A ≡ o A ,1 A ≡ ι A .

(3.9)

Similarly, the dual cobasis of A A will be denoted collectively by A A . By
deﬁnition one has that
 A A  B A = δA B .
It follows from Equation (3.9) and the previous condition that
0 A = −ιA ,1 A = o A .

Using this notation and given two spinors ξ A and ηB , one can writeξ A = ξ A A A ,η B = η B B B ,ξ A ≡ ξ A A A ,η B ≡ η B B B .

where

Hence



[[η, ξ]] = ηA ξ A = ηP P A ξ Q Q A = ηP ξ P .
The components AB of the antisymmetric spinor AB with respect to thebasis A A are given by

 



oA oA oA ιA
0 1
AB(AB ) ≡ AB A B==.
(3.10)ιA oA ιA ιA−1 0
Now, a direct computation shows that

−1 
0
0 1
=
1
−1 0

−1
0


.

Hence, consistent with Equation (3.2) one has that
 AB A B 
0 1AB( ) ≡   A  B =.
−1 0
An alternative way of rewriting the previous discussion isδA B = A A A B ,AB = AB A A B B ,AB = AB A A B B .

72

Spacetime spinors

From the latter it follows that
δA B = o A ι B − ι A o B ,(3.11a)AB = oA ιB − ιA oB ,(3.11b)=o ι −ι o .

(3.11c)



AB

A B

A B

3.1.8 Complex conjugation of spinors
In order to relate spinors with tensors one has to consider the operation ofcomplex conjugation discussed in Section 3.1.1. The convention to denote theoperation of complex conjugation in the abstract index notation is to add a bar
to the kernel symbol and a prime to each of the indices. For example, one has that

ζ A = ζ̄ A ∈ SA .
¯ = ζ.
The operation of complex conjugation is idempotent – given ζ ∈ S, then ζ̄Using abstract index notation one writes the latter as ζ̄ A = ζ A .
A spinor ξ A···CS ···U D···EW  ···Y  with, say, p unprimed contravariant indices,r primed contravariant indices, q unprimed covariant indices and s primed
covariant indices describes the most general type of spinors. It is obtained fromthe S-linear mapξ : SA × · · · × SC × SS  × · · · × SU  × SD × · · · × SE × SW × · · · × SY → C.
p timesq timesr timess times

The algebra S• is then extended to accommodate this more general type ofspinors with unprimed and primed indices.
An important consequence of the fact that the spaces S and S̄ are notisomorphic is that it is not possible to single out 2-spinors which are intrinsicallyreal or imaginary unless one assumes further structure on S• . From a notational
point of view, as S and S̄ are not isomorphic, the relative position of primed andunprimed indices is irrelevant. Thus, one can write expressions like ζAA = ζA A .
Notice, in contrast, that the reordering of groups of primed indices or groups ofunprimed indices is not allowed unless the spinor possesses special symmetries.
The rules for the raising and lowering of indices of valence 1 spinors areextended to higher valence spinors in a natural way. Primed indices are raised and  lowered using the spinors A B ∈ SA B and A B  ∈ SA B  which are related,
respectively, to AB and AB by complex conjugation. That is,¯A B  ≡ AB ,¯A B ≡ AB .


It is conventional to write A B  , A B instead of ¯A B  and ¯A B .
Finally, note that the discussion of Section 3.1.6 concerning the decomposition of spinors in irreducible components, and in particular Lemma 3.1 and
Proposition 3.1, can be directly extended to the case of spinors containing primed

3.1 Algebra of 2-spinors

73

indices or combinations of primed or unprimed indices. In particular, one has thefollowing decomposition of a spinor with two unprimed and two primed indices:
11
ηAA BB  = η(AB)(A B  ) + ηP P (A B  ) AB + η(AB)Q Q A B 22
1+ AB A B  ηQ Q Q Q .
4(3.12)A particular case of the above decomposition is when ζAA BB  is the spinorial
counterpart of an antisymmetric rank-2 tensor ζab = −ζba . In this case one hasthatζAA BB  = ζAB A B  + ζ̄A B  AB ,(3.13)



where ζAB ≡ 12 ζAP  B P , and one has that ζAB = ζ(AB) .
3.1.9 The relation between spinors and tensors
Spinors provide a simple representation of several tensorial operations. Although
every four-dimensional tensor (world tensor ) can be represented in terms ofspinors, the converse is not true. There are spinors which admit no discussion interms of tensors. This observation is based on the fact that 2-spinors are related
to representations of the group of (2×2) complex matrices with unit determinant,SL(2, C), while tensors are related to the Lorentz group. These groups are notisomorphic to each other. The group SL(2, C) covers the Lorentz group in a
2 : 1 way; see, for example, Carmeli (1977); Sexl and Urbantke (2000) for further
discussions on this issue.
Hermitian spinors
The key property to relate 2-spinors to world tensors is hermicity. A spinorξ ∈ S• is said to be Hermitian if and only if ξ = ξ̄, that is, if the spinor isequal to its complex conjugate. For this to be the case, ξ needs to have the same
number of unprimed and primed indices. By raising and lowering the indices asnecessary one can, without loss of generality, assume that the spinor has the same
number of unprimed and primed contravariant indices and the same number of
unprimed and primed covariant indices, for example, ξAA ···DD EE ···HH . In thiscase the hermicity condition reads








ξAA ···DD EE ···HH = ξ¯AA ···DD EE ···HH ,where on the right-hand side it has been used that the position of primed andunprimed indices can be interchanged.
Consider now ξ AA ∈ SAA . If {o, ι} and {ō, ῑ} are, respectively, spin bases of
S and S̄, one can write










ξ AA = aoA ōA + bιA ῑA + coA ῑA + dιA ōA ,(3.14)

74

Spacetime spinors
for some a, b, c, d ∈ C. In other words, a pair AA of indices is associated tofour complex components. If one assumes, in addition, ξ AA to be Hermitian,¯ Thus, the hermicity condition reducesthen it follows that a, b ∈ R and c = d.
the number of independent components to four real ones. Consequently, one can
think of the Hermitian spinor ξ AA ∈ SAA as describing a four-dimensional
vector (world-vector ) ξ a .
The argument described in the previous paragraph can be extended in a
natural fashion to higher valence Hermitian spinors, ξAA ···DD EE ···HH , so that
one can regard each pair of unprimed-primed indices (i.e. AA , EE , · · · ) as
associated to a tensorial index (i.e. a , e , · · · ).
In what follows let
gAA BB  ≡ AB A B  .

(3.15)

A computation then shows that ḡAA BB  = gAA BB  and, in addition, that








g AA BB = AB A B ,












gAA BB  g BB CC = gAA CC ≡ δA C δA C ,gAA BB  g AA BB = 4,
gAA BB  = gBB  AA .
Furthermore, given vAA ∈ SAA it can be readily veriﬁed thatvAA g AA BB = v BB ,

v AA gAA BB  = vBB  .

Hence, the spinor gAA BB  has all the properties of a spinorial counterpart of themetric tensor. These ideas will now be put in more precise terms.
The Infeld-van der Waerden symbols
In order to describe explicitly the correspondence between spinors and tensors at
a point p ∈ M, consider a basis {ea } ⊂ T |p (M) and let gab ≡ g(ea , eb ) denotethe components of the metric g with respect to this basis. Let also {ω a } ⊂T ∗ |p (M) denote the dual basis to {ea } so that ω b , ea  = δa b . It is conventional
to assume that the basis is g-orthogonal ; that is, gab = ηab . Finally, let {A } ⊂ S
denote a spin basis, and let AB denote the components of the spinor AB withrespect to the latter basis. The scalars gab and AB can be put in correspondencewith each other via an equation of the form
AB A B  = σ a AA σ b BB  ηab ,(3.16)

where σ a AA are the so-called Infeld-van der Waerden symbols. These canbe regarded as the entries of four (2 × 2) matrices (σ a AA ), a = 0, . . . , 3.
Unprimed indices denote the rows and the primed indices the columns of thematrix. Given σ a AA , one deﬁnes the inverse symbol σb BB via the relations
σa AA σ b AA = δa b ,

σa AA σ a BB  = δB A δB  A .

(3.17)

3.1 Algebra of 2-spinors

75

From these expressions it follows that the correspondence (3.16) can be inverted
to yieldηab = σa AA σb BB AB A B  .

(3.18)

Using Equation (3.18) and observing that ηab = ηab , it follows that

σa AA = σa AA .

(3.19)

Hence, (σa AA ) and (σ a AA ) describe Hermitian matrices. An explicit computation shows that the matrices






1
1
1 0
0 1
(σ0 AA ) ≡ √,(σ1 AA ) ≡ √
,
0 1
1 0
2
2




1
1
0 i
1 0
AA
AA
)≡ √
)≡ √
(σ2
, (σ3
,
−i 0
0 −1
2
2
and
(σ

0

AA

1
)≡ √
2

1
(σ AA ) ≡ √
2




2



1
0

0
1

0
i

−i
1

,

(σ


,

1

AA

1
)≡ √
2



1
(σ AA ) ≡ √
2
3

0
1


1
0
1
0


,
0
−1


,

satisfy the relations (3.16), (3.17), (3.18) and (3.19). The above matrices
correspond, up to a normalisation factor, to the so-called Pauli matrices.
Now, consider arbitrary v ∈ T |p (M) and α ∈ T ∗ |p (M). In terms of the bases{ea } and {ω a }, v and α can be written asv = v a ea ,v a ≡ ω a , v,α = αa ω a ,αa ≡ α, ea .

The components v a and αa can be put in correspondence with Hermitian spinors
using the Infeld-van der Waerden symbols via the rules

v a → v AA = v a σa AA ,(3.20a)αa → αAA = αa σ AA .

(3.20b)a

In terms of arrays of explicit components and matrices one has 01v + v 3 v 1 + iv 2
0 1 2 3(v , v , v , v ) → √,v 1 − iv 2 v 0 − v 321
α0 + α3 α1 − iα2(α0 , α1 , α2 , α3 ) → √.
α1 + iα2 α0 − α32A quick computation shows thatα, v = v a αa = v AA αAA








= v 00 α00 + v 01 α01 + v 10 α10 + v 11 α11= v 0 α0 − v 1 α1 − v 2 α2 − v 3 α3 .

76Spacetime spinors

Thus, one has that the assignments deﬁned in (3.20a) and (3.20b) are consistent
with the inner product deﬁned on T |p (M) by the metric g.
The assignment given by (3.20a) and (3.20b) can be extended to tensors ofarbitrary rank. For example, given the tensor Tab c , denote its components withrespect to {ea } and {ω b } by Tab c . One then has the assignment

Tab c → TAA BB  CC ≡ σ a AA σ b BB  σc CC Tab c .
The object TAA BB  CC will be called the spinorial counterpart of the tensor
components Tab c .
3.1.10 The spinorial representation of null vectors
As already mentioned in the introduction to this chapter, one of the key
advantages of the use of spinors is the convenient representation of null vectors
they provide. More precisely, one has the following result:
Proposition 3.3 (spinorial counterpart of null vectors) The spinorial
counterpart of a non-vanishing real null vector k a can be written as

k AA = ±κA κ̄A ,(3.21)

for some valence 1 spinor κA .


Proof A direct computation shows that k AA as given by Equation (3.21) is
indeed the spinorial counterpart of a null vector. Conversely, a computation
yieldsg(k, k) = AB A B  k AA k BB











= 2(k 00 k 11 − k 01 k 10 ) = det(k AA ).


Thus, the requirement g(k, k) = 0 implies that k AA , regarded as a (2×2) matrix,
has rows/columns which are linearly dependent. Accordingly, there exist valence
1 spinors κA and λB such that k AA = κA λ̄A . As, k is non-zero, it follows that
κA , λB = 0. From the reality of k, it follows that its spinor counterpart k AAmust be Hermitian; that is, k AA = k̄ AA . Hence, κA λ̄A = κ̄A λA . Contracting
the latter with κA one has that κA λA = 0, so that κA and λA must be
proportional to each other. The proportionality factor can be absorbed into κA
by means of a redeﬁnition of the spinor. The sign in Equation (3.21) is that ofthe proportionality constant.
Remark. A null vector constructed using the positive sign in Equation (3.21)
will be said to be future pointing , while one using the negative sign will becalled past pointing .
From Proposition 3.3 it follows that every valence 1 spinor κA deﬁnes a null
vector k. However, this is not a one-to-one correspondence. More precisely, a3.1 Algebra of 2-spinors

77

spinor diﬀering from κA by a complex phase, that is, eiϑ κA , with ϑ ∈ R will giverise to the same null vector. The phase change is said to be right-handed if
ϑ > 0. This phase does not aﬀect the construction of the vector k. Nevertheless,it contains some geometric information. To see this, consider a further spinor
μA such that κA μA = 1 so that {κA , μA } constitute a spin basis. Now, one canreadily verify that
1sAA ≡ √ (κA μ̄A + μA κ̄A ),
2




i
tAA = √ (κA μ̄A − μA κ̄A ),2

are the spinorial counterparts of two unit spacelike vectors s and t and that they
are both orthogonal to k. At each point p ∈ M, s and t span a subspace ofT |p (M) which is orthogonal to k. This subspace is called the ﬂag of the spinorκA ; the pole of the ﬂag is the vector k.
Now, suppose κA is subject to a phase change such that
κA → eiϑ κA .

(3.22)In order to retain the normalisation κA μA = 1, the transformation (3.22) implies
the transformation μA → e−iϑ μA . Furthermore, one has thats → cos 2ϑs + sin 2ϑt,t → − sin 2ϑs + cos 2ϑt,so that a phase change of ϑ in κA implies a change of 2ϑ in its ﬂag; the ﬂagpole,
however, remains unchanged.
3.1.11 Null tetrads
Inspection of Equation (3.14) shows that every spin basis {o, ι} gives rise toan associated vector basis consisting of null vectors. This null tetrad has
the peculiarity of consisting of two real null vectors and two complex null
vectors which are the complex conjugates of each other. In order to analyse
this further, let

lAA ≡ oA ōA ,nAA ≡ ιA ῑA ,mAA ≡ oA ῑA ,m̄AA ≡ ιA ōA .

Furthermore, let la , na , ma and m̄a (or l, n, m, m̄) denote the tensorial
counterparts of the above spinors. Using the above deﬁnitions one can verify
that
la na = −ma m̄a = 1,

(3.23)

while all the other remaining contractions vanish. Using relations (3.11a)–(3.11c)
it can be readily shown thatgab = 2l(a nb) − 2m(a m̄b) ,g ab = 2l(a nb) − 2m(a m̄b) .

78Spacetime spinors

An orthonormal tetrad {ea } can be readily obtained from the null tetrad{l, n, m, m̄}. Namely, let
1e0 = √ (l + n),21e1 = √ (m + m̄),2ie2 = √ (m − m̄),21
e3 = √ (l − n).
2(3.24a)(3.24b)(3.24c)(3.24d)Using the relations in (3.23) it can be veriﬁed that the latter vectors indeed
constitute an orthonormal tetrad. Furthermore, it can be readily checked thate0 is timelike while e1 , e2 and e3 are spacelike. The vector e0 is said to be
future pointing as both l and n are future pointing in the sense of Section 3.1.10.
Moreover, a right-handed phase change (i.e. ϑ > 0) in the spin basis of the formoA → eiϑ oA , ιA → e−iϑ ιA leads to the right-handed rotations
e1 → cos 2ϑe1 + sin 2ϑe2 ,e2 → − sin 2ϑe1 + cos 2ϑe2 ,while at the same time leaving e0 and e3 unchanged. Accordingly, the triadof spacelike vectors {e1 , e2 , e3 } deﬁned by (3.24b)–(3.24d) is said to be righthanded. The inverse relations to (3.24a)–(3.24d) are given by1
l = √ (e0 + e3 ),21
m = √ (e1 − ie2 ),21
n = √ (e0 − e3 ),21m̄ = √ (e1 + ie2 ).
2

The spinorial counterpart of the volume form
The spinorial counterpart of the volume 4-form abcd is given by
AA BB  CC  DD = i(AB CD A C  B  D − AC BD A B  C  D ).

(3.25)

Using the Jacobi identity (3.5) it can be veriﬁed that the above expression isindeed totally antisymmetric under interchange of the pairs AA , BB  , CC  andDD  . Moreover, one has








AA BB  CC  DD AA BB CC DD = 24,
and








σ0 AA σ1 BB σ2 CC σ3 DD AA BB  CC  DD = 1;
compare Section 2.5.3. The expression (3.25) can be deduced applying a
decomposition in irreducible components to AA BB  CC  DD and exploiting its
antisymmetry properties.

3.1 Algebra of 2-spinors

79

3.1.12 Changes of basis and SL(2, C) transformations
Let {A A } and {˜A A } denote two spin bases for S. The spinors of one basis canbe expressed as linear combinations of the spinors of the other basis. This canbe conveniently be written as
˜A A = ΛA P P A ,(3.26)

where (ΛA P ) denotes an invertible (2 × 2) matrix. The associated spinor cobases
A A } are related in a similar way:{A A } and {˜˜A A = ΛA P P A ,(3.27)

where (ΛA P ) is another invertible (2 × 2) matrix. Now, one has that

 


δA B = ˜A P ˜B P = ΛA P P Q ΛB Q Q Q = ΛA P ΛB Q P Q Q Q
= Λ A P Λ B Q δP Q = Λ A P Λ B P .
Hence, the matrices (ΛA P ) and (ΛA P ) are inverses of each other.
Now, given a contravariant valence 1 spinor κA , one can expand it in terms of
A A } as
the bases {A A } and {˜κA = κA A A = κ̃A ˜A A .
As a consequence of the change of basis (3.26), the coeﬃcients κA and κ̃A are
related to each other via
κ̃A = ΛA P κP .
Similarly, from the transformation rule (3.27), the components μA and μ̃A of a
A A }
valence 1 covariant spinor μA with respect to the spin cobasis {A A } and {˜can be found to be related via
μ̃A = ΛA P μP .
The transformation rules given in the previous paragraph can be extended
in a natural way to higher valence spinors and to spinors with primed indices.
For example, if v AA and ṽ AA denote the components of the spinor v AA withrespect to the two diﬀerent sets of bases, one has thatṽ AA = ΛA P Λ̄A P  v P P .
A case of special importance is that of the antisymmetric spinor AB for which
the transformation rule between bases is given by˜AB = ΛA P ΛB Q P Q .

(3.28)

Earlier in the chapter, the notion of simplectic transformations was introduced. The properties of these transformations can be investigated from
Equation (3.28). As a consequence of the discussion of Section 3.1.7 the matrices

80Spacetime spinors(AB ) and (˜AB ) both have the form given by Equation (3.10). It follows from
Equation (3.28) that2det (˜AB ) = det ΛA B
det(AB ).
Furthermore as det (˜AB ) = det(AB ) = 1, one concludes that det ΛA B = ±1.
Hence, if one restricts attention to the transformations with positive determinant,one ﬁnds that the set of transformations that preserve the antisymmetric product[[·, ·]] is given by the group SL(2, C).
Relation to the Lorentz transformations
Following the discussion of the previous paragraphs, the components gAA BB  ofthe spinorial counterpart of the metric transform under a change of spin basis asg̃AA BB  ≡ ˜AB ˜A B  = ΛA P Λ̄A P ΛB Q Λ̄B  Q P Q P  Q .
Using the Infeld-van der Waerden symbols, the latter can be rewritten asη̃ab = Λa c Λb d ηcd ,
with




Λa c ≡ σa AA σ c P P  ΛA P Λ̄A P .
The above expression provides the relation between SL(2, C) and Lorentz
transformations; see, for example, Sexl and Urbantke (2000) for more details.
3.1.13 Soldering forms
The connection between spinors and world tensors has been implemented interms of the components with respect to some vector and spin bases. There is adiﬀerent perspective of this translation in terms of so-called soldering forms.
The metric tensor g can be written in terms of the orthonormal cobasis {ω a } asg = ηab ω a ⊗ ω b .
This last expression can be rewritten, using the correspondence (3.18), as








g = AB A B  σa AA σb BB ω a ⊗ ω b = AB A B  ω AA ⊗ ω BB ,

(3.29)



where ω AA ≡ σa AA ω a . The four covectors {ω AA } are called the solderingforms . In terms of abstract index notation one writes the soldering form asω AA a . A similar discussion can be made with the contravariant metric g  . Fromg  = η ab ea ⊗ eb , together with (3.16), one can writeg  = AB A B eAA ⊗ eBB  ,(3.30)

3.2 Calculus of spacetime spinors

81

where eAA ≡ σ a AA ea . In abstract index notation one would write eAA ainstead of eAA . In view of the above, given a vector v ∈ T |p (M) and a covector
α ∈ T ∗ |p (M), one can writev = v AA eAA ,

α = αAA ω AA .

As a ﬁnal remark concerning the connection between spinors and world tensors,
it is observed that ea = δa b eb . Thus, δa b can be interpreted as the components
ea b of the frame vector ea with respect to the frame {ea }. Contracting ea b withσb BB one ﬁndsea BB ≡ ea b σb BB = σa BB .
3.2 Calculus of spacetime spinors
The discussion of the previous section has been restricted to spinors at a given
point of the spacetime manifold M. It is now assumed that a spinorial structure
can be constructed in a consistent way on the whole of M – the conditions
ensuring this are discussed in Section 3.3, and essentially amount to requiring
the spacetime to be orientable. The spinorial structure over M (also called aspin bundle) will be denoted by S(M). Consistent with this notation, thespinorial structure at a point p ∈ M will be denoted by S|p (M).
As is the case with tensors, the idea of relating spinors deﬁned at diﬀerent
points of the spacetime manifold requires the use of the notion of a connection
and its associated covariant derivative. Thus, it is necessary to extend thenotion of a connection in such a way that it applies to spinor ﬁelds. In what
follows, by a spinor ﬁeld it is understood a smooth assignment of a spinor,say, ξA···CD ···F  G···LP ···N , to each point of the spacetime manifold. The setsof spinorial ﬁelds over M will be denoted in a similar manner to the sets ofspinors at a point, that is, S• (M), SA (M), SA (M), SAA B (M), and so on.
3.2.1 The spinorial covariant derivative
A spinor covariant derivative ∇AA is a map∇AA : SB···C D···E  (M) → SB···C AD···A E  (M).
Given an arbitrary spinor ζ B···C D···E  , its spinorial covariant derivative will bedenoted by ∇AA ζ B···C D···E  . The mapping deﬁned by ∇AA is required to satisfy
the following properties:(i) Linearity. Given ζ B···C D···E  , η B···C D···E  ∈ SB···C D···E  (M),








∇AA (ζ B···C D···E  + η B···C D···E  ) = ∇AA ζ B···C D···E  + ∇AA η B···C D···E  .

82Spacetime spinors








∈

(ii) Leibnitz rule. Given ﬁelds ζ B···C D···E ξ F ···G H···I  ∈ SF ···G H···I  (M),

SB···C D···E  (M) and

∇AA (ζ B···C D···E  ξ F ···G H···I  ) = ξ F ···G H···I  ∇AA ζ B···C D···E + ζ B···C D···E  ∇AA ξ F ···G H···I  .
(iii) Hermicity. Given ζ B···C D···E  ∈ SB···C D···E  (M),∇AA ζ B···C  D···E  = ∇AA ζ̄ B ···C D ···E .
(iv) Action on scalars. Given a scalar φ, then ∇AA φ is the spinorial
counterpart of ∇a φ.
(v) Representation of derivations. Given a derivation D on spinor ﬁelds,
there exists a spinor ξ AA such thatDζ B···C D···E  = ξ AA ∇AA ζ B···C D···E  ,

for all ζ B···C D···E  ∈ S• (M).
Remark. The above list of properties is more general than the ones given in, say,Penrose and Rindler (1984) and Stewart (1991), as the present discussion does
not assume that the spinor covariant derivative is compatible with the -spinor;
that is, ∇AA BC = 0.
For completeness, the following result proved in Penrose and Rindler (1984)
is recalled:
Theorem 3.1 (existence of the spinorial covariant derivative) Everycovariant derivative ∇ over M has a spinorial counterpart ∇AA .
3.2.2 Spin connection coeﬃcients
In speciﬁc computations, given a spin basis {A A }, it is convenient to introduce
the notion of the spin connection coeﬃcients associated to a certain
connection. The direct spinorial counterparts of the connection coeﬃcients Γa c b
are given after suitable contraction with the Infeld-van der Waerden symbols by
the spinor componentsΓAA BB CC  ≡ ω BB BB  ∇AA eCC  BB ,(3.31)



where ∇AA ≡ eAA AA ∇AA denotes the directional covariant derivativein the direction of eAA . Now, using that

ω BB BB  = B B ¯B B  ,eCC  CC = C C ¯C  C ,it follows that








ΓAA BB CC  = B B ¯B B  ¯C  B ∇AA C B + B B ¯B B  C B ∇AA ¯C  B= B B δC  B ∇AA C B + ¯B B  δC B ∇AA ¯C  B .



3.2 Calculus of spacetime spinors

83

Hence, deﬁning the spin connection coeﬃcientsΓAA B C ≡ B B ∇AA C B ,(3.32)

one obtains
ΓAA BB CC  = ΓAA B C δC  B + Γ̄AA B C  δC B .

(3.33)

Using δC B = C Q B Q , the deﬁnition of ΓAA B C and requiring that
∇AA δC B = 0
one also has that
ΓAA B C = −C Q ∇AA B Q .
The spin connection coeﬃcients provide a way of computing the covariant
derivative of spinors without a tensorial counterpart. Given κA = κA A A ∈SA (M) one has that
∇AA κB ≡ B Q ∇AA κQ= B Q ∇AA (κP P Q )= B Q eAA (κP )P Q + κP ∇AA P Q
= eAA (κB ) − ΓAA P B κP .
Similar computations show, for example, that
∇AA ζ B = eAA (ζ B ) + ΓAA B P ζ P ,∇AA ξB  CC = eAA (ξB  CC ) − Γ̄AA Q B  ξQ CC








+ΓAA C Q ξB  QC + Γ̄AA C Q ξB  CQ .
The generalisation to spinors of arbitrary valence and number of primed indicescan be readily obtained from the above examples.
Metric and Levi-Civita spin connection coeﬃcients
So far, the discussion of the spin connection coeﬃcients has been completely
general. In the present section it is assumed that the connection is metric.
The spinorial counterpart of the metric compatibility condition ∇a gbc = 0 is
given by∇AA (BC B  C  ) = B  C  ∇AA BC + BC ∇AA B  C  = 0.
Regarding the second equality as a (partial) decomposition in irreducible terms,
one has that
∇AA BC = 0,∇AA B  C  = 0.

84Spacetime spinors

In order to investigate the implications of a metric connection on its associatedspin connection coeﬃcients, it is convenient to compute∇AA BC = eAA (BC ) − ΓAA Q B QC − ΓAA Q C BQ= − ΓA A CB + ΓAA BC = 0
as eAA (BC ) = 0; again, the components BC are constants. Hence, one
concludes that
ΓAA BC = ΓAA (BC) .
3.2.3 Spinorial curvature
The spinorial counterpart of the curvature tensors can be introduced in a natural
way by looking at the commutator of spinorial covariant derivatives. Moreprecisely, one can write∇AA , ∇BB  ξ CC = RCC P P  AA BB  ξ P P

(3.34)with∇AA , ∇BB   ≡ ∇AA ∇BB  − ∇BB  ∇AA − ΣAA P P BB  ∇P P  ,consistent with the notation of Section 2.4.3 and with ΣAA CC BB  representing the spinorial counterpart of the torsion tensor of ∇. The spinor
RCC DD AA BB  is the spinorial counterpart of the Riemann curvaturetensor Rc dab . In the following discussion it is assumed that the connection ∇is completely general – in particular, it could have torsion and be non-metric,
so that ∇AA BC = 0. As a consequence, the curvature spinor has only thesymmetry

RCC DD AA BB  = −RCC DD BB  AA .
The curvature spinor in terms of the spin connection coeﬃcients
In order to obtain a simpler representation of the curvature spinor it is convenient
to look ﬁrst at its expression in terms of spin connection coeﬃcients. To thisend, one can consider the frame expression (2.31) for the Riemann tensor, andcontract it with the Infeld-van der Waerden symbols. One readily obtains


RCC DD AA BB  = eAA (ΓBB  CC DD ) − eBB  (ΓAA CC DD )




















+ΓF F  CC DD ΓBB  F F AA − ΓF F  CC DD ΓAA F F BB 
+ΓBB  F F DD ΓAA CC F F  − ΓAA F F DD ΓBB  CC F F −ΣAA F F BB  ΓF F  CC DD .

3.2 Calculus of spacetime spinors

85

Now, making use of the decomposition (3.32) for the spin connection coeﬃcients,
one obtains after a lengthy, but straightforward calculation that



RCC DD AA BB  = RC DAA BB  δD C + R̄C D AA BB  δD C ,(3.35)

where
RC DAA BB  ≡ eAA (ΓBB  C D ) − eBB  (ΓAA C D )−ΓF B  C D ΓAA F B − ΓBF  C D Γ̄AA F B  + ΓF A C D ΓBB  F A+ΓAF  C D Γ̄BB  F A + ΓAA C F ΓBB  F D − ΓBB  C F ΓAA F D−ΣAA F F BB  ΓF F  C D .
This last expression can be regarded as the spinorial counterpart of the ﬁrst
Cartan structure equation; see Equation (2.31).
The commutator of covariant derivatives on arbitrary spinors
The commutator expression (3.34) applies only to spinors arising from a tensorial
counterpart. In this section this commutator expression is applied to arbitrary
valence spinors. In order to do this, observe that Equation (3.35) also holds if
expressed in terms of abstract spinorial indices. More precisely, one has that



RCC DD AA BB  = RC DAA BB  δD C + R̄C D AA BB  δD C ,(3.36)

where, in general RCDAA BB  = R(CD)AA BB  .
Applying the commutator (3.34) to the particular case when ξ CC = D C D C
one obtains, after taking into account the split (3.36), thatD C ∇AA , ∇BB  D C + D C ∇AA , ∇BB  D C








= D C RC DAA BB  D D + D C R̄C D AA BB  D D .
From the latter one can conclude that
∇AA , ∇BB  D C = RC QAA BB  D Q ,∇AA , ∇BB  D C = R̄C Q AA BB  D Q .
Now, using that P C Q C = δP Q , and that ∇AA , ∇BB  δP Q = 0, one ﬁnds that
P C ∇AA , ∇BB  Q C = −Q C ∇AA , ∇BB  P C(3.37a)= −

(3.37b)Q

CR

C

DAA BB DP .

Multiplying the previous expression by P D and using that P D P C = δD C one
obtainsδD C ∇AA , ∇BB  Q C = −Q C RC QAA BB  δD Q .

86Spacetime spinors

Finally, using that ∇AA , ∇BB  δD C = 0, one concludes that
∇AA , ∇BB  Q D = −RC DAA BB  Q C .

(3.38)

A similar argument applied to primed basis spinors yields
∇AA , ∇BB  Q D = −R̄C D AA BB  Q C  .

(3.39)

Now, using that ∇AA , ∇BB   applied to a scalar is zero, one has that
Equations (3.37a), (3.37b), (3.38) and (3.39) render the following formulae forarbitrary valence 1 spinors:
∇AA , ∇BB  μC = RC QAA BB  μQ ,
∇AA , ∇BB  λ̄

C

= R̄

C

Q AA BB  λ̄Q

(3.40a),(3.40b)∇AA , ∇BB  κC = −RQ CAA BB  κQ ,
∇AA , ∇BB  ν̄C  = −R̄Q(3.40c)C  AA BB  ν̄Q .

(3.40d)The extension to higher valence spinors follows from the Leibnitz rule. For
example, one has that∇AA , ∇BB  ξCD E = −RQ CAA BB  ξQD E − RQ DAA BB  ξCQ E
+R̄E Q AA BB  ξCD Q .
3.2.4 Decomposition of a general curvature spinor
Expression (3.36) is a convenient starting point to analyse the decomposition of
the curvature spinor in terms of irreducible components. Lowering the index pairCC  using the -spinor one obtains:
RCC  DD AA BB  = RCDAA BB  D C  + R̄C  D AA BB  DC= −RCDAA BB  C  D − R̄C  D AA BB  CD .

(3.41)For the curvature spinor of a general connection one has that RCDAA BB  =
R(CD)AA BB  . However, one still has that
RCDAA BB  = −RCDBB  AA .
This antisymmetry can be exploited using the split (3.13) in such a way that theindices CD are not touched. Accordingly, one obtains
RCDAA BB  = XCDAB A B  + YCDA B  AB ,
where

1
RCDAQ B Q ,2
1
YCDA B  = YCD(A B  ) ≡ RCDA QB  Q .
2

XCDAB = XCD(AB) ≡(3.42)

3.2 Calculus of spacetime spinors

87

To complete the decomposition of the curvature spinor in irreducible components
one can apply the decomposition formulae (3.8) and (3.12) for valence 4 spinors
to XCDAB and YCDA B  . This idea will not be pursued any further here. However,it will be convenient to single out certain components of the decomposition in
irreducible terms of XCDAB . It is conventional to set
ΨABCD ≡ X(ABCD) ,Λ≡1
XP Q P Q .
6

Let CCC  DD AA BB  denote the spinor obtained from the split (3.41) of thecurvature spinor by setting XABCD = X(ABCD) and YCDA B  = 0. One has that
CCC  DD AA BB  = −ΨABCD A B  C  D − Ψ̄A B  C  D AB CD .

(3.43)As a consequence of the total symmetry of ΨABCD it can be readily veriﬁed that
CCC  DD AA BB  is the spinorial counterpart of a trace-free tensor. Following thediscussion in Section 2.5.2, it must be the spinorial counterpart of the Weyltensor Ccdab .
Decomposition of the curvature spinor of a torsion-free connection
The decomposition of the curvature spinor is now particularised to the caseof a torsion-free connection. In this case, the Riemann curvature tensor has
the cyclic symmetry of the Bianchi identity. The latter is best exploited using
the alternative expression of the identity given by Equation (2.23) involving theright-dual of the Riemann tensor. Using the spinorial counterpart of the volume
form given by Equation (3.25) one has that




i
(δC E δD F δC  F δD E − δC F δD E δC  E δD F )RAA BB  EE  F F 
2= iRAA BB  CD DC  ,∗RAA BB  CC  DD  =so that the spinorial counterpart of Equation (2.23) is given by

RCC  QQ A Q Q A = 0.
A direct evaluation of the above condition using the splits (3.41) and (3.42)
shows that

XCQA Q C  A − X̄C  Q A Q CA + YCAC  A − ȲC  A AC = 0,
so that
XP Q P Q = X̄P Q P Q ,ȲC  A AC = YCAC  A .

Hence, one has that XP Q P Q (i.e. Λ) is a real scalar, while YABA B  is a Hermitian
tensor, and, thus, it is the spinorial counterpart of a rank 2 tensor.

88Spacetime spinors
Decomposition on the curvature spinor of a metric connection

As already seen, a connection which is compatible with a metric g satisﬁes∇AA CD = 0. It follows then that ∇AA , ∇BB  CD = 0. However, one also
has that∇AA , ∇BB  CD = −RQ CAA BB  QD + RQ DAA BB  CQ ,from which one concludes that
RCDAA BB  = R(CD)AA BB  .
The latter can be reexpressed in terms of the following symmetries of the spinors
XABCD and YABA B  :XABCD = X(AB)CD ,YABA B  = Y(AB)A B  .

Decomposition of the curvature spinor of a Levi-Civita connection
Finally, one can collect the results of the previous subsections to obtain the wellknown irreducible decomposition of the spinorial counterpart of the Riemanntensor of a Levi-Civita connection. As the Levi-Civita connection associated tothe metric g is both torsion-free and metric, it follows then thatXABCD = X(AB)(CD) ,XCQA Q = 0.

It follows from (3.8) that XABCD = XCDAB and that
1
XABCD = X(ABCD) − A(C D)B XP Q P Q
3= ΨABCD + Λ(DB CA + CB DA ).
Similarly, for YABA B  one has that
YABA B  = Y(AB)(A B  ) ,so that according to the general split (3.12) YABA B  corresponds to a trace-freerank 2 tensor.
To conclude the analysis, it is convenient to compute the Ricci tensor andscalar in terms of the spinors XABCD and YABA B  . From Equations (3.41) and(3.42) it follows directly that

RAA BB  = −XQA Q B A B  − X̄Q A Q B  AB + 2YABA B 
R = −4XP Q P Q ,where RAA BB  denotes the spinorial counterpart of the Ricci tensor Rab
and it has been used that for a Levi-Civita connection YABA B  = ȲA B  AB and 
XP Q P Q = X̄P  Q P Q . In particular, one has thatR = −24Λ.

3.2 Calculus of spacetime spinors

89

As YABA B  is trace-free, it has to be related to Φab , the symmetric trace-freepart of the Ricci tensor. Indeed, a calculation for its spinorial counterpart showsthat
1
2ΦABA B  ≡ RAA BB  − RAB A B 
4= 2YABA B  .
It can be veriﬁed that ΦABA B  satisﬁes the symmetries
ΦABA B  = ΦBAA B  = ΦABB  A = ΦBAB  A .

(3.44)

Putting together the discussion of this section, one ﬁnds that the spinor
counterpart of the Riemann curvature tensor of a Levi-Civita connection canbe decomposed as
RAA BB  CC  DD = −A B  C  D (ΨABCD + 2ΛA(C D)B )−AB CD (Ψ̄A B  C  D + 2ΛA (C  D )B  )+A B  CD ΦABC  D + AB C  D ΦCDA B  .
Working back from this expression one can recover the decomposition of
the Riemann tensor in terms of the Weyl and Schouten tensor given in
Equations (2.21a) and (2.21b).
3.2.5 The AB -operator
In some applications it is convenient to have a more explicit expression for thecommutator of spinorial covariant derivatives. In the remainder of this section itis assumed that ∇AA is the spinorial counterpart of a Levi-Civita connection.
Exploiting the antisymmetry of Equation (3.40a) with respect to the pairs AA
and BB  one can rewrite it asA B  AB + AB A B  )μC = RC QAA BB  μQ ,(3.45)whereAB ≡ ∇Q (A ∇B) Q ,A B  ≡ ∇Q(A ∇Q B  ) .

It can be veriﬁed that both AB and A B  are linear and satisfy the Leibnitz
rule – one has, for example, that
AB (μC λD ) = (AB μC )λD + μC (AB λD ).


Deﬁning the D’Alembertian operator as  ≡ ∇P P  ∇P P , one obtains thedecomposition∇AQ ∇B Q =1
AB  + AB .
2

90

Spacetime spinors
Now, contracting indices suitably in Equation (3.45) one readily obtains
AB μC = X C QAB μQ ,

A B  μC = Y C QA B  μQ .

Using the explicit expressions for the curvature spinors XABCD and YABA B  fora Levi-Civita connection, as given in Section 3.2.4 one concludes thatAB μC = ΨABCD μD − 2Λμ(A B)C ,A B  μC = ΦCDA B  μD .

(3.46)

The above expressions can be extended to higher order valence spinors by meansof the Leibnitz rule.
The expressions in (3.46) can be extended to the case of connections withtorsion; see Penrose (1983) for the general theory and Gasperı́n and Valiente
Kroon (2015) for explicit expressions and applications.
3.3 Global considerations
The discussion on null vectors and their ﬂagpoles in Section 3.1.10 makes anatural connection with the notion of orientability and the assumptions needed
to ensure the existence of spinorial structures on a region of spacetime.
As seen in Proposition 3.3, every non-vanishing null vector is either future
pointing or past pointing, in accordance with the choice of sign made in
Equation (3.21). Thus, the existence of spinors on a region of spacetime provides
a way to deﬁne a time orientation. In a similar way, the idea of a right-handed
phase change of a triad of orthonormal vectors {e1 , e2 , e3 }, as discussed in
Section 3.1.10, can be used to deﬁne a notion of space orientation. Thus, at leastat an intuitive level, the existence of a spinorial structure over a spacetime seems
to imply that the spacetime is time orientable and space orientable. It turns outthat the converse is also true: time and space orientability ensure the existenceof a spinorial structure. More precisely, one has the following result proved in
Geroch (1968):
Theorem 3.2 (orientability and the existence of a spinor structure) A
non-compact spacetime (M, g) has a spinor structure if and only if there exists
on M a global system of orthonormal tetrads.
Part IV of this book will be concerned with the construction of spacetimes fromsuitably posed initial value problems. Thus, it is convenient to have a criterion
to encode the existence of a spinorial structure in an initial value problem.
An example of this is the following result in Geroch (1970c):
Proposition 3.4 (global hyperbolicity and the existence of a spinor
structure) Every globally hyperbolic spacetime has a spinor structure.
The notion of global hyperbolicity is discussed in Section 14.1.

3.4 Further reading91

An orientable spacetime may have several spinorial structures. One can ensure
uniqueness of the spinorial structure if one restricts further the topology of thespacetime. More precisely, one has that (see Geroch (1968)):
Proposition 3.5 (uniqueness of the spinorial structure) The spinorial
structure of a spacetime is unique if and only if M is simply connected.
3.4 Further reading
Further details on the various topics covered in the present chapter can befound in Penrose and Rindler (1984), Stewart (1991) and O’Donnell (2003). The
discussion in these references leads, in a natural way, to the Newman-Penrose
formalism and applications like the Petrov algebraic classiﬁcation of the Weyltensor. Some discussion on the use of spinors in the construction and analysisof exact solutions to the Einstein ﬁeld equations can be found in Stephani et al.
(2003) and Griﬃths and Podolský (2009). The relation between Dirac spinors
and 2-spinors is presented in Penrose and Rindler (1984) and Stewart (1991).
A pure mathematics perspective can be found, for example, in Petersen (1991);
see also Choquet-Bruhat et al. (1982).
A more general perspective of the discussion of the present chapter can beobtained by making use of the notion of ﬁbre bundles; see, for example, Ashtekar
et al. (1982). In terms of this language, the spinorial structure arises as a principal
ﬁbre bundle over the spacetime manifold M with structure group SL(2, C).
This point of view is convenient for computer algebra implementations; see,for example, Martı́n-Garcı́a (2014). The ﬁbre bundles are useful in analyses that
require the blowing up of particular points of spacetime – as in the analysis
of caustics in Friedrich and Stewart (1983) or the so-called problem of spatial
inﬁnity of Friedrich (1998c).
Appendix: the Newman-Penrose formalism
The idea of a spinor-based null tetrad formalism was introduced in the seminal
article by Newman and Penrose (1962); see also Newman and Penrose (1963).
This so-called Newman-Penrose (NP) formalism was ﬁrst used as a wayof analysing the asymptotics of gravitational radiation. The potential of theformalism to obtain exact solutions to the Einstein ﬁeld equations, in particular,
ones having an algebraically special Weyl tensor, was quickly realised; see,for example, Stephani et al. (2003) for an entry point to the literature ofexact solutions. Reﬁnements of the formalism which are adapted to speciﬁc
conﬁgurations or types of problems are available in the literature, most noticeably
Geroch et al. (1973); see also Machado and Vickers (1995, 1996).
The key aspects of a generic spinor-based null tetrad formalism have alreadybeen covered in this book. One of the peculiarities of the formalism, as introduced
in Newman and Penrose (1962), is the use of speciﬁc symbols to denote

92

Spacetime spinors

directional derivatives and the spin coeﬃcients. This notation will not be usedin this book as the Newman-Penrose (NP) formalism assumes, from the onset, aLevi-Civita connection. However, the discussion in this book will very often use
more general connections. Hence, one has more independent spin coeﬃcients.
Moreover, the labelling of spin coeﬃcients through indices lends itself betterfor a systematic analysis of the properties of the relevant equations. Additionaldiﬃculties with the NP formalism arise with the space spinor formalism; see thenext chapter.
The purpose of this appendix is to provide a guide to the translation, wheneverpossible, between NP objects and the ones used in this book.
The directional derivatives
Let {o, ι} denote, as usual, a spin basis. Also, let {l, n, m, m̄} denote the nulltetrad constructed from the spin basis, as described in Section 3.1.9. The NP
convention for the directional derivatives along the directions given by the nulltetrad isD ≡ la ∇a = oA ōA ∇AA = ∇00 ,Δ ≡ na ∇a = ιA ῑA ∇AA = ∇11 ,δ ≡ ma ∇a = oA ῑA ∇AA = ∇01 ,δ̄ ≡ m̄a ∇a = ιA ōA ∇AA = ∇10 .
The spin coeﬃcients
In what follows, it is assumed that the connection ∇ is Levi-Civita so that
∇AA BC = 0. The NP convention for the spin coeﬃcients of ∇ is given by: = Γ00 0 0 = −Γ00 1 1 = Γ00 10 ,α = Γ10 0 0 = −Γ10 1 1 = Γ10 10 ,β = Γ01 0 0 = −Γ01 1 1 = Γ01 10 ,γ = Γ11 0 0 = −Γ11 1 1 = Γ11 10 ,π = Γ00 0 1 = Γ00 11 ,

κ = −Γ00 1 0 = Γ00 00 ,λ = Γ10 0 1 = Γ10 11 ,

ρ = −Γ10 1 0 = Γ10 00 ,μ = Γ01 0 1 = Γ01 11 ,σ = −Γ01 1 0 = Γ01 00 ,

ν = Γ11 0 1 = Γ11 11 ,

τ = −Γ11 1 0 = Γ11 00 .

The above spin coeﬃcients can be expressed entirely in terms of the directional
derivatives D, Δ, δ, δ̄ applied to the null frame vectors or, alternatively, applied tothe spin basis {o, ι}. See O’Donnell (2003) and Stewart (1991) for details on this.
Explicit expressions of the spin coeﬃcients in terms of curls (antisymmetrised
derivatives) have been worked out in Cocke (1989).

3.4 Further reading

93

The Ricci and Weyl tensors
The NP conventions to denote the components of the Weyl spinor ΨABCD withrespect to {o, ι} are:
Ψ0 ≡ ΨABCD oA oB oC oD ,

Ψ1 ≡ ΨABCD oA oB oC ιD ,

Ψ3 ≡ ΨABCD oA ιB ιC ιD ,

Ψ2 ≡ ΨABCD oA oB ιC ιD ,

Ψ4 ≡ ΨABCD ιA ιB ιC ιD .

The conventions for the components of the trace-free Ricci spinor ΦAA BB  are:

Φ01 ≡ ΦAA BB  oA oB ōA ῑB ,

Φ10 ≡ ΦAA BB  oA ιB ōA ōB ,Φ12 ≡ ΦAA BB  oA ιB ῑA ῑB ,

Φ21 ≡ ΦAA BB  ιA ιB ōA ῑB ,Φ00 ≡ ΦAA BB  oA oB ōA ōB ,Φ02 ≡ ΦAA BB  oA oB ῑA ῑB ,Φ11 ≡ ΦAA BB  oA ιB ōA ῑB ,Φ20 ≡ ΦAA BB  ιA ιB ōA ōB ,





















Φ22 ≡ ΦAA BB  ιA ιB ῑA ῑB .
Notice that in both lists of deﬁnitions the value of the index denotes the numberof contractions with the spinor ι.
The NP formalism makes use of the symbol Λ to denote a multiple of the traceof the Ricci tensor. The relation to the Ricci scalar is
R = −24Λ.
The Newman-Penrose ﬁeld equations
Newman and Penrose (1962) provided explicit expressions of the Ricci and the
Bianchi identities in terms of their notation for the spin connection coeﬃcientsand the components of ΨABCD and ΦAA BB  . These equations are collectively
called the Newman-Penrose ﬁeld equations. Explicit expressions are available
in O’Donnell (2003), Penrose and Rindler (1986) and Stewart (1991). Besides
the NP ﬁeld equations, the formalism consists also of explicit expressions for thecommutators of the directional derivatives D, Δ, δ, δ̄. Expressions for the sourcefree Maxwell equations are available in the literature as well; see, for example,
the appendix in Stewart (1991).

4Space spinors

This chapter discusses a framework for spinors in which a further structure is
introduced – a so-called Hermitian inner product. The resulting formalism will bereferred to as space spinors or SU (2, C)-spinors. The space spinor formalism
can be used to describe the geometry of three-dimensional Riemannian manifolds
and, more generally, foliations of spacetime. Moreover, it can also be used toprovide a description of the hyperplanes associated to a congruence of timelike
curves.
The notion of space spinors was ﬁrst introduced in Sommers (1980); see alsoSen (1981). It provides a systematic approach to the construction of evolution
equations which can be regarded as a spinorial version of the 1 + 3 formalism
for tensors. Space spinors are used in several other areas of relativity suchas quantum gravity (see e.g. Ashtekar (1991)), the construction of quasi-local
notions of energy (see e.g. Szabados (2009)) and global aspects of the geometryof 3-manifolds (see e.g. Bäckdahl and Valiente Kroon (2010a); Beig and Szabados(1997); Tod (1984)).
4.1 Hermitian inner products and 2-spinors
Let (M, g) denote a four-dimensional Lorentzian manifold. As in Chapter 3, it isassumed that at each point p ∈ M one has a two-dimensional simplectic vector
space S|p (M) as given by Deﬁnition 3.1. One has the following deﬁnition:
Deﬁnition 4.1 (Hermitian inner product) A Hermitian inner producton a simplectic two-dimensional vector space S is a function ·, · : S × S → C
which is:(i) Hermitian; that is, given ξ, η ∈ Sξ, η = η, ξ4.1 Hermitian inner products and 2-spinors

95(ii) linear in the second entry; that is, given ξ, η, ζ ∈ S, z ∈ Cξ, η + zζ = ξ, η + zξ, ζ(iii) positive deﬁnite; that is, given ξ ∈ Sξ, ξ ≥ 0
and ξ, ξ = 0 if and only if ξ = 0.
From conditions (i) and (ii) it follows that a Hermitian inner product isantilinear in the ﬁrst entry; that is, given ξ, η, ζ ∈ S, z ∈ C, one hasξ + zζ, η = ξ, η + z̄ζ, η.
4.1.1 Hermitian conjugation
In what follows, given a spacetime (M, g), assume that for each point p ∈ M, thevector space S|p (M) is endowed with a Hermitian inner product which changes
smoothly from point to point. The Hermitian inner product can be expressed interms of a Hermitian spinor AA ∈ SAA (M) such thatξ, η = AA ξ¯A η A .

(4.1)

It can be veriﬁed that the right-hand side of the above expression satisﬁes conditions (i) and (ii) of Deﬁnition 4.1. Given a spinor basis {A A }, the componentsof AA with respect to the basis are given by AA ≡ AA A A ¯A A . The
components AA can be thought of as the entries of a (2 × 2) matrix (AA ).
The positivity condition (iii) of Deﬁnition 4.1 requires the above matrix to bediagonalisable and to have positive eigenvalues. Thus, it is natural to considera (not necessarily normalised) basis {A A } for which (AA ) takes the diagonal
form


00
0
(AA ) =.
0
11
The scaling of the basis {A A } can be ﬁxed, without loss of generality, so that
(AA ) is the identity matrix. In the rest of the book, whenever a Hermitian
inner product is discussed, it will be assumed that a spin basis {A A } has beenchosen so that

1 0(AA ) =
.
(4.2)
0 1
A direct consequence of the above normalisation condition is that one can
write




AA = oA ōA + ιA ῑA = 1 A ¯1 A + 0 A ¯0 A ,(4.3a)

96

Space spinors


















A A = oA ōA + ιA ῑA = 1 A ¯0 A − 0 A ¯1 A ,AA = oA ōA + ιA ῑA = 0 A ¯0 A + 1 A ¯1 A .

(4.3b)(4.3c)

From these expressions it follows thatAA BA = oA ιB − ιA oB = 1 A 1 B + 0 A 0 B .
Thus,AA A B = δA B .

(4.4)



Notice, in particular, that AA AA = 2.
The spinor AA induces an operation of Hermitian conjugation + :•S (M) → S• (M). Given μA ∈ S(M), we deﬁne its Hermitian conjugate μ+
A
via


A
μ+
A ≡ A μ̄A .

(4.5)

It follows then that one can writeξ, η = −ξ + A η A = ηA ξ +A .
Observe that as a consequence of the see-saw rule, Equations (3.4a) and (3.4b),

one has μ+A ≡ −A A μ̄A . The Hermitian conjugation is extended to higher
valence spinors by requiring(μλ)+ = μ+ λ+for μ, λ ∈ S• (M). It is a consequence of the normalisation condition (4.2) thatkμ++A1 ···Ak = (−1) μA1 ···Ak .

Furthermore, μ+A μA = 0 if and only if μA = 0, the latter as a result of condition(iii) of Deﬁnition 3.1. Using the representation of A A given by (4.3b) one ﬁnds
that

o

o+
A = ιA ,ι+
A = −oA ,(4.6a)+A+A= −o .

(4.6b)

A

=ι ,

ι

A

Hence, the normalisation leading to (4.3a)–(4.3c) is equivalent to the normalisation conditionoA o+A = 1,
for the spinor o. Notice also that, as a consequence of the previous discussion, a
non-zero spinor and its Hermitian conjugate are linearly independent. Finally, acalculation using the expression of AB in terms of oA and ιA , yields

+ AB = A A B B A B  = AB .

4.2 The space spinor formalism

97

Remark. In the rest of the book, when working with spinor structures endowedwith a Hermitian product, it will always be assumed that a spin basis satisfying
relations (4.6a) and (4.6b) has been chosen.
4.2 The space spinor formalism
A consequence of the existence of an operation of Hermitian conjugation is that,given a spinor ξ ∈ S• (M), its complex conjugate ξ̄ and its Hermitian conjugate
ξ + contain the same information. This observation allows the introduction ofa spinorial formalism based entirely on spinors with unprimed indices bycontracting all the primed indices in the spinors with A A . Given ξA1 ···Ap A1 ···Aq ,we deﬁne its space spinor counterpart ξA1 ···Ap B1 ···Bq as

ξA1 ···Ap B1 ···Bq ≡ B1 A1 · · · Bq Aq ξA1 ···Ap A1 ···Aq .

(4.7)

The above expression can be inverted by recalling the normalisation condition(4.4) to yieldξA1 ···Ap A1 ···Aq = (−1)q B1 A1 · · · Bq Aq ξA1 ···Ap B1 ···Bq .
Thus, the information contained in a spinor with primed indices and its spacespinor counterpart is equivalent.
4.2.1 The Hermitian product and three-dimensional vectors
The operation of Hermitian conjugation gives rise to a notion of reality forspinors. More precisely, a spinor μA1 B1 ···Ak Bk with an even number of indiceswill be said to be real ifkμ+A1 B1 ···Ak Bk = (−1) μA1 B1 ···Ak Bk ,and imaginary if
k+1
μA1 B1 ···Ak Bk .
μ+
A1 B1 ···Ak Bk = (−1)

Consider now a symmetric valence-2 spinor v AB ∈ S• (M). Given a spacespinor basis {o, ι} such that ι ≡ o+ , one can write
v AB = aoA oB + bιA ιB + co(A ιB) ,(4.8)

with a, b, c ∈ C. The Hermitian conjugate of v AB is given by
v +AB = āιA ιB + b̄oA oB − c̄ι(A oB) .
If v AB is real, that is, v +AB = −v AB , then b̄ = −a and c̄ = c. Thus, the real
spinor v AB has only three real components so that it describes a three-dimensional
vector v i . This argument can be extended to higher valence real space spinors
so that
ξA1 B1 ···Ak Bk C1 D1 ···Cm Dm = ξ(A1 B1 )···(Ak Bk ) (C1 D1 )···(Cm Dm ) ,98Space spinors

if real, can be regarded as the space spinor counterpart of a three-dimensional
tensor ξi1 ···ik j1 ···jm – every pair of symmetric spinor indices is associated to a
spatial tensor index. One can summarise the previous discussion in the following:
Lemma 4.1 (the distribution associated to a Hermitian product) A
Hermitian spinor AA on S• (M) induces a three-dimensional distribution Π
on M.
Observation. The distribution Π may not possess integrable manifolds.
The consequences of Lemma 4.1 can be further elaborated by considering the
spinorial counterpart of the projector ha b associated to the distribution Π. To
this end let

1
hAA BB ≡ δA B δA B − AA BB .
2
It can be readily veriﬁed that

hAA BB hBB  CC = hAA CC ,

hAA BB BB  = 0.

Now, given vAA ∈ S• (M) denote by vAB its space spinor counterpart. A
calculation then shows that

v(AB) = B A hAA CC vCC  = (A A vB)A .


Thus, the spinor hAA BB is the projector associated to the distribution induced
by AA . The space spinor version of hAA BB is given by hABCD ≡ B A D C
hAA CC  . It can be readily veriﬁed that hABCD = h(AB)(CD) . Using the Jacobi
identity (3.5) one can show that
hABCD ≡ −A(C D)B .

(4.9)

It can also be veriﬁed thathABP Q hP QCD = hAB CD ≡ A (C B D) ,hABCD hABCD = hP Q P Q = 3.

In addition, one has
h+ABCD = hABCD ,so that hABCD is a real space spinor. Moreover, given v AB = v (AB) and uAB =u(AB) one has thatv AB hABCD = vCD ,uAB hABCD = uCD ,uAB hCD AB = uCD .

Finally, it is observed that if the spinor v AB is real, then using the decompositionof Equation (4.8) one readily ﬁnds that
1
1
hABCD v AB v CD = vAB v AB = 2ab − c2 = −2|b|2 − c2 ≤ 0.
2
2

4.2 The space spinor formalism

99

Thus, given p ∈ M, the spinor hABCD gives rise to a negative deﬁnite innerproduct on Π|p ⊂ T |p (M). Accordingly, the vectors in Π are spatial with respect
to the metric g. As Π may not possess integrable submanifolds, hABCD is notnecessarily the spinorial counterpart of a (negative deﬁnite) three-dimensional
Riemannian metric of a spacelike submanifold of M.
4.2.2 Spatial Infeld-van der Waerden symbols
The relation between space spinors and three-dimensional vectors can beformalised by means of suitable soldering objects. At each point p ∈ M consider ag-orthonormal basis {ei } of Π|p and let {ω i } denote the associated cobasis. One
has then that g(ei , ej ) = −δij . In addition, let {A A } be a spin basis satisfyingA= 0 A ; compare Equation (4.6b). Spatial Infeld-van der Waerden+1symbols can be deﬁned from the spacetime Infeld-van der Waerden symbols
σa AA and σ a AA through the relations

σi AB ≡ −(A A σi B)A ,σ i AB ≡ (A A σ i B)A .

hABCD = −σ i AB σ j CD δij ,σi AB σ j AB = δi j .

(4.10)The explicit expressions for the spatial Infeld-van der Waerden symbols are
given by






1
1
1
−1 0
−i 0
0 1
AB
AB
AB
=√
=√
=√
σ1
, σ2
, σ3
,
0 1
2
2 0 −i2 1 0
and1
σ 1 AB = √
2



−1 0
0 1


,

1
σ 2 AB = √
2



i 0
0 i


,

1
σ 3 AB = √
2



0
1

1
0


.

Given the above, the components of a three-dimensional vector v and a threedimensional covector ξ can be put in correspondence with symmetric valence-2
real spinors via the formulaev i → v AB = v i σi AB ,ξi → ξAB = ξi σ i AB ,or more explicitly
1(v , v , v ) → √
2
1

2



3

1
(ξ1 , ξ2 , ξ3 ) → √2

−v 1 − iv 2v3v3
1v − iv 2−ξ1 + iξ2ξ3ξ3
ξ1 + iξ2,(4.11a).

It can be veriﬁed thatξ, v = ξi v i = ξAB v AB = −ξ1 v 1 − ξ2 v 2 − ξ3 v 3 .

(4.11b)

100

Space spinors

The above correspondence between space spinors and three-dimensional vectors
can be readily extended to higher rank tensors. For example, given thecomponents Tij k of a tensor Tij k one has the correspondence
Tij k → TABCD EF ≡ σ i AB σ j CD σk EF Tij k .
Finally, it is observed that σ i AB and σi AB can be used to deﬁne an alternative
set of basis and cobasis {eAB }, {ω AB } for Π|p through the relationseAB ≡ σ i AB ei ,ω AB ≡ σi AB ω i .

(4.12)In terms of these deﬁnitions the symbols σ i AB are the components of the frame{eAB } with respect to itself. It can be veriﬁed thatω AB , eCD  = hCD AB ,g(eAB , eCD ) = hABCD .

4.2.3 Changes of basis and SU (2, C) transformations
To characterise the class of transformations preserving the structure of theHermitian inner product it is convenient to consider a change of spin basis˜A A = OA P P A ,˜A A = OA P P A ,where (OA P ) and (OA P ) are SL(2, C) matrices such thatOA P O B P = δA B .
It follows that


˜A A

˜ AA = AA ˜A A ¯






= AA OA P P A ŌA P ¯P  A = P P  OA P ŌA P .
˜ AA are the identity
Hence, if one requires OA P to be such that both AA and matrix – compare Equation (4.2) – then OA P and ŌA P have to be inversesof each other; that is, the transformation described by the matrix (OA B ) is
an SU (2, C) transformation. This property explains the alternative name ofSU (2, C) spinors used to describe spinorial structures endowed with a Hermitian
product; see, for example, Ashtekar (1991). It is a direct consequence of theprevious discussion that the notions of real and imaginary space spinors as
discussed in Section 4.2.1 are invariant under SU (2, C) transformations.
It can be readily veriﬁed that SU (2, C) transformations are related tothree-dimensional rotations, that is, O(3) transformations. As SU (2, C)transformations are a special case of SL(2, C) transformations, it follows thatAB = OA C OB D CD ; that is, ˜AB = AB . From the latter one has that
hABCD = OA E OB F OC G OD H hEF GH .

4.2 The space spinor formalism

101

Contracting the above expression with spatial Infeld-van der Waerden symbols
one obtains
δij = Oi k Oj l δkl ,where the matrix (Oi k ) with elements given by
Oi k ≡ σi AB σ k EF OA E OB F
is an O(3)-transformation, that is, a three-dimensional real matrix preserving
the identity matrix.
4.2.4 Spinors on three-dimensional manifolds
In this section it is assumed that the distribution Π associated to the Hermitianspinor AA has an integral manifold S. It follows that S is a spacelike
hypersurface of the spacetime manifold M, and the restriction of Π to S coincides
with the tangent bundle T (S). Vectors and covectors in T (S) are associated tosymmetric valence-2 real spinors.
Under the assumptions of the previous paragraph, consistently with thediscussion of Section 4.2.1, one has that the spinor hABCD = −A(C D)B is the
spinorial counterpart of the three-dimensional (negative deﬁnite) Riemannian
metric h induced on S by g. One can writeh = −δij ω i ⊗ ω j = hABCD ω AB ⊗ ω CDwith the coframe {ω AB } deﬁned as in Equation (4.12). In a natural manner, thespinor AA can be identiﬁed with the normal to S.
An alternative point of viewThe discussion of spinors on three-dimensional manifolds outlined in the previous
paragraphs assumes that S is a spacelike hypersurface of a spacetime (M, g).
A more intrinsic perspective can be obtained by postulating the existence of aspinorial structure over S, to be denoted by S(S), endowed with an operationof Hermitian conjugation + : S(S) → S(S) satisfying the properties discussed
in Section 4.1.1. This point of view leads one to consider conditions ensuring
the existence of this space spinor structure. An example of a suﬃcientcondition is that the vacuum Einstein constraint equations (see Chapter 11)
can be solved on S. If this is the case, the three-dimensional manifold can be
regarded as a spacelike hypersurface of a spacetime (M, g); see Chapter 14. The
spacetime (M, g) is globally hyperbolic and, thus, admits a spinor structure;see Proposition 3.4. The g-normal to S in M induces the required operation of
Hermitian conjugation.
In what follows, the notational conventions of Section 3.1.4 are adopted, andone writes S• (S), SA (S), SA (S), . . . to denote the various bundles associatedto S(S).

102

Space spinors
Totally symmetric spinors

Spinors provide a simple representation of the operation of taking the symmetrictrace-free part of a three-dimensional tensor.
Proposition 4.1 (space spinor representation of trace-free threedimensional tensors) Let TA1 B1 ···Ap Bp denote the spinorial counterpart of athree-dimensional real tensor Ti1 ···ip . One has that
TA1 B1 ···Ap Bp = T(A1 B1 ···Ap Bp )if and only if

Ti1 ···ip = T{i1 ···ip } .

Proof Any possible contraction of T(A1 B1 ···Ap Bp ) with hABCD must vanish sothat Ti1 ···ip must be trace free. Conversely, if Ti1 ···ip = T{i1 ···ip } , then one has
that
hA1 B1 A2 B2 TA1 B1 A2 B2 ···Ap Bp = T P Q P Q···Ap Bp = 0.
Using the decomposition (3.8) together with the symmetries of TA1 B1 A2 B2 ···Ap Bp
in the indices A1 B1 A2 B2 one concludes that
TA1 B1 A2 B2 ···Ap Bp = T(A1 B1 A2 B2 )···Ap Bp .
Now, considering the contraction of the pair A2 B2 with pairs outside the symmetrisation bracket and repeating the previous argument as many times as necessary one concludes that TA1 B1 A2 B2 ···Ap Bp must be completely symmetric.
4.2.5 Timelike congruences and Hermitian products
Assume now that the spacetime (M, g) has some privileged future directed
timelike vector τ with parameter τ .1 The vector τ does not need to behypersurface orthogonal. Let τ AA denote the spinorial counterpart of τ andconsider the normalisation g(τ , τ ) = 2. As discussed in Section 2.7.1, τ deﬁnes
a distribution on M. Let Sτ denote the hyperplanes generated by τ ; as τ is
not hypersurface orthogonal, the hyperplanes are not, in general, the tangent
bundles to the leaves of a foliation of M. The timelike spinor τAA induces aHermitian product ξ, η = τAA ξ¯A η A for ξ A , η A ∈ S(S). Indeed, as τ AA is
the spinorial counterpart of a spacetime vector, it is a Hermitian spinor, so thatτAA ξ¯A η A = τAA η̄ A ξ A .
Furthermore, given that τ AA is timelike future directed and ξ A ξ¯A describes a
future-directed null vector, it follows that τAA ξ A ξ¯A ≥ 0. Thus, formula (4.1)

1 It is possible to construct a “space spinor” formalism adapted to spacelike congruences with


tangent vector ρAA ; see e.g. Szabados (1994). This requires adapting some of the formulae
given in the preceding sections. In particular, the associated Hermitian product needs to benegative deﬁnite. Moreover, the analogue of Equation (4.13) is given by ρAA ρBA = B A so
that ρAA ρAA = −2.

4.2 The space spinor formalism

103

and the various subsequent expressions in Section 4.1 can be used for the choiceAA = τAA .
From the discussion in Section 4.1.1 it follows that there exists a spin basis{A A } such that

τ AA = 0 A 0 A + 1 A 1 A .
In particular, one has thatτAA τ BA = A B .

(4.13)Space spinor split of general spacetime spinors
The tensorial counterpart of a spinor μA1 A1 ···Ak Ak can be expanded in terms of
the spatial frame {eAB } if and only if it is spatial with respect to τ , that is, if
the k conditions···τ A1 A1 μA1 A1 ···Ak Ak = 0,

τ Ak Ak μA1 A1 ···Ak Ak = 0

hold. In this case, its space spinor counterpart is given by

μA1 B1 ···Ak Bk = τB1 A1 · · · τBk Ak μA1 A1 ···Ak Ak = μ(A1 B1 )···(Ak Bk ) .
To deal with the spinorial counterparts of tensors which are not spatial in thesense described above, one makes use of the projector1hBB AA ≡ A B A B − τAA τ BB ,2

which takes a spinor ξA1 A1 ···Ak Ak onto the spatial spinorξA1 A1 ···Ak Ak hA1 A1 B1 B1 · · · hAk Ak Bk Bk .
The space spinor version of the above spatial spinor is obtained by contracting
the primed indices with τA A as in formula (4.7). In particular, this procedure
applied to the projector hAA BB  yields hABCD . The non-spatial components of
ξA1 A1 ···Ak Ak can be obtained by a full contraction of a primed-unprimed pair of
indices with τ AA .
An alternative way of looking at the projection procedure described in theprevious paragraphs is the following: given the spinorial counterpart ξA1 A1 ···A1 Akof a (in principle non-spatial) tensor, deﬁne

ξA1 B1 ···Ak Bk ≡ τB1 A1 · · · τBk Ak ξA1 A1 ···A1 Ak .
Then ξ(A1 B1 )···(Ak Bk ) encodes the spatial part of ξA1 A1 ···A1 Ak , while ξP1 P1 · · · Pk Pk
corresponds to its pure time component. Mixed time-spatial components havethe form ξP P (A2 B2 )···(Ak Bk ) , and so on.

104

Space spinors

As a particular example of the previous discussion one has that for a Hermitianspinor vAA ∈ S• (M) it holds thatvAA =

1
τAA v − τ Q A v(QA)
2

where v ≡ vP P  τ P P and vAB ≡ τB A vAA . Observing that v = vQ Q one canwrite, alternatively, thatvAB =1
AB v + v(AB) .
2

The 1 + 3 split of frame and the metricGiven a g-orthonormal frame {eAA } and its coframe {ω AA }, the discussion of
the previous paragraphs implies that they can be written as
eAA =
ω AA =

1
τAA e − τ B A eAB ,
2

1 AA
τ
ω + τC A ω CA ,2

where the various vectors and covectors in the decomposition are given byeAB ≡ τ(A P eB)P  ,e ≡ τ P P eP P  ,ω ≡ τP P  ω P P ,ω AB ≡ −τ (A P  ω B)P .


Now, recalling that ω AA , eBB   = B A B  A , one obtains thatω, e = 2,ωAB, e = 0,ω, eAB  = 0,
ω AB , eCD  = hAB CD .

Using the above pairings together with expression (3.29) one obtains thefollowing 1 + 3 split of g:g=1ω ⊗ ω + hABCD ω AB ⊗ ω CD .
2

(4.14)

In particular, one has that
hABCD ≡ g(eAB , eCD ) = −A(C D)B .
If τ is hypersurface orthogonal, then the vectors {eAB } and the covectors{ω AB } are intrinsic to the hypersurfaces Sτ orthogonal to τ ; thus, they can be
regarded as belonging to T (Sτ ) and T ∗ (Sτ ), respectively. In addition,h ≡ hABCD ω AB ⊗ ω CD
corresponds to the (negative deﬁnite, Riemannian) metric induced by g on Sτ .
Let ijk denote the volume form of the three-dimensional metric h, and let

4.3 Calculus of space spinors

105

ABCDEF be its spinorial counterpart. Using the antisymmetry properties of
ABCDEF it can be expressed in terms of AB asi
ABCDEF = √ (AC BE DF + BD AF CE ).
2(4.15)

Furthermore, it can be checked that
ABCDEF ABCDEF = −6.
Alternatively, Equation (4.15) can be obtained from Equation (3.25), by suitable
contactions with τ AA . More precisely, one has that




1
CDEF GH = √ τ AA τD C τF E τH G AA CC  EE  GG .
2

4.3 Calculus of space spinors
This section discusses the notion of covariant derivative in the context of thespace spinor formalism. For simplicity of the presentation, it is assumed that
one has a situation as described in Section 4.2.5 where the spinor AA is givenby the spinorial counterpart τAA of the tangent vector τ to a timelike congruencein (M, g). Moreover, it is also assumed that ∇AA is the spinorial counterpartof the Levi-Civita connection of the metric g.
4.3.1 The Sen connection

The spinor τ AA can be used to obtain a space spinor version of the spacetime
spinorial covariant derivative ∇AA . More precisely, one can deﬁne∇AB ≡ τB A ∇AA .
The latter, in turn, can be written in terms of its irreducible components as∇AB =1
AB P + DAB ,2(4.16)whereP ≡ τ AA ∇AA ,

DAB ≡ τ(B A ∇A)A .

The operator P is the directional derivative of the connection ∇ in the
direction of τ . The diﬀerential operator DAB is the so-called Sen connectionof ∇ relative to the vector ﬁeld τ . In view of these deﬁnitions one can also write∇AA =

1
τAA P − τA Q DAQ .
2

The timelike vector τ is completely arbitrary; in particular, it is not assumed
to be hypersurface orthogonal. This has several consequences; most notably, theSen connection has, in general, a non-vanishing torsion which can be expressed interms of the covariant derivative of τ AA . Furthermore, even in the case when τ is106Space spinors

hypersurface orthogonal, DAB does not coincide with the Levi-Civita connection
D of the intrinsic 3-metric of the hypersurfaces Sτ orthogonal to τ . Finally,
it is pointed out that DAB is not a real diﬀerential operator in the sense that+
= −DAB .
DAB
The derivative of τ AAFor future use, it is convenient to deﬁne1
χABCD ≡ √ τD C ∇AB τCC  .
2(4.17)

Using the split (4.16) one obtains the decomposition
χABCD =

1AB χCD + χ(AB)CD
2

where

1
χAB ≡ √ τB A PτAA ,21
χ(AB)CD ≡ √ τD C DAB τCC  .
2

It can be veriﬁed that the above spinors satisfy the following symmetry andreality properties:
χAB = χ(AB) = −χ+AB ,χ(AB)CD = χ(AB)(CD) = χ+(AB)CD .

The spinor χAB corresponds to the acceleration vector of τ , while χ(AB)CD
is related to the Weingarten tensor of the distribution deﬁned by τ . It can bechecked that the distribution is integrable if and only if χQ (BC)Q = 0. In this case
τ is hypersurface orthogonal, and χABCD corresponds to the extrinsic curvatureof the orthogonal hypersurfaces Sτ .
The hypersurface orthogonal caseIf τ is hypersurface orthogonal, given a spinor μC , the covariant derivative DAB
deﬁned by1(4.18)DAB μC ≡ DAB μC + √ χ(AB)C Q μQ
2
can be veriﬁed to be torsion-free. As DAB CD = 0 and using that χABCD =χAB(CD) , one concludes that DAB CD = 0. Thus, DAB is metric and must coincide with the (spinorial counterpart of the) Levi-Civita connection of the leavesof the foliation deﬁned by τ . It can be further veriﬁed from Equation (4.18) that+
DAB μC = −DAB μ+C,
so that DAB is a real diﬀerential operator in the sense of Section 4.2.1.
Remark. The derivative DAB as deﬁned in Equation (4.18) provides an explicit
example of the notion of space spinor covariant derivative to be introduced in
Section 4.3.3.

4.3 Calculus of space spinors

107

4.3.2 Space spinor split of the spacetime connection coeﬃcients
Following the notation of Section 3.2.2, let ΓAA CD denote the spin connection
coeﬃcients of a Levi-Civita connection ∇AA with respect to some spin basis{A A }. Its space spinor counterpart ΓABCD is deﬁned by
ΓABCD ≡ τB A ΓAA CD .
The spin coeﬃcients ΓABCD satisfy no speciﬁc reality conditions. However,sometimes it is convenient to have a split of ΓABCD into real and imaginaryparts. One has that












τB A (∇AA τCC  )τD C = −τB A ΓAA Q C τQC  τD C − τB A Γ̄A A Q C  τCQ τD CC= −ΓABCD + τB A τC Q τA E τQ F τC  G Γ+
EAF G τD= −ΓABCD − δB E δC F δD G Γ+
EAF G
= −ΓABCD − Γ+BACD ,
where it has been used that eAA (τCC  ) = 0, the identityΓ̄A AB  C  = −τA E τB  F τC  G Γ+
EAF G ,and the identity (4.13). Hence, it follows that χABCD corresponds, essentially,to the real part of ΓABCD ; that is,1
χABCD = − √ (ΓABCD + Γ+
ABCD ).
2
The reality of the above expression follows from Γ++
ABCD = ΓABCD . The
imaginary part of ΓABCD is given by1
ξABCD = √ (ΓABCD − Γ+
ABCD ).
2
Inverting the deﬁnitions of χABCD and ξABCD it follows then that1
ΓABCD = √ (ξABCD − χABCD ),211= √ (ξABCD − χ(AB)CD ) − √ AB χCD .
2
2 2
Observe the symmetry conditions
χABCD = χAB(CD) ,ξABCD = ξ(AB)(CD) .

108

Space spinors
4.3.3 Intrinsic derivatives

When working with a three-dimensional Riemannian (S, h) it is convenient
to make use of an intrinsic notion of covariant derivative, a so-called spacespinor covariant derivative DAB , compatible with operation of Hermitian
conjugation, which is the spinorial counterpart of the Levi-Civita connection Dof the metric h. One regards DAB as a map
DAB : SC···D E···F (S) → SC···D ABE···F (S).
The properties of the operator DAB have to be consistent with those of theoperator deﬁned in Equation (4.18). It is required to satisfy:(i) Symmetry. Given ζ C···D E···F ∈ S• (S) one has
DAB ζ C···D E···F = D(AB) ζ C···D E···F .
(ii) Linearity. Given ζ C···D E···F , η C···D E···F ∈ S• (S) one has
DAB (ζ C···D E···F + η C···D E···F ) = DAB ζ C···D E···F + DAB η C···D E···F .
(iii) Leibnitz rule. Given ζ C···D E···F , ξ G···H P ···Q ∈ S• (S) one has
DAB (ζ C···D E···F ξ G···H P ···Q ) = ξ G···H P ···Q DAB ζ C···D E···F+ ζ C···D E···F DAB ζ G···H P ···Q .
(iv) Reality. Given ζ C···D E···F ∈ S• (S) one has

DAB ζ C···D E···F

+= −DAB ζ +C···D E···F(v) Action on scalars. Given a scalar φ ∈ X(S), then DAB φ is the spinorial
counterpart of Di φ.
(vi) Representation of derivations. Given a derivation D, there exists a
spinor ξ AB ∈ S• (S) such that
Dζ C···D E···F = ξ P Q DP Q ζ C···D E···F
for all ζ C···D E···F ∈ SC···D E···F .
(vii) Compatibility with the -spinor. The operator DAB satisﬁes
DAB CD = 0 so that, in addition, DAB hCDEF = 0.
(viii) No torsion. For φ ∈ X(S) one has that DAB DCD φ = DCD DAB φ.

4.3 Calculus of space spinors

109

The space spinor spin coeﬃcientsLet {ei } and {ω i } denote, respectively, an h-orthonormal basis and cobasis on S.
One deﬁnes the spatial connection coeﬃcients γi k j via the equation
Di e j = γi k j e k .

(4.19)In what follows, it will be assumed that the connection D has spinorial
counterpart DAB satisfying conditions (i)–(viii) of the previous section. Let{eAB } and {ω AB } denote, respectively, the vector basis and cobasis obtained
from {ei } and {ω i } through the correspondences in (4.12) and let DAB denotethe associated covariant directional derivative.
The spinorial counterpart of the spatial connection coeﬃcients γAB CD EF
can be obtained by contraction of γi k j with the spatial Infeld-van der Waerden
symbols so that the reality condition
+ CD
CD
γAB
EF = −γABEF(4.20)

holds. Now, deﬁning the space spinor directional covariant derivative
DAB ≡ σ i AB Di , the spinorial counterpart of (4.19) can be written as
DAB eEF = γAB CD EF eCD .
Hence, one has
γAB CD EF = ω CD , DAB eEF ,
so that γAB CD EF has the symmetries
γAB CD EF = γ(AB) (CD) (EF ) .
Now, as DAB hCDEF = 0, it follows then from
DAB hCDEF = eAB (hCDEF ) − γAB P Q CD hP QEF − γAB P Q EF hCDP Q ,thatγABCDEF = −γABEF CD .
This antisymmetry can be exploited to obtain the decomposition
1
1
γAB P D P F δE C + γAB CP EP δF D
2
2= γAB D F δE C + γAB C E δF D ,γAB CD EF =where the space spinor spin coeﬃcients, γAB D F = γ(AB) D F , have been
deﬁned byγAB D F ≡1
γAB P D P F .
2(4.21)

Observing the reality condition (4.20) and that +
AB = AB , it follows that
+ C
C
γAB
D = −γAB D ;110Space spinorsthat is, the space spinor connection coeﬃcients are imaginary. A computation
similar to the one performed in Section 3.2.2 to express the spacetime spin
coeﬃcients in terms of derivatives of the spin basis shows that
γAB C D = C Q DAB D Q = −D Q DAB C Q .
From these expressions, it can be shown that given spinors κA and μA withcomponents κA and μA with respect to the space spinor basis {A A }, one has
DAB κC = eAB (κC ) − γAB Q C κQ ,DAB μC = eAB (μC ) + γAB C Q μQ ,where DAB κC ≡ C Q DAB κQ and DAB μC ≡ C Q DAB μQ .
The three-dimensional curvature spinors
As D is being assumed to be the Levi-Civita connection of a three-dimensional
negative deﬁnite metric h, it follows that the spinorial counterpart rABCDEF GH
of the Riemann tensor rijkl of D satisﬁes the antisymmetry propertyrABCDEF GH = −rCDABEF GH .
Hence, one has the decompositionrABCDEF GH = −rACEF GH BD − rBDEF GH AC ,(4.22)

withrACEF GH ≡1
rAQC Q EF GH ,2rACEF GH = r(AC)EF GH .

Now, as rABCDEF = −rABEF CD one has further thatrABCDEF = rABCE DF + rABDF CE ,withrABCE =

1
rABCQE Q ,2rABCE = rAB(CE) .

As a consequence of the symmetry rABCDEF GH = rEF GHABCD , the spinorrABCD inherits the symmetry rABCD = rCDAB . Taking into account allthese symmetries in the general decomposition for a general valence-4 spinor,
Equation (3.8), one concludes that
1
rABCD = r(ABCD) + rP Q P Q hABCD .
3
In what follows let sABCD and r denote, respectively, the spinorial counterpart of the trace-free Ricci tensor and the Ricci scalar of theconnection D. One has thatsABCD = r(ABCD) ,r = −4rP Q P Q .

4.4 Further reading

111

Hence, one ﬁnds that rABCDEF can be written as




1
1
1
1
sABCE − rhABCE DF +sABDF − rhABDF CE .
rABCDEF =2
122
12(4.23)
Using an argument similar to the one employed in Section 3.2.3 one ﬁnds thatthe commutator of the covariant derivative DAB satisﬁes(DAB DCD − DCD DAB )μE = rE F ABCD μF .
Finally, for completeness it is noticed that the three-dimensional second Bianchi
identity takes, in the present context, the form
DP Q sP QAB =1
DAB r.
6

This last expression can be obtained from multiplying by ijk the tensorial
Bianchi identity
Di rjklm + Dj rkilm + Dk rijlm = 0,
and considering its spinorial counterpart using Equations (4.22) and (4.23).
4.4 Further reading
The notions of space spinor and space spinor splits were originally introduced in
Sommers (1980); see also Sen (1981). A monograph on space spinors is Torres
del Castillo (2003). An alternative discussion, having applications in quantum
gravity in mind, is given in an appendix of Ashtekar (1991). The space spinor
formalism was ﬁrst used in Friedrich (1988, 1991) to analyse the conformal ﬁeld
equations. Further developments can be found in Friedrich (1995, 1998c), and aslightly diﬀerent perspective on these ideas is given in Frauendiener (1998a).

5
Conformal geometry

Conformal geometry is concerned with the properties of angle-preserving geometric transformations. Conformal geometry, as a branch of diﬀerential geometry,has a long story going back to the work of Cotton, Schouten and Weyl; see,for example, Cotton (1899), Schouten (1921) and Weyl (1918, 1968). It remains
an active area of research; compare the monograph by Feﬀerman and Graham(2012).
The approach to the use of conformal methods in general relativity followedin this book goes back to the seminal work by R. Penrose in the 1960s; see
Penrose (1963, 1964). Penrose’s ideas allowed to reformulate, in a geometricmanner, the study of the asymptotic behaviour of the gravitational ﬁeld. Since
then, conformal methods have provided a valuable tool for the analysis of global
aspects of the Einstein ﬁeld equations and their solutions. Conformal methods
have also been useful in the construction of exact solutions to the Einstein ﬁeld
equations; see Stephani et al. (2003).
This chapter provides an introduction to the notions of conformal geometry
to be used in the later parts of this book. The organisation of this chapter isgeared towards applications.

5.1 Basic concepts of conformal geometry
This section discusses the basic notions of conformal geometry that will be usedthroughout this book.

5.1.1 Conformal rescalings and transformations
The key notion in conformal geometry is that of a conformal rescaling . In
what follows, let g̃ and g denote two metrics on a manifold M̃. The metrics

5.1 Basic concepts of conformal geometry113g̃ and g are said to be conformally related (or simply conformal ) to eachother if there exists a positive Ξ ∈ X(M̃) such thatg = Ξ2 g̃.

(5.1)

The scalar Ξ is called the conformal factor. Throughout this book, the symbolΞ will be used to denote a generic conformal factor on a four-dimensional
manifold.
The conformal rescaling in Equation (5.1) gives rise to an equivalence relation
among the set of metrics over M̃. The conformal class of a metric g̃, to be
denoted by [g̃], is the collection of metrics conformally related to g̃. A conformalclass is also called a conformal structure. From Equation (5.1) it follows thatthe contravariant metrics g̃  and g  are related byg  = Ξ−2 g̃  ;that is, g ab = Ξ−2 g̃ ab , so as to ensure that g̃ab g̃ bc = δa c and gab g bc = δa c .
Closely related to the notion of conformally related metrics is the concept
of conformal transformations. To discuss this idea, let M̃ and M denote two
manifolds with metrics g̃ and g, respectively. A conformal transformation(also called conformorphism) is a diﬀeomorphism ϕ : M̃ → M such that thepull-back of g is conformal to g̃. That is, one has thatϕ∗ g = Ξ2 g̃.

(5.2)

Notice that as ϕ is a diﬀeomorphism, then (ϕ∗ )−1 is well deﬁned
 the last 2and∗ −1Ξ g̃ .
expression could have been written, alternatively, as g = (ϕ )A special case of the previous discussion occurs when g̃ is a ﬂat metric – in theLorentzian four-dimensional case the Minkowski metric η and in the Riemannian
three-dimensional case the Euclidean metric δ. In these cases one then says thatg is conformally ﬂat. Determining whether a given conformal class contains
the ﬂat metric is a classical problem in conformal geometry; see Section 5.2.3.
The conformal group
As before, let [g̃] denote the conformal class of a Lorentzian metric g̃ on amanifold M̃. Consider a frame {ẽa } which is orthonormal with respect to g̃. If{ω̃ a } denotes the associated coframe, one has thatg̃ = ηab ω̃ a ⊗ ω̃ b ,that is,g̃(ẽa , ẽa ) = ηab .

In order to investigate the type of transformations of {ω̃ a } which lead to anothermetric g ∈ [g̃], writeω̃ a = K a c ω c ,114

Conformal geometry

with (K a c ) denoting some transformation matrix and {ω a } another orthonormal
frame. The condition on the matrix (K a c ) so that it leads to another memberof the conformal class, say, g = Ξ2 g̃, is then given byg̃ = ηab K a c K b d ω c ⊗ ω d = Ξ−2 ηcd ω c ⊗ ω d .
The latter expression suggests writing
K a b = Ξ−1 Λa b ,where (Λa b ) is a Lorentz transformation; that is, Λa c Λb d ηab = ηcd . The
group of (four-dimensional) Lorentz transformations will be denotedby O(1, 3). It follows that at a point p ∈ M̃ the group of transformations
taking a g̃-orthonormal frame to a frame which is orthonormal with respect
to another metric in the conformal class [g], the so-called conformal group
CO(1, 3), is given by CO(1, 3) = R+ × O(1, 3). The previous discussion canbe adapted to the case of three-dimensional Riemannian metrics. In that case,the conformal group, denoted by CO(3), is given by CO(3) = R+ × O(3), whereO(3) denotes the group of three-dimensional orthogonal transformations(rotations).
5.1.2 Conformal extensions and conformal compactiﬁcations
If a smooth mapping ϕ : M̃ → M satisfying condition (5.2) is injective
but not surjective (i.e. ϕ(M̃)  M), then one says that M is a conformal
extension of M̃. An important type of conformal extensions are the so-called
conformal compactiﬁcations. A conformal compactiﬁcation of a manifold M̃with metric g̃ is a conformal transformation ϕ : M̃ → U where U is a relatively
compact (i.e. the closure of U is compact), connected, open set of a manifold M
such thatg = (ϕ∗ )−1 (Ξ2 g̃)in U ,

with a conformal factor Ξ such that:(i) Ξ > 0 in U .
(ii) Ξ = 0 on ∂U , the boundary of the open set U . The set ∂U is called theconformal boundary of M̃.
Examples of conformal extensions will be discussed in Chapter 6.
5.2 Conformal transformation formulae
The discussion of Section 2.4.4 can be applied to obtain the transformation
˜ and ∇
formulae relating the curvature tensors of the Levi-Civita connections ∇of two metrics g̃ and g related to each other by Equation (5.1).

5.2 Conformal transformation formulae

115

5.2.1 Transformation formulae for the connection
As a ﬁrst step, one needs to ﬁnd the speciﬁc form of the transition tensor Qa c b
˜ and– see Equation (2.13). The ﬁrst observation is that as the connections ∇∇ are torsion free, it follows from Equation (2.15) that the transition tensor issymmetric; that is, one has that
Qa c b = Q(a c b) .
Using formula (2.14) one has that˜ a gbc = −Qa d b gdc − Qa d c gbd .
∇a gbc − ∇˜ a gbc = ∇˜ a (Ξ2 g̃bc ) = 2Ξ∇˜ a Ξg̃bc (as ∇˜ a g̃bc = 0) one ﬁnds
From ∇a gbc = 0 and ∇that2(Ξ−1 ∇a Ξ)gbc = Qa d b gdc + Qa d c gbd .
Two further companion equations can be obtained from the latter by permuting
cyclically the indices abc . Adding two of them and subtracting the third one, one
can solve for Qa c b to ﬁnd
Qa c b = Ξ−1 (∇a Ξδb c + ∇b Ξδa c − ∇d Ξg dc gab ).
This last expression can be rewritten in a more concise form as
Qa c b = Sab cd (Ξ−1 ∇d Ξ),

(5.3)

where
Sab cd ≡ δa c δb d + δa d δb c − gab g cd .
To simplify the presentation of the various transformation formulae, letΥa ≡ Ξ−1 ∇a Ξ,

Υa c b ≡ Sab cd Υd .

Hence, one can write schematically that˜ = S(Υ),
∇−∇

(5.4)

and Equation (5.3) yields Qa c b = Υa c b . The tensor S appeared in Section
2.5.2 in the decomposition of the Riemann tensor; see Equation (2.21b). Using
Equation (5.1) one ﬁnds thatδa c δb d + δa d δb c − gab g cd = δa c δb d + δa d δb c − g̃ab g̃ cd .
Hence, the tensor S is independent of the representative of the conformal class;that is, it is an invariant of [g̃].

116

Conformal geometry
5.2.2 Transformation formulae for the curvature

Combining the results of Section 2.4.4 with the expression for the transition
tensor of Equation (5.3) one obtains a transformation rule for the Riemanntensor:Rc dab − R̃c dab = 2(∇[a Υb] c d + Υ[a c |e| Υb] e d ).

(5.5)

Some of the transformation formulae for the various concomitants of the
Riemann tensor are dimension dependent; thus, they are analysed separately.
The 4-dimensional caseIn the four-dimensional case one has that the Ricci and Schouten tensors and˜ and ∇ are related to each other, respectively,
Ricci scalar of the connections ∇by the expressions


1
2
3
∇c ∇d Ξ − 2 ∇c Ξ∇d Ξ , (5.6a)Rab − R̃ab = − ∇a ∇b Ξ − gab g cd
Ξ
Ξ
Ξ
1
1
Lab − L̃ab = − ∇a ∇b Ξ +∇c Ξ∇c Ξ gab ,
Ξ
2Ξ2
1
6
12
R − 2 R̃ = − ∇c ∇c Ξ + 2 ∇c Ξ∇c Ξ.
ΞΞΞ

(5.6b)(5.6c)

Using the tensor Sab cd , one can rewrite the transformation formula for the
Schouten tensor, Equation (5.6b), in the alternative form
1
Lab − L̃ab = ∇a Υb + Sab cd Υc Υd .
2

(5.7)

By letting ϑ ≡ Ξ−1 , the transformation rule for the Ricci tensor can be
rewritten as6∇a ∇a ϑ − Rϑ = −R̃ϑ3 .
Using the irreducible decomposition of the Riemann tensor, Equation (2.21b), asa deﬁnition for the Weyl tensor, together with Equations (5.5) and (5.6b), one
ﬁnds thatC c dab = C̃ c dab .
In other words, the Weyl tensor is an invariant of the conformal class [g̃]. Using
this invariance and the transformation law for the connection, a calculation leads
to the important identity
˜ a C a bcd .
∇a (Ξ−1 C a bcd ) = Ξ−1 ∇(5.8)

A further tensor which will play a role in the present treatment of conformal
˜ This tensor is deﬁned asgeometry is the so-called Cotton tensor of ∇.
˜ a L̃bc − ∇˜ b L̃ac .
Ỹabc ≡ ∇

5.2 Conformal transformation formulae

117

Notice that by construction Ỹabc = Ỹ[ab]c . The Cotton tensor is closely related
to the Weyl tensor. To see this, consider the second Bianchi identity
˜ [e R̃a |b|cd] = 0,
∇

(5.9)

satisﬁed by the Riemann tensor of the metric g̃; see Section 2.4.3. Now, as seenin Section 2.5.2, for a Levi-Civita connection, the Riemann tensor R̃a bcd can be
decomposed in terms of the Weyl tensor C a bcd and the Schouten tensor L̃ab asR̃a bcd = C a bcd + 2(g̃ a [c L̃d]b − g̃b[c L̃d] a ).

(5.10)

Substituting the latter into Equation (5.9) one obtains
˜ e L̃d] a − g̃ a [c ∇˜ e L̃d]b ) = ∇˜ [e C a |b|cd] .
2(g̃b[c ∇Contracting the indices a and e one obtains
˜ c L̃db − ∇˜ d L̃cb = ∇˜ a C a bcd .
∇(5.11)

˜ a C a bcd .
Ỹcdb = ∇(5.12)

That is,In particular, one sees that if C a bcd = 0, then Ỹcdb = 0. Moreover, as a consequenceof the ﬁrst Bianchi identity for the Weyl tensor, Ỹ[abc] = 0. The Riemann tensor
Ra bcd of the connection ∇ satisﬁes equations analogous to (5.9) and (5.10). It
follows by the same computation described above that∇c Ldb − ∇d Lcb = ∇a C a bcd .

(5.13)

Alternatively, deﬁning the Cotton tensor of ∇, Ycdb ≡ ∇c Ldb − ∇d Lcb , one canwrite
Ycdb = ∇a C a bcd .

(5.14)

Combining Equations (5.8), (5.12) and (5.14) one ﬁnds that the transformation
rule for the Cotton tensor is given by:
Ycdb − Ỹcdb = Υa C a bcd .

(5.15)The three-dimensional caseIn the case of a three-dimensional manifold, let h = Ω2 h̃ – throughout, thesymbol Ω will be used to denote a generic conformal factor on a manifold ofdimension three. One has that


1
1
2
Dk Dl Ω − 2 Dk ΩDl Ω ,(5.16a)rij − r̃ij = − Di Dj Ω − hij hkl
Ω
Ω
Ω
1
1
lij − ˜lij = − Di Dj Ω +Dk ΩDk Ω hij ,
Ω
2Ω2
1
4
6
r − 2 r̃ = − Di Di Ω + 2 Di ΩDi Ω.
ΩΩΩ

(5.16b)(5.16c)118

Conformal geometry

where Di denotes the Levi-Civita covariant derivative of the metric h, andrij , lij , r correspond to its Ricci and Schouten tensors and its Ricci scalar,respectively. The transformation law of the Schouten tensor is of particularinterest. Comparing Equations (5.6b) and (5.16b) one sees that although thedeﬁnition of the Schouten tensor is dimension dependent, its transformation
formula is not.
Letting ϑ ≡ Ω−1/2 , the transformation law for the Ricci scalar can be recast as
8Di Di ϑ − rϑ = −r̃ϑ5 .

(5.17)

This expression plays an important role in the discussion of the Einsteinconstraint equations; see Chapter 11.
Given the three-dimensional Schouten tensor ˜lij , its associated Cotton tensorỹijk is given byỹijk ≡ D̃i ˜ljk − D̃j ˜lik .

(5.18)

Using the transformation rule (5.16b), a computation shows thatyijk = ỹijk .
That is, in three dimensions the Cotton tensor is conformally invariant.
Sometimes it is more convenient to work with its Hodge dual, the so-called
Cotton-York tensor , given by1ỹij = − ỹklj i kl .
2
It can be readily veriﬁed thatyij = yji ,yi i = 0,

Di yij = 0.

Moreover, the Cotton-York tensor satisﬁes the transformation ruleyij = Ω−1 ỹij .

(5.19)

5.2.3 Characterising conformal ﬂatness
Given a conformal class [g] on a manifold M, an important question is whether
the ﬂat metric belongs to it, so that g is conformally ﬂat. Conformally ﬂat metrics
are a source of geometric intuition in general relativity as they have a simpler
curvature tensor depending on the Schouten tensor only. Conformal ﬂatness ischaracterised by the following classical result:
Theorem 5.1 (Weyl-Schouten theorem)(i) Let (M, g) be a manifold with metric of dimension n ≥ 3. The metric g is
conformally ﬂat if and only if the Cotton tensor of g vanishes.
(ii) Let (M, g) be a manifold with metric of dimension n ≥ 4. The metric g is
conformally ﬂat if and only if the Weyl tensor of g vanishes.

5.3 Weyl connections

119

Proof A direct computation shows that if a metric is conformally ﬂat, thenboth its Cotton and Weyl tensors vanish; this proves the if part.
In order to prove the only if part, one uses the fact that if the Weyl tensor
vanishes then, for dimensions n ≥ 4, the Cotton tensor vanishes; compare
Equation (5.14). In view of Equation (5.15) one concludes that the vanishingof the Cotton tensor holds for any metric in the conformal class. From this point
onwards, the proofs for the various dimensions are similar. For simplicity, only
the four-dimensional case is considered.
Given a metric g in the conformal class, one needs to ﬁnd a conformal factor
Ξ such that g = Ξ2 η where η is the ﬂat Minkowski metric. Motivated by
the transformation law for the Schouten tensor, Equation (5.6b), consider theequation1∇a αb + αa αb − αc αc gab = −Lab .
2(5.20)The latter can be read as an overdetermined partial diﬀerential equation for the
covector αa . Given a solution to Equation (5.20), an antisymmetrisation yields
that ∇[a αb] = 0 so that αa is a closed covector. Thus, locally αa is exact andcan be written as αa = ∇a (ln Ξ) = Ξ−1 ∇a Ξ for some function Ξ. Comparing
Equation (5.20) with (5.6b) one concludes that the Schouten tensor of the metricg̃ = Ξ−2 g must vanish. As C a bcd = 0, the whole Riemann tensor of g̃ mustvanish. Consequently, one concludes that g̃ = η.
Hence, to conclude the proof one needs to show that Equation (5.20) admits
a solution under the assumption that C a bcd = 0 and Yabc = 0. Applying ∇cto Equation (5.20), antisymmetrising on ca and ﬁnally using the commutator ofcovariant derivatives, one ﬁnds the integrability condition
Rd bca αd + 2α[a ∇c] αb + 2αe ∇[c αe ga]b = 0.

(5.21)

Now, as C a bcd = 0 one has that
Ra bcd = 2(δ a [c Ld]b − gb[c Ld] a ).
Using the latter expression for the Riemann tensor together with Equation (5.20),
one ﬁnds that the integrability condition (5.21) is automatically satisﬁed.
A general version of the Frobenius theorem ensures the existence of a solution
αb to Equation (5.20); see, for example, Choquet-Bruhat et al. (1982) or Spivak(1970).
5.3 Weyl connections
˜ denote the Levi-Civita connection of a metric
As in the previous sections, let ∇g̃ on M̃. Some of the applications of conformal geometry to be considered in thisbook give rise to connections which are not necessarily the Levi-Civita connection

120

Conformal geometryof a metric but, nevertheless, respect the conformal class. A Weyl connection
ˆ such thatis a torsion-free connection ∇ˆ a g̃bc = −2 f˜a g̃bc ,∇(5.22)

for some arbitrary covector f˜a .
˜ and ∇
ˆ can be obtained
The transition tensor Qa c b relating the connections ∇using an argument similar to the one employed in Section 5.2.1 to compute thetransition tensor of a conformal rescaling. One ﬁnds that
Qa c b = Sab cd f˜d .
Schematically one writesˆ −∇˜ = S(f̃ ).
∇If the covector f̃ is exact, so that on suitable open sets it can be written in theform f = −Ξ−1 dΞ with some smooth function Ξ > 0, then the Weyl connectionˆ is, in fact, the Levi-Civita connection of the metric g = Ξ2 g̃.
∇ˆ a δb c = 0 satisﬁed by a generic connection together with
The condition ∇cthe relation δb = g̃bd g̃ dc and the deﬁning property of a Weyl connection,
Equation (5.22), show that
ˆ a g̃ bc = 2 f˜a g̃ bc .
∇Using the above expressions one readily obtains thatˆ e Sab cd = ∇ˆ e (δa c δb d + δb c δa d − g̃ab g̃ cd )∇
ˆ e g̃ cd = 0.
ˆ e (g̃ab g̃ cd ) = −∇
ˆ e g̃ab g̃ cd − g̃ab ∇= −∇In what follows, let R̂a bcd denote the Riemann tensor of the Weyl connection
ˆ This tensor possesses the basic symmetry R̂a bcd = −R̂a bdc . As the connection∇.
ˆ has vanishing torsion, it follows that R̂a bcd satisﬁes the ﬁrst and second∇
Bianchi identities in the form:R̂a [bcd] = 0,(5.23a)ˆ [e R̂ |b|cd] = 0.
∇(5.23b)a5.3.1 Weyl propagation
To investigate the relation between Weyl connections and the conformal class [g̃],consider a curve γ with parameter s ∈ I ⊆ R on (M̃, g̃) with tangent ẋ ∈ T (M̃).
A vector u ∈ T (M̃) is said to be Weyl propagated along γ if it is parallelyˆ that is, u satisﬁes the
propagated along γ with respect to a Weyl connection ∇;equationˆ ẋ u = 0.
∇

5.3 Weyl connections121

˜ one has that
Writing the latter in terms of the Levi-Civita connection ∇˜ a ub = −ẋa Sac be uc f˜e ,ẋa ∇= g̃cd uc ẋd f˜b − uc f˜c ẋb − ẋc f˜c ub .
In index-free notation one has that˜ ẋ u = g̃(u, ẋ)f̃  − f̃ , uẋ − f̃ , ẋu.
∇
Let {ea } denote an arbitrary frame which is Weyl propagated along γ so thatˆ ẋ ea = 0. Letting g̃ab ≡ g̃(ea , eb ), a computation then shows that∇ˆ ẋ g̃ab = ∇ˆ ẋ (g̃(ea , eb )) = −2f̃ , ẋg̃ab .
∇(5.24)

ˆ ẋ (ln g̃ab ) = −2f̃ , ẋ. The latter equation can beConsequently, one obtains ∇solved to give
  s


f̃ , ẋdsg̃ab (η) = g̃ab (η ) exp −2s
along the curve x(s). Thus, one ﬁnds that Weyl connections respect the conformalclass in the sense that parallel propagation of a metric using a Weyl connection
leads to a metric in the same conformal class. Notice also that Equation (5.24)
allows one to conclude that if the frame is orthogonal at some point along the
curve, then it is orthogonal elsewhere on γ – the normalisation, however, is lost.
5.3.2 Transformation formulae for the curvature
The transformation formulae between the curvature tensors of the Levi-Civita
˜ and the Weyl connection ∇ˆ follow directly from the generalconnection ∇discussion of Section 2.4.4.
ˆIn what follows let f˜a c b ≡ Sab cd f˜d . If R̂a bcd denotes the Riemann tensor of ∇,then one has that˜ [c f˜d] a b + f˜e a [c f˜d] e b ),R̂a bcd − Ra bcd = 2(∇
˜ d] f˜b + ∇˜ [c f˜d]
˜ [c f˜a g̃d]b − δ a b ∇= 2(δ a [c ∇−δ a [c f˜d] f˜b + g̃b[c f˜d] f˜a + δ a [c g̃d]b f˜e f˜e ).

(5.25a)(5.25b)Note that the above transformation law involves both the symmetric and˜ a f˜b .
antisymmetric parts of the covariant derivative ∇A transformation formula for the Ricci tensor R̂bd ≡ R̂a bad can be obtained
directly from Equation (5.25b):
˜ e f˜e + 2 f˜e f˜e .
˜ d f˜c + ∇˜ c f˜d + 2 f˜c f˜d − g̃cd ∇R̂cd − R̃cd = −3∇(5.26)

122

Conformal geometry

Now, as there is no canonical metric to lower or raise indices in expressions
involving a Weyl connection, it is conventional to choose a representative of theconformal class, say, g̃, and use it to compute traces. In this spirit one deﬁnes
the Ricci scalar of the Weyl connection via R̂ ≡ g̃ ab R̂ab . It can then be directlycomputed that˜ a f˜a − 6 f˜a f˜a .
R̂ − R̃ = −6∇(5.27)

Combining the transformation formula for the Riemann tensor, Equation (5.25a), with the irreducible decomposition of the Riemann tensor R̃a bcd
given by Equation (2.21b), one can ﬁnd an analogous decomposition for theˆRiemann tensor R̂a bcd of ∇:R̂c dab = C c dab + 2Sd[a ce L̂b]e ,= C c dab + 2(g̃ c [a L̂b]d − δ c d L̂[ab] − g̃d[a L̂b] c ),where
1L̂ab =
2(5.28a)11R̂(ab) − R̂[ab] − g̃ab R̂26

ˆ This deﬁnition isis the Schouten tensor of the Weyl connection ∇.
independent of the choice of the representative of the conformal class. Making
use of the transformation laws for the Ricci tensor and scalar, Equations (5.26)
and (5.27), one ﬁnds that˜ a f˜b − f˜a f˜b + 1 g̃ab f˜c f˜c ,L̃ab − L̂ab = ∇21
˜ a f˜b − Sab cd f˜c f˜d ,=∇2
ˆ a f˜b + 1 Sab cd f˜c f˜d .
=∇2(5.29a)(5.29b)(5.29c)Finally, it is observed that letting R̂abcd ≡ g̃ae R̂e bcd , it follows from thediscussion in the previous paragraphs that
ˆ [c fd] ,R̂abcd = R̂[ab]cd + 2g̃ab ∇(5.30a)= R̂[ab]cd − 2g̃ab L̂[cd] .

(5.30b)

These formulae show in an explicit way how the usual symmetries of thecurvature tensor are obstructed by the covector deﬁning a Weyl connection.
5.4 Spinorial expressions
This section discusses the spinorial counterparts of the tensorial expressions
obtained in the previous sections of this chapter.

5.4 Spinorial expressions

123

5.4.1 Conformal rescalings
As in previous sections, let g̃ and g denote two metrics on M̃ related to eachother by the conformal rescaling (5.1). Following the discussion of Chapter 3,
the spinorial counterparts of g̃ and g are given byg̃AA BB  = ˜AB ˜A B  ,gAA BB  = AB A B  ;compare Equation (3.15). Hence, it is natural to consider the transformationlaws
AB = Ξ˜AB ,

AB = Ξ−1 ˜AB ,A B  = Ξ˜ A B  ,A B = Ξ−1 ˜A B .









Let {õA , ι̃A } and {oA , ιA } denote two spin bases satisfying, respectively, theconditions
˜AB = õA ι̃B − ι̃A õB ,AB = oA ιB − ιA oB .

There are several possible transformation rules between the two spin bases which
are consistent with the above equations and with the rescaling (5.1). Namely,
one has:oA = õA ,ιA = Ξι̃A ,oA = Ξ−1 õA ,oA = ΞõA ,ιA = ι̃A ,oA = õA ,oA = Ξ1/2 õA ,ιA = Ξ1/2 ι̃A ,ιA = ι̃A ,(5.31a)ιA = Ξ−1 ι̃A ,(5.31b)oA = Ξ−1/2 õA ,ιA = Ξ−1/2 ι̃A .

(5.31c)

The choice of the most convenient transformation rule depends on the nature of
the application at hand; see, for example, Chapter 10.
Transformation rules for the connection and curvature
In what follows let ΥAA ≡ Ξ−1 ∇AA Ξ denote the spinorial counterpart of thecovector Υa . Let also Υa c b ≡ Sab cd Υd . Its spinorial counterpart is given by








ΥAA CC BB  = δA C δA C ΥBB  + δB C δB  C ΥAA − AB A B  ΥCC .
By rewriting












δA C δA C ΥBB  + δB C δB  C ΥAA = δA C A C B  D ΥBD + δB C B  C A D ΥAD ,and using the Jacobi identity (3.5), one ﬁnds thatΥAA CC BB  = ΥAA C B δB  C + ῩA A C B  δB C ,whereΥAA C B ≡ δA C ΥBA .

124

Conformal geometry

The reduced coeﬃcient ΥAA C B can be used to obtain the transformation laws
relating the covariant derivatives of spinors. In particular, one has for arbitraryspinors κA , μA , ξ A and η A that˜ AA κB = ∇AA κB + ΥBA κA ,∇˜ AA μB  = ∇AA μB  + ΥAB  μA ,∇˜ AA ξ B = ∇AA ξ B − δA B ΥCA ξ C ,∇˜ AA η B  = ∇AA η B  − δA B  ΥAC  η C  .
∇These expressions can be extended, in a direct way, to higher valence spinors.
For the curvature spinors, it can be veriﬁed thatΨ̃ABCD = ΨABCD ,Φ̃AA BB  = ΦAA BB  + Ξ−1 ∇A(A ∇B  )B Ξ.
5.4.2 Weyl connections
ˆ AA denote the spinorial counterpart of the Weyl connection
In what follows, let ∇ˆˆ AA ˜BC and∇ deﬁned by Equation (5.22). To determine expressions for ∇BCˆ∇AA ˜
one notices that the spinorial version of Equation (5.22) isˆ AA (˜∇BC ˜B  C  ) = −2 f˜AA ˜BC ˜B  C  ,so thatˆ AA ˜BC + ˜BC ∇ˆ AA ˜B  C  = −2 f˜AA ˜BC ˜B  C  .
˜B  C  ∇The latter is satisﬁed if one setsˆ AA ˜BC = −f˜AA ˜BC .
∇ˆ AA δB C = 0, one can readily compute
From this expression and using that ∇ˆ AA ˜BC . One ﬁnds that∇
ˆ AA ˜BC = f˜AA ˜BC .
∇Decomposition of the spin connection coeﬃcients of a Weyl connection
Let {˜
A A } denote a spin basis with respect to ˜AB . Following the general discussion on spin connection coeﬃcients of Section 3.2.2 – compare Equation (3.33)– the spinorial counterparts of the connection coeﬃcients Γ̃a b c and Γ̂a b c admit
the decompositions¯  B  δ B ,Γ̃AA BB CC  = Γ̃AA B C δC  B + Γ̃
AA
C C



¯
BB
B
B
B
Γ̂AA
+ Γ̂AA C  δC B .
CC  = Γ̂AA C δC 

5.4 Spinorial expressions

125

The spinorial counterpart of the equationΓ̂a b c = Γ̃a b c + δa b f˜c + δc b f˜a − ηac f˜b
is given byΓ̂AA BB CC  = ΓAA BB CC  + δA B δA B f˜CC +δC B δC  B f˜AA − AC A C  f˜BB .
Now, by rewriting








δA B δA B f˜CC  = δA B A B C  D f˜CD ,δC B δC  B f˜AA = δC B C  B A D f˜ADand using the Jacobi identity (3.5), one ﬁnds that

δA B δA B f˜CC  + δC B δC  B f˜AA − AC A C  f˜BB= δA B δC  B f˜CA + δC B δA B f˜AC  .
Hence,
Γ̂AA BB CC  = (Γ̃AA B C + δA B f˜CA )δC  B¯  B  + δ  B f˜  )δ B ,
+(Γ̃
C
AA
C
A
AC




so thatΓ̂AA B C = Γ̃AA B C + δA B f˜CA .

(5.32)

In particular, as Γ̃AA BC = Γ̃AA (BC) , it follows that
Γ̂AA Q Q = f˜AA .
Decomposition of the curvature tensors
The discussion of the decomposition of the spinorial counterpart of a general
Riemann tensor given in Section 3.2.3 can be applied to the case of a Weylconnection. In particular, if R̂AA BB  CC  DD denotes the spinorial counterpartˆ one has that Equation (3.35)of the Riemann tensor of a Weyl connection ∇,gives, in the present context, the decomposition¯
R̂AA BB  CC  DD = A B  R̂ABCC  DD + AB R̂A B  CC  DD ,where1
ˆ CC  f˜DD − ∇ˆ DD f˜CC  ),R̂ABCC  DD ≡ R̂(AB)CC  DD + AB (∇2
1= R̂(AB)CC  DD − AB (L̂CC  DD − L̂DD CC  ),2

126

Conformal geometry

ˆand L̂AA BB  denotes the spinorial counterpart of the Schouten tensor of ∇.
A more detailed expression is given by
R̂ABCC  DD = −ΨABCD C  D + L̂BC  DD AC − L̂BD CC  AD .

(5.33)The spinorial counterpart L̂AA BB  of the Schouten tensor admits, in turn, thedecomposition
L̂AA BB  = ΦAA BB  −1
RAB A B  + ΦAB A B  + Φ̄A B  AB
24

where ΦAA BB  represents the trace-free part of 12 R̂(ab) , while ΦAB describes theantisymmetric tensor 14 R̂[ab] .
5.5 Conformal geodesics
This section discusses a class of invariants of the conformal structure of aspacetime (M̃, g̃). To motivate the discussion let x(s), s ∈ I ⊂ R, denote a
curve on M̃ with tangent given by x ≡ dx/ds. The curve x(s) is a geodesic
˜ x x = 0. The transformation rule of the covariant
if it satisﬁes the equation ∇˜derivative ∇ under the conformal rescaling (5.1) implies, in turn, the equation∇x x = 2Υ, x x − g(x , x )Υ .

(5.34)

Let τ = τ (s) denote a new parameter. Writing ẋ ≡ dx/dτ and τ  ≡ dτ /ds, thechain rule yields x = τ  ẋ, so that Equation (5.34) impliesτ 2 ∇ẋ ẋ = (2Υ, ẋτ 2 − τ  )ẋ − τ 2 g(ẋ, ẋ)Υ .
This last expression suggests choosing the parameter τ so that it satisﬁes theconditionτ  = 2Υ, ẋτ 2 .
As Υ is known along the curve, this equation can be read as a second-order
ordinary diﬀerential equation for τ . Thus, it can always be solved locally so thatτ 2 ∇ẋ ẋ = −τ 2 g(x , x )Υ .
It follows that only when the curve x(s) is null (i.e. g(x , x ) = 0) is it possible
to reparametrise so that x(s) is a geodesic. Hence, timelike or spacelike geodesics
are not, in general, conformal invariants.

5.5 Conformal geodesics

127

5.5.1 Basic deﬁnitions
A conformal geodesic on a spacetime (M̃, g̃) is a pair (x(τ ), β(τ )) consistingof a curve x(τ ) on M̃, τ ∈ I ⊂ R, with tangent ẋ(τ ) and a covector β(τ ) alongx(τ ) satisfying the equations
˜ ẋ ẋ = −2β, ẋẋ + g̃(ẋ, ẋ)β  ,∇˜ ẋ β = β, ẋβ − 1 g̃  (β, β)ẋ + L̃(ẋ, ·),∇2(5.35a)(5.35b)˜ Associwhere L̃ denotes the Schouten tensor of the Levi-Civita connection ∇.
ated to a conformal geodesic, it is convenient to consider a frame {ea } which is
Weyl propagated along x(τ ) so that˜ ẋ ea = −β, ea ẋ − β, ẋea + g̃(ea , ẋ)β  .
∇(5.36)

Initial data for the conformal geodesic Equations (5.35a) and (5.35b) consistof an initial position, an initial direction for the curve and an initial value forthe covector:x ∈ M̃,ẋ ∈ T |x (M̃),β  ∈ T ∗ |x (M̃).

(5.37)

Piccard’s theorem – see, for example, Hartman (1987) – ensures the existence ofa unique conformal geodesic (x(τ ), β(τ )) near x satisfying for given τ ∈ R
x(τ ) ≡ x ,ẋ(τ ) ≡ ẋ ,β(τ ) ≡ β  .

A direct computation using Equations (5.35a) and (5.35b) yields the relations
˜ ẋ (g̃(ẋ, ẋ)) = −2β, ẋg̃(ẋ, ẋ),∇˜ ẋ β, ẋ = −β, ẋ2 + 1 g̃(ẋ, ẋ)g̃  (β, β) + L̃(ẋ, ẋ),
∇
2
 


˜
∇ẋ g̃ (β, β) = β, ẋg̃ (β, β) + 2L̃(ẋ, β  ).

(5.38a)(5.38b)(5.38c)

In particular, from Equation (5.38a) it follows that if g̃(ẋ, ẋ) = 0 at some point
along the conformal geodesic, one has that g̃(ẋ, ẋ) = 0 everywhere else. This
null conformal geodesic can, in turn, be reparametrised so that it coincides witha null geodesic of g̃.
Expressions in abstract index notation
For later use, it is observed that the conformal geodesic equations can be written
in abstract index notation using the tensor Sab cd as
˜ c ẋa = −Sef ac ẋe ẋ f βc ,ẋc ∇˜ c βa =ẋc ∇1
Sca ef βe βf ẋc + L̃ca ẋc ,2

˜ c ea a = −Scd af ea d ẋc βf .
ẋc ∇

128

Conformal geometry
5.5.2 Conformal geodesics and changes of connection

The motivation behind the notion of conformal geodesics is not directly apparent
from the deﬁning Equations (5.35a) and (5.35b). Their relevance becomesapparent only once one considers their transformation rules under conformal
rescalings and transitions to Weyl connections.
As in the previous section, let (x(τ ), β(τ )) denote a solution to the conformal
geodesic Equations (5.35a) and (5.35b) on a spacetime (M̃, g̃). Given f̌ ∈ T ∗ (M̃)ˇ via the relation
one can deﬁne a Weyl connection ∇ˇ ≡∇˜ + S( f̌ ).
∇(5.39)

A computation using Equations (5.35a) and (5.35b) shows that (x(τ ), β̌(τ )) withβ̌(τ ) ≡ β(τ ) − f̌ (τ )(5.40)

ˇis a solution to the ∇-conformal
geodesic equations:ˇ ẋ ẋ = −2β̌, ẋẋ + g̃(ẋ, ẋ)β̌  ,∇ˇ ẋ β̌ = β̌, ẋβ̌ − 1 g̃  (β̌, β̌)ẋ + Ľ(ẋ, ·),∇2
ˇ The latter iswhere Ľ denotes the Schouten tensor of the Weyl connection ∇.
given by
ˇ a fˇb − 1 Sab cd fˇc fˇd .
Ľab = L̃ab − ∇2Thus, one concludes that conformal geodesics are invariants of [g̃]. Notice, inparticular, that one could have chosen f̌ = −Ξ−1 dΞ for some positive Ξ ∈ X(M̃)
so that the change of connections given by Equation (5.39) corresponds, in fact,
to a conformal rescaling of g̃.
ˇNow, choosing f̌ (τ ) = β(τ ) one has that β̌(τ ) = 0, so that the ∇-conformal
geodesic equations reduce to:ˇ ẋ ẋ = 0,∇Ľ(ẋ, ·) = 0.

(5.41)

Moreover, the frame propagation Equation (5.36) yields
ˇ ẋ ea = 0.
∇Hence, given a congruence of conformal geodesics on (M̃, g̃), there exists a Weyl
ˇ on [g̃] with respect to which the curves x(τ ) are (aﬃne) geodesics
connection ∇and the frame {ea } is parallely propagated. This observation justiﬁes the nameconformal geodesics given to a solution to Equations (5.35a) and (5.35b). Thus,
conformal geodesics not only are an invariant of the conformal structure, butalso single out a particular Weyl connection on the conformal class [g].

5.5 Conformal geodesics

129

5.5.3 Reparametrisations
Given two solutions to the conformal geodesic Equations (5.35a) and (5.35b),(x(τ ), β(τ )) and (x̄(τ̄ ), β̄(τ̄ )), it is natural to ask under which conditions x(τ )and x̄(τ̄ ) coincide locally (as sets of points) so that τ = τ (τ̄ ) and x(τ (τ̄ )) = x̄(τ̄ ).
Let ẋ ≡ dx/dτ and x̄ ≡ dx̄/dτ̄ denote the corresponding tangent vectors andassume that g̃(ẋ, ẋ) = 0 and g̃(x̄ , x̄ ) = 0. By deﬁnition, the tangent vector xsatisﬁes
˜ x̄ x̄ = −2β̄, x̄ x̄ + g̃(x̄ , x̄ )β̄  ,∇˜ x̄ β̄ = β̄, x̄ β̄ − 1 g̃  (β̄, β̄)x̄ + L̃(x̄ , ·).
∇2(5.42a)(5.42b)Now, letting τ  ≡ dτ /dτ̄ one has thatx̄ = τ  ẋ,

˜ ẋ ẋ.
˜ x̄ x̄ = τ  ẋ + τ 2 ∇∇˜ ẋ ẋSubstituting the latter into Equation (5.42a) and using (5.35a) to eliminate ∇one obtains
τ  ẋ + 2τ 2 β̄ − β, ẋẋ + τ 2 g̃(ẋ, ẋ)(β  − β̄ ) = 0.

(5.43)

It follows from this last equation that the diﬀerence β̄ − β  has components
only along ẋ. Hence, one can writeβ̄ − β = αẋ ,(5.44)

for some scalar α. Substituting into Equation (5.43) one obtains the diﬀerential
equationτ  + ατ 2 g̃(ẋ, ẋ) = 0.

(5.45)

Combining Equations (5.35a), (5.35b), (5.42b) and (5.44) one obtains
1α̇ = 2β, ẋα + g̃(ẋ, ẋ)α2 .
2(5.46)

Equations (5.44), (5.45) and (5.46) encode the requirement that the curves x(τ )and x̄(τ̄ ) coincide as sets. Using Equation (5.38a) together with Equation (5.46)
one ﬁnds that˜ ẋ (αg̃(ẋ, ẋ)) = 1 (αg̃(ẋ, ẋ))2 .
∇2This last equation can be solved to giveαg̃(ẋ, ẋ) =2α g̃(ẋ , ẋ ),1 − α g̃(ẋ , ẋ )(τ − τ )

130

Conformal geometry

where α ≡ α(τ ), ẋ ≡ ẋ(τ ) and τ denotes some ﬁducial value of theparameter τ . Using Equations (5.44) and (5.45) one ﬁnally ﬁnds that:
x̄ =

4κ
ẋ,
1 + 2κα g̃(ẋ , ẋ )(τ − τ )(5.47a)β̄ = β +2α g̃(ẋ , ẋ )ẋ ,(1 − α g̃(ẋ , ẋ )(τ − τ )) g̃(ẋ, ẋ)(5.47b)τ = τ +4κ(τ̄ − τ̄ ),1 + 2κα g̃(ẋ, ẋ)(τ̄ − τ̄ )(5.47c)with κ a non-zero real constant. One can summarise the previous discussion in
the following lemma:
Lemma 5.1 (admissible reparametrisations of conformal geodesics)
The admissible reparametrisations taking (non-null) conformal geodesics into
(non-null) conformal geodesics are given by fractional transformations of the
form
τ →

aτ + b,cτ + d

(5.48)

with a, b, c, d ∈ R.
If α = 0, then Equation (5.47c) shows that the reparametrisation reduces toan aﬃne parameter transformation. Notice also, that with a suitable choice ofconstants, it is always possible to choose a parametrisation such that τ → ∞ fora given value of τ̄ . This property of conformal geodesics is in stark contrast to
the behaviour of standard geodesics.
A ﬁnal remark concerning the reparametrisation of conformal curves follows
from evaluating Equations (5.47a) and (5.47b) at τ . One ﬁnds that x̄ = 4κ ẋ
and β̄  = β̄  + α ẋ . Consequently, the transformations of initial data given byẋ → 4κ ẋ ,β  → β  + α ẋ ,(5.49)

preserve the set of points covered by the conformal geodesics. From the discussion
in the previous paragraphs it follows that the transformation of initial data (5.49)
implies a reparametrisation of the resulting curves.
5.5.4 Geodesics as conformal geodesics
It is of natural interest to investigate the relation between conformal geodesics
and metric geodesics. For a null conformal geodesic this relation can be readily
established. If (x̄(τ̄ ), β̄(τ̄ )) denotes a null conformal geodesic, it follows readily
from Equation (5.42a) that˜ x̄ x̄ = −2β, x̄ x̄ .
∇5.5 Conformal geodesics

131

Thus, using an argument similar to the one discussed at the beginning of Section
5.5, one ﬁnds that null conformal geodesics are, up to a reparametrisation, nullgeodesics.
The situation for non-null conformal geodesics is more complicated andrequires restrictions of the Schouten tensor of the spacetime. One has thefollowing result (see Friedrich and Schmidt (1987)):
Lemma 5.2 (standard geodesics as conformal geodesics) Any non-nullg̃-geodesic in an Einstein spacetime (M̃, g̃) is, up to a reparametrisation, a nonnull conformal geodesic.
˜ ẋ ẋ = 0.
Proof Let x(τ ) denote a solution to the metric geodesic equation ∇Consider a reparametrisation of the curve of the form τ = τ (τ̄ ). The analysis
in Section 5.5.3 suggests completing x(τ̄ ) to a conformal geodesic using anansatz of the form β̄ = α(τ̄ )ẋ . Writing, as in the previous section, x̄ = τ  ẋ,Equation (5.42a) readily leads to the conditionτ  + ατ 2 g̃(ẋ, ẋ) = 0,
where it is noticed that g̃(ẋ, ẋ) is constant along the curve as it is a g̃-geodesic.
To obtain an equation for α one substitutes the ansatz for β̄ into (5.42b) and˜ ẋ β̄ = α ẋ so that
notices that ∇α ẋ =1 2 
α τ g̃(ẋ, ẋ)ẋ + τ  L̃(ẋ, ·).
2

The solvability of this equation depends on the available information about L̃.
In the case of an Einstein space one has that L̃(ẋ, ·) = 16 λẋ so that one obtains
α =

1
1 2 
α τ g̃(ẋ, ẋ) + λτ  ,26which can always be solved – at least locally.
A partial converse of Lemma 5.2 is given by:
Lemma 5.3 (conformal geodesics as metric geodesics) Let (M̃, g̃) be a
Einstein spacetime and let g = Ξ2 g̃ be a further metric on M̃. A conformal
geodesic (x̄(τ̄ ), β̄(τ̄ )) with respect to the metric g is, up to a reparametrisation,a g̃-geodesic if there exists a function α(τ̄ ) such thatβ̄ = −Υ + αx̄ .
˜ ẋ ẋ = 0 implies, under the conformal rescaling
Proof The geodesic equation ∇2
g = Ξ g̃, the equation∇ẋ ẋ = 2Υ, ẋẋ − g(ẋ, ẋ)Υ .

(5.50)

132

Conformal geometry

It follows from the analysis of Section 5.5.3 that, to reparametrise the conformal
geodesic equations for the metric g to yield Equation (5.50), one needs to havea parameter α such that β̄ = −Υ + αx̄ .
5.5.5 Conformal factors associated to congruencesof conformal geodesics
In what follows, for simplicity it will be assumed that the spacetime (M̃, g̃)
can be covered by a non-intersecting congruence of conformal geodesics. The
congruence of conformal geodesics can be used to single out a metric g ∈ [g̃] bymeans of a conformal factor Θ such thatg(ẋ, ẋ) = 1,g = Θ2 g̃.

(5.51)

That is, the tangent vector ﬁeld of the congruence of conformal geodesics is gnormalised – accordingly, the parameter τ of the geodesics corresponds to the˜ ẋ to the ﬁrst equation in (5.51) and using
g-proper time. It follows by applying ∇the conformal geodesic Equation (5.35a) thatΘ̇ = β, ẋΘ,(5.52)

˜ ẋ Θ. Thus, by prescribing Θ ≡ Θ(τ ) at some ﬁduciary value τ ∈ R
where Θ̇ ≡ ∇along the conformal geodesic one ﬁnds that the value of Θ is fully determinedby Equation (5.52). If the initial value Θ is chosen to vary smoothly along the
curves on the congruence, one readily obtains a conformal factor for the whole ofthe spacetime. It is important to remark that this conformal factor depends on
the particular congruence of conformal geodesics; a diﬀerent choice of congruence
would lead to a diﬀerent Θ and, hence, to a diﬀerent conformal metric g. Thus, if
the congruence of conformal geodesics is speciﬁed by a prescription of initial dataof the form given in (5.37) on an initial hypersurface S, then g is determined in
an implicit way by the initial data for the congruence and by Θ . In the remainderof this section it will be shown that for metrics g̃ satisfying the vacuum Einsteinequations this correspondence can be made explicit.
A direct consequence of Equations (5.38a) and (5.52) is that˜ ẋ (g(ẋ, ẋ)) = 0.
∇Hence, one sees that a conformal geodesic that is, respectively, timelike, nullor spacelike at a given point in M̃ preserves its causal character throughout the whole curve. Further computations using the conformal geodesic
Equations (5.35a) and (5.35b) and the relations (5.38a)–(5.38c) and (5.52) showthat1Θg̃(ẋ, ẋ)g̃  (β, β) + ΘL̃(ẋ, ẋ),2...  ˜
Θ = ∇ẋ (L̃(ẋ, ẋ)) + L̃(ẋ, β )g̃(ẋ, ẋ) + β, ẋL̃(ẋ, ẋ) Θ.
Θ̈ =(5.53a)(5.53b)5.5 Conformal geodesics

133

Moreover, if {ea } denotes a g-orthonormal frame, that is, g(ea , eb ) = ηab ,propagated according to Equation (5.36) with e0 = ẋ, one readily ﬁnds that˜ ẋ (Θβ, ea ) = ΘL̃(ẋ, ea ) + 1 Θg̃  (β, β)g̃(ẋ, ea ).
∇(5.54)
2˜ ẋ (g(ea , eb )) = 0. The
Notice that for the frame {ea } one has, in addition, that ∇expressions discussed in the previous paragraph lead to the following result ﬁrst
proven in Friedrich (1995):
Proposition 5.1 (the canonical conformal factor associated to a
conformal geodesic) Let (M̃, g̃) denote an Einstein spacetime. Suppose that(x(τ ), β(τ )) is a solution to the conformal geodesic equations (5.35a) and (5.35b)
and that {ea } is a g-orthonormal frame propagated along the curve according toEquation (5.36). If g = Θ2 g̃ is such that g(ẋ, ẋ) = 1, then the conformal factor
Θ satisﬁes1
Θ(τ ) = Θ + Θ̇ (τ − τ ) + Θ̈ (τ − τ )2 ,2

(5.55)

where the coeﬃcients Θ ≡ Θ(τ ), Θ̇ ≡ Θ̇(τ ) and Θ̈ ≡ Θ̈(τ ) are constantalong the conformal geodesic and are subject to the constraintsΘ̇ = β  , ẋ Θ ,Θ Θ̈ =1 1g̃ (β  , β  ) + λ.
2
6(5.56)

Furthermore, along each conformal geodesicΘβ0 = Θ̇,Θβi = Θ βi ,(5.57)

where βa ≡ β, ea .
Proof For an Einstein spacetime the Schouten tensor is given by L̃ = 16 λg̃.
...
Substituting this expression into Equation (5.53b), one ﬁnds that Θ = 0 so that
Equation (5.55) follows. The constraints (5.56) follow from Equations (5.52) and(5.53a). Finally, the relations in (5.57) follow from (5.52) and (5.54).
5.5.6 The g̃-adapted equations
As a consequence of the normalisation condition (5.51), the parameter τ is the
g-proper time of the curve x(τ ). In some computations it is more convenient toconsider a parametrisation in terms of a g̃-proper time τ̃ . To this end, consider
the parameter transformation τ̃ = τ̃ (τ ) given by
 τ
ds
dτ
= Θ,
so thatτ̃ = τ̃ +,(5.58)dτ̃τ Θ(s)with inverse τ = τ (τ̃ ). In what follows, write x̃(τ̃ ) ≡ x(τ (τ̃ )). It can then be
veriﬁed that
x̃ ≡

dτ dx
dx̃
=
= Θẋ,dτ̃dτ̃ dτ

(5.59)

134

Conformal geometry

so that g̃(x̃ , x̃ ) = 1. Hence, τ̃ is, indeed, the g̃-proper time of the curve x̃. Now,
consider, consistent with Equation (5.47b), the split≡

β = β̃ + ẋ ,β, ẋ
,g̃(ẋ, ẋ)(5.60)

where the covector β̃ satisﬁesβ̃, ẋ = 0,g  (β, β) = β, ẋ2 + g  (β̃, β̃).

(5.61)

It can be readily veriﬁed thatg̃(ẋ, ẋ) = Θ−2 ,β, ẋ = Θ−1 Θ̇, = ΘΘ̇.

(5.62)

Using the split (5.60) in Equations (5.35a) and (5.35b) and taking into accountthe relations in (5.59), (5.61) and (5.62), one obtains the following g̃-adapted
equations for the conformal geodesics:
˜ x̃ x̃ = β̃  ,∇(5.63a)










˜ x̃ β̃ = β x̃ + L̃(x̃ , ·) − L̃(x̃ , x̃ )x̃ ,∇2(5.63b)with β 2 ≡ −g̃  (β̃, β̃) – observe that as a consequence of (5.61) the covector β̃is spacelike, and, thus, the deﬁnition of β 2 makes sense. The Weyl propagation
Equation (5.36) can also be cast in a g̃-adapted form. A calculation shows that˜ x̃ (Θea ) = −β̃, Θea x̃ .
∇Equation (5.63a) provides a clear-cut interpretation of the covector β̃ – itcorresponds to the physical acceleration of the conformal curve. Recalling thatg̃ = Θ2 g and using (5.61) together with Equation (5.57) of Proposition 5.1 one
ﬁnds thatβ 2 = −g̃  (β̃, β̃) = −Θ2 g  (β̃, β̃) = Θ2 δ ij βi βj = Θ2 δ ij βi βj .

(5.64)

That is, β 2 is a constant along the conformal geodesic. Using Equation (5.63a) toeliminate β̃ in Equation (5.63b), one obtains a third-order diﬀerential equation
for the curve x̃(τ̃ ):˜ x̃ x̃ = β 2 x̃ + L̃ (x̃ , ·) − L̃(x̃ , x̃ )x̃ .
˜ x̃ ∇∇

(5.65)

A computation making use of the expressions derived in this section shows that˜ x̃ g̃(β̃, β̃) = 2L̃(x̃ , β̃).
∇Consequently, unless (M̃, g̃) is an Einstein spacetime the acceleration of thecurve cannot be constant. This is related to an open question concerning the
behaviour of conformal geodesics discussed in Tod (2012): if a conformal geodesicγ enters every neighbourhood of a point p, does γ necessarily pass throughp with a ﬁnite limiting velocity and acceleration? This potential pathologicalbehaviour is known as spiralling ; see Figure 5.1. This does not happen for5.5 Conformal geodesics

γ
U

135

γ
U

Figure 5.1 Spiralling of conformal geodesics: (left) a standard geodesic γ
entering a geodesically convex ball U must leave it in ﬁnite proper time; (right)
by contrast, a conformal geodesic γ  may not leave U and spiral towards apoint.

standard geodesics, for if a geodesic enters a geodesically convex ball, then it
must leave it too; see Section 11.6.2 for a discussion of the notion of geodesicallyconvex ball. Using Piccard’s existence theorem for ordinary diﬀerential equations– see, for example, Hartman (1987) – on Equation (5.65), it follows that spiralling
can occur only if either β̃ or x̃ diverge.
5.5.7 The conformal geodesic deviation equations
An important issue arising in applications involving congruences of conformal
geodesics is that of deciding whether the congruence develops caustics, thatis, points where it becomes singular. To address this one needs to consider theconformal geodesic deviation equations for the congruence. The deviation of theseequations is analogous to the one leading to the geodesic deviation equation forstandard geodesics; see Section 2.4.5.
In what follows let(xη (τ ), β η (τ )) ≡ (x(τ, η), β(τ, η))denote a family of conformal geodesics depending smoothly on a parameter η ∈R. Following the notation used in previous sections for ﬁxed η, let ẋ denotethe tangent vector to the curves of the congruence. The deviation vector anddeviation covector are deﬁned, respectively, by
˜ z β.
ζ≡∇(5.66)

˜ ẋ z − ∇˜ z ẋ = 0,[ẋ, z] = ∇(5.67)z ≡ ∂η x,
A short computation shows thatso that z is a well-deﬁned deviation vector ; compare Section 2.4.5. Moreover,making use of the deﬁnition of the Riemann tensor given by Equation (2.9), one
has that

136

Conformal geometry
˜ z ẋ − ∇˜ ẋ ẋ.
˜ ẋ ∇
˜ z∇
Riem[g̃](ẋ, z)ẋ = ∇(5.68)

Hence,
˜ ẋ z = ∇˜ z ẋ = ∇˜ ẋ ẋ + Riem[g̃](ẋ, z)ẋ,˜ ẋ ∇
˜ ẋ ∇
˜ z∇∇
as a consequence of Equations (5.67) and (5.68). Now, using the conformal
˜ ẋ ẋ = −S(β; ẋ, ẋ), where S(β; ẋ, ẋ)geodesic equation (5.35a) in the form ∇cd a bcorresponds to Sab ẋ ẋ βc in abstract index notation, one ﬁnds that˜ ẋ ∇
˜ ẋ z = −∇˜ z (S(β; ẋ, ẋ)) + Riem[g̃](ẋ, z)ẋ∇˜ z β; ẋ, ẋ) − 2S(β; ∇˜ z ẋ, ẋ) + Riem[g̃](ẋ, z)ẋ. (5.69)= −S(∇
A similar computation shows that˜ ẋ ζ = ∇˜ zβ = ∇˜ ẋ β − β · Riem[g̃](ẋ, z)
˜ ẋ ∇
˜ z∇
∇
1˜
˜
= ∇
z (β · S(β; ẋ, ·)) + ∇z (L̃(ẋ, ·)) − β · Riem[g̃](ẋ, z)2
˜ z L̃(ẋ, ·) + L̃(∇˜ z β · S(β; ẋ, ·)˜ z ẋ, ·) + 1 ∇= −β · Riem[g̃](ẋ, z) + ∇21
˜ z β; ẋ, ·) + 1 β · S(β; ∇˜ ẋ z, ·)+ β · S(∇(5.70)
22
where, in the third line, Equation (5.35b) in the form
˜ ẋ β = 1 β · S(β; ẋ, ·) + L̃(ẋ, ·)∇2
has been used. Finally, taking into account the deﬁnitions in (5.66) inEquations (5.69) and (5.70), one obtains the conformal geodesic deviation
equations:˜ ẋ z = Riem[g̃](ẋ, z)ẋ − S(ζ; ẋ, ẋ) − 2S(β; ẋ, ∇˜ ẋ z),˜ ẋ ∇∇(5.71a)˜ ẋ ζ = −β · Riem[g̃](ẋ, z) + ∇˜ z L̃(ẋ, ·) + L̃(∇˜ z ẋ, ·) + 1 ζ · S(β; ẋ, ·)∇2
1
1
˜ ẋ z, ·),+ β · S(ζ; ẋ, ·) + β · S(β; ∇(5.71b)22
whereS(β; u, v) ≡ β, uv + β, vu − g̃(u, v)β  ,α · S(β; u, ·) ≡ α, uβ + β, uα − g̃  (α, β)u ,for u, v ∈ T (M̃) and α ∈ T ∗ (M̃). In standard abstract index notation S(β; u, v)
corresponds to the expression Sab cd ua v b βc , while α · S(β; u, ·), to Sab cd ua βc αd .
A caustic in a conformal geodesic is a point along the curve for which z = 0.
Caustics of conformal geodesics are more complicated than caustics of metricgeodesics since, for a given tangent vector, there exists a three-parameter familyof conformal geodesics with the same tangent vector. Moreover, the analysis

5.6 Further reading

137

of Equation (5.71a) requires the simultaneous consideration of the evolution
equation of the deviation covector ζ, Equation (5.71b). This feature can be
useful in applications: Equation (5.71a) has two extra terms, −S(ζ; ẋ, ẋ) and˜ ẋ z), not appearing in the standard geodesic deviation equation;−2S(β; ẋ, ∇under suitable circumstances these terms may be used to counteract the naturaltendency of the curvature to develop caustics.
The g̃-adapted conformal geodesic deviation equations
Following the strategy discussed in Section 5.5.6, one can rewrite the conformal
geodesic deviation equations in a way adapted to the metric g̃. To this end deﬁne
the g̃-adapted deviation vector and covectorz̃ ≡ ∂λ x̃,˜ z̃ β̃.
ζ̃ ≡ ∇

Now, observing that [x̃ , z̃] = 0, a computation taking into account the g̃adapted conformal geodesic Equations (5.63a) and (5.63b) and the commutatorof covariant derivatives leads to the following g̃-adapted conformal geodesic
deviation equations:˜ x̃ z̃ = Riem[g̃](x̃ , z̃)x̃ + ζ̃  ,˜ x̃ ∇∇

(5.72a)˜ x̃ z̃  .
˜ x̃ ζ̃ = −β̃ · Riem[g̃](x̃ , z̃) + (∇˜ z̃ β 2 )x̃ + β 2 ∇∇(5.72b)˜ is assumed to be torsion
A computation exploiting the fact that the connection ∇free gives
˜ z̃ β 2 = ∇˜ x̃ β 2 = 0,
˜ z̃ ∇
˜ x̃ ∇∇
where the last equality follows from the fact that β 2 is constant along a given
conformal geodesic; see Equation (5.64). Hence, the components of the terms
˜ x̃ z̃  in Equation (5.72b) are constant and can be evaluated atwith x̃ and ∇some ﬁducial time.
5.6 Further reading
Basic references for applications of conformal geometry in general relativity
are Penrose and Rindler (1984, 1986) and Stewart (1991). A discussion of theproperties of the Weyl and Cotton tensor can be found in Garcı́a et al. (2004).
The ﬁrst systematic treatments of conformal geodesics in the context of generalrelativity can be found in Schmidt (1986) and Friedrich and Schmidt (1987).
A discussion of Weyl connections making use of the more general language ofﬁbre bundles is given in Friedrich (1995); a brief presentation of the subject
in the spirit of this chapter can be found in Friedrich (2002). A discussion of
the properties of conformal geodesics in the context of general relativity can befound in Friedrich (2003a); a more technical discussion can be found in the earlier

138

Conformal geometryreference Friedrich (1995). Properties of conformal geodesics have been exploredfrom a diﬀerent perspective in Tod (2012).
The results of Proposition 5.1 strongly depend on the hypothesis that (M̃, g̃) is
an Einstein space – in other words, g̃ satisﬁes the vacuum Einstein equations. To
get around this restriction, a more general class of curves has been introduced inLübbe and Valiente Kroon (2012). These curves are a suitable generalisationof the conformal geodesics which allow the recovery of the conclusions of
Proposition 5.1 for general spacetimes and, thus, provide a systematic way ofidentifying the conformal boundary of non-vacuum spacetimes. A discussionof the associated deviation equations with explicit expressions for the case of
warped-product spacetimes is given in Lübbe and Valiente Kroon (2013a).
A detailed mathematical theory of conformal connections can be found in
Ogiue (1967) and Kobayashi (1995). A more recent monograph on the subject
is Feﬀerman and Graham (2012). Conformal geometry is naturally related totwistor theory; a discussion of this and related topics such as tractors can befound in Eastwood (1996).
The reader interested in surveys on research in conformal geometry is referred
to Kulkarni and Pinkall (1988), Chang et al. (2007) and Branson et al. (2004)
as suitable entry points to the literature in the subject.

Part II
General relativity and conformal geometry

6
Conformal extensions of exact solutions

Exact solutions to the Einstein ﬁeld equations are the prime source of geometricand physical intuition in general relativity. This chapter revisits some of theclassical exact solutions of general relativity (the Minkowski, de Sitter, antide Sitter and Schwarzschild spacetimes) from the point of view of conformal
geometry. In addition, a general discussion of the construction of Penrose
diagrams of static spherically symmetric spacetimes is provided. Most of thematerial in this chapter can be considered as classic – complementary discussions
can be found in, for example, Hawking and Ellis (1973) and Griﬃths andPodolský (2009). In view of the applications in the later parts of the book,particular emphasis is given to the construction of explicit congruences ofconformal geodesics in the exact solutions.
6.1 Preliminaries
6.1.1 Spherical symmetry
In what follows, let SO(3) denote the group of homogeneous linear transformations of R3 onto itself which preserve the Euclidean length of vectors andthe orientation of the space. A spacetime (M, g) is said to be sphericallysymmetric if the group SO(3) acts by isometry on (M, g) with simply
connected, complete, spacelike two-dimensional orbits; see, for example, Ehlers(1973). Two points p, q ∈ SO(3) are said to be in the same orbit if there isan element of the group SO(3) taking p to q. Given a spherically symmetricspacetime it is natural to introduce the quotient manifold Q ≡ M/SO(3),that is, the manifold obtained from M by identifying points on the same orbit.
The manifold Q inherits from (M, g) a two-dimensional Lorentzian metric γ,
the quotient metric. Let Γ denote the subset of Q corresponding to the ﬁxed
points of the action of SO(3). If Γ is non-empty, then it can be shown that it isa connected timelike boundary of Q – the centre of symmetry . A sphericallyspacetime can have none, one or two centres; see Künzle (1967).

142

Conformal extensions of exact solutions

Given a spherically symmetric spacetime (M, g), there exists a function  :Q → R such that the spacetime metric g can be written in the warped productform
g = γ + 2 σ,(6.1)

where σ is the standard metric of S2 given, in the usual spherical coordinates(θ, ϕ), by
σ = dθ ⊗ dθ + sin2 θdϕ ⊗ dϕ.
The function  is not necessarily an areal coordinate.
6.1.2 The 3-sphere
The unit 3-sphere S3 is the three-dimensional submanifold of R4 deﬁned byS3 ≡ {(w, x, y, z) ∈ R4 | w2 + x2 + y 2 + z 2 = 1}.
The standard Euclidean metric in R4 induces, in a natural way, a 3-metrich̄ on S3 , the standard metric of S3 . The metric h̄ is best expressed using
spherical coordinates (ψ, θ, ϕ) such thatw = cos ψ,

x = sin ψ cos θ,

y = sin ψ sin θ cos ϕ,z = sin ψ sin θ sin ϕ,taking the range 0 ≤ ψ ≤ π, 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. For simplicity ofpresentation, in what follows the degeneracy of the spherical coordinate system(ψ, θ, ϕ) will be ignored as it can be dealt with by introducing further coordinate
charts. In terms of these coordinates one hash̄ = dψ ⊗ dψ + sin2 ψσ.
Conventionally, the point given by ψ = 0 will be called the north pole, while
the one with ψ = π will be called the south pole.
At every point p ∈ S3 the restriction of the coordinates (w, x, y, z) can beused to construct suitable local coordinates. For example, if w(p) > 0, then thecoordinates (xα ) = (x, y, z) constitute a well-deﬁned system of local coordinates
on the northern hemisphere of S3 .
A frame on S3
A direct computation shows that the vector ﬁelds on T (R4 )
∂
∂
∂
∂
−z
+x
−y
,
∂z
∂w
∂y
∂x
∂
∂
∂
∂
c2 ≡ w
−y
+z
−x ,
∂y
∂w
∂x
∂z
∂
∂
∂
∂
c3 ≡ w
−x
+y
−z
,
∂x
∂w
∂z
∂y

c1 ≡ w(6.2a)(6.2b)(6.2c)

6.1 Preliminaries

143

are linearly independent and tangent to S3 ; hence, they can be regarded as
globally deﬁned vectors on T (S3 ). This point of view will be used systematicallyin this book. As ci = 0 on S3 one has, in fact, a globally deﬁned frame on the
3-sphere. Moreover, it can be shown thath̄(ci , cj ) = δij .
Accordingly, the vectors {ci } are h̄-orthogonal. A direct calculation shows that
[c1 , c2 ] = 2c3 ,[c2 , c3 ] = 2c1 ,[c3 , c1 ] = 2c2 .

The above expressions can be more concisely written as[ci , cj ] = 2ij k ck ,where ijk denotes the components of the volume form in R3 . In particular,
one has that 123 = 1. The above commutators can be combined with
the Cartan structure equations – see Equations (2.41) and (2.42) – to computethe connection coeﬃcients γi j k with respect to the frame {ci }. One obtains theconcise expression
γi j k = −i j k .
The compactiﬁcation of R3 into S3
An important example of conformal compactiﬁcation is the so-called pointcompactiﬁcation of the Euclidean space R3 into S3 . Letδ = dr ⊗ dr + r2 dθ ⊗ dθ + r2 sin2 θdϕ ⊗ dϕdenote the standard (negative deﬁnite) three-dimensional Euclidean metric in
spherical coordinates with 0 ≤ r < ∞, 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π.
An explicit computation shows that the Cotton tensor – see Equation (5.18) –of the metrics δ and h̄ vanish so that they must be conformally related; compare
Theorem 5.1. In order to make this correspondence explicit, writeh̄ = ω 2 δ(6.3)

where ω is a conformal factor to be determined. Expressing the radial coordinate
as r = r(ψ), one ﬁnds from Equation (6.3) the conditions
ω 2 r2 = 1,

r2 ω 2 = sin2 ψ,(6.4)

where  denotes the derivative with respect to ψ. A solution to the equations in(6.4) is given byω=ψ2
sin2 ,
α2r(ψ) = α cot

ψ
,
2

(6.5)

where α is a real constant. Notice that r → ∞ as ψ → 0. Thus, the transformation
given by (6.5) is a compactiﬁcation of R3 sending the north pole of S3 to the144

Conformal extensions of exact solutions

point at inﬁnity in R3 , while the south pole of S3 is sent to the origin of R3 . An
alternative solution to equations (6.4), sending the south pole to the point at
inﬁnity and the north pole to the origin, is given byω=ψ2
cos2 ,
α2r(ψ) = α tan

ψ
,
2

(6.6)

as can be veriﬁed by an explicit computation.
6.1.3 The Einstein static universe
The Einstein static universe – sometimes also called the Einstein cosmos
or Einstein cylinder – is the spacetime (ME , g E ) given byME ≡ R × S3 ,g E ≡ dT ⊗ dT − h̄.

(6.7)

It can be readily veriﬁed that ∂ T is a timelike Killing vector of g E so that thesolution is indeed static. Moreover, as (S3 , σ) is a homogeneous and isotropic
Riemannian manifold, it follows that (ME , g E ) is spatially homogeneous andisotropic.
A computation shows thatR[g E ] = −6,
1
Schouten[g E ] = (dT ⊗ dT + h̄).
2
W eyl[g E ] = 0,

(6.8a)(6.8b)

Hence, one sees that (ME , g E ) is conformally ﬂat. A discussion of the propertiesof the Einstein static universe as a solution to the Einstein ﬁeld equations with aperfect ﬂuid matter source can be found in, for example, Griﬃths and Podolský
(2009) and Hawking and Ellis (1973).
Finally, it is observed that the Einstein static universe is spherically symmetric.
Comparing the metric g E in (6.7) with the warped product metric (6.1) it isnatural to set
γ E ≡ dT ⊗ dT − dψ ⊗ dψ,E ≡ sin ψ,

so that (T, ψ) can be used as coordinates of the quotient manifold QE ≡ (R ×
S3 )/SO(3) ≈ R × [0, π].
A class of conformal geodesics in the Einstein static universe
In what follows, consider the congruence of curves on (ME , g E ) given byx(τ ) = (τ, x ),τ ∈ R,(6.9)

with x ∈ S3 ﬁxed. Varying x over S3 one obtains a non-intersecting timelike
congruence covering the whole of ME . It can be veriﬁed that the curves (6.9)
are geodesics for g E with proper time τ and tangent vector ẋ = ∂ T .

6.2 The Minkowski spacetime

145

The curves (6.9) can be recast as conformal geodesics. To see this, one follows
the argument of Lemma 5.2 and introduces a parameter τ̄ such that τ = τ (τ̄ )and makes use of the ansatzβ̄ ≡ α(τ̄ )ẋ = α(τ̄ )dT.
Substituting the above expression into the conformal geodesic Equations (5.42a)and (5.42b) and taking into account formula (6.8b) for the Schouten tensor of
the Einstein universe one ﬁnds the equationsτ  + ατ 2 = 0,

α =

1  2
τ (α + 1),
2

with  denoting diﬀerentiation with respect to τ̄ . A solution to the above
equations is given byτ̄τ = 2arctan ,
2

α=

τ̄
.
2

(6.10)

Now, one has that β̄, x  = ατ  so that the conformal factor Θ̄ satisfying
the condition Θ̄2 g E (x , x ) = 1 obeys the equation Θ̄ = β̄, x Θ̄ with initialcondition Θ̄ = 1. The diﬀerential equation for Θ̄ can be solved to give1Θ̄ = 1 + τ̄ 2 .
4(6.11)

It can be veriﬁed thatβ̄ =1τ̄ dT = Θ̄−1 dΘ̄.
2

Using the conformal factor Θ̄ one obtains a conformal representation of theEinstein universe with metric ḡ E ≡ Θ̄2 g E so that2
1 2ḡ E = dτ̄ ⊗ dτ̄ − 1 + τ̄h̄,
4where the parameter τ̄ has been introduced as the new time coordinate.
This conformal representation of the Einstein cylinder will be known as the
expanding Einstein cylinder. Notice that x = Θ̄−1 ẋ so that ḡ E (x , x ) = 1.
It can be readily veriﬁed that the congruence is integrable and that the curves
are orthogonal to the surfaces of constant τ̄ .
6.2 The Minkowski spacetime
The Minkowski solution (M̃, η̃) is the spacetime given by M̃ = R4 andη̃ = ημν dxμ ⊗ dxν ,(6.12)

where (xμ ) = (t, x, y, z) and ημν ≡ diag(1, −1, −1, −1). Alternatively, using
spherical coordinates one can writeη̃ = dt ⊗ dt − r2 dr ⊗ dr − r2 σ.

(6.13)

146

Conformal extensions of exact solutions

Using the expression of the Minkowski metric in Cartesian coordinates one
readily sees that Riem[η̃] = 0 so that, in particular, Ric[η̃] = 0; that is, η̃is a solution to the vacuum Einstein ﬁeld equations with vanishing cosmologicalconstant. Moreover, one has that W eyl[η̃] = 0 so that η̃ is conformally ﬂat and,thus, conformal to the metric of the Einstein cylinder. This relation is analysedin the next section.
6.2.1 The compactiﬁcation into the Einstein cylinder
A standard procedure for the construction of conformal extensions of Lorentzianmanifolds is to make use of pairs of so-called null coordinates. In the present casea convenient choice is given byu ≡ t − r,v ≡ t + r.

(6.14)

Conventionally, the coordinate is called a retarded time, while v is an advanced
time. It can be readily veriﬁed that η̃  (du, du) = η̃  (dv, dv) = 0. It follows that11(du ⊗ dv + dv ⊗ du) − (v − u)2 σ.
24In order to have r ≥ 0 one has the restriction u ≤ v. The present analysis ismainly concerned with the behaviour at inﬁnity; thus, it is natural to introduce
a further transformation of coordinates:η̃ =u ≡ tan U,v ≡ tan V,

U, V ∈ (− 12 π, 12 π),

From the relations
1
dU = (1 + u2 )dU,du =cos2 Uand the identity

dv =v − u = tan V − tan U =U ≤ V.

1dV = (1 + v 2 )dV,
cos2 Vsin(V − U ),cos U cos V

one obtains
η̃ =



1
2(dU ⊗ dV + dV ⊗ dU ) − sin2 (U − V )σ .
4 cos2 U cos2 VThis last expression suggests deﬁning the unphysical metric η ≡ Ξ2M η̃ whereΞM ≡ 2 cos U cos V,(6.15)

so thatη = 2(dU ⊗ dV + dV ⊗ dU ) − sin2 (U − V )σ.
The conformal factor ΞM vanishes whenever U = ± 12 π or V = ± 21 π. In order
to investigate the situation in more detail one introduces the ﬁnal change of
coordinatesψ ≡ V − U,

T ≡ V + U.

6.2 The Minkowski spacetime147

i+
R × S3
I+

M

i0

Γ

I−

i−

Figure 6.1 Conformal extension of the Minkowski spacetime. Left, conformal
embedding of the Minkowski spacetime in the Einstein cylinder: the shaded
region corresponds to the set MM of equation (6.21). Right, Penrose diagramof the Minkowski spacetime: the line Γ corresponds to the axis of symmetry,the points i0 , i+ and i− are spatial inﬁnity, future timelike inﬁnity and pasttimelike inﬁnity, respectively. Finally, I + and I − are future and past nullinﬁnity; see main text for further details.

Using standard trigonometric identities, one can rewrite the conformal factor(6.15) in terms of the coordinates T and ψ to obtainΞM = cos T + cos ψ.

(6.16)

Thus, one ends up with the metricη = dT ⊗ dT − dψ ⊗ dψ − sin2 ψσ.

(6.17)

Thus, one has that η = g E . Consequently, the rescaling procedure described in
the previous paragraphs compactiﬁes the Minkowski spacetime into a region of
the Einstein cylinder ; see Figure 6.1, left panel. The standard coordinates (t, r)
on the Minkowski spacetime are related to the (T, ψ) coordinates on the Einstein
cylinder via the formulae:
t=

sin T
,
cos T + cos ψ

r=

sin ψ
.
cos T + cos ψ(6.18)

It follows from the previous discussion that the Minkowski spacetime (R4 , η̃)is conformal to the domainM̃M ≡ p ∈ ME | 0 ≤ ψ(p) < π, ψ(p) − π < T (p) < π − ψ(p) ,on which ΞM > 0. In addition to M̃M , it is convenient to single out a number ofsubsets of ME playing a special role in the discussion of the asymptotic behaviourof the Minkowski spacetime:

148

Conformal extensions of exact solutions(a) Future and past null inﬁnity are deﬁned as the hypersurfacesI ± ≡ p ∈ ME | 0 < ψ(p) < π, T (p) = ±(π − ψ(p)) ,(6.19)

on which ΞM = 0. A calculation shows thatdΞM = − sin T dT − sin ψdψ,(6.20)

so that dΞM = 0 on I ± . It can, however, be veriﬁed thatg E (dΞM , dΞM )|I ± = 0,
so that I ± are null hypersurfaces.
(b) Spatial inﬁnity is deﬁned by
i0 ≡ p ∈ ME | ψ(p) = π, T (p) = 0 .
Inspection of expression (6.7) for the metric g E shows that the radius of the
2-sphere deﬁned by T = 0 and ψ = 0 vanishes. Accordingly, i0 consists of asingle point. Evaluating the diﬀerential (6.20) at i0 one ﬁnds thatdΞM |i0 = 0,

Hess ΞM |i0 = −g E |i0 .

(c) Future and past timelike inﬁnity is deﬁned as
i± ≡ p ∈ ME | ψ(p) = 0, T (p) = ±π .
Again, from the metric (6.7) it follows that the 2-spheres deﬁned by T = ±π
and ψ = 0 have vanishing radius so that both i+ and i− correspond to points.
Using (6.20) one ﬁnds thatdΞM |i± = 0,

Hess ΞM |i± = g E |i± .

The motivation for the above deﬁnitions follows from the analysis of geodesics;/ I ± . Finally,see below. It is important to point out that by convention i0 , i± ∈it is convenient to deﬁne the manifold with boundary
MM = M̃M ∪ I + ∪ I − ∪ i+ ∪ i− ∪ i0 ,(6.21)

which will be called the conformally extended Minkowski manifold.
The Penrose diagram of the Minkowski spacetime
The spherical symmetry of the Minkowski spacetime can be exploited to provide
a diagrammatic representation of the global structure of the spacetime known
as a Penrose (or Penrose-Carter) diagram. It follows from the discussion
in Section 6.1.1 that the action of the group SO(3) on MM gives rise to thequotient manifold with coordinates (T, ψ) given by
QM ≡ p ∈ QE | 0 ≤ ψ(p) ≤ π, ψ(p) − π ≤ T (p) ≤ π − ψ(p) .
In what follows, in a slight abuse of notation, the projections of I ± , i± , i0 ⊂MM on the quotient manifold QM will be denoted, again, by the same symbols.

6.2 The Minkowski spacetime

149

Clearly, I + , i± , i0 ⊂ ∂QM ; however, ∂QM has a further component consistingof the centre of symmetryΓ ≡ p ∈ QM | ψ(p) = 0, −π < T (p) < π}.
As the conformal metric η is the standard one on the Einstein cylinder, it follows
from the discussion in Section 6.1.3 that the quotient metric inherited by η on
QM is the two-dimensional Minkowski metricγ M = dT ⊗ dT − dψ ⊗ dψ.
Given the above, the Penrose diagram of the Minkowski spacetime is simply
the depiction of QM as a subset of R2 as shown in Figure 6.1, right panel. A
discussion of the construction of Penrose diagrams for more general spacetimes
is given in Section 6.5.2.
Analysis of the behaviour of geodesics
Intuition on the various features of the construction described in the previous
paragraphs can be obtained by analysing the behaviour of various types ofphysical metric geodesics. To this end one notices the following formulae that canbe veriﬁed using the coordinate transformations taking the original Minkowski
metric of Equation (6.13) into the metric (6.17):
sin T = 

2t
(1 + (t − r)2 ) (1 + (t + r)2 )cos T = cos ψ = 

1 − t2 + r 2(1 + (t − r)2 ) (1 + (t + r)2 )1 + t2 − r 2(1 + (t − r)2 ) (1 + (t + r)2 ),(6.22a),(6.22b).

(6.22c)(a) Spacelike geodesics. Radial spacelike geodesics in the Minkowski spacetime can be described using the radial coordinate r as a parameter. It follows
then that the time coordinate of the curves is given byt = ar + t ,t ∈ R.

a2 < 1,

For r → ∞ it follows from (6.22a)–(6.22c) thatsin T → 0,

cos T → 1,cos ψ → −1.

Hence one concludes that T → 0 and ψ → π as the curve escapes to inﬁnity.
Thus, in the unphysical picture, spacelike radial geodesics ﬁnish at the same
point, spatial inﬁnity i0 , independently of the value of a.

150

Conformal extensions of exact solutions(b) Timelike geodesics. For concreteness, consider the family of geodesics
described by|a| > 1.

t = ar + t ,It can be veriﬁed that as r → ∞ one has the limits
sin T → 0,

cos T → −1,

cos ψ → 1.

Depending on whether sin T approaches 0 from the right or the left, thelatter limits correspond to either T → π and ψ → 0 or T → −π and ψ → 0.
Thus, the timelike geodesics start and ﬁnish, respectively, at i− and i+ .
(c) Null geodesics. Consider, for example, the family of outgoing null geodesics
described by the condition u = u , where u is a constant and u is the nullcoordinate deﬁned in (6.14). Now, taking the limit v → ∞ one ﬁnds thatsin T → 1,
1 + u2

cos T → u,
1 + u2cos ψ → − u
.
1 + u2

Thus, in the limit one has that T = π − ψ, corresponding to future nullinﬁnity I + . Similarly, for incoming geodesics described by the conditionv = v , v a constant and with v as deﬁned in (6.14), one ﬁnds that thelimit points lie on the line T − ψ = π, corresponding to past null inﬁnity
I − . Summarising, incoming null geodesics start at I − while outgoing nullgeodesics end at I + .
6.2.2 Compactiﬁcations adapted to spatial inﬁnity
The discussion of the structure of spatial inﬁnity is better carried out inan alternative conformal representation. Intuitively, the region of spacetime
associated with the spatial inﬁnity of the Minkowski spacetime (R4 , η̃) is
contained in the domain D̃ ≡ {p ∈ R4 | ημν xμ (p)xν (p) < 0}, the complement of
the light cone through the origin, where (xμ ) denote the standard Cartesian
coordinates. Now, consider the coordinate inversion deﬁned byyμ = −xμ,X2

xμ = −yμ,Y2

where X 2 ≡ ημν xμ xν and Y 2 ≡ ημν y μ y ν . This coordinate transformation mapsD̃ onto itself. Moreover, a computation yields
dy μ = −

1
X2


δμλ −


2 μ
ν
x
η
x
dxλ ,
λν
X2

so thatημν dy μ ⊗ dy ν = X −4 ημν dxμ ⊗ dxν .

6.2 The Minkowski spacetime

151

I+
i0

I−

Figure 6.2 The Minkowski spacetime close to null and spatial inﬁnity: theconformal boundary corresponds to the surface of the cones, while the interiorof the spacetime corresponds to the exterior of the light cones; see main text
for further details.

This computation suggests introducing the conformal factor Ξ = 1/X 2 . Hence,
one concludes thatη = Ξ2 η̃ = Ξ2 ημν dxμ ⊗ dxν .
Hence, one has a conformal representation of the Minkowski spacetime which isalso ﬂat. An inspection shows that the boundary ∂ D̃ decomposes into the sets
I + = {p ∈ R4 | y 0 (p) > 0, ημν y μ (p)y ν (p) = 0},
I − = {p ∈ R4 | y 0 (p) < 0, ημν y μ (p)y ν (p) = 0},i0 = {p ∈ R4 | (y μ (p)) = (0, 0, 0, 0)};see Figure 6.2. An analysis similar to the one carried out in Section 6.2.1 showsthat these sets admit the interpretation of future null inﬁnity, past null inﬁnity
and spatial inﬁnity, respectively. More precisely, I + (I − ) can be thought of as
being generated by the end points of future (past) directed null geodesics while
all spatial geodesics eventually run into the point i0 . The null hypersurfaces I +form the null cone through the point i0 . Deﬁning the manifold with boundary
D ≡ D̃ ∪ ∂ D̃ one observes that the conformal metric η extends smoothly throughthe boundary.
6.2.3 Conformal geodesics in the Minkowski spacetime
Conformal geodesics in the Minkowski spacetime can be computed using theversion of the equations adapted to the physical metric; see Section 5.5.6. Using
standard Cartesian coordinates so that all the Christoﬀel symbols vanish, thethird-order Equation (5.65) reduces tox̃ = β 2 x̃ ,β 2 ≡ −η̃  (β̃, β̃),(6.23)

152

Conformal extensions of exact solutions

where it is recalled that β 2 is constant along the conformal geodesic and  denotes
diﬀerentiation with respect to the physical proper time τ̃ . Regarding (6.23) as asecond-order equation for x̃ one has thatx̃ = v 1 cosh(β τ̃ ) + v 2 sinh(β τ̃ ),where v 1 and v 2 are two constant vectors on the Minkowski spacetime. Making
use of the initial conditions x̃ (0) = β̃  and x̃ (0) = x one ﬁnds thatx̃ = x cosh(β τ̃ ) + β −1 β̃  sinh(β τ̃ ).
A ﬁnal integration taking into account the initial condition x̃(0) = x andEquation (5.63a) yieldsx̃(τ̃ ) = x + β −1 x sinh(β τ̃ ) + β −2 β̃  cosh(β τ̃ ) − β 2 β̃  ,β̃(τ̃ ) = βx sinh(β τ̃ ) + β̃  cosh(β τ̃ ),

where in the ﬁrst expression, in an abuse of notation, the vectors x and β̃  areunderstood as describing points in R4 . To rewrite this general solution in termsof the unphysical proper time τ one makes use of formula (5.58). A computation
yields the formulae:
1(6.24a)x(τ ) = x + Θ Θ−1 (τ ) ẋ τ + η̃(ẋ , ẋ )β  τ 2 ,2
1(6.24b)β(τ ) = 1 + τ β  , ẋ  β  − η̃  (β  , β  )ẋ τ,
2
where


Θ(τ ) = Θ1
2
1 + β  , ẋ τ + η̃(ẋ , ẋ )η̃ (β  , β  )τ .
4

Conformal geodesics which satisfy η̃(ẋ, ẋ) = 0 at some point coincide, following
the discussion of Section 5.5.4, with null geodesics. Those with β  = 0 are
standard geodesics of the Minkowski spacetime. Now, if ẋ is spacelike ortimelike one can assume, without loss of generality, that β  , ẋ  = 0 –following the discussion of Section 5.5.3 this can always be achieved through areparametrisation of the form given by Equation (5.48) of Lemma 5.1. If ẋ andβ  generate a timelike 2-surface and ẋ is timelike, then the conformal geodesic
is a hyperbola in the plane tangent to that 2-surface. An example of such typeof curve is given by the expression


4τ
2aτ 2
1
2
(6.25), +, 0, 0 ,|τ | ≤x(τ ) =2222
4−a τ a 4−a τawhere a−2 ≡ −η̃(ẋ, ẋ). Examples of these conformal geodesics are depicted in
Figure 6.3.

6.2 The Minkowski spacetime

153

i+
t
I+

a=2

a=1

a=

4
5
r

i0

Γ

I−
i−

Figure 6.3 Examples of conformal geodesics in the Minkowski spacetime: left,
plot in (t, r)-coordinates of the curve (6.25) for the parameter choices a =4, 1, 2; right, location of the curves in the Penrose diagram of the Minkowski
5
spacetime. Notice that the curves intersect the conformal boundary at thesame points. The diagram is quantitatively correct.

A special class of conformal geodesics
As a consequence of the transformation properties of the conformal geodesic
equations, the family of curves on the Einstein cylinder given by Equation (6.9)
deﬁnes a congruence of conformal geodesics on the Minkowski spacetime.
Recalling that ḡ E ≡ Θ̄2 g E and g E ≡ Ξ2M η̃ where Θ̄ and ΞM are the conformal
factors given, respectively, by Equations (6.11) and (6.16), one has that ḡ E =Θ2M η̃ with ΘM ≡ ΞM Θ̄. Along the conformal curves one has that
4 − τ̄ 2
ΞM = cos τ + cos ψ =+ cos ψ,4 + τ̄ 2
where the second equality is obtained from the reparametrisation formula(6.10) and standard trigonometric identities. Moreover, one ﬁnds that along theconformal geodesics


1
2ψ
2ψ 2
τ̄ .
1 − tanΘM = 2 cos
24
2
To obtain the covector β M associated to the solution of the η̃-conformal geodesic
equations letΥM ≡ Ξ−1
M dΞM = −sin τ dT + sin ψdψ.
cos τ + cos ψ

Recalling that τ = 2arctan 12 τ̄ it follows, using standard trigonometric identities,
that

154

Conformal extensions of exact solutions

Figure 6.4 A class of conformal geodesics ruling the Einstein cylinder; seemain text for further details.





1 2
τ̄
ΥM = −Θ−1
τ̄
dT
+
1
+
sin
ψdψ
.
M
4
Finally, deﬁning β M ≡ β̄+ΥM , one concludes from the transformation formulae
for the solutions of the conformal geodesic equations that the pair (x(τ̄ ), β M (τ̄ ))
with


τ̄
x(τ̄ ) = 2arctan , x ,
2




τ̄
1 2
−1
−1
β M (τ̄ ) = arctan − ΘM τ̄ dT − ΘM 1 + τ̄ sin ψdψ,2
4
is a solution to the η̃-conformal geodesic equations. Notice, in particular, that
β M (0) = −

sin ψ
dψ.
1 + cos ψA depiction of the above class of conformal geodesics is given in Figure 6.4.
6.2.4 Hyperboloids in the Minkowski spacetime
An important class of spacelike hypersurfaces in the Minkowski spacetime isgiven by the standard hyperboloids
Hk = {p ∈ R4 | t2 (p) − r2 (p) = k},k > 0.

(6.26)A direct computation reveals that the unit normal vector to these hypersurfaces
is given by1ν  = √ (t∂ t + r∂ r ).
kUsing this expression one can verify that the extrinsic curvature of thehyperboloids is pure trace, that is, proportional to the intrinsic metric of Hk .
The mean curvature (i.e. the trace of the extrinsic curvature) is given by6.3 The de Sitter spacetime

155

i+

t

I+

i0

Γ

k=1
k =1
2
k =1
4
k= 1
10

I−
r

i−

Figure 6.5 Examples of hyperboloids in the Minkowski spacetime: left, the1standard hyperboloids Hk with k = 1, 12 , 14 , 10in (t, r)-coordinates (seeEquation (6.26)); right, location of the hyperboloids in the Penrose diagram.
The diagram is quantitatively correct.

3
K̃ = √ .
kThat is, the standard hyperboloids are surfaces of constant mean curvature.
Making use of the coordinates (U, V ) introduced in Section 6.2.1, the deﬁning
equation for the hyperboloids can be rewritten as tan U tan V = k. The
hyperboloids intersect null inﬁnity whenever V = 12 π. It follows that, in thiscase, U = 0. Thus, the hyperboloids Hk intersect null inﬁnity at the same points
independent of the value of k. The hypersurfaces diﬀer from each other by theangle α at which they intersect null inﬁnity. One can compute thattan α = −

dU
= k.
dV

In particular, if k 2 = 1, one has that α = 14 π. This particular hyperboloid
corresponds to a horizontal line T = 12 π in the Penrose diagram of the Minkowski
spacetime; see Figure 6.5.
A more general class of hyperboloids can be obtained by translating thestandard hyperboloids (6.26). To this end, one considers the deﬁning equationk = (t−t )2 −r2 for ﬁxed k and t . Varying t one obtains a family of translated
hyperboloids Ht ,k . The intersection of the Ht ,k now depends on the value oft : if V = 12 π, it follows that U = arctan t .
6.3 The de Sitter spacetime
The de Sitter spacetime (M̃dS , g̃ dS ) is the solution to the vacuum Einsteinﬁeld equations Ric[g̃] = λg̃ with negative constant Ricci scalar, in the signature

156

Conformal extensions of exact solutions

conventions of this book. The spacetime manifold is given by M̃dS = R × S3 andthere exist coordinates where the metric is given by

3
2
2
,
−∞ < t < ∞,(6.27)a≡g̃ dS = dt ⊗ dt − a cosh (t/a) h̄,|λ|with h̄ denoting the standard metric of the 3-sphere. Alternatively, there exist
further coordinates (t̄, r̄) in terms of which the metric of the de Sitter spacetime
takes the form



−1
1 2
1 2
g̃ dS = 1 + λr̄ dt̄ ⊗ dt̄ − 1 + λr̄
dr̄ ⊗ dr̄ − r̄2 σ.
(6.28)
3
3
This metric is static for r̄2 > − 13 λ. A discussion of the relation between various
systems of coordinates can be found in Griﬃths and Podolský (2009).
To construct a conformal extension of the de Sitter spacetime it is convenient
to introduce a new coordinate T via the condition
dt = a cosh(t/a) dT.
Fixing the constant of integration by requiring that T = 0 if t = 0 one obtains


T
1
1
t = ln tan+ π ,T = 2a arctan et − aπ,22a 4
or, equivalently tan(T /2) = tanh(t/2a). Using standard trigonometric identities
the latter can be recast as
1.
cos T =
cosh(t/a)Thus, one concludes thatg̃ dS = a2 cosh2 t (dT ⊗ dT − h̄) .
The latter expression suggests introducing the conformal factor
ΞdS =

1
1
= cos T,a cosh(t/a)
a(6.29)

so that the conformal metric Ξ2dS g̃ is, again, that of the Einstein cylinder. It
follows that the locus of points for which ΞdS = 0 corresponds to T = ± 12 π;
notice that T → ± 12 π as t → ±∞. In view of the latter, one deﬁnes future andpast conformal inﬁnity, respectively, asπ±≡ p ∈ ME T (p) = ±.
(6.30)IdS
2
The terminology for these sets will be justiﬁed in the next section. From theprevious discussion it follows that the de Sitter spacetime is conformal to thedomainππM̃dS ≡ p ∈ ME − < T (p) <22

6.3 The de Sitter spacetime

157
I+
dS

R × S3

M

Γ2

Γ1

I−
dS

Figure 6.6 Conformal extension of the de Sitter spacetime. Left, conformal
embedding of the de Sitter spacetime in the Einstein cylinder: the shaded
region corresponds to the set MdS of Equation (6.31). Right, Penrose diagramof the de Sitter spacetime: the lines Γ1 and Γ2 correspond to the axes of+−and IdS
denote, respectively, future and past conformal
symmetry, while IdS
inﬁnity; see main text for further details.

of the Einstein cylinder. Moreover, one sets+−
MdS ≡ M̃dS ∪ IdS∪ IdS.

(6.31)

To construct the Penrose diagram of the de Sitter spacetime one considers thequotient domain QdS ≡ MdS /SO(3). The boundary ∂QdS of the quotientmanifold consists of the projection of the conformal boundary (to be denoted±) and the two centres of symmetry Γ1 and Γ2 given, respectively,
again by IdS
by the conditions ψ = 0 and ψ = π. A depiction of the Penrose diagram of thede Sitter spacetime is given in Figure 6.6, right panel.
6.3.1 Behaviour of geodesics
As in the case of the Minkowski spacetime, intuition about the conformal
representation of the de Sitter spacetime can be obtained through the analysisof the behaviour of geodesics. For simplicity set λ = −3 so that a = 1 in the line
element (6.27). The geodesic equations can be found to bet2 − cosh2 t ψ 2 = ,

ψ =


,
cosh2 twhere  is a constant,  takes the values 1, 0 or −1 depending on whether one
considers timelike, null or spacelike geodesics and  denotes diﬀerentiation withrespect to an aﬃne parameter s. As in the case of the Minkowski spacetime one
distinguishes the following three cases:

158

Conformal extensions of exact solutions(a) Spacelike geodesics. The geodesic equations can be solved to yieldt(s) = arcsinh
2 − 1 sin(s − s ) ,ψ(s) = ψ + arctan  tan(s − s ) ,with s and ψ real constants. Thus, it follows that the range of thecoordinates is bounded and the geodesics remain in a compact region
following the same path over and over.
(b) Timelike geodesics. For concreteness, consider future-pointing geodesics– the past pointing case is similar. The geodesic equations can be solved to
give:


2 + 1 sinh(s − s ) ,t(s) = arcsinhψ(s) = ψ + arctan  tanh(s − s ) .
It follows that for s → ∞ one has t → ∞. Thus, one obtains the limit points
T =

π
,
2

ψ = ψ + arctan .

+
The geodesics approach a deﬁnite point on the spacelike hypersurface IdS
deﬁned in (6.30).
(c) Null geodesics. In this case, the geodesic equations give the solutiont(s) = arcsinh s,
ψ(s) = ψ + arctan(s).
From these expressions it follows, on the one hand, that if s → ∞, thent → ∞,T →π,2

ψ → ψ +π,2while on the other, if s → −∞, thent → −∞,π
T →− ,
2

ψ → ψ −

π
.
2

−
+
Hence, the null geodesics start and ﬁnish, respectively, at IdS
and IdS, in
3
antipodal points on S .

6.3.2 Conformal geodesics in the de Sitter spacetime
As a consequence of the conformal invariance of conformal geodesics, the curvesin the Einstein universe discussed in Section 6.1.3 are also conformal geodesicsof the de Sitter spacetime.
Making use of the relations g E = Ξ2dS g̃ dS and ḡ E = Θ̄2 g E where Θ̄ and ΞdS
are the conformal factors given, respectively, by Equations (6.11) and (6.29), one

6.4 The anti-de Sitter spacetime

159

ﬁnds that ḡ E = Θ2dS g̃ dS with ΘdS ≡ Θ̄ΞdS . A calculation using the ﬁrst of theequations in (6.10) shows that
4 − τ̄ 2
ΞdS = a cos τ = a,
4 + τ̄ 2
where in a slight abuse of notation the coordinate T has been replaced by theparameter of the curves τ . Hence, one ﬁnds that1
ΘdS = a 1 − τ̄ 2 ,4
so that ΘdS vanishes at τ̄ = ±2. To construct the covector associated to thecongruence of conformal geodesics consider ΥdS ≡ ΞdS −1 dΞdS . A calculation
then shows that16τ̄4τ̄dτ̄ = −dτ.
ΥdS = −
416 − τ̄4 − τ̄ 2
Letting β dS ≡ β̄ + ΥdS , it follows from the transformation laws for conformal
geodesics in Section 5.5.2 that the pair (x(τ̄ ), β ds (τ̄ )), with




2τ̄
τ̄
β dS (τ̄ ) = −x(τ̄ ) = 2arctan , x ,dτ̄ ,(6.32)
24 − τ̄ 2
is a solution to the g̃ dS -conformal geodesic equations with parameter τ̄ . Notice,in particular, that at the Cauchy surface given by τ̄ = 0 one has that β dS (0) = 0.
Following the discussion from the previous paragraph, the surface given by the−. In some applications, one
condition τ̄ = −2 represents past null inﬁnity IdS−.
needs to prescribe initial data for the congruence of conformal geodesics at IdS
In this case, it is convenient to introduce the further reparametrisation τ̂ = τ̄ + 2
so that2τ̂ − 4
1 2
β dS = −ΘdS = τ̂ − τ̂ ,dτ̂ .
44τ̂ + τ̂ 2
6.4 The anti-de Sitter spacetime
The anti-de Sitter spacetime is given by the manifold M̃adS ≈ R4 equippedwith the metric



3
2
2
2
,
a≡
g̃ adS = cosh r dt ⊗ dt − a dr ⊗ dr + sinh r σ ,λwith t ∈ R and r ∈ (0, ∞). Strictly speaking, this spacetime is the so-called
universal covering space of the anti-de Sitter spacetime – the classicalanti-de Sitter spacetime has a periodic time coordinate and, thus, closed timelike
curves. As in the case of the de Sitter spacetime, it is possible to introduce
coordinates (t̄, r̄) in terms of which the metric takes the form



−1
1
1
dr̄ ⊗ dr̄ − r̄2 σ.
(6.33)g̃ adS = 1 + λr̄2 dt̄ ⊗ dt̄ − 1 + λr̄2
3
3

160

Conformal extensions of exact solutions

As (in the signature conventions used in this book) λ > 0, there are no horizonsin the anti-de Sitter spacetime; see, for example, Griﬃths and Podolský (2009).
To obtain a conformal representation of this spacetime, it is convenient toconsider a new radial coordinate via the condition
dr = cosh rdψ,so that ψ = 2 arctan er − 12 π. This condition is equivalent to tan ψ = sinh r.
Hence, one has that ψ ∈ [0, 12 π]. Setting T = t/a, a calculation then shows thatg̃ adS = a2 cosh2 r (dT ⊗ dT − h̄) .
The latter suggests introducing the conformal factor
ΞadS =a= a cos ψ.
cosh r(6.34)

Thus, the conformal metric Ξ2adS g̃ is, again, that of the Einstein cylinder. Fromthe previous discussion it follows that the anti-de Sitter spacetime is conformal
to the domain
πM̃adS = p ∈ ME 0 ≤ ψ(p) <,2of the Einstein cylinder. Notice, however, that in contrast to the conformal
representation of the de Sitter spacetime, the conformal anti-de Sitter spacetime
does not cover the whole spatial sections of the cylinder. In particular, one hasthat the conformal factor ΞadS vanishes at ψ = 12 π; that is, the conformalboundary is, in this case, a timelike hypersurface. Following standard usage,
deﬁne the conformal inﬁnity of the anti-de Sitter spacetime asπ
IadS ≡ p ∈ ME ψ(p) =.
2
One also deﬁnes
MadS ≡ M̃adS ∪ IadS .

(6.35)The Penrose diagram for the anti-de Sitter spacetime is constructed by considering the quotient domain QadS ≡ MadS /SO(3) with boundary ∂QadS = IadS ∪Γwhere IadS denotes the projection of null inﬁnity onto QE and Γ denotes thecentre of symmetry given by the condition ψ = 0. A depiction of the Penrose
diagram is given in Figure 6.7.
6.4.1 Geodesics in the anti-de Sitter spacetime
In what follows, for simplicity assume that λ = 3 so that a = 1. Radial geodesics
in the anti-de Sitter spacetime are described by the equations
cosh2 r T 2 − r2 = ,

T =


.
cosh2 r6.4 The anti-de Sitter spacetime

161

R × S3

MadS

Γ

IadS

Figure 6.7 Conformal extension of the anti-de Sitter spacetime. Left, conformal embedding of the anti-de Sitter spacetime in the Einstein cylinder: theshaded region corresponds to the set MadS of Equation (6.35). Right, Penrose
diagram of the anti-de Sitter spacetime: the line Γ corresponds to the axis ofsymmetry, while IadS denotes conformal inﬁnity; see main text for further
details.

These equations can be obtained from those of the de Sitter spacetime by the
replacements t → r, ψ → T and  → −; see Section 6.3.1. Again, one has threecases to consider:(a) Spacelike geodesics. In this case, the geodesic equations can be solved to
give


2 + 1 sinh(s − s ) ,r(s) = arcsinhT (s) = T + arctan  tanh(s − s ) ,with s and T real constants. Thus, for s → ∞ one obtains the limitsπr → ∞, ψ → , T → T + arctan .
2
As a consequence, in the conformal representation, radial spacelike geodesics
approach the conformal boundary IadS .
(b) Timelike geodesics. For simplicity only future-oriented geodesics are
considered. The solution to the geodesic equations is then given by2 − 1 sin(s − s ) ,r(s) = arcsinhT (s) = T + arctan  tan(s − s ) .
Accordingly, the coordinate r is periodic while τ grows unbounded – thelimit points of these curves are not in the Einstein cylinder.
(c) Null geodesics. In this case, the solution to the geodesic equations isr(s) = arcsinh s,
T (s) = T + arctan s.

162

Conformal extensions of exact solutions
As a consequence of these equations one has the limitsr → ∞,

ψ→π,
2

T → T +π,2

as s → ∞. Thus, in the conformal representation, the null geodesics end at
the conformal boundary IadS .
6.4.2 Conformal geodesics in the anti-de Sitter spacetime
The methods used to construct conformal geodesics in the Minkowski and the deSitter spacetimes can also be used in the anti-de Sitter spacetime. For conciseness,the discussion is restricted to the class of conformal geodesics arising from thecurves (6.9) in the Einstein universe.
Using that g E = Ξ2adS g̃ adS and that ḡ E = Θ̄2 g E where Θ̄ and ΞadS are theconformal factors given by Equations (6.11) and (6.34), one ﬁnds that ḡ E =Θ2ads g̃ adS , where
1 2ΘadS ≡ Θ̄ΞadS = a cos ψ 1 + τ̄ .
4
LettingΥadS ≡ Ξ−1
adS dΞadS = − tan ψdψ,
one ﬁnds that the associated covector is given by
β adS ≡ β̄ + ΥadS ==1
τ̄ dT − tan ψdψ22τ̄dτ̄ − tan ψdψ.
4 + τ̄ 2

The expression for the actual curve is, as in the case of the de Sitter spacetime,
given byτ̄
(6.36)x(τ̄ ) = 2arctan , x .
2
An important property of this non-intersecting congruence of conformal geodesics
is that curves that for some value of the parameter τ̄ are at the conformalboundary IadS remain in it for all values of τ̄ ; this observation follows from thefact that the curve given by Equation (6.36) is constant in the spatial directions.
Remark. As arctan 12 τ̄ → 12 π as τ̄ → ∞ and τ = 2arctan 12 τ̄ , the parameter τ̄does not cover the whole Einstein cylinder and only exhausts the slab [−π, π]×S3 .
In order to continue the conformal geodesic to other portions of the anti-de Sitter
spacetime one has to introduce a reparametrisation of the curve by means of afractional transformation as discussed in Lemma 5.1.

6.5 Conformal extensions of static and stationary black hole spacetimes 163
6.5 Conformal extensions of static and stationary
black hole spacetimes
A natural extension of the discussion of the previous sections is the analysisof the conformal structure of spacetimes describing black holes. The more
complicated topology of these spacetimes and the presence of singularities andhorizons make this analysis a much more challenging endeavour. In fact, several
aspects of the conformal structure of static and stationary black holes are open
research questions.
6.5.1 The Schwarzschild spacetime
The Schwarzschild spacetime, being static and spherically symmetric, is thesimplest type of black hole spacetime. The Birkhoﬀ theorem states that any
spherically symmetric solution to the vacuum Einstein ﬁeld equations withvanishing cosmological constant is, in fact, isometric to the Schwarzschildspacetime; see, for example, Misner et al. (1973). Moreover, the black holeuniqueness theorems show that the Schwarzschild spacetime is the only static
black hole spacetime; see, for example, Chruściel et al. (2012b) for an entry point
to the extensive literature on this topic.
The Schwarzschild metric is given in standard (t, r) coordinates by the line
elementg̃ S =

2m
1−
r





2m
dt ⊗ dt − 1 −r−1dr ⊗ dr − r2 σ,(6.37)

with m the so-called mass parameter. The reader interested in a discussion of
the various aspects of the Schwarzschild spacetime is referred to, for example,
Griﬃths and Podolský (2009) and Hawking and Ellis (1973).
To obtain a conformal extension of the Schwarzschild spacetime it is convenient
to make use of coordinates adapted to the light-cone structure of the spacetime.
Accordingly, one introduces the advanced and retarded null EddingtonFinkelstein coordinatesu ≡ t − r − 2m log |r − 2m| ,v ≡ t + r + 2m log |r − 2m| ,so that the line element (6.37) transforms into
1g̃ S =
2



2m
1−
r


(du ⊗ dv + dv ⊗ du) − r2 σ,

where the relation between r and the coordinates (u, v) is given implicitly by thecondition1
r + 2m log |r − 2m| = (v − u).
2

164

Conformal extensions of exact solutions

The singular behaviour of the metric at r = 2m is then removed by means of a
reparametrisation of the null coordinates. Namely, one setsU ≡ −4me−u/4m ,V ≡ 4mev/4m ,so that one obtainsg̃ S =

m −r/2me(dU ⊗ dV + dV ⊗ dU ) − r2 σ,
r(6.38)

where r is now given implicitly by the condition
U V = −8m(r − 2m)er/2m .
The horizon is then given by the condition U V = 0 while the singularity
corresponds to U V = 16m2 . The line element in Equation (6.38) is the socalled Kruskal-Székeres form of the Schwarzschild spacetime. It provides the
maximal analytic extension of the Schwarzschild metric (6.37). Inspection of
the admissible range of coordinates in Equation (6.38) shows that the resulting
maximal manifold has the topology of R × R × S2 .
To compactify the Kruskal-Székeres form of the Schwarzschild metric one
introduces a further coordinate transformation:




U
V
Ū ≡ arctan,V̄ ≡ arctan
4m
4m
where
1
1
− π < Ū < π,2211
− π < V̄ < π,2211
− π < Ū + V̄ < π.
22

It follows then thatdU = 4m sec2 Ū dŪ ,dV = 4m sec2 V̄ dV̄ ,so that the line element (6.38) transforms into
16m3 −r/2m
2
2
2
2
2
g̃ S = sec Ū sec V̄
e(dŪ ⊗ dV̄ + dV̄ ⊗ dŪ ) − r cos Ū cos V̄ σ .
r
It is, therefore, natural to consider a conformal factor of the form
ΞS = cos Ū cos V̄ ,so that g S = Ξ2S g̃ S is given bygS =16m3 −r/2me(dŪ ⊗ dV̄ + dV̄ ⊗ dŪ ) − r2 cos2 Ū cos2 V̄ σ,rwhere r = r(Ū , V̄ ). This conformal metric is singular at r = 0 (the singularity).
In order to discuss the structure of the conformal boundary of the Schwarzschildspacetime, it is convenient to introduce the coordinates
T ≡ V̄ + Ū ,

ψ ≡ V̄ − Ū ,

6.5 Conformal extensions of static and stationary black hole spacetimes 165
so that T ∈ [−π, π], ψ ∈ [−π, π]. One sees that the maximal analytic extensionof the Schwarzschild spacetime is conformal to the interior of the domain MS ⊂(−π, π) × [−π, π] × S2 with boundary given by+−−∂MS = I1+ ∪ I2+ ∪ I1− ∪ I2− ∪ i01 ∪ i02 ∪ i+
1 ∪ i2 ∪ i1 ∪ i2 ,where by analogy with the analysis of the conformal boundary of the Minkowski
spacetime one deﬁnes the various components of null inﬁnity as




1
1
+
+
I2 ≡ Ū = π ,I1 ≡ V̄ = π ,
2
2




1
1
−
−
I2 ≡ V̄ = − π ,I1 ≡ Ū = − π ,22
and the two components of spatial inﬁnity asi01 ≡ {T = 0, ψ = π},i02 ≡ {T = 0, ψ = −π}.

Finally, the timelike inﬁnities are given by




1
1
±
±
i2 ≡ T = ±π, ψ = − π .
i1 ≡ T = ±π, ψ = π ,22An analysis of the geodesics on the Schwarzschild spacetime justiﬁes the namegiven to the various components of ∂MS . Observe that the singularities at r = 0
are not included as part of the boundary ∂MS . In this representation, the spatial
inﬁnities i01 and i02 can be seen to correspond to two points on the conformal
manifold. Further properties of the conformal structure of the Schwarzschildspacetime – in particular, the nature of i0 – will be analysed in the context of theconformal Einstein ﬁeld equations in Chapter 20. Finally, the Penrose diagram of
the Schwarzschild spacetime can be readily obtained by considering the quotientmanifold QS = MS /SO(3); the resulting diagram is given in Figure 6.8.
Conformal geodesics in the Schwarzschild spacetime
A detailed analysis of a class of conformal geodesics in this spacetime can befound in Friedrich (2003a) where it is shown that the Schwarzschild spacetime
can be completely covered by a (non-singular) congruence of conformal geodesics.
This congruence is adapted to the spherical symmetry of the spacetime.
6.5.2 Conformal extensions of other static, sphericallysymmetric spacetimes
The procedure to construct a conformal extension of the Schwarzschild spacetime
discussed in Section 6.5.1 can be generalised to include a wide class of static,
spherically symmetric spacetimes. In this section, an adaptation of a generalprocedure given on Walker (1970) is discussed. This discussion illuminates theconformal diagram of a number of spacetimes.

166

Conformal extensions of exact solutions
+

+

i2

i1

+

+

I2

+

H2

I1

+

H1

i02

i01
H−
2

I−
2

i−
2

H−
1

I−
1

i−
1

Figure 6.8 Penrose diagram of the Schwarzschild spacetime. The null hypersurfaces I1± and I2± correspond to the four diﬀerent components of null±inﬁnity, while the points i01 , i02 and i±
1 , i2 denote, respectively, the various
locations of spatial and timelike inﬁnities. The serrated lines denote thesingularities, and H1± and H2± correspond to the various components of thehorizon; see the main text for further details.

In what follows, let (M̃, g̃) denote a spherically symmetric spacetime endowedwith a further Killing vector ∂ t . The following considerations will be independentof the matter content of the spacetime; hence, the spacetime is not assumed to
be a vacuum. Attention will be restricted to spacetimes in which it is possible
to ﬁnd coordinates (t, r) such that the metric g̃ takes the formg̃ = F (r)dt ⊗ dt − F (r)−1 dr ⊗ dr − r2 σ.
The coordinate r is an areal coordinate; that is, the area of a 2-sphere describedby the conditions t = constant, r = constant is 4πr2 . The function F (r) is thenorm of the Killing vector ∂ t . When F (r) > 0 the Killing vector ∂ t is timelike,
and, thus, the metric g̃ is static.
To simplify the presentation, the subsequent analysis will make use of thequotient manifold Q̃ = M̃/SO(3). The two-dimensional quotient metric γ̃
induced by g̃ on Q̃ is given byγ̃ = F (r)dt ⊗ dt − F (r)−1 dr ⊗ dr.
The Levi-Civita connection associated to the Lorentzian metric γ̃ will bedenoted by D
 ˜ . Let ẋ = (ṫ, ṙ) denote the tangent vector to an aﬃnely parametrised
geodesic in Q̃; here, and in what follows, a dot ( ˙ ) denotes diﬀerentiation withrespect to an aﬃne parameter. The geodesic equation D ˜ ẋ ẋ = 0 can be integrated
once to yieldṫ = κF,ṙ = κ 2 − F ,where κ is a constant and  ≡ γ̃(ẋ, ẋ). As γ̃ is a two-dimensional metric, theonly invariant of the curvature of γ̃ is the Ricci scalar R[γ̃] = F  , where  denotes6.5 Conformal extensions of static and stationary black hole spacetimes 167
diﬀerentiation with respect to r. In the remainder of this section it will be shownthat if F and F  are ﬁnite for all r ∈ R, then every geodesic in Q̃ can be extended
until it is complete. If, on the other hand, F or F  become unbounded for somevalue r , then only those geodesics along which r = r within a ﬁnite aﬃnedistance from some point in Q̃ are incomplete and inextendible; hence, Q̃ andalso M̃ are singular. The extensions obtained from the following considerations
are maximal.
Elementary blocks. In what follows, assume that F has a ﬁnite number ofzeros, to be denoted by ai , i = 1, . . . , n with a1 < · · · < an . If F approaches aconstant ﬁnite value as r → ∞, so that R[γ̃] = F  → 0, then one can redeﬁnecoordinates so that limr→∞ F = ±1. In this case Q̃ is asymptotically ﬂat and anull conformal boundary similar to that of the Minkowski spacetime can beconstructed; an analogous discussion can be made in the case r → −∞. A
diﬀerent possible asymptotic behaviour occurs when F becomes unbounded as
r → ∞. Using the de Sitter and the anti-de Sitter metrics in static form asgiven by Equations (6.28) and (6.33) one sees that the behaviour F → −∞ andF → ∞ as r → ∞ corresponds, respectively, to de Sitter-like and anti-de Sitterlike asymptotic regions. These regions can be compactiﬁed to obtain conformalboundaries similar to those of the de Sitter and anti-de Sitter spacetimes, thatis, given, respectively, by spacelike and timelike hypersurfaces.
When F vanishes, the orbits of the timelike Killing vector become null; thatis, one has a Killing horizon. This suggests dividing Q̃ into n + 1 regions(blocks). Each of these regions is bounded by two of the Killing horizons, by a
Killing horizon and conformal inﬁnity, or by a Killing horizon and a singular line
at r = r for r ﬁxed. The maximal extension of Q̃ is found by gluing together
elementary blocks along their boundaries (seams). In what follows, for a nonsingular seam it will be understood one where F = 0 and F  is ﬁnite, whilea singular seam will be one where F or F  (or both) are unbounded. Blocks
can be glued together only along non-singular seams across which F  is smooth.
In each regionQ̃i ≡ (t, r) ∈ Q̃ | t ∈ R, r ∈ [ai , ai+1 ] ,ﬁx some value ri ∈ (ai , ai+1 ) of r and deﬁne null coordinates via
 r
 rui ≡ t −F −1 (s)ds,
vi ≡ t +F −1 (s)ds.
riri

In terms of these new coordinates the metric γ̃ takes the formγ̃ =1F (r)(dui ⊗ dvi + dvi ⊗ dui ).
2This form of the metric is smooth for ui , vi ∈ R if F (r) is smooth. The coordinate
r will be regarded as a function of (ui , vi ) given, implicitly, as the solution to theequations

168

Conformal extensions of exact solutions
1(vi − ui ) =2

 r

F −1 (s)ds,

ri1(ui + vi ) = t.
2

(6.39)

The construction can be extended to singular blocks by setting ri = r .
In the non-singular case, the integrals ai+1
 r−1
F (s)ds,
F −1 (s)ds,
airi

are divergent as the points r = ai , ai+1 are poles of the integrand. From thisobservation together with the formulae in (6.39), assuming F > 0 in Q̃i , one
deduces the limits:(a) If r → ai+1 and vi is ﬁnite, then ui → −∞ and t → −∞.
(b) If r → ai+1 and ui is ﬁnite, then vi → +∞ and t → +∞.
(c) If r → ai and vi is ﬁnite, then ui → +∞ and t → +∞.
(d) If r → ai and ui is ﬁnite, then vi → −∞ and t → −∞.
The setting described by the above limits is depicted in Figure 6.9, left panel.
The coordinates (ui , vi ) can be compactiﬁed via
Ui ≡ arctan ui ,Vi ≡ arctan vi ,with Ui , Vi ∈ [− 12 π, 12 π] so that γ̃ can be rewritten asγ̃ =1
F (r) sec2 Ui sec2 Vi (dUi ⊗ dVi + dVi ⊗ dUi ).
2(6.40)In what follows, for simplicity, given a regular block Q̃i with coordinate r ∈[ai , ai+1 ], it is assumed that the zeros of F (r) are such that
F (r) sec2 Ui sec2 Vi < ∞as1
Ui → ± π2or1
Vi → ± π.
2v = −∞

u = +∞

u = const.
r = air = const.

r = ai+1v = const.
r = ai+1

r = ai+1

u = −∞

r = const.

v = +∞
r = air = ai+1

r = air = aiv = const.

v = −∞
F >0

v = +∞
u = +∞

u = const.

u = −∞
F<0

Figure 6.9 Coordinates in a regular block without asymptotic regions; seemain text for details.

6.5 Conformal extensions of static and stationary black hole spacetimes 169
(a)

(b)

(c)

(e)

(f)

(d)

Figure 6.10 Elementary blocks for the construction of Penrose diagrams: (a)non-singular block without asymptotic regions; (b) non-singular block with anull I ; (c) non-singular block with a spacelike I ; (d) non-singular block witha timelike I ; (e) block with a timelike singularity; (f) block with a spacelike
singularity.

In the case of blocks Q̃i with r ∈ [ai , ±∞) corresponding to asymptotic regions,the metric (6.40) allows us to read out a conformal factor. The particular formof the conformal factor depends on the particular nature of the asymptotic end:⎧cos Ui cos Vi
if F (r) → 1 as r → ∞
⎪
⎪
⎨
Ξi =
cos Ui cos Vi⎪⎪

if F (r) → ∞ as r → ∞.
⎩F (r)The resulting construction can be depicted in a conformal diagram.
The discussion of the case F < 0 in Q̃i is analogous. In this case the orbitsof the Killing vector ∂ t are spacelike, and the hypersurfaces of constant r are
timelike. The behaviour of the various coordinates in a regular elementary block
is summarised in Figure 6.9, right panel. A depiction of the various elementary
blocks is given in Figure 6.10.
Flipping of blocks. The convention used in drawing the diagrams in Figure 6.9
is that the coordinate r increases from left to right (if F > 0) and from bottom totop (if F < 0). As this is a mere convention, it is possible to ﬂip the blocks about
ri . This operation eﬀectively interchanges the roles of u and v. In addition, as
the metric γ̃ is independent of the coordinate t, one has the discrete symmetryt → −t which allows further ﬂipping of blocks with respect to the surfaces of
constant t – vertically if F > 0 and horizontally if F < 0.
Gluing blocks. All geodesics such that r = ai for some ﬁnite value of the aﬃneparameter are incomplete. These geodesics can be extended by gluing blocks
along non-singular seams. The convention followed in gluing the blocks is thatthe time coordinate in each block, t if F (r) > 0 and r if F (r) < 0, changes

170

Conformal extensions of exact solutions
(b)
1

1

Q̃i

Q̃i+1

2

1

0

0
(a)

1

2

Figure 6.11 The two ways of gluing elementary blocks as discussed in the maintext. In the conﬁguration (1) the block Q̃i+1 has been ﬂipped about ri , while
in conﬁguration (2) it is necessary to invert its time orientation; see the main
text for further details.

vertically. Consider, for example, the gluing of a block Q̃i+1 with F (r) > 0
and a block Q̃i with F (r) < 0. By suitably ﬂipping the blocks, they can beglued together in the two ways shown in Figure 6.11. Under the assumptionthat F  is smooth at r = ai+1 (and hence also the curvature), the gluing ofblocks is equivalent to showing that the null hypersurfaces are continuous along
the seams, that is, that there is a coordinate system covering both blocks in aneighbourhood of r = ai+1 in a smooth fashion. This construction is implemented
through Eddington-Finkelstein type coordinates.
As a ﬁrst example consider conﬁguration (1) of Figure 6.11 where the blockQ̃i is glued to a block Q̃i+1 which has been ﬂipped about ri . Direct inspection
reveals that while advanced null coordinates exhaust at the gluing seam (i.e.
they become inﬁnite), a null retarded coordinate extends to the two blocks Q̃iand Q̃i+1 . Accordingly, one setsdu = dt − F −1 (r)dr,so thatγ̃ = F (r)du ⊗ du + (du ⊗ dr + dr ⊗ du).

(6.41)

Now, allowing r ∈ [ai , ai+2 ] one ﬁnds that the coordinates (u, r) cover both
blocks in conﬁguration (1) of Figure 6.11 – the resulting combined block is shownin conﬁguration (1) of Figure 6.12. In particular, the coordinate u is ﬁnite atr = ai+1 and the metric (6.41) is smooth for r ∈ (ai , ai+2 ).
In order to perform the gluing in conﬁguration (2) of Figure 6.11, one needs to
ﬂip the block Q̃i+1 about t. Direct inspection shows that for this conﬁguration
retarded null coordinates exhaust at the gluing seam. Accordingly, one introduces
advanced null coordinates
dv = dt + F −1 (r)dr,so as to obtainγ̃ = F (r)dv ⊗ dv − (dv ⊗ dr + dr ⊗ dv).

(6.42)

6.5 Conformal extensions of static and stationary black hole spacetimes 171
(a)

2

(b)

1

1

2

0

0
1

1

2
1

1

1
2

0

0

Figure 6.12 The two composite blocks obtained from the the gluing proceduresin Figure 6.11.

1

0
2
1

2

1

0
1

0

Figure 6.13 Gluing applied to three blocks simultaneously. A fourth block can
be glued using the Kruskal construction – see the main text for further details.

The coordinate v is ﬁnite at r = ai+1 . Thus, the pair (v, r) covers the composite
block. One can verify that the metric (6.42) is smooth for r ∈ (ai , ai+2 ); theresulting combined block is shown in conﬁguration (2) of Figure 6.12.
Gluing à la Kruskal. The gluings discussed in the last paragraphs can be
performed simultaneously; the resulting conﬁguration is shown in Figure 6.13.
In addition, one could also glue a further region Q̃i+1 obtained from Q̃i+1 by
applying the reﬂection in the time coordinate. Notice, however, that the point
p is not covered by either of the coordinates (u, r) and (v, r). Depending on theparticular form of F (r) it may be possible to obtain a single coordinate patch forthe four blocks; this is the case, for example, in the Schwarzschild spacetime. The
procedure to do this makes use of a generalisation of the Kruskal coordinates;
see Figure 6.13. The general strategy is to ﬁnd coordinates (U, V ) such that the
metric takes the form

172

Conformal extensions of exact solutionsγ̃ = G(r)(dU ⊗ dV + dV ⊗ dU ),where G is bounded and non-zero at r = ai+1 . Since U = U (t, r) and V = V (t, r)
one readily ﬁnds the conditions
G∂t U ∂t V = F,∂r U ∂t V + ∂t U ∂r V = 0,G∂r U ∂r V = −F −1 .

It can be veriﬁed that a solution to the above is given by




dr
dr
1
U (t, r) = a exp bt + b,
V (t, r) = exp −bt + b
,
F (r)
a
F (r)



dr
F (r)
,
G(r) = 2 exp −2b
b
F (r)




where a and b are constants. The function G(r) given by the above formulae
can be singular. The key point in the construction is to analyse whether theconstant b can be chosen so that G(r) is bounded and non-zero at r = ai+1 . Asan illustration, in the case of the Schwarzschild spacetime one has that
F (r) = 1 −2mr

so thatG(r) =

1(r − 2m)1−4mb e−2br .
b2 rHence, choosing b = 1/4m one has
G(r) =

16m2 −r/2me,rwhich is bounded and non-zero at r = 2m, the location of the horizon.
By contrast, in the case of the extremal Reissner-Nordström spacetime – see
Equation (6.43) – one hasm 2
F (r) = 1 −r

so thatG(r) =



1
6m2 b
2−4mb
(r
−
m)
exp
−2br
+
.
b2
r−m

In this case one cannot ﬁnd a value of b which makes G(r) ﬁnite and non-zeroat r = m. In particular, the choice b = 1/2m yields


3m
r
1
exp − +
G(r) =,
4m2
m r−m
which is singular at r = m. More generally, if F (r) is a rational function, it canbe shown that the value b can be chosen so that G(r) is ﬁnite and non-zero atr = ai+1 if ai+1 is a non-repeated zero of F (r); see Walker (1970) for the detailsof the proof.

6.5 Conformal extensions of static and stationary black hole spacetimes 173
Some examples
The procedure described in the previous paragraphs can be employed toconstruct the Penrose diagrams and conformal compactiﬁcations of a number ofwell-known spherically symmetric spacetimes. In particular, one has the following(details can be found in the given references):
The non-extremal Reissner-Nordström spacetime. This is the solution to
the Einstein-Maxwell ﬁeld equations given by the metric

g̃ =

1−

2m q 2
+ 2
r
r




−1
2m q 2+ 2
dr ⊗ dr − r2 σ,
dt ⊗ dt − 1 −rrwith q 2 < m2 . In this case F (r) = 1−2m/r+q 2 /r2 has two zeros. One can identify
three elementary blocks: an asymptotically ﬂat region, a standard regular blockand a block with a timelike singularity. In this case Kruskal’s construction canbe employed to glue four blocks simultaneously. The resulting Penrose diagram
is given in Figure 6.14; see Carter (1973).

Figure 6.14 Penrose diagram of the Reissner-Nordström spacetime in the nonextremal (q 2 < m2 ) case. The points i0 correspond to the various spatial
inﬁnities, the points i± to future and past timelike inﬁnity, respectively, andthe lines I ± to the various components of null inﬁnity. The dashed lines
H ± correspond to the various horizons. Finally, the serrated lines denote thesingularities.

174

Conformal extensions of exact solutions

Figure 6.15 Penrose diagram of the Reissner-Nordström spacetime in theextremal case (q 2 = m2 ). The point i0 corresponds to spatial inﬁnity, thepoints i± to future and past timelike inﬁnity, respectively, and the lines I ±to the various components of null inﬁnity. The dashed lines labeled by H
correspond to the various horizons. Finally, the serrated lines denote thesingularities.

The extremal Reissner-Nordström spacetime. This is the particular caseof the Reissner-Nordström spacetime for which q 2 = m2 . The metric is given by


m 2
m −2
g̃ = 1 −dt ⊗ dt − 1 −dr ⊗ dr − r2 σ.
(6.43)rr
In this case, one has a double zero of F (r) = (1 − m/r)2 . Thus, one cannot
make use of Kruskal’s construction. One can identify two elementary blocks: an
asymptotically ﬂat one and a block with a timelike singularity. The resulting
Penrose diagram is given in Figure 6.15; see Carter (1966a, 1971). An interesting
property of the extremal Reissner-Nordström spacetime is that it is conformallyinvariant under a certain spatial inversion; see Couch and Torrence (1984). This
discrete conformal symmetry can be used to relate properties of null inﬁnity withproperties of the horizon; see Bizon and Friedrich (2012). A similar symmetry
exists for a particular combination of the parameters in the Reissner-Nordströmde Sitter spacetime; see Brännlund (2004).
The Schwarzschild-de Sitter and Schwarzschild-anti de Sitter spacetimes. The metric for these spacetimes is given by


−1

2m 1 22m 1 2+ λr dt ⊗ dt − 1 −+ λr
dr ⊗ dr − r2 σ,g̃ = 1 −r
3
r
3
where it is assumed that m > 0. If λ > 0 (the anti-de Sitter case), then it can be
veriﬁed that F (r) = 1 − 2m/r − λr2 /3 has only one real root corresponding to a
black hole-type horizon. The resulting diagram is given in Figure 6.16. If λ < 0(the de Sitter case) and 0 < −9λm2 < 1, then F (r) can be shown to have two6.5 Conformal extensions of static and stationary black hole spacetimes 175

Figure 6.16 Penrose diagram of the Schwarzschild-anti de Sitter spacetime.
The vertical lines I denote the two components of conformal inﬁnity, while
the dashed lines labeled by H ± denote the various components of the horizon.
The serrated lines denote the singularities.

Figure 6.17 Penrose diagram of the Schwarzschild-de Sitter spacetime. The
horizontal lines labeled by I ± correspond to the various components ofconformal inﬁnity. The dashed lines Hc and Hb denote, respectively, thecosmological and black hole horizons. The serrated lines indicate the locationof the singularities.

positive real roots corresponding, respectively, to a black hole-type horizon anda cosmological type horizon. The resulting blocks can be arranged in a periodic
diagram as given in Figure 6.17; see, for example, Griﬃths and Podolský (2009).
In this case it is also possible to make topological identiﬁcations; see Beig and
Heinzle (2005). The cases −9λm2 = 1 and −9λm2 > 1 correspond, respectively,
to the so-called extremal and hyperextremal cases.
Other examples of spacetimes amenable to the general construction describedin this section are the Nariai solution, the Reissner-Nordström-de Sitter and theReissner-Nordström-anti de Sitter solutions; see, for example, Brill and Hayward(1994) for a detailed discussion.
6.5.3 Extending across the conformal boundary
In Schmidt and Walker (1983) it has been observed that the conformal representations of some spacetimes can be extended across the conformal boundary. In
the case of the Schwarzschild solution this is best seen by considering the metricwritten in terms of a retarded null coordinate:2m(6.44)du ⊗ du + (du ⊗ dr + dr ⊗ du) − r2 σ,g̃ S = 1 −r

176

Conformal extensions of exact solutions

where, in particular, u ∈ R. Deﬁning  ≡ 1/r, a calculation yields that
2 g̃ S = 2 (1 − 2m) du ⊗ du − (du ⊗ d + d ⊗ u) − σ.

(6.45)In this representation, future null inﬁnity I + is given by the condition  = 0.
The key observation is that the metric (6.45) can be analytically extended byallowing  to take negative values. To identify the spacetime on the other sideof I + one undoes the conformal rescaling to obtainḡ S = (1 − 2m) du ⊗ du −11
(du ⊗ d + d ⊗ du) − 2 σ,2

where  ∈ (−∞, 0). To bring the metric to a more familiar form one introduces
new coordinates r̄ = −1/, v̄ = u and deﬁnes m̄ = −m, so that2m̄ḡ S = 1 −(6.46)dv̄ ⊗ dv̄ − (dv̄ ⊗ dr̄ + dr ⊗ dv̄) − r̄2 σ,r̄with r̄ ∈ (0, ∞) and v̄ ∈ R. The metric (6.46) corresponds to the negative
mass Schwarzschild spacetime in advanced null coordinates. The null
hypersurface I + of the conformal extension of the original (positive mass)
Schwarzschild spacetime corresponds to the null hypersurface I − of the negative
mass Schwarzschild spacetime. It is important to point out that the spacetimes
described by the metrics (6.44) and (6.46) are causally disconnected; however,at the level of the conformal structure, they are an extension of each other. Thissituation is depicted, at the level of Penrose diagrams, in Figure 6.18.
The ideas described in the previous paragraphs can be used to construct socalled maximal conformal extensions of the Schwarzschild spacetime.
Further details can be found in Schmidt and Walker (1983). The construction
described in this section can be adapted to other spacetimes, for example, theReissner-Nordström solution.

Figure 6.18 Extending the conformal structure of the Schwarzschild spacetime
through null inﬁnity. The future null inﬁnity of the positive mass Schwarzschildspacetime is identiﬁed with the past null inﬁnity of the negative mass
Schwarzschild spacetime, see main text for further details. The points denotedby white dots are excluded from the discussion.

6.6 Further reading

177

6.6 Further reading
The discussion presented in this chapter has been restricted to the analysis of theconformal structure of static, spherically symmetric spacetimes. Some aspects ofthis discussion can be adapted to the analysis of other exact solutions like theKerr and Kerr-Newman spacetimes; this is discussed in, for example, Carter(1973), Hawking and Ellis (1973) and Griﬃths and Podolský (2009). Anotherclass of spacetimes amenable to an explicit discussion of its conformal structure isthat of the Friedman-Lemaı̂tre-Robertson-Walker (FLRW) cosmological models;see again Hawking and Ellis (1973) and Griﬃths and Podolský (2009).
An alternative discussion of Penrose diagrams of spherically symmetricspacetimes which allows for dynamic conﬁgurations can be found in appendix Cof Dafermos and Rodnianski (2005). The idea of a Penrose diagram can beadapted to the analysis of suitable two-dimensional timelike totally geodesic
hypersurfaces of non-spherically symmetric spacetimes. This idea has been
particularly fruitful in the case of the Kerr and Kerr-Newman spacetime; see,for example, Carter (1966b, 1968, 1973), Hawking and Ellis (1973) and Griﬃths
and Podolský (2009). This strategy, has been adapted to a variety of situationsin Chruściel et al. (2012a).

7
Asymptotic simplicity

The analysis of the conformal structure of exact solutions carried out in
Chapter 6 exhibited a number of common features among the various spacetimes
considered. The most conspicuous one is that they all admit a smooth conformal
extension which attaches a boundary to the spacetime. This conformal boundary
represents points at inﬁnity. It is natural to ask whether this property isshared by a larger class of spacetimes. This question leads to the notion of
asymptotic simplicity. In formulating this notion one tries to strike a delicate
balance: the deﬁnition should be strong enough so that it excludes clearly
pathological situations, but at the same time it should leave enough room toinclude interesting spacetimes that go beyond the obvious explicit examples.
The original deﬁnition of asymptotic simplicity is due to Penrose (1963, 1964,1965). This deﬁnition has had a lasting inﬂuence on the ﬁeld of mathematical
relativity, in general, and in the applications of conformal methods to the analysisof global properties of spacetimes, in particular.
7.1 Basic deﬁnitions
The following deﬁnition of asymptotic simplicity is adapted from Hawking andEllis (1973):
Deﬁnition 7.1 (asymptotically simple spacetimes) A spacetime (M̃, g̃) is
said to be asymptotically simple if there exists a smooth, oriented, timeoriented, causal1 spacetime (M, g) and a smooth function Ξ on M suchthat:(i) M is a manifold with boundary I ≡ ∂M.
(ii) Ξ > 0 on M \ I , and Ξ = 0, dΞ = 0 on I .

1 A causal spacetime is one in which there exist no closed timelike or null (i.e. causal) curves;see also Chapter 14.

7.1 Basic deﬁnitions

179(iii) There exists an embedding ϕ : M̃ → M such that ϕ(M̃) = M \ I andϕ∗ g = Ξ2 g̃.
(iv) Each null geodesic of (M̃, g̃) acquires two distinct endpoints on I .
The spacetime (M̃, g̃) is called the physical spacetime, while (M, g) is
known as the unphysical spacetime. The boundary I is generally known as
conformal inﬁnity. In the cases where I corresponds to a null hypersurface
one calls it null inﬁnity. More informally, I is also called scri 2 – a shortened
version of script I. In a slight abuse of notation, one usually identiﬁes M̃ and
M \ I so that one writes g = Ξ2 g̃; see, for example, the examples discussed in
Chapter 6. In what follows phrases like “at inﬁnity ” are to be understood asmeaning in a suitable neighbourhood of I in M.
Deﬁnition 7.1 allows for a non-vanishing matter content. Spacetimes for which,in addition, one has that Rab = 0 in a neighbourhood of I in ϕ−1 (M) aresometimes called asymptotically empty and simple.
Remarks(a) Restriction on the conformal class. Deﬁnition 7.1 imposes restrictions
only on the conformal class of the admissible spacetimes (M̃, g̃). It does not
single out any speciﬁc conformal representation; that is, it does not provide
a canonical unphysical spacetime (M, g).
(b) Conformal inﬁnity is a hypersurface. The boundary I as introducedin point (i) in Deﬁnition 7.1 is a well-deﬁned three-dimensional hypersurfaceof M with normal given by dΞ. In particular, sets where dΞ = 0 – such as
spatial inﬁnity i0 and the timelike inﬁnities i± of the Minkowski spacetime –are excluded from I . Points of this type, if present, will still be regarded as
belonging to the conformal boundary but will be treated separately.
(c) Conformal inﬁnity is at inﬁnity. Points (ii) and (iii) of Deﬁnition 7.1
ensure that the boundary I shares the key properties of the null inﬁnity of
the Minkowski, de Sitter and anti-de Sitter spacetimes. To see that this is thecase one needs to analyse the behaviour of null geodesics. The transformation
behaviour of null geodesics under conformal rescalings has already been
discussed in Section 5.5. In what follows, let s̃ and s denote, respectively,g̃-aﬃne and g-aﬃne parameters of a null geodesic γ ⊂ M̃. It follows thenthat s̃ and s are related to each other by the equation
1
ds̃
= 2.
ds
Ξ
Without loss of generality, one can choose the unphysical aﬃne parameters to vanish at I ; that is, Ξ = O(sα ) along the null geodesic with α > 0.
2 Remarkably, the word scri is pronounced in the same way as the Polish word scraj meaningboundary.

180

Asymptotic simplicity

Now, as dΞ = 0 at I , one concludes that, in fact, α ≥ 1. Hence, s̃ → ∞ as
Ξ → 0 – that is, from the physical point of view (as measured by the aﬃneparameter s̃) the null geodesic never reaches the conformal boundary I .
Thus, the conformal boundary lies at inﬁnity from the perspective of thephysical metric g̃.
(d) Smoothness of the conformal extension and decay. As will be
discussed in Chapter 10 the smoothness assumption in Deﬁnition 7.1 imposesa sharp decay behaviour on the gravitational ﬁeld at inﬁnity – in particular,
it leads to what is known as the peeling behaviour of the Weyl tensor.
There are variations of Deﬁnition 7.1 in which the smoothness requirement
is relaxed to admit conformal extensions of class C k for some suitable
positive integer k; see Penrose and Rindler (1986). The physical relevanceof these weaker regularity conditions is a delicate technical point whichcannot be satisfactorily assessed by just looking at speciﬁc examples. These
weaker regularity conditions lead to a diﬀerent asymptotic behaviour of thegravitational ﬁeld.
(e) Matter and causal nature of null inﬁnity. As already mentioned,
Deﬁnition 7.1 allows for the presence of matter in the spacetime. If theenergy-momentum tensor of the matter models has a suitable decay at
inﬁnity, then the causal nature of I is ﬁxed by the sign of the cosmologicalconstant λ: it is spacelike if λ < 0, null if λ = 0 and timelike if λ > 0; see
Theorem 10.1.
(f) The completeness requirement. Point (iv) in Deﬁnition 7.1 is a completeness condition which, in particular, excludes spacetimes such as the
Schwarzschild solution in which there exist null geodesics which do not reachI – not only those falling into the black hole region, but also those lying in
the photon sphere at r = 3m; see Wald (1984).
(g) Regular solutions which are not asymptotically simple. That aspacetime is smooth and geodesically complete is not a guarantee that it
admits a smooth conformal extension. An example of this is given by theso-called Nariai spacetime described byM̃ = R × (S1 × S2 ),g̃ = dt ⊗ dt − cosh2 tdψ ⊗ dψ − σ,

(7.1)

which is a solution to the vacuum Einstein ﬁeld equations with λ = −1.
In addition to being geodesically complete, the Nariai spacetime is also
globally hyperbolic; see Section 14.1. Remarkably, the Nariai spacetime does
not even admit a patch of a conformal boundary. To see this, assume one has
a conformal extension with the required properties. The standard conformaltransformation laws imply thatC̃abcd C̃ abcd = Ξ4 Cabcd C abcd .
Thus, if the solution admits a smooth conformal extension, thenC̃abcd C̃ abcd = 0. On the other hand, a direct computation with the line

7.2 Other related deﬁnitions

181

element (7.1) shows that C̃abcd C̃ abcd = constant = 0. This is a contradiction,
and accordingly there cannot exist a piece of conformal boundary which
is C 2 . This argument is adapted from Friedrich (2015a); an alternative
topological argument has been given in Beyer (2009a).
In order to consider spacetimes for which the completeness condition (iv)in Deﬁnition 7.1 does not hold, one introduces the further notion of weakly
asymptotically simple spacetimes, that is, spacetimes whose asymptotic
region is diﬀeomorphic to that of an asymptotically simple spacetime. Moreprecisely, one has the following:Deﬁnition 7.2 (weakly asymptotically simple spacetimes) A spacetime(M̃, g̃) is said to be weakly asymptotically simple if there exists an asymptotically simple spacetime (M̃ , g̃  ) and a neighbourhood U  of I  ≡ ∂M suchthat ϕ−1 (U  ) ∩ M̃ is isometric to an open subspace Ũ of M̃.
Basic examples of asymptotically simple spacetimes have been given in
Chapter 6. Notoriously, all the given examples are time independent. Moregenerally, it can be shown that stationary solutions to the vacuum equations
Rab = 0 with a suitable behaviour at inﬁnity are at least weakly asymptoticallysimple; see Damour and Schmidt (1990) and Dain (2001b). Thus, it is natural
to ask whether there are dynamic solutions to the Einstein ﬁeld equations. At
the level of exact solutions, the closest examples are given by the spacetimes
known as boost-rotation symmetric spacetimes – see, for example, Bičák
and Schmidt (1989), Bičák (2000) and Griﬃths and Podolský (2009) – and inparticular the so-called C-metric – see Ashtekar and Dray (1981). All theseexact solutions contain some pathologies (e.g. naked singularities, piercing ofnull inﬁnity) which prevent them from being true examples of asymptoticallysimple spacetimes.
A detailed discussion of the properties of asymptotic simple spacetimes
requires the conformal Einstein ﬁeld equations and is deferred to Chapter 10.
7.2 Other related deﬁnitions
The deﬁnition of asymptotic simplicity makes neither reference to nor restricts
the behaviour of the conformal spacetime (M, g) at spatial inﬁnity. Several
authors have introduced more reﬁned deﬁnitions of asymptotic simplicity inwhich further requirements on the behaviour of the gravitational ﬁeld at nullinﬁnity are prescribed, as in, for example, Persides (1979), or at spatial inﬁnity,
as in the concept of asymptotically empty and ﬂat spacetime at null andspatial inﬁnity of Ashtekar and Hansen (1978); see also Persides (1980). Similarideas have been pursued by a number of authors in an attempt to analyse thestructure of timelike inﬁnity; see, for example, Persides (1982a,b), Porrill (1982),
Cutler (1989) and Herberthson and Ludwig (1994).

182

Asymptotic simplicity

The aim of the deﬁnitions mentioned in the previous paragraph and similar
other proposals for the analysis of the asymptotic structure of spacetime is to
identify a minimal number of assumptions on the asymptotic structure which, inturn, can be used to develop a formalism to construct physical and geometricalnotions of interest. A critique to this approach is that, a priori, they do notprovide any information on the genericity of the assumptions or on the size of theclass of spacetimes they contain. Moreover, it is not clear how these spacetimes
can be constructed. As pointed out in Geroch (1976), pages 3–4:
Conditions too strong will have the eﬀect of eliminating solutions which
would seem clearly to represent isolated systems; conditions too weak may
have the eﬀect of admitting too many solutions, or what is worse, may
result in a structure which is so weak that potentially useful aspects of theasymptotic behaviour of one’s ﬁelds are lost in a sea of bad behaviour · · ·There are no correct or incorrect deﬁnitions, only more or less useful ones.
The point of view pursued in this book is that rather than making assumptions
on the nature of spatial, null or timelike inﬁnity, the structures of the conformalboundary of a spacetime should arise as a result of the evolution of some initial
data set for the Einstein ﬁeld equations.
7.3 Penrose’s proposal
Asymptotically empty and simple spacetimes (i.e. spacetimes with a vanishing cosmological constant and matter suitably decaying at inﬁnity) play animportant role in the approach to the analysis of isolated systems in generalrelativity put forward by Penrose (1963, 1964). The notion of an isolated system
is a convenient idealisation of astrophysical systems where the eﬀects of thecosmological expansion are ignored. This notion allows one to deﬁne concepts ofclear physical interest such as the total energy of a system or its mass loss
due to gravitational radiation. Intuitively, an isolated system should behave
asymptotically like the Minkowski spacetime. Penrose, based on earlier work by
Bondi et al. (1962) and Sachs (1962b), takes this idea further; see also Friedrich(2002, 2004):
Penrose’s proposal. Far ﬁelds of isolated gravitating systems behave like thoseof asymptotically simple spacetimes in the sense that they can be smoothly
extended to null inﬁnity after a suitable conformal rescaling.
In other words, if a spacetime (M̃, g̃) describes an isolated system, then it
should be weakly asymptotically simple.
As pointed out in Remark (d) earlier in the chapter, the requirement ofsmoothness results in a very deﬁnite decay behaviour of the gravitational ﬁeld at
inﬁnity. Whether this behaviour is actually realised in solutions to the Einsteinequations, and if so to what extent, is a delicate question which will be analysedin later chapters of this book.

7.4 Further reading

183

7.4 Further reading
The literature on asymptotic simplicity and other deﬁnitions of asymptotic
ﬂatness is dauntingly vast and is best accessed through reviews. There are a goodnumber of references covering various periods and aspects of the topic. Penrose(1964, 1967) gives an overview of the early ideas and results on asymptotic
simplicity; Geroch (1976) provides a good discussion on the physical motivationof the study of isolated systems in general relativity, the notion of conserved
quantities and asymptotic symmetries; Schmidt (1978), Newman and Tod (1980)
and Ashtekar (1980, 1984) provide alternative discussions of these topics andFriedrich (1992, 1998a, 1999) provides reviews of the notion of asymptotic
simplicity from the point of view of the conformal Einstein ﬁeld equations andthe construction of global solutions. More recent reviews of the topic can befound in Frauendiener (2004), Ashtekar (2014) and Friedrich (2015a).

8
The conformal Einstein ﬁeld equations

To use conformal rescalings to analyse the global existence of asymptoticallysimple spacetimes one requires a suitable conformal representation of theEinstein ﬁeld equations. The naive direct approach to this problem is to make useof the transformation law of the Ricci tensor. However, this leads to equations
which are singular at the conformal boundary, so that the standard theoryof partial diﬀerential equations (PDEs) cannot be applied. Remarkably, by
introducing new variables, it is possible to obtain a system of equations forvarious conformal ﬁelds which is regular even at the conformal boundary andwhose solutions imply, in turn, solutions to the Einstein ﬁeld equations – thisconstruction was ﬁrst done in Friedrich (1981b). These equations are known as
the conformal Einstein ﬁeld equations.
This chapter provides derivations of two versions of the conformal ﬁeld
equations introduced by Friedrich: the so-called standard conformal Einstein ﬁeld
equations written in terms of the Levi-Civita connection of a conformally rescaled(unphysical) spacetime, and the extended conformal ﬁeld equations which are
given in terms of a Weyl connection. These two versions of the conformal
equations can be expressed in tensorial, frame or spinorial form. The presentationin this chapter allows for the presence of general classes of matter models. It also
provides a discussion of some basic properties of the equations, in particular,
their conformal covariance and their relation to the Einstein ﬁeld equations.
8.1 A singular equation for the conformal metric
Assume one has two spacetimes (M̃, g̃) and (M, g) which are related to eachother by means of a conformal transformation given by
gab = Ξ2 g̃ab .

(8.1)

Following the conventions of Section 7.1, (M̃, g̃) is called the physical spacetime, while (M, g) is known as the unphysical spacetime. For simplicity, thediscussion in this section is restricted to the case R̃ab = 0.

8.2 The metric regular conformal ﬁeld equations

185

From the discussion in Chapter 5, the conformal rescaling (8.1) implies thetransformation law


(8.2)
Rab = R̃ab − 2Ξ−1 ∇a ∇b Ξ − gab g cd Ξ−1 ∇c ∇d Ξ − 3Ξ−2 ∇c Ξ∇d Ξ
for the Ricci tensor. Combining this expression with the vacuum Einstein ﬁeld
equations one obtains the following conformal vacuum Einstein ﬁeld equation:
1Rab − Rgab = −2Ξ−1 (∇a ∇b Ξ − ∇c ∇c Ξgab ) − 3Ξ−2 ∇c Ξ∇c Ξgab .
2

(8.3)

The latter equation can be interpreted as an Einstein ﬁeld equation for theunphysical metric g with an unphysical matter with energy-momentum tensor
Tab given by
Tab ≡ −2Ξ−1 (∇a ∇b Ξ − ∇c ∇c Ξgab ) − 3Ξ−2 ∇c Ξ∇c Ξgab .
Equation (8.3) contains factors of Ξ−1 which become singular at Ξ = 0. Following
the discussion of Chapter 7, such points correspond to the conformal boundary of
the spacetime – a region of the unphysical spacetime (M, g) for which one would
like to be able to make analytic statements. This is not possible for Equation (8.3)
as the standard theory of PDEs assumes equations which are formally regular. It
is important to observe that multiplying Equation (8.3) by Ξ2 does not improve
the state of aﬀairs as one has then an equation whose principal part (i.e. theterms containing the higher order derivatives) vanishes at Ξ = 0.
8.2 The metric regular conformal ﬁeld equations
In what follows, it will be shown that by introducing new variables andreinterpreting old ones, it is possible to obtain a set of equations which is regulareven at the conformal boundary. Under suitable conditions, a solution of thissystem implies a solution to the physical Einstein ﬁeld equations.
The analysis of this section assumes a general matter content of the spacetime
so that1(8.4)
R̃ab − R̃g̃ab + λg̃ab = T̃ab2
and˜ a T̃ab = 0.
∇
From the above it follows directly thatR̃ = 4λ − T̃ ,L̃ab ≡

1
1
1
1
R̃ab + R̃g̃ab = T̃ab + (λ − T̃ )g̃ab ,
2
12
2
6

where T̃ ≡ g̃ ab T̃ab and L̃ab denotes the physical Schouten tensor.

(8.5a)(8.5b)

186

The conformal Einstein ﬁeld equations
8.2.1 The regularisation of the transformation law
for the Schouten tensor

The starting point of the construction is the singular transformation law for
the Ricci tensor given by Equation (8.2). In practice, it is more convenient towork with the Schouten tensor than with the Ricci tensor. The analogue ofEquation (8.2) for the Schouten tensor is given by1
Lab = L̃ab − Ξ−1 ∇a ∇b Ξ + Ξ−2 ∇c Ξ∇c Ξ gab .
2

(8.6)

Formally, the most singular term in this equation is 12 Ξ−2 ∇c Ξ∇c Ξ. From thetransformation law
R = Ξ−2 R̃ − 6Ξ−1 ∇c ∇c Ξ + 12Ξ−2 ∇c Ξ∇c Ξ,
it follows thatΞ−2 ∇c Ξ∇c Ξ = 11 
R − Ξ−2 R̃ + Ξ−1 ∇c ∇c Ξ.
12
2

(8.7)

(8.8)

1 −2
Ξ R̃.
The right-hand side of the last expression contains the singular term − 12
Yet substituting Equation (8.8) into (8.6), some cancellations occur. Making useof Equation (8.5b) one obtains

1
1
1 
1
R − Ξ−2 R̃ gab + Ξ−1 ∇c ∇c Ξgab .
Lab = T̃ab + (λ − T̃ )g̃ab − Ξ−1 ∇a ∇b Ξ +2
6
24
4
Now, deﬁning the Friedrich scalar

s≡

1 c
1
∇ ∇c Ξ + RΞ,
4
24

(8.9)

and writing Ξ−2 R̃gab = R̃g̃ab , one obtains


1
1
1
1
Lab = T̃ab +λ − T̃ − R̃ g̃ab − Ξ−1 ∇a ∇b Ξ + Ξ−1 sgab ,
2
6
6
24
1
1
(8.10)= T̃ab − T̃ g̃ab − Ξ−1 ∇a ∇b Ξ + Ξ−1 sgab ,28where in the last expression, Equation (8.5a) has been used. The last expression
brings about the question of the transformation law for the energy-momentum
tensor T̃ab upon the conformal rescaling g = Ξ2 g̃. As T̃ab is not a geometric object
derived from the metric g̃ and concomitants thereof, one is free to choose thetransformation law which best suits the analysis. As will be further elaborated in
Chapter 9, a convenient choice is to deﬁne the unphysical energy-momentum
tensor Tab as
Tab ≡ Ξ−2 T̃ab .
It follows then that



1
1
1
1
1
T̃ab − T̃ g̃ab = Ξ2
Tab − T gab = Ξ2 T{ab} ,
2
8
2
8
2

(8.11)

8.2 The metric regular conformal ﬁeld equations

187

where T ≡ g ab Tab so that T̃ = Ξ4 T and T{ab} denotes the g-trace-free part of
Tab . Substituting Equation (8.11) into Equation (8.10) one obtains
Lab =

1 2
Ξ T{ab} − Ξ−1 ∇a ∇b Ξ + Ξ−1 sgab .
2(8.12)

This last equation still contains formally singular terms. To get around thisproblem, one reads it not as determining the components of the conformal metricg contained in Lab , but as conditions on the second covariant derivative of theconformal factor Ξ. Adopting this point of view, and multiplying Equation (8.12)
by Ξ one obtains
1∇a ∇b Ξ = −ΞLab + sgab + Ξ3 T{ab} .
2(8.13)

Equation (8.13) promotes the ﬁelds s and Lab to the level of unknowns for which
suitable equations need to be constructed. This will be done in the following
sections.
8.2.2 The equation for s
In order to construct an equation for s, one applies ∇c to Equation (8.13) andobtains∇c ∇a ∇b Ξ = −∇c ΞLab − Ξ∇c Lab + ∇c sgab
3
1
+ Ξ2 ∇c ΞT{ab} + Ξ3 ∇c T{ab} .
22(8.14)

By commuting covariant derivatives, the right-hand side of this equation can be
rewritten as∇c ∇a ∇b Ξ = ∇a ∇c ∇b Ξ − Rd bca ∇d Ξ.
Hence, contracting the indices b and c one ﬁnds that Equation (8.14) implies∇a (∇c ∇c Ξ) + Rca ∇c Ξ = −Lca ∇c Ξ − Ξ∇c Lac + ∇a s1
3
+ Ξ2 ∇c ΞT{ac} + Ξ3 ∇c T{ac} .
22(8.15)

Now, the deﬁnition of the ﬁeld s, Equation (8.9), implies that11∇a (∇c ∇c Ξ) = 4∇a s − Ξ∇a R − R∇a Ξ.
6
6
Using this expression in (8.15) and observing that1
Rab = 2Lab + Rgab ,6(8.16)

one obtains
13
1
3∇a s − Ξ∇a R = −3Lac ∇c Ξ − Ξ∇c Lac + Ξ2 ∇c ΞT{ac} + Ξ3 ∇c T{ac} .
6
2
2

188

The conformal Einstein ﬁeld equations

Now, let Gab ≡ Rab − 12 Rgab denote the Einstein tensor of the metric g. One hasthat ∇a Gab = 0. This last equation can be rewritten in terms of the Schouten
tensor as
1(8.17)∇c Lca − ∇a R = 0.
6
Making use of this last expression one obtains
1
1∇a s = −Lac ∇c Ξ + Ξ2 ∇c ΞT{ac} + Ξ3 ∇c T{ca} .
26(8.18)

This is a suitable equation for s.
8.2.3 The equations for the curvature
Equation (8.13) brings the unphysical Schouten tensor Lab into play. Thus, one
needs to obtain an equation which can be regarded as a diﬀerential conditionon Lab . The natural place to look for such an equation is the second Bianchi
identity; see Section 2.4.3. In Section 5.2.2, it has been shown that the second
Bianchi identity together with the decomposition of the Riemann tensor in termsof the Weyl and Schouten tensors lead to the expressions
˜ d L̃cb = ∇˜ a C a bcd ,˜ c L̃db − ∇∇∇c Ldb − ∇d Lcb = ∇a C a bcd ;compare Equations (5.11) and (5.13). As it stands, the second of the above
equations is not a satisfactory diﬀerential condition for Lab as it contains, in
its right-hand side, the divergence of the Weyl tensor. One needs to ﬁnd an
expression for the latter in terms of undiﬀerentiated ﬁelds. Observe that theright-hand side of this equation can be expressed in terms of the physical energymomentum tensor T̃ab using formula (8.5b). This will not be done at this point.
Instead, it is more convenient to expresses it in terms of the physical Cotton
˜ c L̃db − ∇˜ d L̃cb , so that
tensor Ỹcdb ≡ ∇˜ a C a bcd = Ỹcdb .
∇(8.19)

˜ a C a bcd in terms of an expression
Now, one would like to express the divergence ∇involving the covariant derivative ∇. For this, one makes use of the identity
˜ a C a bcd ;∇a (Ξ−1 C a bcd ) = Ξ−1 ∇see Equation (5.8). Making use of the latter in Equation (8.19) one obtains∇a (Ξ−1 C a bcd ) = Ξ−1 Ỹcdb .
This equation seems to lead to a dead end because of the Ξ−1 terms appearing on
both sides, and which do not cancel out. However, deﬁning the rescaled Weyltensorda bcd ≡ Ξ−1 C a bcd ,(8.20)

8.2 The metric regular conformal ﬁeld equations

189

and the rescaled Cotton tensor
Tcdb ≡ Ξ−1 Ỹcdb ,(8.21)

one obtains the formally regular equation∇a da bcd = Tcdb .

(8.22)

This last equation suggests that the Weyl tensor C a bcd be replaced by the rescaled
Weyl tensor da bcd in the construction of a regular set of conformal ﬁeld equations.
In Chapter 10 it will be seen that the deﬁnitions of da bcd and Tcdb are justiﬁed in
the sense that under suitable assumptions the tensors da bcd and Tcdb are regularat the points where Ξ = 0; see, in particular, Theorem 10.3.
One is now in the position of returning to the analysis of the equation for the
Schouten tensor. Writing C a bcd in terms of da bcd one obtains∇c Ldb − ∇d Lcb = ∇a (Ξda bcd )= ∇a Ξda bcd + Ξ∇a da bcd .
Finally, using Equation (8.22) in the last term yields∇c Ldb − ∇d Lcb = ∇a Ξda bcd + ΞTcdb ,(8.23)

which, again, is formally regular if Ξ = 0.
8.2.4 The regularised transformation rule for the Ricci scalar
To relate solutions of the conformal ﬁeld equations to solutions of the Einsteinﬁeld equations, one also needs to consider a regularised version of the transformation rule for the Ricci scalar, Equation (8.7). Multiplying this transformation
law by Ξ2 and rearranging the various terms one obtains
R̃ = Ξ2 R + 6Ξ∇c ∇c Ξ − 12∇c Ξ∇c Ξ.
Finally, using Equations (8.5a) and (8.9) one concludes that1
λ = 6Ξs − 3∇c Ξ∇c Ξ + Ξ4 T.
4(8.24)To understand the role of this equation it is useful to compute the derivativeof its right-hand side. One has that1∇a 6Ξs − 3∇c Ξ∇c Ξ + Ξ4 T
41
= 6∇a Ξs + 6Ξ∇a s − 6∇a ∇c Ξ∇c Ξ + Ξ3 ∇a ΞT + Ξ4 ∇a T4
= Ξ4 (∇c Tca + Ξ−1 ∇a ΞT ),where in the second equality Equations (8.13) and (8.18) have been used toremove, respectively, the terms ∇a ∇c Ξ and ∇a Ξ.

190

The conformal Einstein ﬁeld equations

As will be further discussed in Chapter 9, the tensors T̃ab and Tab satisfy the
relation˜ a ΞT̃bc ).
˜ b T̃ca − Ξ−1 ∇g bc ∇b Tca = Ξ−4 g̃ bc (∇˜ b T̃ba = 0 it follows that
Hence, if ∇∇c Tca + Ξ−1 ∇a ΞT = 0.

(8.25)

This last relation implies that


1 4
c
∇a 6Ξs − 3∇c Ξ∇ Ξ + Ξ T = 0.
4
One has the following result:Lemma 8.1 (propagation of the cosmological constant) If Equations(8.13), (8.18) and (8.25) are satisﬁed on M and, in addition, Equation (8.24)
holds at a point p ∈ M, then Equation (8.24) is also satisﬁed on M.
Thus, Equation (8.24) plays the role of a constraint which is preserved, uponevolution, by virtue of the other conformal ﬁeld equations.
8.2.5 Properties of the metric conformal ﬁeld equations
The discussion of the previous sections is summarised in the following list ofequations:1∇a ∇b Ξ = −ΞLab + sgab + Ξ3 T{ab} ,211∇a s = −Lac ∇c Ξ + Ξ2 ∇c ΞT{ac} + Ξ3 ∇c T{ca} ,26∇c Ldb − ∇d Lcb = ∇a Ξda bcd + ΞTcdb ,(8.26a)(8.26b)(8.26c)∇a d bcd = Tcdb ,(8.26d)16Ξs − 3∇c Ξ∇c Ξ + Ξ4 T = λ.
4(8.26e)a

These are known as the (regular) metric conformal Einstein ﬁeld equations. Equations (8.26a)–(8.26e) should be read as diﬀerential conditions for theﬁelds Ξ, s, Lab , da bcd . As already mentioned, Equation (8.26e) plays the role ofa constraint. At the points where Ξ = 0, these equations are complemented by
˜ a T̃ab = 0 expressed in terms of conformal
the physical conservation equation ∇quantities:∇c Tca + Ξ−1 ∇a ΞT = 0.

(8.27)

Observe that in contrast to Equations (8.26a)–(8.26e), Equation (8.27) is notformally regular at Ξ = 0. This equation will be analysed in more detail in
Chapter 9.

8.2 The metric regular conformal ﬁeld equations

191

In what follows, for a solution to the metric conformal Einstein ﬁeld
equations it will be understood that a collection of ﬁelds(gab , Ξ, s, Lab , da bcd , Tab )satisﬁes Equations (8.26a)–(8.26e) and (8.27).
Remark. The discussion so far has not considered an equation for thecomponents of the metric gab . To obtain the required condition assume that
the Schouten tensor Lab is determined through Equation (8.26c) and consider
the relation (8.16) expressed in terms of some local coordinates (xμ ):1
Rμν = 2Lμν + Rgμν .
6
Recalling that the components Rμν can be expressed in terms of second-orderderivatives of the components of the metric, one can read the previous expression
as a diﬀerential condition for gμν . To cast this equation in the form of somerecognisable type of PDE one needs to make a particular choice of coordinates;
see the discussions in Section 13.5.1 and in the Appendix of Chapter 13 on the
reduced Einstein ﬁeld equations.
The conformal vacuum Einstein ﬁeld equations
An important particular case of Equations (8.26a)–(8.26e) occurs when T̃ab = 0
on the whole of M̃. Then one also has that Tabc = 0, and the conformal ﬁeld
equations reduce to:∇a ∇b Ξ = −ΞLab + sgab ,(8.28a)∇a s = −Lac ∇ Ξ,

(8.28b)c∇c Ldb − ∇d Lcb = ∇a Ξd bcd ,(8.28c)∇a d bcd = 0,(8.28d)aa6Ξs − 3∇c Ξ∇ Ξ = λ.
c(8.28e)Equations (8.28a)–(8.28e) are known as the conformal vacuum Einstein ﬁeld
equations.
The metric conformal ﬁeld equations and the Einstein ﬁeld equations
Any solution to the Einstein ﬁeld equations satisﬁes Equations (8.26a)–(8.26e)
for any (smooth) choice of conformal factor Ξ. The converse of this observation
is given in the following result.
Proposition 8.1 (solutions of the conformal Einstein ﬁeld equations as
solutions to the Einstein ﬁeld equations) Let(gab , Ξ, s, Lab , da bcd , Tab )192

The conformal Einstein ﬁeld equations

denote a solution to Equations (8.26a)–(8.26d) and (8.27) such that Ξ = 0 on an
open set U ⊂ M. If, in addition, Equation (8.26e) is satisﬁed at a point p ∈ U ,then the metric g̃ab = Ξ−2 gab is a solution to the Einstein ﬁeld Equations (8.4)
on U .
Proof It will be ﬁrst shown that the Schouten tensor L̃ab of the metric g̃ab =Ξ−2 gab satisﬁes Equation (8.5b). Notice that the metric g̃ab is well deﬁned on
U as Ξ = 0. The transformation law for the Schouten tensor under conformal
rescalings gives
1L̃ab = Lab + Ξ−1 ∇a ∇b Ξ − Ξ−2 ∇c Ξ∇c Ξgab .
2
Using Equations (8.26a) and (8.26b) the latter simpliﬁes toL̃ab =

1
1
T̃ab + (λ − T̃ )g̃ab ,2
6

as required. In order to conclude that the Einstein ﬁeld equations hold, one also
needs to compute the Ricci scalar of the metric g̃ab . As a consequence of Lemma
8.1 one has that Equation (8.26e) holds on the whole of U . From the latter, again
using (8.26a) and (8.26b) and recalling that T̃ = Ξ4 T , it follows that R̃ = 4λ− T̃ .
Combining the obtained expressions for L̃ab and R̃ one readily concludes that
(8.4) is indeed satisﬁed.
Conformal freedom and conformal gauge
Consider a solution (gab , Ξ, s, Lab , da bcd , Tab ) to the metric conformal ﬁeld
Equations (8.26a)–(8.26e) and (8.27). As a consequence of Proposition 8.1, one
has that g̃ = Ξ−2 g and T̃ab = Ξ2 Tab , give rise to a solution of the Einsteinﬁeld equations as long as Ξ = 0. Consider now another conformal factor Ξ́.
From Ξ́, together with the physical ﬁelds g̃ab and T̃ab , one can construct, bydirect computation using the deﬁnitions of Sections 8.2.1–8.2.3, a collection ofconformal ﬁelds (ǵab , Ξ́, ś, Ĺab , d´a bcd , T́ab ). In particular, one has that ǵab = Ξ́2 g̃aband T́ab = Ξ−2 Tab . These ﬁelds constitute, in turn, a solution to the metricconformal ﬁeld equations. That is, they satisfy´ a∇´ b Ξ́ = −Ξ́Ĺab + śǵab + 1 Ξ́3 T́{ab} ,
∇
2
1
c
2
c
´ Ξ́ + Ξ́ ∇´ Ξ́T́{ac} + 1 Ξ́3 ∇´ c T́{ca} ,´ a ś = −Ĺac ∇∇2
6´ c Ĺdb − ∇´ d Ĺcb = ∇´ a Ξd´a bcd + Ξ́T́cdb ,∇´ a d´a bcd = T́cdb ,∇´ c Ξ́ + 1 Ξ́4 T́ = λ,´ c Ξ́∇6Ξ́ś − 3∇4
´ c T́ca + Ξ́−1 ∇´ a Ξ́T́ = 0.
∇

8.2 The metric regular conformal ﬁeld equations

193

The unphysical metrics g and ǵ are conformally related to each other: one hasthat ǵ = κ2 g with κ ≡ Ξ́Ξ−1 , Ξ = 0. Using the transformation formulae ofChapter 5, one can express the solution (ǵab , Ξ́, ś, Ĺab , d´a bcd , T́ab ) in terms of(gab , Ξ, s, Lab , da bcd , Tab ) and κ. One has thatΞ́ = κΞ,ǵab = κ2 gab ,(8.29a)1ś = κ−1 s + κ−2 ∇c κ∇c Ξ + κ−3 Ξ∇c κ∇c κ,
2
1Ĺab = Lab − κ−1 ∇a ∇b κ + κ−2 ∇c κ∇c κ gab ,2d´a bcd = κda bcd ,(8.29d)T́ab = κ−2 Tab .

(8.29e)(8.29b)(8.29c)The two sets of solutions to the metric conformal ﬁeld equations are said to be
conformally related.
From the discussion of the previous paragraphs it follows that there exists aninﬁnite number of solutions to the metric conformal ﬁeld equations giving riseto the same solution of the Einstein ﬁeld equations. This is a manifestation of
the conformal invariance of the equations. This conformal invariance is tied
to a conformal freedom (or gauge) which, in turn, manifests itself in theproperties of the unphysical metric g. This conformal freedom has to be ﬁxed
in some way if one is to apply the theory of PDEs to the metric conformal ﬁeld
equations.
The issue of the conformal gauge discussed in the previous paragraph isclosely related to the Ricci scalar R of the unphysical metric g. The scalar
R does not explicitly appear in the conformal Equations (8.26a)–(8.26e) and(8.27). Hence, it is not determined by the equations. Of course, given a solution(gab , Ξ, s, Lab , da bcd , Tab ), one can readily compute R. In general, conformally
related solutions to the metric conformal ﬁeld equations will give rise to diﬀerent
Ricci scalars. In order to understand better the connection between the conformal
gauge and the Ricci scalar, consider a metric ǵ conformally related to g via ǵ =
κ2 g. The transformation law for the Ricci scalar under conformal transformations
implies that6∇a ∇a κ − Rκ = −Ŕκ3 ,(8.30)

from where Ŕ can be determined. Alternatively, if Ŕ is an arbitrary scalar on M,
then Equation (8.30) can be read as a linear wave equation for κ. Given suitable
initial data for this equation, it can be solved locally. The solution κ gives, inturn, the metric ǵ = κ2 g. From this point of view, the scalar ﬁeld Ŕ plays therole of a conformal gauge source function. In particular, one could chooseŔ = 0. As will be seen in later chapters of this book, this choice, despite its
simplicity, is not necessarily the best one.

194

The conformal Einstein ﬁeld equations

8.3 Frame and spinorial formulation of the conformal ﬁeld equations
8.3.1 The frame formulation
This section provides a discussion of a frame formulation of the conformal
Einstein ﬁeld equations. This version of the ﬁeld equations is more ﬂexible thanthe metric one.
General deﬁnitions, frame ﬁelds
In what follows, consider a set of frame ﬁelds {ea }, a = 0, . . . , 3 which isorthonormal with respect to the metric g. Frames of this type will be said to be
g-orthonormal. One has thatg(ea , eb ) = ηab = diag(1, −1, −1, −1).
Following the conventions of Chapter 2, let Γa c b = ω c , ∇a eb  denote theconnection coeﬃcients of the connection ∇. As a consequence of the metriccompatibility of ∇ one has thatΓa d b ηdc + Γa d c ηbd = 0.
The components, Σa c b of the torsion of ∇ are given by the relation
Σa c b ec = [ea , eb ] − (Γa c b − Γb c a )ec .
In the case of ∇ one naturally has that Σa c b = 0.
The geometric and the algebraic curvature
The discussion of the conformal ﬁeld equations in terms of a frame formalism
requires the expression of the components Rc dab of the Riemann tensor Rc dab
with respect to the frame {ea }; see Equation (2.31). Let P c dab denote the righthand side of equation (2.31), namely,
P c dab ≡ ea (Γb c d ) − eb (Γa c d )+ Γf c d (Γb f a − Γa f b ) + Γb f d Γa c f − Γa f d Γb c f .
In what follows, P c dab will be known as the geometric curvature. To complete
the discussion one also needs to consider the decomposition of the Riemanntensor in terms of the Weyl tensor C c dab and the Schouten tensor Lab ; see
Equation (2.21b). The frame version of the decomposition is given by
Rc dab = C c dab + 2Sd[a ce Lb]e ,where, consistent with the general conventions of Chapter 2, C c dab and Lab
denote, respectively, the components of the tensors C c dab and Lab with respectto {ea }. The components ρc dab of the algebraic curvature are given byρc dab ≡ Ξdc dab + 2Sd[a ce Lb]e .

8.3 Frame and spinorial formulation of the conformal ﬁeld equations 195
In the above deﬁnition it has been used that C c dab = Ξdc dab . The geometricand algebraic curvature serve as useful shorthands of expressions which will be
repeatedly used. Observe, in particular, that the equation P c dab = ρc dab encodes
the idea that the ﬁelds C c dab and Lab correspond to the components of the Weyland Schouten tensor of the connection deﬁned by Γa b c .
The frame zero quantities and the frame conformal ﬁeld equations
The frame version of the conformal ﬁeld Equations (8.26a)–(8.26e) and (8.27)
are readily obtained by contraction with the frame {ea } and the coframe {ω a }.
One obtains
1∇a ∇b Ξ = −ΞLab + sηab + Ξ3 T{ab} ,211∇a s = −Lac ∇c Ξ + Ξ2 ∇c ΞT{ac} + Ξ3 ∇c T{ca} ,26∇c Ldb − ∇d Lcb = ∇a Ξda bcd + ΞTcdb ,∇a da bcd = Tcdb ,16Ξs − 3∇c Ξ∇c Ξ + Ξ4 T = λ,4
and∇c Tca + Ξ−1 ∇a ΞT = 0,
where the directional derivative ∇a acts on components of tensorial ﬁelds
according to the rules in (2.28). The above frame conformal ﬁeld equations will
be complemented by the structure equations
Σa c b ec = 0,P c dab = ρc dab ,which express that for the connection ∇, its torsion must vanish and its geometricand algebraic curvature must coincide.
For convenience of the subsequent discussion one introduces a set of zero
quantities:Σab ≡ Σa c b ec ,(8.31a)Ξ dab ≡ P dab − ρ dab ,
c

c

c

1
Zab ≡ ∇a ∇b Ξ + ΞLab − sηab − Ξ3 T{ab} ,
2
1 2 c
1
c
Za ≡ ∇a s + Lac ∇ Ξ − Ξ ∇ ΞT{ac} − Ξ3 ∇c T{ca} ,2
6Δcdb ≡ ∇c Ldb − ∇d Lcb − ∇a Ξda bcd − ΞTcdb ,Λbcd ≡ ∇a d bcd − Tcdb ,a1
Z ≡ 6Ξs − 3∇c Ξ∇ Ξ + Ξ4 T − λ,4Ma ≡ ∇c Tca + Ξ−1 ∇a ΞT.
c(8.31b)(8.31c)(8.31d)(8.31e)(8.31f)(8.31g)(8.31h)196

The conformal Einstein ﬁeld equations

In terms of the above zero quantities, the frame version of the conformal
ﬁeld equations can be compactly written asΣab = 0,Ξc dab = 0,

Zab = 0,Za = 0,(8.32a)Δcdb = 0,Λbcd = 0,

Z = 0,

Ma = 0.

(8.32b)Accordingly, a solution to the frame conformal Einstein ﬁeld equations is a collection (ea , Γa b c , Ξ, s, Lab , da bcd , Tab ) satisfying Equations (8.32a)and (8.32b). The equations associated to the zero quantities Σab and Ξc dab
provide diﬀerential conditions for the components of the frame vectors {ea } andfor the connection coeﬃcients Γa b c . The role of the equations associated to thezero quantities Zab , Za , Δcdb , Λbcd , Z and Ma is similar to that of their metriccounterparts in Section 8.2.
By considering a frame version of the conformal ﬁeld equations, one introduces
a further gauge freedom into the system. This gauge freedom corresponds to theLorentz transformations preserving the g-orthonormality of the frame vectors{ea }. In this case one speaks of a frame gauge freedom. As in the case of theconformal freedom discussed in Section 8.2.5, this freedom needs to be ﬁxed inorder to be able to apply the methods of the theory of PDEs. These issues will
be discussed further in Chapter 13.
The frame conformal ﬁeld equations and the Einstein ﬁeld equations
As in the case of the metric conformal ﬁeld equations, a solution to the frame
conformal ﬁeld equations implies, under suitable conditions, a solution to the
Einstein ﬁeld equations; see Proposition 8.1. More precisely, one has:
Proposition 8.2 (solutions to the frame conformal ﬁeld equations as
solutions to the Einstein ﬁeld equations) Let(ea , Γa b c , Ξ, s, Lab , da bcd , Tab )denote a solution to the frame conformal ﬁeld Equations (8.32a) and (8.32b)with Γa c b satisfying the metric compatibility condition
Γa d b ηdc + Γa d c ηbd = 0,
and such thatΞ = 0,det(η ab ea ⊗ eb ) = 0,

on an open set U ⊂ M. Then the metric g̃ = Ξ−2 ηab ω a ⊗ ω b , where {ω a } isthe dual frame to {ea }, is a solution to the Einstein ﬁeld Equations (8.4) on U .
Proof As a consequence of the metric compatibility assumption and Σab = 0,
the coeﬃcients Γa c b can be interpreted as the connection coeﬃcients of aLevi-Civita connection with respect to the frame {ea }. By the uniqueness of8.3 Frame and spinorial formulation of the conformal ﬁeld equations 197
the Levi-Civita connection, g = ηab ω a ⊗ ω b is the metric associated to thisconnection. Notice that by assumption g is well deﬁned on U . Furthermore,because of Ξc dab = 0 and exploiting the uniqueness of the decomposition of the
Riemann tensor in terms of the Weyl and the Schouten tensors, it follows that Lab
are the components, with respect to the frame {ea }, of the Schouten tensor of themetric g. From here, following arguments analogous to those used in the proof of
Proposition 8.1 one concludes that g̃ = Ξ−2 ηab ω a ⊗ ω b and T̃ab = Ξ2 ω a a ω b b Tabgive a solution to the Einstein ﬁeld Equations (8.4) on U .
8.3.2 Spinorial formulation of the conformal ﬁeld equations
The frame conformal ﬁeld equations lead, in a natural way, to a spinorial
formulation. This formulation of the equations reveals in a more clear fashion
the inherent algebraic structure of the equations and provides a systematic
procedure for the construction of evolution equations. The formulation discussed
in this section is not an abstract spinor formulation, but rather a frame spinor
formulation.
General remarks concerning the spinorial formulation
Following the discussion in Section 3.1.13, the g-orthonormal frame {ea } gives
rise to a frame {eAA } such that g(eAA , eBB  ) = AB A B  ; that is, {eAA } isa null tetrad. In what follows, letΣAA CC BB  ,P CC DD AA BB  ,LAA BB  ,

d

AA



ρCC DD AA BB  ,BB  CC  DD  ,TAA BB  ,TAA BB  CC  ,denote, respectively, the spinorial counterparts of the ﬁelds
P c dab ,Σa c b ,ρc dab ,Tab ,da bcd ,Lab ,Tabc .

The spinorial counterpart of the geometric curvature, P CC DD AA BB  , is
expressed in terms of the spinorial connection coeﬃcients ΓAA BB CC  . These,in turn, can be expressed in terms of the reduced spin connection coeﬃcientsΓAA B C ; see formula (3.33). As the connection ∇ is metric, it follows that
ΓAA BC = ΓAA (BC) ; compare Section 3.2.2. By analogy to the split of the
spinorial counterpart of the curvature tensor – Equation (3.35) – one can split
the geometric curvature as

P CC DD AA BB  = P C DAA BB  δD C + P̄ C D AA BB  δD C .
In what follows, the discussion will make use only of the reduced spinorial
geometric curvature
P C DAA BB  ≡ eAA (ΓBB  C D ) − eBB  (ΓAA C D )− ΓF B  C D ΓAA F B − ΓBF  C D Γ̄AA F B  + ΓF A C D ΓBB  F A
+ ΓAF  C D Γ̄BB  F A + ΓAA C F ΓBB  F D − ΓBB  C F ΓAA F D .

198The conformal Einstein ﬁeld equations

The spinorial algebraic curvature has a similar split. Its information is encodedin the ﬁeld
ρC DAA BB  ≡ −ΨC DAB A B  + LDB  AA δB C − LDA BB  δA C ,where it is recalled that ΨABCD is the Weyl spinor; see Equation (3.43). One
then introduces the totally symmetric rescaled Weyl spinor φABCD as
φABCD ≡ Ξ−1 ΨABCD .
Consistent with Equation (3.43), φABCD is related to the spinorial counterpartof da bcd via
dAA BBCC  DD = −φABCD A B  C  D − φ̄A B  C  D AB CD .

(8.33)

Hence, the reduced spinorial algebraic curvature can be written as
ρC DAA BB  ≡ −ΞφC DAB A B  + LDB  AA δB C − LDA BB  δA C .
The spinorial counterpart of T{ab} , the symmetric trace-free part of Tab ,is given by T(AB)(A B  ) ; compare Equation (3.12). Finally, exploiting theantisymmetry Tcdb = −Tdcb of the rescaled Cotton tensor, one has the split
TCC  DD BB  = TCDBB  C  D + T̄C  D BB  CD ,(8.34)



where TCDBB  ≡ 12 TCQ D Q BB  . Observe that TCDBB  = T(CD)BB  .
The spinorial zero quantities
The spinorial counterparts of the frame conformal Einstein ﬁeld equations are
obtained by suitable contraction with the Infeld-van der Waerden symbols.
Simpler expressions are obtained if one takes into account the remarks madein the previous subsection. It is convenient to introduce the following spinorial
zero quantities:ΣAA BB  ≡ [eAA , eBB  ] − (ΓAA CC BB  − ΓBB  CC AA )eCC  , (8.35a)
ΞC DAA BB  ≡ P C DAA BB  − ρC DAA BB  ,ZAA BB  ≡ ∇AA ∇BB  Ξ + ΞLAA BB  − sAB A B 1− Ξ3 T(AB)(A B  ) ,2
ZAA ≡ ∇AA s + LAA CC  ∇CC Ξ1
1− Ξ2 ∇CC ΞT(AC)(A C  ) − Ξ3 ∇CC T(AC)(A C  ) ,2
6
ΔCC  DD BB  ≡ ∇CC  LDD BB  − ∇DD LCC  BB 
− ∇AA ΞdAABB  CC  DD  − ΞTCC  DD  BB  ,(8.35b)(8.35c)(8.35d)(8.35e)(8.35f)

8.3 Frame and spinorial formulation of the conformal ﬁeld equations 199ΛBB  CC  DD ≡ ∇AA dAA BB  CC  DD − TCC  DD BB  ,
1
Z ≡ 6Ξs − 3∇CC  ∇CC + Ξ4 T − λ,4MAA ≡ ∇CC TCC  AA + Ξ−1 ∇AA ΞT.

(8.35g)(8.35h)(8.35i)Hence, the spinorial conformal Einstein ﬁeld equations are given in terms of theabove zero quantities as
ΣAA BB  = 0,

ΞC DAA BB  = 0,

ΔCC  DD BB  = 0,

ZAA BB  = 0,

ΛBB  CC  DD = 0,

Z = 0,ZAA = 0,(8.36a)MAA = 0.

(8.36b)A reduced set of zero quantities can be obtained by explicitly making use of theantisymmetry of several of the spinorial zero quantities. In particular, it is noticed
that as ΔCC  DD BB  = −ΔDD CC  BB  and ΛBB  CC  DD = −ΛBB  DD CC 
one can write
ΔCC  DD BB  = ΔCDBB  C  D + Δ̄C  D BB  CD ,ΛBB  CC  DD = ΛBB  CD C  D + Λ̄B  BC  D CD ,where
ΔCDBB  ≡

1
ΔCQ D Q BB  ,2ΛBB  CD ≡

1
ΛBB  CQ D Q .
2

A direct computation using the splits (8.33) and (8.34) yields


ΔCDBB  = ∇(C Q LD)Q BB  + ∇Q B  ΞφCDBQ + ΞTCDBB  , (8.37a)
ΛBB  CD = ∇Q B  φBCDQ + TCDBB  .

(8.37b)Thus, an equivalent spinorial formulation of the conformal ﬁeld equations isgiven byΣAA BB  = 0,

ΞC DAA BB  = 0,

ΔCDBB  = 0,

ΛBB  CD = 0,

ZAA BB  = 0,
Z = 0,ZAA = 0

MAA = 0.

(8.38a)(8.38b)

The antisymmetry of the zero quantities ΣAA BB  and ΞC DAA BB  can also
be exploited to obtain reduced zero quantities ΣAB = Σ(AB) and ΞC DAB =ΞC D(AB) . This strategy will not be pursued further.
The spinorial conformal ﬁeld equations and the Einstein ﬁeld equations
As a consequence of the equivalence between spinorial and frame expressions
discussed in Section 3.1.9, it follows that each of the two spinorial formulationsof the conformal ﬁeld Equations (8.36a) and (8.36b) or (8.38a) and (8.38b) is
equivalent to the frame conformal ﬁeld Equations (8.32a) and (8.32b). Thus, ananalogue of Proposition 8.2 holds for the spinorial conformal ﬁeld equations withthe metricg̃ = Ξ−2 AB A B  ω AA ⊗ ω BB ,200

The conformal Einstein ﬁeld equations

yielding the required solution to the Einstein ﬁeld equations. In this last
expression {ω AA } denotes the duals of the frame {eAA }.
8.3.3 Conformal freedom in the frame and spinorialconformal ﬁeld equations
The transformation laws for the various conformal ﬁelds under a conformal gaugechange follow from the tensorial version given in (8.29a)–(8.29e). As before,
assume that one has two metrics g and ǵ such that g = κ2 ǵ. Consider now
a g-orthonormal frame {ea } with associated coframe {ω a }. Fromg(ea , eb ) = κ2 ǵ(ea , eb ) = ηab ,it follows that {éa } and {ώ a }, withéa ≡ κea ,ώ a ≡ κ−1 ω a ,are a ǵ-orthonormal frame and a ǵ-orthonormal coframe, respectively. As a consequence, the tensorial transformation formulae (8.29a)–(8.29e) may pick up factors of κ depending on whether they are contracted with ea or éa . For, exampled´a bcd ≡ ώ a a éb b éc c éd d d´a bcd= κ3 ω a a eb b ec c ed d da bcd = κ3 da bcd .
Similar considerations lead to1
Ĺab = κ2 Lab + κ2 ∇a (κ−1 ∇b κ) − Sab cd ∇c κ∇d κ,
2T́ab = κ2 Tab ,where
Sab cd ≡ ea a eb b ω c c ω d d Sab cd≡ δa c δb d + δa d δb c − ηab η cd .
The spinorial counterparts of the conformal ﬁelds obey similar transformaA A } denote the spin dyads associated, respectively, to thetions. If {A A } and {´frame vectors {ea } and {éa }, then´A A = κA A .
As a consequence one has, for example, thatΨ́ABCD = κ3 ΨABCD .
8.4 The extended conformal Einstein ﬁeld equations
The conformal Einstein ﬁeld equations discussed in the previous sections are
expressed in terms of the Levi-Civita connection of the unphysical metric g.
This section provides a more general version of the equations by rewriting them

8.4 The extended conformal Einstein ﬁeld equations

201

in terms of a Weyl connection. The resulting system of equations is known as theextended conformal Einstein ﬁeld equations. The use of Weyl connections
introduces a further freedom in the equations. This freedom can be exploited toincorporate conformally privileged gauges. The idea of reexpressing the vacuum
conformal ﬁeld equations in terms of a Weyl connection was ﬁrst introducedin Friedrich (1995). Further discussions can be found in Friedrich (1998c, 2002,
2004). The extension of these ideas to the matter case has been given in Lübbe
and Valiente Kroon (2012, 2013b).
In what follows, for ease of presentation, the discussion in this section isrestricted to the vacuum case.
Basic setting
As in the previous sections of this chapter, let g denote an unphysical Lorentzian
metric related to a physical metric g̃ via g = Ξ2 g̃. The metric g̃ is assumed to˜ denote, respectively,
satisfy the vacuum Einstein ﬁeld equations. Let ∇ and ∇the Levi-Civita connections of the metrics g and g̃.
ˆ deﬁned via
In what follows, consider a Weyl connection ∇ˆ − ∇ = S(f ),∇(8.39)

where f is a smooth covector. As
˜ = S(Ξ−1 dΞ),∇−∇ˆ −∇˜ = S(f + Ξ−1 dΞ). Hence, deﬁning
it follows that ∇β ≡ f + Ξ−1 dΞ,one has thatˆ −∇˜ = S(β).
∇
It is convenient to deﬁned ≡ Ξf + dΞ,(8.40)

so that d = Ξβ.
ˆ is torsion free, it follows that its Riemann curvature
As the Weyl connection ∇tensor R̂c dab can be decomposed in terms of its Schouten tensor L̂ab and the Weyltensor of the conformal class of C c dab ; see Equation (5.28a). Using the latter andrecalling the deﬁnition of the rescaled Weyl tensor dc dab , Equation (8.20), one
obtains the equationR̂c dab = 2Sd[a ce L̂b]e + Ξdc dab .
Consistent with the discussion in Section 5.3.2, Equations (5.29a)–(5.29c), the˜ ∇ and ∇ˆ are related to each other via
Schouten tensors of the connections ∇,202

The conformal Einstein ﬁeld equations

1
L̂ab − Lab = ∇a fb − Sab cd fc fd ,2
ˆ a βb + 1 Sab cd βc βd ,L̂ab − L̃ab = ∇2
1−1
Lab − L̃ab = ∇a (Ξ ∇b Ξ) + Ξ−2 Sab cd ∇c Ξ∇d Ξ.
2
ˆ = 0 one has
Taking into account the above expressions and recalling that ∇Sthatˆ c L̂db − ∇ˆ d L̂cb = (∇ˆ c Ldb − Scd ef fe Lf b − Scb ef fe Ldf )∇ˆ d Lcb − Sdc ef fe Lf b − Sdb ef fe Lcf )−(∇ˆd − ∇ˆ c )fb
ˆ d L̂cb + (∇
ˆ c∇
ˆ d∇
ˆ c L̂db − ∇=∇efefˆ c ff − Lcf ) − Scb fe (∇ˆ d ff − Ldf ).
+Sdb fe (∇A further computation using Equation (5.29a) and the deﬁnition of the tensor
S yields
ˆ c ff − Lcf ) − Scb ef fe (∇ˆ d ff − Ldf ) = (Scb ef L̂df − Sdb ef L̂cf )fe
Sdb ef fe (∇= 2Sb[c ef L̂d]f fe .
Hence, recalling the split (5.28a) of the Riemann tensor one obtains
ˆ c L̂db − ∇ˆ d L̂cb = ∇c Ldb − ∇d Lcb + fa C a bcd .
∇Thus, the Weyl connection version of the vacuum Cotton equation is given byˆ a L̂bc − ∇ˆ b L̂ac = de de cab .
∇Now, for the Bianchi Equation (8.22) one has that
ˆ a da bcd = ∇a da bcd − Sah f a ff dh bcd + Sab f h ff da hcd∇+Sac f h ff da bhd + Sad f h ff da bch= ∇a da bcd − fa da dcb + fa da cdb= ∇a da bcd − fa da bcd ,where in the last line it has been used that da bcd satisﬁes the ﬁrst Bianchi identity
da bcd + da cdb + da dbc = 0. Hence, Equation (8.22) expressed in terms of the Weylˆ takes the form:
connection ∇ˆ a da bcd = fa da bcd .
∇As a summary of this section one has the two equations:ˆ a L̂bc − ∇ˆ b L̂ac = dd dd cab ,∇ˆ d ddcab = fd dd cab .
∇(8.41a)(8.41b)

These two equations will be regarded as the core of the extended conformal
Einstein ﬁeld equations. They provide diﬀerential conditions on the Schouten
tensor of the Weyl connection and the rescaled Weyl tensor.

8.4 The extended conformal Einstein ﬁeld equations

203

8.4.1 The frame version of the extended conformal ﬁeld equations
Equations (8.41a) and (8.41b) need to be supplemented with equations which
provide information about the metric g associated to the conformal factor Ξ andwhich also allow to determine the covector f giving rise to the Weyl connection
ˆ The most convenient way of doing this is to make use of a frame formalism.
∇.
As in Section 8.3.1 let {ea }, a = 0, . . . , 3 denote a frame ﬁeld which is
g-orthonormal so that g(ea , eb ) = ηab . As ∇ is the Levi-Civita connection of g,its connection coeﬃcients, Γa c b = ω c , ∇a eb , satisfy the metric compatibility
condition of Equation (2.29).
ˆ denote the Weyl connection constructed from the Levi-Civita
Let now ∇ˆ a eb connection ∇ and the covector f using Equation (8.39). If Γ̂a c b = ω c , ∇ˆdenotes the connection coeﬃcients of ∇ with respect to the frame {ea }, one hasthatΓ̂a c b = Γa c b + Sab cd fd ,(8.42a)= Γa b + δa fb + δb fa − ηab η fd .
c

c

c

fa =

1 b
Γ̂a b ,
4cd

(8.42b)

In particular, one has that
(8.43)as Γa b b = 0 in the case of a metric connection.
ˆ Using the transformation
Let Σ̂a c b denote the torsion of the connection ∇.
formula for the torsion under change of connections, Equation (2.15), together
with Equation (8.42a), one obtains
Σ̂a c b − Σa c b = −2S[ab] cd fd = 0.
Thus,Σ̂a c b = 0,
as Σa c b = 0. As in Section 8.3.1 it is convenient to distinguish between thegeometric curvature P̂ c dab – that is, the expression for the components of theˆ in terms of the connection coeﬃcients Γ̂a c b
Riemann tensor of the connection ∇c– and the algebraic curvature ρ̂ dab – that is, the expression of the Riemanntensor in terms of the Schouten and Weyl tensors. These are given byP̂ c dab ≡ ea (Γ̂b c d ) − eb (Γ̂a c d )+Γ̂f c d (Γ̂b f a − Γ̂a f b ) + Γ̂b f d Γ̂a c f − Γ̂a f d Γ̂b c f ,ρ̂c dab ≡ Ξdc dab + 2Sd[a ce L̂b]e .
In analogy to the discussion of Section 8.3.1, it is convenient to introduce a setof geometric zero quantities associated to the various equations. In the present
case let:

204

The conformal Einstein ﬁeld equationsΣ̂ab ≡ [ea , eb ] − (Γ̂a c b − Γ̂b c a )ec ,(8.44a)Ξ̂c dab ≡ P̂ c dab − ρc dab ,ˆ c L̂db − ∇ˆ d L̂cb − da da bcd ,Δ̂cdb ≡ ∇(8.44b)ˆ a d bcd − fa d bcd .
Λ̂bcd ≡ ∇(8.44d)aa(8.44c)Now, taking into account Equation (8.43) one has thatΞ̂c cab = ea (Γ̂b c c ) − eb (Γ̂a c c ) + Γ̂d c c (Γ̂b d a − Γ̂a d b ) − 2Sc[a ce L̂b]e ,= 4 ea (fb ) − eb (fa ) − fd (Γ̂b d a − Γ̂a d b ) − L̂ba + L̂ab ,ˆ a fb − ∇ˆ b fa − L̂ba + L̂ab .
=4 ∇In view of the latter, it is convenient to deﬁneΞ̂ab ≡1 c
ˆ a fb − ∇ˆ b fa − L̂ab + L̂ba .
Ξ̂ cab = ∇4(8.45)In terms of the zero quantities discussed in the previous paragraphs, one deﬁnes
the extended conformal vacuum Einstein ﬁeld equations as the conditions
Ξ̂c dab = 0,Σ̂ab = 0,Δ̂cdb = 0,Λ̂bcd = 0.

(8.46)

These equations yield diﬀerential conditions, respectively, for the componentsof the frame {ea }, the spin coeﬃcients Γ̂a c b (including the components fa of
the covector f ), the components of the Schouten tensor L̂ab and the componentsof the rescaled Weyl tensor da bcd . In contrast to the standard conformal ﬁeld
Equations (8.32a) and (8.32b), there are no equations which can be regarded as
diﬀerential conditions on the conformal factor Ξ and the components da of thecovector d. As will be seen in Chapter 13, these objects will be ﬁxed throughgauge conditions.
In order to relate the extended conformal ﬁeld Equations (8.46) to the Einsteinﬁeld equations, one introduces further zero quantities:ˆ a Ξ,
δa ≡ da − Ξfa − ∇ˆ a βb − 1 Sab cd βc βd + 1 λΞ−2 ηab ,γab ≡ L̂ab − ∇26ˆςab ≡ L̂[ab] − ∇[a fb] .

(8.47a)(8.47b)(8.47c)

The associated equations
δa = 0,γab = 0,

ςab = 0,(8.48)

will be treated as constraints. The ﬁrst equation expresses the relation between
the covectors d, f and the conformal factor Ξ. The second equation encodes therelation between the components of the Schouten tensor of the Weyl connectionL̂ab and the physical Schouten tensor via the Einstein ﬁeld equations – thisconstraint is the analogue of the standard conformal equation Zab = 0. The role

8.4 The extended conformal Einstein ﬁeld equations

205of the equations in (8.48) is similar to that of the equation Z = 0 of the standard
conformal ﬁeld equations.
In the particular case when fa = 0 it follows from (8.40) that da = ∇a Ξ.
ˆ = ∇. Under these circumstances the extended conformalHence, one has that ∇ﬁeld Equations (8.46) reduce toΣab = 0,Ξc dab = 0,

Δcdb = 0,Λbcd = 0,

where the zero quantities Σab , Ξc dab , Δcdb and Λbcd are as deﬁned in Section
8.3.1.
The conformal covariance of the equations
As in the case of the standard conformal ﬁeld equations, the extended conformal
ﬁeld equations discussed in the previous section are conformally covariant. To
make this statement more precise, consider a spacetime (M̃, g̃) and two metrics
g and ǵ conformally related to g̃ via
g = Ξ2 g̃,ǵ = Ξ́2 g̃,´ denote, respectively, the Levi-Civita
so that g = κ2 ǵ with κ ≡ ΞΞ́−1 . Let ∇, ∇connections of the metrics g and ǵ. One has that˜ = S(Ξ−1 dΞ),
∇−∇

´ −∇
˜ = S(Ξ́−1 dΞ́),∇(8.49)

and, furthermore,´ = S(κ−1 dκ).
∇−∇In addition, consider the covectors f and f́ and deﬁne by means of these twoˆ and ∇
ˇ via
the Weyl connections ∇ˆ − ∇ = S(f ),∇ˇ −∇´ = S(f́ ).
∇(8.50)

Combining Equations (8.49) and (8.50) one ﬁnds that the relation between the
˜ and the Weyl connections ∇ˆ and ∇
ˇ is given byphysical connection ∇ˆ −∇˜ = S(β),∇ˇ −∇˜ = S(β́),∇β ≡ f + Ξ−1 dΞ,β́ ≡ f́ + Ξ́−1 dΞ́.

whereCombining these expressions one ﬁnds thatˆ −∇
ˇ = S(β − β́)∇= S(k + κ−1 dκ),with k ≡ f − f́ . Hence, letting d = Ξβ and d́ = Ξ́β́, one concludes thatd = κ−1 d́ + κΞ́k + Ξ́dκ.

(8.51)

206

The conformal Einstein ﬁeld equations

Assume now that the ﬁelds (ea , Γ̂a b c , L̂ab , da bcd , Ξ, da ) constitute a solution to the extended conformal ﬁeld Equations (8.46). Then, proceedingin analogy to the discussion in Section 8.3.3, one ﬁnds that the ﬁelds(éa , Γ̌a b c , Ľab , d´a bcd , Ξ́, d´a ) withéa = κea
ˆ a κ − κSac bd (kd + κ−1 ∇ˆ d κ),
Γ̌a b c = κΓ̂a b c + δc b ∇ˆ a (kb + κ−1 ∇ˆ b κ)Ľab = κ2 L̂ab − κ2 ∇1
ˆ c κ)(kd + κ−1 ∇ˆ d κ),− κ2 Sab cd (kc + κ−1 ∇2d´a bcd = κ3 da bcd ,Ξ́ = κ−1 Ξ,ˆ a κ − κΞka ,d´a = κda − Ξ∇are also a solution of the extended conformal equations. Observe that theˆ∇-quantities
are components with respect to the frame {ea } which isˇg-orthonormal, while the ∇-quantities
are components on the {éa } frame whichis ǵ-orthonormal.
8.4.2 The spinorial version of the extended conformal ﬁeld equations
The frame formulation of the extended conformal ﬁeld equations discussed in
the previous subsection leads directly to its spinorial counterpart. The strategy
is analogous to the one adopted in Section 8.3.2.
The spinorial counterpart of the g-orthogonal frame {ea } is given by the null
ˆ AA ≡
tetrad {eAA } satisfying g(eAA , eBB  ) = AB A B  . Furthermore, let ∇aˆ
eAA ∇a . Similarly, letΣ̂AA CC BB  ,P̂ CC DD AA BB  ,L̂AA BB  ,dAA ,ρ̂CC DD AA BB  ,fAA ,

denote, respectively, the spinorial counterparts of the ﬁelds
Σa c b ,P c dab ,ρc dab ,L̂ab ,

da ,

fa .



The spinorial counterpart of the geometric curvature, P̂ CC DD AA BB  , is givenin terms of the spinorial connection coeﬃcients Γ̂AA BB CC  . These, in turn, canbe expressed in terms of the reduced spin connection coeﬃcients Γ̂AA B C by¯Γ̂AA BB CC  = Γ̂AA B C δC  B + Γ̂A A B C  δC B ,(8.52)
consistent with formula (3.33). The reduced Weyl spin connection coeﬃcientsare related to the unphysical spin connection coeﬃcients via
Γ̂AA B C = ΓAA B C + δA B fCA ,see Equation (5.32).

Γ̂AA Q Q = fAA ;8.4 The extended conformal Einstein ﬁeld equations

207

The geometric and algebraic Weyl curvature admit, respectively, the splitsP̂ CC DD AA BB  = P̂ C DAA BB  δD C + P̄ C D AA BB  δD C ,ρ̂CC DD AA BB  = ρ̂C DAA BB  δD C + ρ̂¯C D AA BB  δD C .
The formula giving the reduced geometric curvature P̂ C DAA BB  in terms of
the reduced spin connection coeﬃcients is identical to that for a Levi-Civita
connection. Namely, one has thatP̂ C DAA BB  ≡ eAA (Γ̂BB  C D ) − eBB  (Γ̂AA C D )¯− Γ̂F B  C D Γ̂AA F B − Γ̂BF  C D Γ̂AA F B  + Γ̂F A C D Γ̂BB  F A¯+ Γ̂AF  C D Γ̂BB  F A + Γ̂AA C E Γ̂BB  E D − Γ̂BB  C E Γ̂AA E D .
In particular, it can be veriﬁed that¯P̂ Q QAA BB  = eAA (fBB  ) − eBB  (fAA ) + Γ̂AA Q B fQB  + Γ̂AA Q B  fBQ¯− Γ̂BB  Q A fQA − Γ̂BB  Q A fAQˆ AA fBB  − ∇ˆ BB  fAA .
=∇Hence, one can write1ˆ CC  fDD − ∇ˆ DD fCC  ).
P̂ABCC  DD = P̂(AB)CC  DD + AB (∇2
The reduced algebraic curvature spinor satisﬁes a similar expression. Namely,
one has that1ρ̂ABCC  DD = ρ̂(AB)CC  DD − AB (L̂CC  DD − L̂DD CC  ),2withρ̂(AB)CC  DD = −ΞφABCD C  D + L̂BC  DD AC − L̂BD CC  AD ;compare Equation (5.33).
The objects discussed in the previous paragraphs can be used, in turn, todeﬁne the zero quantities:Σ̂AA BB  ≡ [eAA , eBB  ] − (Γ̂AA CC BB  − Γ̂BB  CC AA )eCC  , (8.53a)
Ξ̂C DAA BB  ≡ P̂ C DAA BB  − ρ̂C DAA BB  ,ˆ CC  L̂DD BB  − ∇ˆ DD L̂CC  BB 
Δ̂CC  DD BB  ≡ ∇− dAA dAA

BB  CC  DD  ,(8.53b)(8.53c)(8.53d)ˆ AA dAA BB  CC  DD
Λ̂BB  CC  DD ≡ ∇

− fAA dAA BB  CC  DD

(8.53e)

208

The conformal Einstein ﬁeld equations

One can exploit the symmetries of some of the above zero quantities to obtain
reduced zero quantities. In particular, one can write¯
Λ̂BB  CC  DD = Λ̂BB  CD C  D + Λ̂B  BC  D CD ,withΛ̂BB  CD ≡


1Λ̂BB  CQ D Q .
2

A similar idea can be applied to Σ̂AA BB  , Ξ̂C DAA BB  and Δ̂CC  DD BB  . Thisidea will not be pursued here.
The spinorial version of the extended conformal Einstein ﬁeld equations isexpressed in terms of the above zero quantities as:Σ̂AA BB  = 0,

Ξ̂C DAA BB  = 0,

Δ̂CC  DD BB  = 0,Λ̂BB  CD = 0.

(8.54a)(8.54b)Finally, let δAA , ςAA BB  and γAA BB  denote the spinorial counterparts of thezero quantities δa and γab . One then requires that
δAA = 0,

ςAA BB  = 0,γAA BB  = 0.

(8.55)

8.4.3 The extended conformal Einstein ﬁeld equations and theEinstein ﬁeld equations
As in the case of the other versions of the conformal ﬁeld equations discussed inthis chapter, it is important to analyse the precise relation between the extended
conformal ﬁeld equations and the (physical) Einstein ﬁeld equations. One has thefollowing:
Proposition 8.3 (solutions to the extended conformal ﬁeld equations
as solutions to the Einstein ﬁeld equations) Let(ea , Γ̂a b c , L̂ab , da bcd )denote a solution to the extended conformal ﬁeld Equations (8.46) for somechoice of the conformal gauge ﬁelds (Ξ, da ) satisfying the supplementary
Equations (8.48). Furthermore, supposeΞ = 0,det(η ab ea ⊗ eb ) = 0,

on an open subset U ⊂ M. Then the metric g̃ = Ξ−2 ηab ω a ⊗ ω b , where {ω a } isthe dual frame to {ea }, is a solution to the Einstein ﬁeld equations (8.4) on U .
Proof As a consequence of the conformal equation Σ̂ab = 0, the ﬁelds Γ̂a b c canbe interpreted as the connection coeﬃcients, with respect to the frame ﬁeld {ea },ˆ In order to show that ∇ˆ is a Weyl connection, oneof a torsion-free connection ∇.

8.5 Further reading209ˆ a ηbc . This is best done using spinors. As eAA (BC ) = 0,
needs to compute ∇one has that
ˆ AA BC = −Γ̂AA Q B QC − Γ̂AA Q C BQ = −Γ̂AA CB + Γ̂AA BC∇= −Γ̂AA Q Q BC = −fAA BC .
Recalling that AB A B  is the spinorial counterpart of ηab and observing theˆ is a Weyl connection.
ˆ a ηbc = −2fa ηbc ; that is, ∇split (8.52) one concludes that ∇ccNow, as Ξ̂ dab = 0, the ﬁelds L̂ab and Ξd dab obtained as a solution to theextended conformal ﬁeld equations correspond to, respectively, the Schouten
ˆ as a consequence of thetensor and the Weyl tensor of the connection ∇,uniqueness of the decomposition in terms of irreducible components.
ˆ one can deﬁne a new connection ∇ via ∇ ≡Given the Weyl connection ∇,ˆ − S(f ). By construction, this connection is metric. The Schouten tensor of ∇∇is then given by1
Lab = L̂ab − ∇a fb + Sab cd fc fd .
2
As Ξ̂c dab = 0, it follows that Ξ̂ab as deﬁned by Equation (8.45) also vanishes. Asˆ is torsion free, so is ∇. Hence, one concludes that ∇ must be the Levi-Civita∇
connection of the metric g ≡ ηab ω a ⊗ ω b . The latter expression is a well-deﬁned
Lorentzian metric on U as its determinant is, by hypothesis, non-vanishing.
˜ via ∇˜ ≡ ∇ − S(Ξ−1 dΞ). As
Finally, one deﬁnes a physical connection ∇˜ is the Levi-Civita connection ofˆ a Ξ so that ∇δa = 0 it follows that da = fa + ∇−2
ab
the metric g̃ ≡ Ξ ηab ω ⊗ ω . The latter is well deﬁned as long as Ξ = 0. The
˜ is given by
Schouten tensor of ∇L̃ab = Lab − ∇a (Ξ−1 ∇b Ξ) − Ξ−2 Sab cd ∇c Ξ∇d Ξ.
As a consequence of δa = 0 and γab = 0 one concludes that1 −2λΞ ηab .
6
Thus, g̃ is a solution to the vacuum Einstein ﬁeld equations on U .
L̃ab =

Remark. Given the equivalence between the frame and spinorial versions of theextended conformal ﬁeld equations, the latter result also provides the connection
between the spinorial extended conformal ﬁeld equations and the Einstein ﬁeld
equations.
8.5 Further reading
The standard conformal Einstein ﬁeld equations were ﬁrst introduced in Friedrich(1981a,b, 1982). General aspects of the Cauchy problem of the conformal
equations were ﬁrst discussed in Friedrich (1983); see also Friedrich (1984).
A systematic discussion of gauge issues and hyperbolic reductions of theequations has been given in Friedrich (1985). A discussion of the conformal ﬁeld

210

The conformal Einstein ﬁeld equations

equations with trace-free matter was ﬁrst given in Friedrich (1991). The extended
conformal ﬁeld equations were introduced in Friedrich (1995). Reviews discussing
various aspects of the conformal ﬁeld equations can be found in Friedrich (2002,
2004). The extended conformal ﬁeld equations with (trace-free) matter were ﬁrst
discussed in Lübbe and Valiente Kroon (2012).
The conformal ﬁeld equations can be related to other geometrical objects(twistors); see Frauendiener and Sparling (2000). The extended conformal ﬁeld
equations can be set in a more geometrical framework involving the language ofdiﬀerential forms. A discussion of this has been given in Friedrich (1995). Certain
applications of the conformal equations require the use of a lift of the equations
to a suitable ﬁbre bundle. A discussion of this type of procedure can be foundin Friedrich (1986b, 1998c).
A diﬀerent approach to the construction of regular conformal ﬁeld equations
based on the Feﬀerman-Graham obstructions – see, for example, Graham and
Hirachi (2005) – has been elaborated in Anderson (2005a) and Anderson andChruściel (2005). This approach gives rise to suitable ﬁeld equations for an
arbitrary number of odd space dimensions and has been used to prove (global andsemi-global) existence and stability results of higher dimensional asymptoticallysimple spacetimes.
In the context of conformal geometry, given a metric g, it is natural to ask
whether there exists a further metric in the conformal class [g] which is an
Einstein space. This problem has been addressed in, for example, Baston and
Mason (1987), Kozameh et al. (1985) and Mason (1986).

9
Matter models

This chapter provides a discussion of various matter models amenable to atreatment by means of conformal techniques. These matter models can be usedas matter sources for the conformal Einstein ﬁeld equations discussed in Chapter
8. The matter models to be considered are the electromagnetic ﬁeld, radiation
perfect ﬂuids and the conformally invariant scalar ﬁeld. These matter models
share the property of having an energy-momentum tensor which is trace free.
This property leads to simple transformation laws for the equations satisﬁed by
the matter models. Moreover, the unphysical equations obtained by means ofthese transformations are regular at points where the conformal factor vanishes.
9.1 General properties of the conformal treatmentof matter models
The fundamental object in the description of a matter model in general relativity
is its energy-momentum tensor T̃ab . The equations describing the model arethen given by
˜ a T̃ab = 0.
∇

(9.1)

The energy-momentum tensor is related to the curvature of the spacetime via
the Einstein ﬁeld equations; see Equation (8.4). Despite this connection, aconformal transformation g = Ξ2 g̃ does not directly imply a transformation
rule for the physical energy-momentum tensor T̃ab . Nevertheless, it isconvenient to deﬁne an unphysical energy-momentum tensor Tab whenrewriting Equation (9.1) in terms of geometric quantities derived from therescaled metric g.
9.1.1 The unphysical energy-momentum tensor
There is considerable freedom in a possible deﬁnition of Tab . Guiding principles
are simplicity in both the deﬁnition and the resulting form of the unphysical212

Matter models

version of Equation (9.1). Arguably, the simplest deﬁnition of the unphysicalenergy-momentum tensor is one which is homogeneous with respect to theconformal factor Ξ. Accordingly, set
Tab = Ξ−2 T̃ab .

(9.2)

˜ a T̃bc − Ξ−5 ∇˜ c Ξg̃ ab T̃ab .
g ab ∇a Tbc = Ξ−4 g̃ ab ∇(9.3)

It follows then that

Hence, Equation (9.1) implies the equation ∇a Tab = 0 only if
T̃ = 0,T̃ ≡ g̃ ab T̃ab .

This observation justiﬁes deﬁnition (9.2) and the importance given in this chapter
to trace-free matter models. As a result of the homogeneous nature of thetransformation law in Equation (9.2), Tab is trace free if T̃ab is trace free.
In the case of matter models with T̃ab = 0, deﬁne T ≡ g ab Tab , so that T =Ξ−4 T̃ . It follows from Equations (9.1) and (9.3) that∇a Tab = Ξ−1 ∇b Ξ T.
This is an equation which is formally singular at the points where Ξ = 0. Dealing
with this singularity is the essential problem faced in the analysis of generalmatter models by means of conformal methods.
9.1.2 The rescaled Cotton tensor
As discussed in Chapter 8, the matter ﬁelds couple to the conformal Einstein ﬁeld
equations through the rescaled Cotton tensor Tabc ; compare Equations (8.22)
and (8.23). Recall that the physical Cotton tensor is given by Tabc = Ξ−1 Ỹabc
where
˜ a L̃bc − ∇˜ b L̃ac ;Ỹabc = ∇compare Equation (8.21). One readily ﬁnds that




1
˜ a T̃bc − ∇˜ a T̃ g̃bc − ∇˜ b T̃ac − 1 Ξ−1 ∇˜ b T̃ g̃ac ,Tabc = Ξ−1 ∇2
6where it has been used that the physical Schouten tensor L̃ab is related to thephysical energy-momentum tensor via Equation (8.5b). In what follows, attention
will be restricted to the trace-free matter case so that1
˜ a T̃bc − ∇˜ b T̃ac .
Tabc = Ξ−1 ∇2The latter can be reexpressed in terms of the unphysical connection ∇ and theunphysical energy-momentum tensor Tab . A computation using the methods ofChapter 5 yields
Tabc = Ξ∇[a Tb]c + ∇[a ΞTb]c + gc[a Tb]e ∇e Ξ.

(9.4)

9.2 The Maxwell ﬁeld

213

From the above expression it follows that Tabc is regular whenever Ξ = 0 if Tab
is smooth at the conformal boundary.
Equation (9.4) can be expressed in terms of derivatives of the (conformal)
matter ﬁelds. This feature complicates the construction of suitable conformal
evolution equations as it introduces further derivatives of the ﬁelds into theprincipal part of the equations. This diﬃculty can be overcome by introducing
evolution equations for the derivatives of the matter ﬁelds which cannot be
eliminated with the equation ∇a Tab = 0. This analysis depends on the speciﬁc
properties of the matter model under consideration.
9.2 The Maxwell ﬁeld
The electromagnetic or Maxwell ﬁeld is the prototype of a relativistic mattermodel that can be treated by means of conformal methods. The Maxwell ﬁeld is
described by an antisymmetric tensor F̃ab – the Faraday tensor . In terms of
the latter, the source-free Maxwell equations are given by
˜ a F̃ab = 0,∇
˜ [a F̃bc] = 0.
∇

(9.5a)(9.5b)

Multiplying Equation (9.5b) by the volume form ˜dabc , one obtains the alternative
expression∗
˜ a F̃ab
= 0,
∇

(9.6)

∗
where F̃ab≡ − 12 ˜ab cd F̃cd denotes the dual Faraday tensor . Now, introducing
the self-dual Faraday tensor∗,F̃ab ≡ F̃ab + iF̃ab

it follows from (9.5a) and (9.6) that˜ a F̃ab = 0.
∇

(9.7)

This last equation contains the same information as Equations (9.5a) and (9.5b).
The energy-momentum tensor of the electromagnetic ﬁeld is quadratic in the
Faraday tensor. It is given by1T̃ab = F̃ac F̃b c − g̃ab F̃cd F̃ cd .
4∗
It can be readily veriﬁed that T̃ = 0. Making use of the dual F̃ab
one obtains thealternative expressions1
∗T̃ab =(9.8a)
F̃b∗ cF̃ac F̃b c + F̃ac
2
1
= F̃ac F̃¯b c .
(9.8b)
2
It can be readily veriﬁed that the Maxwell Equations (9.5a) and (9.5b) imply
˜ a T̃ab = 0.
that ∇

214

Matter models
Conformal transformation properties

The source-free Maxwell Equations (9.5a), (9.5b) and (9.6) are conformallyinvariant. In order to see this, assume that (M, g) is a conformal extensionof a spacetime (M̃, g̃) with g = Ξ2 g̃, and deﬁne the conformal (unphysical)
Faraday tensor via
Fab ≡ F̃ab .

(9.9)

∗
∗
It follows from this deﬁnition that Fab= F̃ab. Moreover, using the transfor˜mation laws between the connections ∇ and ∇ one ﬁnds that Equations (9.5a),(9.5b) and (9.6) imply∇a Fab = 0,∇[a Fbc] = 0,∗
∇a Fab
= 0,(9.10)

ˆ be a Weyl
which shows the conformal invariance of the equations. Let ∇ˆconnection deﬁned via ∇ − ∇ = S(f ) with f a covector. A further computation
yieldsˆ a Fab = 0,∇

ˆ [a Fbc] = 0,∇∗
ˆ a Fab∇= 0.

Consistent with the transformation law (9.2) for the energy-momentum tensor
one ﬁnds that11∗
Tab = Fac Fb c − gab Fcd F cd = (Fac Fb c + Fac
Fb∗ c ) .
4
2
Substituting the last expression in Equation (9.4) for the rescaled Cotton tensor
one obtains∗∗
Fc∗ d + Fd[a∇b] Fc∗ d
2Tabc = ∇[a Fb]d Fc d + Fd[a ∇b] Fc d + ∇[a Fb]d∗
+ ∇[a ΞFb]d Fc d + ∇[a ΞFb]dFc∗ d∗+ gc[a Fb]e Fd e ∇d Ξ + gc[a Fb]e
Fd∗ e ∇d Ξ.

A direct inspection shows that the ﬁrst four terms of the right-hand side
contain derivatives of the Faraday tensor which cannot be eliminated using
the (conformal) Maxwell Equations (9.10). Thus, it is necessary to consider
equations for the derivatives of Fab . A suitable equation can be obtained from thecommutator of covariant derivatives applied to Fab . More precisely, one has that∇a ∇b Fcd − ∇b ∇a Fcd = −Re cab Fed − Re dab Fce .
In view of this equation one introduces the auxiliary ﬁeld Fabc ≡ ∇a Fbc so that∇a Fbcd − ∇b Facd = −Re cab Fed − Re dab Fce .
By construction one has that
Fabc = Fa[bc] ,F[abc] = 0,

F a ac = 0.

(9.11)

9.2 The Maxwell ﬁeld

215

9.2.1 The spinorial form of the Maxwell equations
The spinorial treatment of the Maxwell ﬁeld is a direct consequence of thedecomposition of spinors in irreducible components; see Section 3.1.6. The
spinorial formulation of the Maxwell equations oﬀers a number of computational
advantages and makes more evident the similarities between the gravitationaland electromagnetic ﬁelds.
In what follows, let F̃AA BB  denote the spinorial counterpart of the Faraday tensor F̃ab . By exploiting the antisymmetry of F̃ab , it follows from Equation(3.13) that there exists a symmetric spinor φ̃AB , the Maxwell spinor, such that¯
F̃AA BB  = φ̃AB ˜A B  + φ̃A B  ˜AB ,φ̃AB =
1
F̃AQ B Q .
2(9.12)

Using the decomposition (9.12) it follows that
F̃AA BB  = 2φ̃AB ˜A B  ,where F̃AA BB  is the spinorial counterpart of the self-dual Faraday tensor F̃ab .
Taking into account Equation (9.7) one obtains
˜ A A φ̃AB = 0.
∇(9.13)

This last equation is known as the spinorial Maxwell equation. A further
computation using Equation (9.8b) shows that the spinorial counterpart of theenergy-momentum tensor takes the simple form¯
T̃AA BB  = φ̃AB φ̃A B  .
Behaviour under conformal rescalings
The deﬁnition of the unphysical (conformal) Faraday tensor given in Equation(9.9) suggests introducing the unphysical Maxwell spinor φAB as
φAB ≡ Ξ−1 φ̃AB .

(9.14)

The factor Ξ−1 in the above deﬁnition is necessary to compensate for the factor
Ξ picked up by the spinor ˜AB in Equation (9.12). It follows from Equation (9.13)
and the transformation law of the connection upon conformal rescalings g = Ξ2 g̃given in Section 5.4 that
∇Q A φBQ = 0.

(9.15)

That is, the transformation rule (9.14) makes the spinorial Maxwell Equation(9.13) conformally invariant – this result was to be expected in view of theequations in (9.10). One readily sees the similarities between Equation (9.15) and
the spinorial Bianchi identity ∇AA φABCD = 0. Consistent with Equation (9.2)
one ﬁnds that the spinorial counterpart of the unphysical energy-momentum
tensor is given by
TAA BB  = φAB φ̄A B  .

216

Matter models

ˆ = ∇+S(f ) as
Equation (9.15) can be expressed in terms of a Weyl connection ∇ˆ Q A φBQ = f Q A φBQ .
∇In order to write a spinorial version of Equation (9.11) it is observed that theaction of the commutator of covariant derivatives on the spinor φAB is given by∇AA ∇BB  φCD − ∇BB  ∇AA φCD = −φQD RQ CAA BB  − φCQ RQ DAA BB  .
Letting ψAA BC ≡ ∇AA φBC , one obtains∇AA ψBB  CD − ∇BB  ψAA CD = −2φQ(C RQ D)AA BB  .
By construction, the auxiliary spinor ψAA BC possesses the symmetries
ψAA BC = ψAA CB ,

ψ Q A BQ = 0.

9.3 The scalar ﬁeld
A scalar ﬁeld φ̃ satisfying the wave equation
˜ a∇
˜ a φ̃ = 0∇(9.16)

is a convenient matter model to couple to the Einstein ﬁeld equations. It provides
a way of incorporating dynamical degrees of freedom in spherically symmetricconﬁgurations; see, for example, Choptuik (1993) and Gundlach and Martı́nGarcı́a (2007). This idea has been exploited in a number of analyses of cosmic
censorship and the formation of black holes through gravitational collapse; see,for example, Christodoulou (1986) and Dafermos (2003, 2005).
Unfortunately, as a direct computation shows, Equation (9.16) does nothave good conformal transformation properties. This diﬃculty can be ﬁxed byconsidering a modiﬁed version – the so-called conformally invariant scalarﬁeld equation
˜ a φ̃ − 1 R̃φ̃ = 0,
˜ a∇
∇
6

(9.17)

where R̃ denotes the Ricci scalar of the physical spacetime metric g̃. Letting, as
usual, g = Ξ2 g̃ and deﬁning the unphysical (conformal) scalar φ as
φ ≡ Ξ−1 φ̃,
one ﬁnds, after a calculation using the transformation rule for the Ricci scalar
Equation (5.6c), that1∇a ∇a φ − Rφ = 0,(9.18)
6
where R denotes the Ricci scalar of g. An energy-momentum tensor for Equation(9.17) is given by
˜ c φ̃∇˜ b φ̃ + 1 φ̃2 L̃ab .
˜ a φ̃∇˜ b φ̃ − 1 g̃ab ∇˜ c φ̃ − 1 φ̃∇˜ a∇T̃ab = ∇422(9.19)

9.3 The scalar ﬁeld217

A peculiarity of the above expression is the presence of the curvature terms
L̃ab in the right-hand side of the energy-momentum tensor. Using the Einsteinﬁeld Equations (8.4), the Schouten tensor can be reexpressed in terms of theenergy-momentum tensor, so that Equation (9.19) takes the form

−1 

1 2
1
1 ˜ ˜
1
c
2
˜
˜
˜
˜
∇a φ̃∇b φ̃ − g̃ab ∇c φ̃∇ φ̃ − φ̃∇a ∇b φ̃ + (λ − T̃ )φ̃ g̃ab .
T̃ab = 1 − φ̃
442
12Taking the trace of Equation (9.19) one ﬁnds that


1
1 2
ab
a˜
˜
T̃ ≡ g̃ T̃ab = − φ̃ ∇ ∇a φ̃ − R̃φ̃ .
2
6Thus, the energy-momentum tensor of Equation (9.19) is trace free if andonly if the conformally invariant wave Equation (9.17) is satisﬁed. A lengthiercomputation using the commutator of covariant derivatives and the Bianchi
˜
˜ a L̃ab = 1 ∇identity in the form ∇6 b R̃ shows that




˜ c∇
˜ a T̃ab = ∇˜ c∇
˜ c φ̃ − 1 R̃φ̃ − 1 φ̃∇˜ c φ̃ − 1 R̃φ̃ .
˜ b φ̃ ∇
˜b ∇
∇
6
2
6
One concludes that T̃ab is divergence free if and only if Equation (9.17) holds.
Finally, using the transformation law for the Schouten tensor under conformal
rescalings, Equation (5.6b), one ﬁnds that111
Tab = ∇a φ∇b φ − gab ∇c φ∇c φ − φ∇a ∇b φ + φ2 Lab ,4
2
2
so that Tab = Ξ−2 T̃ab . It follows from the previous discussion that∇a Tab = 0,

g ab Tab = 0.

Spinorial description
The straightforward spinorial counterpart of Equation (9.19) is given by
˜ P P  φ̃∇
˜ AA φ̃∇˜ BB  φ̃ − 1 ˜AB ˜A B  ∇˜ P P  φ̃T̃AA BB  = ∇
4
1 ˜
1
˜
− φ̃∇AA ∇BB  φ̃ + φ̃2 L̃AA BB  .
2
2
˜ BB  φ̃
˜ BB  φ̃ and ∇˜ AA ∇
˜ AA φ̃∇
Applying the decomposition formula (3.12) to ∇one ﬁnds that˜ P P  φ̃∇
˜ AA φ̃∇˜ BB  φ̃ = ∇˜ A(A φ̃∇˜ B  )B φ̃ + 1 ˜AB ˜A B  ∇˜ P P  φ̃,∇4
˜ AA ∇˜ BB  φ̃ = ∇˜ B  )B φ̃ + 1 ˜AB ˜A B  ∇˜ PP∇
˜ P P  φ̃,
˜ A(A ∇∇4
where it has been used that˜ P B  ) φ̃ = 0,˜ P (A φ̃∇∇˜ P B  ) φ̃ = 0.
˜ P (A ∇∇218

Matter models

The above formulae, together with the wave equation (9.17), imply the following
alternative spinorial expression for the energy-momentum tensor:
˜ B  )B φ̃ + 1 φ̃2 Φ̃AA BB 
˜ A(A φ̃∇˜ B  )B φ̃ − 1 φ̃∇˜ A(A ∇T̃AA BB  = ∇22
where Φ̃AA BB  is the (physical) trace-free Ricci spinor. The unphysical spacetime
version of the above equation follows directly by removing the ˜ of the various
ﬁelds.
9.3.1 Equations for the derivatives of the scalar ﬁeld
As in the case of the electromagnetic ﬁeld, the coupling of the conformallyinvariant scalar ﬁeld to the conformal ﬁeld equations through the rescaled Cotton
tensor Tabc involves derivatives of φ. Indeed, a calculation exploiting the fact that∇[a ∇b] φ = 0 shows that∇[a Tb]c =31∇[b φ∇a] ∇c φ − gc[b ∇a] ∇e φ∇e φ + φ∇[a φLb]c2
2
11− φ∇[a ∇b] ∇c φ + φ2 ∇[a Lb]c .
2
2

The terms in the second line of the preceding equation can be rewritten using
the commutator
1∇[a ∇b] ∇c φ = − Re cab ∇e φ2and the Cotton Equation (8.23). Putting everything together in Equation (9.4)
and rearranging one obtains


3
1
1
1 − φ2 Ξ2 Tabc = Ξ∇[b φ∇a] ∇c φ − Ξgc[b ∇a] ∇e φ∇e φ + φ∇[a φLb]c
4
2
2
1
1
+ ΞφRe cab ∇e φ+ Ξφ2 ∇e ΞRe cab +∇[a ΞTb]c +gc[a Tb]e ∇e Ξ.
4
4The above expression contains ﬁrst and second derivatives of φ which cannot be
eliminated using the wave Equation (9.18). Accordingly, ﬁeld equations for thesederivatives need to be constructed.
In what follows, let φa ≡ ∇a φ, φab ≡ ∇a ∇b φ. As ∇ is torsion free one hasthat φ[ab] = 0 and one can write
1
1
φab = φ{ab} + gab φe e = φ{ab} + gab Rφ,4
24(9.20)

where in the second equality one has used Equation (9.18) in the form φe e = 16 Rφ.
Regarding φa and φab as further ﬁeld unknowns one obtains the ﬁeld equations∇a φ − φa = 0,∇a φb − φ{ab} −1
gab Rφ = 0,24

1∇e φe − Rφ = 0.
6

9.4 Perfect ﬂuids

219

To obtain equations for φab one considers the commutator of covariant derivatives
applied to ∇c φ in the form
1∇[a φb]c = − Rd cab φd .
2
Letting ψab ≡ φ{ab} and using the decomposition (9.20) one obtains∇[a ψb]c −1 1Rgc[a φb] + φgc[b ∇a] R = − Rd cab φd .
24
2Finally, an equation for the trace term φe e is obtained by diﬀerentiating Equation(9.18) so that∇a φe e −1
(φa R + φ∇a R) = 0.
6

9.3.2 Relation to other wave equations
Solutions to the conformally invariant wave Equation (9.17) on a spacetime(M̃, g̃) are related to solutions of the standard wave equation on a conformally
related spacetime (M̀, g̀) through a transformation ﬁrst discussed in Bekenstein(1974): the scalar ﬁeld φ̃ can be used to deﬁne a metric ĝ conformally related tog̃ viag̀ = Ξ̀2 g̃,

1Ξ̀ ≡ 1 − φ̃2 .
4

It follows from a direct computation that the scalar ﬁeld
φ̀ ≡

√
1
6 arctan φ̃2is a solution of the equation` a∇` a φ̀ = 0.
∇
As noticed in Bičák et al. (2010), this observation can be turned into a procedure
to construct solutions to the Einstein-scalar ﬁeld equations out of vacuum static
solutions; see also Buchdahl (1959).
9.4 Perfect ﬂuids
Perfect ﬂuids constitute an important class of matter models for the Einsteinﬁeld equations. In the cosmological context, perfect ﬂuids are used to describe
the matter content of the universe at a suitably large scale; see, for example,
Ellis et al. (2012). Given a spacetime (M̃, g̃), the energy-momentum tensor
of a perfect ﬂuid with 4-velocity ũa , pressure p̃ and density ˜ is given byT̃ab = (˜ + p̃)ũa ũb − p̃g̃ab ,(9.21)

220

Matter models

with ũa satisfying the normalisation condition ũa ũa = 1. The equations of motion
˜ a T̃ab = 0. This last equation gives four
for the ﬁelds ũa , ˜ and p̃ are given by ∇equations for six unknowns. The normalisation of ũa can be used to eliminate
one of the components of the 4-velocity (usually the time component). To close
the system a phenomenological constitutive relation linking the pressure and thedensity must be prescribed. A standard assumption made on perfect ﬂuids is tohave the density and the pressure related to each other by means of a barotropic
equation of state p̃ = f (˜) with f a smooth function of the density ˜. From
Equation (9.21) it follows that T̃ = ˜ − 3p̃. Thus, the energy-momentum tensor
of a perfect ﬂuid is trace free if and only if1
˜.
(9.22)
3This constitutive relation is known as the equation of state of radiation.
In what follows, the discussion will be restricted to perfect ﬂuids satisfying theequation of state (9.22). To discuss the perfect ﬂuid in the conformally rescaled
spacetime (M, g) with g = Ξ2 g̃ it is convenient to consider the following
unphysical conformal ﬁeldsp̃ =ua ≡ Ξũa , ≡ Ξ−4 ˜,

p ≡ Ξ−4 p̃.

The above deﬁnitions are consistent with the transformation law for the energymomentum tensor of Equation (9.2). Moreover, it follows that p = 13 , so thatthe unphysical energy-momentum tensor takes the form
41
ua ub − gab with ∇a Tab = 0.
(9.23)
3
3
Moreover, one has that ua ua = 1, so that diﬀerentiating along ua one ﬁnds thatTab =ua ∇a (ub ub ) = 0.
From this expression it follows that if ua ua = 1 at some point in a ﬂow line, thenua ua = 1 everywhere along the ﬂow line. From Equation (9.23) it readily follows
that
1
4(ua uc ∇c  + ua ∇c uc + uc ∇c ua ) − ∇a  = 0.
3
3
Contracting this equation, respectively, with ua and gab − ua ub one obtains
4ua ∇a  + ∇a ua = 0,
3
1
1
4 c
u ∇c ua + ua uc ∇c  − ∇a  = 0.
3
3
3
These equations are the conformal versions of the equation of energy
conservation and the equations of motion; see, for example, ChoquetBruhat (2008). A discussion on how to use these equations to construct suitableevolution equations for the ﬁelds  and the spatial components ui of the ﬂuid
4-velocity can be found in the same reference.

9.5 Further reading

221

9.5 Further reading
A further matter model amenable to a treatment by means of conformal methods
is the Yang-Mills ﬁeld. The Yang-Mills equations can be regarded as a suitable
generalisation of the Maxwell equations; see, for example, Frankel (2003) for a
discussion. The conformal ﬁeld equations with matter source given by a YangMills ﬁeld of arbitrary gauge group have been discussed in Friedrich (1991). The
discussion of the Maxwell ﬁeld presented in this chapter is adapted from thatreference. A treatment of the conformal Einstein-Maxwell system by means ofWeyl connections is given in Lübbe and Valiente Kroon (2012).
The discussion of the conformal ﬁeld equations coupled to the conformallyinvariant wave equation was ﬁrst given in Hübner (1995). An alternative
approach to the analysis of the conformal Einstein ﬁeld equations with a scalarﬁeld can be found in Bičák et al. (2010). In Friedrich (2015b) it has been shownthat the Einstein–massive scalar ﬁeld system has good conformal properties if
the mass of the scalar ﬁeld and the cosmological constant satisfy the relation
3m2 = −2λ.
Finally, the conformal Einstein-Euler equations have been analysed in Lübbe
and Valiente Kroon (2013b) and used to prove the future non-linear stabilityof perturbations of Friedman-Lemaı̂tre-Robinson-Walker cosmological models
with a radiation ﬂuid. Analyses of the Einstein-Euler system not making useof conformal methods can be found in Rodnianski and Speck (2013) and Speck(2012).
The purpose of this chapter has been to present a discussion of mattermodels with properties which make them suitable sources for the conformal ﬁeld
equations. However, conformal methods have also been used for other types of
constructions. As an example, one has Bičák and Krtouš (2001, 2002) where theconformal invariance of the Maxwell equations has been exploited to construct
the analogue of the Born solution (describing the motion of a pair of uniformly
accelerated charges) in the de Sitter spacetime.

10
Asymptotics

This chapter discusses some basic consequences of the notion of asymptotic
simplicity introduced in Chapter 7. As already mentioned, the main motivation
behind this deﬁnition is to provide a characterisation of a broad class ofspacetimes in which universal structures can be identiﬁed. Once this has been
done, the idea is to use these structures to deﬁne in a rigorous manner conceptsof physical interest.
The characterisation of the gravitational ﬁeld through the analysis of its
asymptotic behaviour has a long tradition dating back to the early works by
Bondi et al. (1962), Sachs (1962b) and Newman and Penrose (1962). Thesestudies culminated in the identiﬁcation of gravitational radiation as a real
physical phenomenon. The developments of this classical theory have beentreated extensively in the literature; see, for example, Geroch (1976), Penrose andRindler (1986), Stewart (1991) and Frauendiener (2004). The readers interested
in the historic development of this idea are referred to Kenneﬁck (2007).
Despite the important insights provided by the classical theory of asymptotics
of general relativity, this approach has the weakness of being, to some extent,
formal. More precisely, it relies on a number of assumptions about the natureof solutions to the Einstein ﬁeld equations – say, for example, the regularityof the conformal boundary – which are hard to verify for a suitably large classof spacetimes. This point is key: the theory of asymptotics of the gravitational
ﬁeld comes fully into life when combined with the (conformal) ﬁeld equations
and methods of the theory of partial diﬀerential equations. This remark does
not disown the fundamental insights into the behaviour of the gravitational ﬁeldthat formal asymptotic analyses have produced, but rather insists on the needto carry the subject further.
Arguably, the most important consequence of asymptotic simplicity is theset of results collectively known as peeling – that is, a detailed descriptionof the asymptotic behaviour of the gravitational ﬁeld expressed in terms of thecomponents of the Weyl tensor. The peeling behaviour is the main subject of this10.1 Basic set up: general structure of the conformal boundary

223

chapter. The basic assumptions behind the peeling results are the main subjectof Chapter 20. Complementary to the discussion of the peeling behaviour, thischapter contains a detailed discussion of a gauge prescription for the analysis of
the structure of the gravitational ﬁeld at the conformal boundary of Minkowskilike spacetimes, the so-called NP gauge. The chapter concludes with a brief
overview of other aspects of the theory of the asymptotics of the gravitational
ﬁeld which are sligthly outside the main focus of this book: the Bondi mass, theBMS group and the so-called Newman-Penrose constants.
10.1 Basic set up: general structure of the conformal boundary
In what follows let (M̃, g̃) be an asymptotically simple spacetime in the sense of
Deﬁnition 7.1 and let (M, g, Ξ) denote an associated conformal extension. As in
Section 7.1, let I denote part of the conformal boundary characterised by therequirementsΞ = 0,dΞ = 0.

(10.1)

Much of the analysis of the present chapter is based on the evaluation of thevarious conformal ﬁeld equations at I . In what follows, the notation
will beused to indicate that a certain equality holds at I . In terms of this notation, theconditions in (10.1) can be rewritten as
Ξ

0,

dΞ 

0.

The basic observation concerning the set I is that its causal nature isdetermined by the sign of the cosmological constant λ. This result follows froma direct inspection of the conformal Einstein ﬁeld equations; see, for example,
Equations (8.26a)–(8.26e) in Section 8.2.5. One has that:
Theorem 10.1 (causal nature of the conformal boundary ) Suppose that
the Friedrich scalar s is ﬁnite at I and that T = o(Ξ−4 ). Then I is a null,spacelike or timelike hypersurface, respectively, depending on whether λ = 0,
λ < 0 or λ > 0.
Proof The normal to the hypersurface I is given by ∇a Ξ. From Equation (8.24)
one directly has that∇a Ξ∇a Ξ1− λ,3(10.2)

as by hypothesis Ξ4 T → 0 if Ξ → 0 and s is ﬁnite at I .
A discussion of the order symbols o and O used in the previous and otherresults of this chapter can be found in the Appendix to Chapter 11.
Remark. Spacetimes with λ = 0 will be said to be Minkowski-like, those with
λ < 0 de Sitter-like and those with λ > 0 anti-de Sitter-like.

224

Asymptotics

The regularity of s at I can be rephrased in terms of a suﬃciently rapid decayof the physical energy-momentum tensor T̃ab . Using the conformal ﬁeld Equation(8.13) it follows that a suﬃcient condition for ∇a ∇b Ξ and s to be ﬁnite at I isthat T{ab} = o(Ξ−3 ). In this case one concludes that∇a ∇b Ξsgab .

(10.3)

It follows from the transformation formulae of the energy-momentum tensor,
Equation (9.2), that if T{ab} = o(Ξ−3 ), then, in fact, T̃{ab} = O(Ξ3 ); see also thediscussion in Stewart (1991). If, in addition, one has that R is ﬁnite at I , thenexpression (10.3) reduces to∇a ∇b Ξ1 c
∇ ∇c Ξgab .
4

The spinorial version of the above expression is∇A(A ∇B  )B Ξ

0.

(10.4)The latter is usually known as the asymptotic Einstein condition; see, forexample, Penrose and Rindler (1986).
10.1.1 Topology of the conformal boundary
As will be seen in Chapter 15, there exists considerable freedom in thespeciﬁcation of the topology of de Sitter-like spacetimes. By contrast, the caseof a vanishing cosmological constant is much more restrictive:
Theorem 10.2 (topology of I for asymptotically Minkowskian spacetimes) Let (M̃, g̃) denote an asymptotically simple spacetime with λ = 0 and let(M, g, Ξ) denote a conformal extension thereof. Then I consists of two disjoint
components I − and I + , each one having the topology of R × S2 .
A discussion of the proof of the above theorem goes beyond the scope of thisbook. The interested reader is referred to Newman (1989) for a proof and for a
discussion on pitfalls in earlier arguments in Penrose (1965) and Geroch (1971b,
1976); see also Hawking and Ellis (1973). Remarkably, this result depends on
the satisfactory resolution of the so-called Poincaré conjecture; see, for example,
Gowers (2008) for an introduction to this (now solved) classical problem inmathematics. Vacuum spacetimes with a vanishing cosmological constant and aconformal inﬁnity with sections which are toroidal, that is, having the topology
of R × S × S have been considered in the literature; see Schmidt (1996). Notethat as a consequence of Theorem 10.2 these spacetimes must exhibit some typeof pathology – and, in particular, they cannot be asymptotically simple.
The behaviour of points in the conformal extension of an asymptotically simple
spacetime for which both Ξ = 0 and dΞ = 0 will be analysed from various
perspectives in Chapters 16, 18 and 20.

10.1 Basic set up: general structure of the conformal boundary

225

10.1.2 Further properties of the case λ = 0
In this section let λ = 0 throughout so that the asymptotically simple spacetime(M̃, g̃) has a null conformal boundary. For ease of the exposition, attention isrestricted to the vacuum case.
As a consequence of Theorem 10.1 the physical spacetime manifold M̃ mustlie either towards the past or the future of I – intuitively, this assertion seems
natural; however, a detailed argument requires the ideas of the discussion on
Lorentzian causality in Chapter 14. Consistent with the discussion of conformal
extensions of exact solutions in Chapter 6, I + (i.e. future null inﬁnity ) will
denote the set on which null geodesics attain a future endpoint while I − (i.e.
past null inﬁnity ) corresponds to the set of past endpoints of null geodesics. A
null hypersurface has the property of being generated by null geodesics; that is,each p ∈ I ± lies on exactly one null geodesic which is everywhere tangent
to I ± . Accordingly, each of I + and I − can be regarded as the union ofthese generators (or rays). Complementary to the latter is the notion of acut of null inﬁnity , that is, a two-dimensional surface C which intersects eachgenerator exactly once. As a result of Theorem 10.2 one has that C ≈ S2 .
The subsequent discussion will, for simplicity, be restricted to I + – ananalogous discussion follows, mutatis mutandis, for I − . By deﬁnition, the0, it follows that the vector
normal to I + is given by dΞ. As g  (dΞ, dΞ)N ≡ −g (dΞ, ·) satisﬁes dΞ, N  = 0 and, thus, is tangent to I + – and, inparticular, to its null generators.
As I + is a hypersurface of M, there exists an embedding ϕ : I + → M. Letq ≡ ϕ∗ g denote the metric induced on I + by g. The metric q is degenerate. To
see this, write dΞ in coordinates adapted to I + ; it follows that ϕ∗ (dΞ) = 0 sothat ϕ∗ (N  ) = 0. Thus, from N  = g(N , ·) one concludes that q(N , ·) = 0 as
claimed – observe that as N is tangent to I + , it follows that it has a well-deﬁned
pull-back.
To analyse the behaviour of the metric q along the generators of I + consider
the Lie derivative £N q. To compute it start from£N gab = N c ∇c gab + ∇c Na gcb + ∇c Nb gac= ∇b Na + ∇a Nb = 2∇a Nb ,as ∇a ∇b Ξ = −∇a Nb = −∇b Na . Hence, using Equation (10.3) it follows that
£N q = −sq.

(10.5)The trace-free part of £N q is called the shear tensor ς of the congruenceof generators of I + – it describes the tendency of a sphere of points in the
congruence to be deformed into an ellipsoid with the same volume. As Equation(10.5) is pure trace, it follows that ς = 0. Thus, the congruence of generators of
I + is shear free. This result is a consequence of the conformal ﬁeld equations via
Equation (10.3) so that from the conformal invariance of the equations it follows

226

Asymptotics

that the shear-freeness of the congruence of generators is a property independentof the particular choice of conformal factor.
The conformal gauge freedom inherent in the construction of a conformal
extension can be exploited to gain further insight into the structure of nullinﬁnity. Given a conformal extension (M, g, Ξ) consider ϑ > 0 and deﬁne a
conformally related metric g  via g  = ϑ2 g. The transformation rule of theFriedrich scalar s – see Equation (8.29b) – yields thats

ϑ−1 s − ϑ−2 N c ∇c ϑ.

Thus, if initially s = 0, one can always ﬁnd a further conformal representation(M, g  , Ξ ) for which s = 0 if one imposes the condition
N c ∇c ϑ = ϑs.

(10.6)

Notice that the above equation can be rewritten as £N ϑ = ϑs, and, accordingly,
it can be read as an ordinary diﬀerential equation along the generators of nullinﬁnity. It is important to observe that once condition (10.6) has been imposed,
one is still left with the freedom of specifying a further rescaling g  = κ 2 g  suchthat £N  κ = 0.
The conformal gauge implied by condition (10.6) yields, together with
Equation (10.5), that£N  q  = 0;

(10.7)

that is, the intrinsic metric of I + is Lie dragged along the generators of nullinﬁnity. Each of the cuts C of null inﬁnity inherits from the metric q on I +a metric k which is non-degenerate. As a consequence of Equation (10.7), if
one considers any other cut C  , one obtains the same induced metric k. Now,
any metric on a two-dimensional surface which is topologically S2 is conformal
to the standard metric of S2 , σ – this fact is a consequence of the so-called
Riemann mapping theorem; see, for example, Krantz (2006), chapter 4.
Hence, one can write k = θ2 σ for some conformal factor θ > 0 on S2 . Undera further conformal gauge transformation g  = κ 2 g  such that £N  κ = 0 (seethe previous paragraph), one can then always assume that the gauge has beenchosen so that k = σ. Under these circumstances the conformal gauge freedom
is reduced to a function κ such that κ 1.
10.2 Peeling properties
One of the most important results of the theory of asymptotics of thegravitational ﬁeld is the so-called Peeling theorem – a precise prescriptionof the decay of the Weyl tensor of an asymptotically simple spacetime. The
Peeling theorem is based on the important observation that the Weyl tensor ofan asymptotically simple spacetime must vanish on I . As will be seen in the10.2 Peeling properties

227

following, this observation follows in a quite straight forward manner if λ = 0.
A more subtle argument is required if λ = 0.
In what follows, let ΨABCD denote the Weyl spinor, and recall that ΨABCD =ΞφABCD . The subsequent analysis is best carried out with the spinorialconformal Einstein ﬁeld equations expressed with respect to a spin dyad {A A };see Section 8.3.2. In this formulation of the ﬁeld equations the ﬁelds are scalars.
Hence, they can readily be evaluated at the conformal boundary without the needof pull-backs. One has the following:
Theorem 10.3 (vanishing of the Weyl tensor at I ) Assume that ΨABCD
is smooth at I . If λ = 0 and the physical Cotton tensor satisﬁes Ỹabc = o(Ξ−1 )at I , then ΨABCD = 0 at I . If λ = 0, the same conclusion follows if Ỹabc =o(Ξ−1 ) and ∇d Ỹabc = o(Ξ−1 ).
Proof (case λ = 0)

The starting point of the analysis is the Bianchi equation∇Q A φABCQ + TBCAA = 0;compare the spinorial conformal Einstein Equation (8.37b). Now, recalling that
φABCD = Ξ−1 ΨABCD and TBCAA = Ξ−1 ỸBCAA it follows that
∇Q A ΞΨABCQ − Ξ∇Q A ΨABCQ = ΞỸBCAA .
Hence, using Ỹabc = o(Ξ−1 ) one ﬁnds that ∇Q A Ξ ΨABCQwith ∇DA Ξ one obtains
∇DA Ξ∇Q A Ξ ΨQ ABC(10.8)
0. Contracting

0.

(10.9)

Now, using Equation (10.2) one has∇DA Ξ∇Q A Ξ =
1∇P P  Ξ∇P P Ξ DQ
2

Substituting the latter in (10.9) one ﬁnds that λΨABCD
on I .
Proof (case λ = 0)3
− λ DQ .
2
0. Hence, ΨABCD = 0

Again, one has that∇AA Ξ ΨABCD

0.

(10.10)

In this case, however, ∇AA Ξ is the spinorial counterpart of a null vector. Hence,
there exists a spinor ιA such that∇AA Ξ = ιA ῑA .

(10.11)

It follows from Equation (10.10) that there exists a scalar ﬁeld ψ such that
ΨABCDψιA ιB ιC ιD .

(10.12)

228

Asymptotics

In order to extract further information consider Equation (10.8) – which is also
valid in the case λ = 0 – and apply ∇EE  to both sides. The assumptions on
ỸCDBB  imply that
∇EE  ∇Q B  ΞΨABCQ + ∇Q B  Ξ∇EE  ΨABCQ − ∇EE  Ξ∇Q B  ΨABCQ

0.

Symmetrising on E  and B  , and using the asymptotic Einstein condition (10.4)
one concludes that
∇Q (B  Ξ∇E  )E ΨABCQ − ∇E(E  Ξ∇Q B  ) ΨABCQ

0.

(10.13)

Now, using identity (3.6) to interchange the indices E and Q one obtains∇Q (B  Ξ∇E  )Q ΨABCE − ∇Q(E  Ξ∇Q B  ) ΨABCE− EQ ST ∇Q (B  Ξ∇E  )S ΨABCT + ∇S(E  Ξ∇Q B  ) ΨABCT

0,which in view of Equation (10.13) reduces to∇Q (B  Ξ∇E  )Q ΨABCE

0.

Using the decomposition (10.11) in this last equation one obtainsιQ ῑ(B  ∇E  )Q ΨABCE

0.



Contracting the latter with ῑB and observing that ῑE  = 0, one concludes thatιQ ῑB ∇QB  ΨABCE

0.

(10.14)Thus,
ιQ ∇QE  ΨABCEα ῑE  ζABCE

for some scalar α and a spinor ζABCE = 0. Substituting back into (10.14) one
concludes that α = 0 so that one has
ιQ ∇QE  ΨABCE

0.

(10.15)In order to bring this last result into a more convenient form one completes the
spinor ιA to a spin basis {A A } = {oA , ιA } with oA ιA = 1 so that ιA = δ1 A and
oA = δ0 A . Thus, contracting Equation (10.15) with ōE and substituting (10.12)
into Equation (10.15) one obtainsιQ ōE ∇QE  (ψιA ιB ιC ιE ) = ∇10 (ψιA ιB ιC ιE )0.

(10.16)

The above expression is to be regarded as a diﬀerential equation for ψ over thecuts of I + . To conclude the argument one makes use of the formalism of theð and ð̄ operators as discussed in the Appendix to this chapter. Accordingly,
in what follows it is assumed that one has a conformal representation for which
the cuts are metric unit spheres S2 . Contracting (10.16) with oA oB oC oE one
obtainsð̄ψ0.

10.2 Peeling properties

229

Now, from ψ = ΨABCD oA oB oC oD it follows that ψ has spin-weight 2. Hence,using Lemma 10.1 in the Appendix to this chapter it follows that ψ
0 andthus ΨABCD vanishes at I .
Remark. The above result strongly depends on the fact that for an asymptotically simple spacetime with λ = 0 one has that I ≈ R × S2 . For the spacetimes
with toroidal null inﬁnities considered in Schmidt (1996), the crucial Lemma
10.1 does not hold – see Frauendiener and Szabados (2001) – and the desiredconclusion cannot be obtained.
A more detailed description
To obtain a more detailed description of the peeling behaviour, it is necessary
to introduce further structure. In what follows, consider a null geodesic γ in(M, g) reaching I at a point p and let γ̃ denote the corresponding null geodesicon (M̃, g̃). At a point q ∈ γ̃ one can choose a spin dyad {õ, ι̃} such that the
tangent to γ̃ is given by the vector l̃ with spinorial counterpart ˜lAA = õA õ¯A .
The spin dyad can be naturally propagated along γ̃ by requiringD̃õA = 0,D̃ι̃A = 0,

(10.17)˜ a = õA õ¯A ∇˜ AA in standard Newman-Penrose (NP) notation.
where D̃ ≡ ˜la ∇Now, let r̃ denote an aﬃne parameter along γ̃. It follows that D̃ = d/dr̃. In
order to rewrite the above expressions in terms of quantities deﬁned on theunphysical spacetime (M, g) it is convenient to consider the transformationoA = õA ,oA = Ξ−1 õA ,ιA = Ξι̃A ,ιA = ι̃A ;(10.18)compare Equations (5.31a)–(5.31c) in Chapter 5. Using the transformation laws
under conformal transformations for the covariant derivatives it follows from
(10.17) thatDιA = Ξ−1 δ̄Ξ oA ,DoA = 0,

where δ̄ ≡ m̄a ∇a = ιA ōA ∇AA . The second of the above expressions is
potentially singular at I – observe, however, that as Ξ0, it follows thatδ̄Ξ
0 as m̄ is intrinsic to I . Thus, the spin dyad {o, ι} is well deﬁned andregular at I . Now, from DoA = 0 it follows that the null geodesic γ is aﬃnely
parametrised. Let r denote a possible aﬃne parameter. Its origin and scaling can
be chosen so thatr = 0,

and

DΞ =

dΞ
= −1
dr

From Remark (c) in Section 7.1 it follows that
dr̃
1
= 2
dr
Ξ

at

p ∈ I.

230

Asymptotics

where r̃ is an aﬃne parameter in the physical spacetime (M̃, g̃). Hence, one
concludes thatr̃ = O(Ξ−1 )near I .

(10.19)Making use of the above relations one obtains the following, more detailed,version of the peeling behaviour:
Theorem 10.4 (Peeling theorem) Let (M̃, g̃) denote an asymptoticallysimple spacetime with λ = 0 for which the hypotheses of Theorem 10.3 hold.
Moreover, letψ̃0 ≡ ΨABCD õA õB õC õD ,ψ̃1 ≡ ΨABCD ι̃A õB õC õD ,ψ̃3 ≡ ΨABCD ι̃A ι̃B ι̃C õD ,ψ̃2 ≡ ΨABCD ι̃A ι̃B õC õD ,ψ̃4 ≡ ΨABCD ι̃A ι̃B ι̃C ι̃D ,thenψ̃0 = O(r̃−5 ),ψ̃1 = O(r̃−4 ),ψ̃3 = O(r̃−2 ),Proofψ̃2 = O(r̃−3 )ψ̃4 = O(r̃−1 ).

Letψ0 ≡ ΨABCD oA oB oC oD ,...

ψ4 ≡ ΨABCD ιA ιB ιC ιD .

It follows from Theorem 10.3 that ψk = O(Ξ). Now, using the transformation
rules (10.18) one has thatψ̃k = Ξ4−k ψk .
Thus, recalling (10.19), one ﬁnds the desired result.
Combining the deﬁnitions of the ﬁelds ψ̃k with the corresponding decays given
by Theorem 10.4 one obtains a detailed expression for the asymptotic behaviourof the Weyl spinor. It can be written schematically as
ΨABCD =

[III]ABCD[N ]ABCD[II]ABCD[I]ABCD++++ O(r̃−5 ), (10.20)
r̃
r̃2
r̃3
r̃4

where [N ]ABCD , [III]ABCD , [II]ABCD and [I]ABCD represent, respectively,
totally symmetric spinors of Petrov type N , III, II and I; for a concise
discussion of the Petrov classiﬁcation of the Weyl tensor using spinors,
see Stewart (1991). For Petrov type N Weyl tensors the spinor ΨABCD has
four repeated principal null directions. They are associated to gravitational
plane waves. Similarly, a spacetime with a Weyl spinor of Petrov type III has
three repeated principal null directions; one of Petrov type II has two principal
directions, while one of Petrov type I is algebraically general. The observationthat a repeated principal null direction is lost at each order in the expansion(10.20) justiﬁes the name of peeling in analogy to the peeling of a fruit; see
Figure 10.1.

10.3 The Newman-Penrose gauge

231

Figure 10.1 Schematic representation of the Peeling theorem: the leading
behaviour of the Weyl tensor corresponds to that of a plane wave (Petrov
type N ). More general behaviour is observed as one looks into higher orderterms.

Remark. The key assumption in the derivation of the peeling behaviour is thesmoothness of the Weyl tensor at I + . A careful inspection of the arguments inthe previous sections shows that the smoothness requirement can be relaxed andthat the conclusions of Theorems 10.3 and 10.4 can be recovered if it is assumed
that ΨABCD is of class C k∗ at I + for some positive integer k∗ . A determinationof a sharp value of k∗ will not be pursued here. One of the challenges in theconstruction of spacetimes satisfying the peeling behaviour or, more generally,
spacetimes which are asymptotically simple is to ensure that their Weyl tensor
has the required regularity at the conformal boundary. The latter will be arecurrent idea in the remainder of this book. The analysis of the non-linear
stability of the Minkowski spacetime in Christodoulou and Klainerman (1993)
renders a Weyl tensor with a limited regularity at I + for which only a partial
peeling behaviour of the form
ψ̃0 = O(r̃−1 ),ψ̃1 = O(r̃−2 ),ψ̃3 = O(r̃−7/2 ),ψ̃2 = O(r̃−3 ),ψ̃4 = O(r̃−7/2 ),can be recovered; see, for example, Friedrich (1992) for a discussion.
10.3 The Newman-Penrose gauge
The analysis leading to the Peeling theorem shows the advantages of using agauge which is adapted to the geometry of null inﬁnity. In this section this idea
is further elaborated. The resulting Newman-Penrose gauge allows one toobtain further insights into the properties of asymptotically simple spacetimes.
10.3.1 The construction of the gauge
As in the previous section let (M, g, Ξ) denote a conformal extension of an
asymptotically simple spacetime (M̃, g̃) with λ = 0. For conciseness, thesubsequent discussion will be restricted to future null inﬁnity I + . An analogous
discussion can be readily adapted for I − .

232

Asymptotics

Figure 10.2 Schematic representation of the setting for the construction of
the NP gauge. The NP gauge is based on a ﬁduciary cut C on I + and isvalid in a neighbourhood U of the conformal boundary. The vector e11 istangent to the generators of null inﬁnity, while e00 generates the outgoingnull hypersurfaces Nu• . See the main text for further details.

In what follows, let {eAA } be a frame satisfying g(eAA , eBB  ) = AB A B 
deﬁned in a neighbourhood U of I + . The frame will be said to be adapted to
I + if – see Figure 10.2:(i) The vector e11 is tangent to I + and is parallely propagated along its
generators; that is, one has∇11 e110.

(ii) On U there exists a function u (a retarded time) which can be regarded1.
as an aﬃne parameter of the generators of I + such that e11 (u)The retarded time is constant on null hypersurfaces transverse to I + andsatisﬁes e00 = g  (du, ·). It follows that e00 is tangent to the hypersurfaces
Nu• = {p ∈ U | u(p) = u• },where u• is a constant. Moreover, e00 is tangent to the null generators ofNu• .
(iii) The ﬁelds {eAA } are tangent to the cuts Cu• ≡ Nu• ∩ I + and parallely
propagated along the direction of e00 . That is, one has∇00 eAA = 0

on

Nu• .

Using the deﬁnition of the spin-connection coeﬃcients it follows from the above
requirements that
Γ10 11Γ10 00 = Γ̄1 00 0 ,0,Γ11 110,Γ11 00 = Γ̄1 00 1 + Γ01 01
Γ00 AB = 0

on U .

(10.21a)on U ,(10.21b)(10.21c)10.3 The Newman-Penrose gauge

233

The condition Γ10 11 0 is, in fact, another way of expressing the fact that thecongruence of null generators of I + is shear free. This can be seen by evaluating
the conformal ﬁeld Equation (8.35c)∇AA ∇BB  Ξ = −ΞLAA BB  + sAB A B 
at I + for AA BB  = 10 10 . It follows that Γ10 11 e00 (Ξ)
so that one concludes Γ10 11 0 as claimed.

(10.22)0, but e00 (Ξ) 

0Remark. The discussion of the previous sections shows that an adapted frame
can always be obtained in a neighbourhood U of I + . The key observation isthat N = g  (dΞ, ·) is tangent to the null generators of I + so that one can set
e11 proportional to N . A suitable choice of aﬃne parameter for N renders theretarded time u and hence the frame vector e00 . The rest of the frame is thennaturally completed by looking at a basis on the tangent bundle of the cuts Cu• .
Following the ideas of Section 10.1.2, the gauge can be further specialised byconsidering a suitable conformal rescaling. Accordingly, considerg → g  = ϑ2 g,Ξ → Ξ = ϑΞ.

(10.23)

The above rescaling will be used to obtain an improved adapted frame {eAA }.
For an arbitrary conformal factor ϑ > 0 and an arbitrary function κ > 0 whichis constant along the generators of I + sete11ϑ−2 κe11 ;(10.24)compare the discussion in Section 10.1.2. In addition, deﬁne a further parameteru = u (u) such that du /du = κ −1 ϑ2 . Integrating along the generators of nullinﬁnity one ﬁnds that
1 u 2
ϑ (s)ds + u .
u =
κ u
The real constants u and u are ﬁxed so that they identify a certain ﬁduciary
is used to denote
cut C ≡ Cu . In what follows, for convenience, the symbol
equality at C . It can be veriﬁed that e11 is parallely propagated and thate11 (u ) = 1. The transformation rule (10.24) is supplemented at C by
e00



κ −1 e00 ,

e01



ϑ−1 e01 .

(10.25)

It can be veriﬁed that g (eAA , eBB  ) = AB A B  on C .
As seen in Section 10.1.2, C ≈ S2 so that the metric k induced by g  on C
is conformal to the standard metric σ of S2 . Accordingly, the conformal factorϑ can be chosen on C so that k σ. A calculation using the transformation
laws of Chapter 5 shows that the rescaling (10.23) and the conditions (10.24)
and (10.25) imply on CΓ10 00 = κ −1 Γ10 00 − ϑ−1 e00 (ϑ) ,(10.26a)



−2
−1
Γ01 11 + ϑ e11 (ϑ) .
(10.26b)Γ01 11 = κϑ

234

Asymptotics

Hence, by a suitable choice of dϑ and κ it is possible to ensure thatΓ10 000,

Γ01 11e00 (Ξ )0,constant = 0.

(10.27)

A convenient way of prescribing the conformal factor ϑ oﬀ C follows fromthe transformation law for the trace-free part of the Ricci tensor Φab under the
rescaling (10.23):


1

−1
−1
c
−1
c
Φab − Φab =− 2ϑ (∇a ∇b ϑ − 2ϑ ∇a ϑ∇b ϑ) − gab (∇c ∇ )ϑ − 2ϑ ∇c ϑ∇ ϑ) ;4
see Equation (5.6a). Transvecting this last equation with e11 ⊗ e11 it follows
that if ϑ satisﬁes the equation2ϑΦ22 ,(10.28)e11 (e11 (ϑ)) − 2ϑ−1 e11 (ϑ)
then Φ22
0. By means of the substitution z = ϑ−1 , Equation (10.28) canbe read as a second-order linear ordinary diﬀerential equation for ϑ−1 along
the generators of I + . Thus, this equation can always be solved, at least in aneighbourhood of C on I + to ensure that
Φ22

0.

(10.29)This last construction also ﬁxes the value of e01 (ϑ) on I + .
The initial data for Equation (10.28) on the ﬁduciary cut C is chosen so thate11 (ϑ) −Γ01 11 consistent with Equation (10.27); compare Equation (10.26b).
Now, taking into account Equations (10.21a) and (10.29), one has that the
Ricci identity – compare the conformal ﬁeld Equation (8.35b) of Chapter 8 –gives for the values AA = 11 , BB  = 01 and CD = 11 that2
e11 (Γ01 11 ) + Γ01 11 + Γ01 11 Γ̄1 10 1 = 0.
The latter equation can be interpreted as a homogeneous diﬀerential equation
along the generators of I + for the reduced spin connection coeﬃcient Γ01 11 . Asa consequence of the initial condition (10.27) on C , it follows that Γ01 11 0.
The construction described in the previous paragraphs provides a speciﬁcationof the conformal factor ϑ and of the function κ which ﬁxes the frame vector e11
completely on I + . Notice, however, that the vectors e01 and e10 (tangent tothe cuts Cu• ) are determined up to a rotation of the forme01 → eic e01 ,e10 → e−ic e10 ,(10.30)with c a real phase on I + . A rotation on T (Cu• ) can be exploited to obtain
additional simpliﬁcations in the spin connection coeﬃcients. A calculation using
the deﬁnition of the spin connection coeﬃcients and taking into account that∇11 e11 0 gives that
Γ11 011
− ω 10 , ∇11 e10 .
2

10.3 The Newman-Penrose gauge

235

Under the rotation (10.30) the above relation transforms asΓ11 01 →i 
1e  (c) − Γ11 01 ,2 11
2

on I + .

Thus, given a particular choice of vectors e01 and e10 on I + , by solving theequatione11 (c)i− Γ11 01 ,2

with c

0,along the generators of I + , it is always possible to rotate the basis according
0. In the following, it will be assumedto (10.30) so as to ensure that Γ11 01
that e01 and e10 have been chosen so that the latter is the case.
The choice of vectors e01 and e10 has some further consequences. Evaluating
the primed version of Equation (10.22) at I + for AA = 01 and BB  = 10 one
ﬁnds that ∇01 ∇10 Ξ −s . Now, as Ξ = 0 on I + and e01 is tangent to I + ,it follows from ∇01 ∇10 Ξ = ∇01 e10 (Ξ ) 0 that s 0 and that
∇AA ∇BB  Ξ0.

(10.31)This last expression can be regarded as a strengthened version of the asymptotic
Einstein condition (10.4).In particular,
for AA = 11 and BB  = 00 Equation

0 so that e00 (Ξ ) is constant along the(10.31) implies that ∇11 e00 (Ξ )generators of I + . Moreover, setting AA = 00 and BB  = 01 and using thate01 (Ξ ) 0 one ﬁnds thatΓ01 Q 0 eQ0 (Ξ ) + Γ̄1 0 Q 0 e0Q (Ξ )0.

Expanding and using, again, that e01 (Ξ ) 0 and recalling (10.21b) one ﬁnds
0. However, e00 (Ξ )  0 so that one concludes thatthat Γ11 00 e00 (Ξ )Γ11 00 0.
To conclude, it is observed that although Equation (10.28) ﬁxed the derivativee11 (ϑ) along I + , the derivative e00 (ϑ) still remains free. A convenient way ofﬁxing e00 (ϑ) can be obtained from the transformation law for the Ricci scalar– see Equation (5.6c) – which, in the present context, takes the formR[g  ] = ϑ−2 R[g] + 12ϑ−2 ∇a ϑ∇a ϑ − 6ϑ−1 ∇a ∇a ϑ.
A natural requirement is to set R[g  ] = 0 on I + so that along the generators of
I + one obtains the equatione11 e00 (ϑ) − 2ϑ−1 e11 (ϑ)e00 (ϑ) F  ,(10.32)whereF ≡ Re



e01


 

1 −1


−1 

e10 (ϑ) − 2Γ01 01 e10 (ϑ) − 2ϑ e01 (ϑ)e10 (ϑ) + ϑ R[g] .
12Equation (10.32) can be regarded as a linear diﬀerential equation for e00 (ϑ)
along the generators of I + with a non-homogeneous term F  which consists of236

Asymptotics

quantities which are already known along I + . Equation (10.32) is supplemented
by the condition e00 (ϑ) = κ −1 ϑΓ10 00 0 consistent with Equation (10.27). It
follows thatR[g  ]

0.

(10.33)Using the Ricci identity, Equation (8.35b), taking into account the conformal
gauge condition (10.33) and the conditions on the spin connection coeﬃcients,
gives for the values AA = 11 , BB  = 10 and CD = 00 a homogeneous ordinary
diﬀerential equation for Γ10 00 along the generators of I + . Observing the initialcondition (10.27) the latter implies that Γ10 00 0. Finally, a further use of the
Ricci identities gives Φ12 = Φ21 0.
The construction of the previous paragraphs is rounded up with the introduction of adapted coordinates. On the ﬁduciary cut C ≈ S2 one chooses somecoordinates θ = (θA ) A = 2, 3 and extends them along I + by requiring them
to be constant along the null generators. On the hypersurfaces Nu transverseto I + it is natural to identify an aﬃne parameter r of the null generators ofthese hypersurfaces in such a way that e00 (r ) = 1 and r 0. The coordinatesθ = (θA ) are propagated oﬀ I + in such a way that they are constant along
the generators of Nu . As a result of this construction one obtains Bondi
coordinates x = (u , r , θA ) in the neighbourhood U of I + .
Summary of the construction
The lengthy construction in this section can be summarised in the following
proposition (for ease of presentation the  in the objects associated to theimproved adapted frame has been dropped from the expressions):
Proposition 10.1 (the NP gauge at I + ) Let (M̃, g̃) denote an asymptotically simple spacetime. Locally, it is always possible to ﬁnd a conformal extension(M, g, Ξ) for whichR[g]0and an adapted frame {eAA } such that the associated spin connection coeﬃcients ΓAA BC satisfy

Γ01 11

Γ00 BC0,Γ11 BC0,0,Γ10 000,Γ10 11Γ̄1 00 1 + Γ01 01

0.

Φ12

0,

In addition, one has thatand e00 (Ξ) is constant on I + .

0,

Φ22

0

10.3 The Newman-Penrose gauge

237

A quick inspection reveals that in the gauge associated to Proposition 10.1
the only non-zero spin connection coeﬃcients on I + are given by Γ01 00 , Γ00 01
and Γ10 01 which in standard NP notation correspond, respectively, to σ, α, β.
0 and describe,
On I + the connection coeﬃcients α and β satisfy α + β̄essentially, the connection of the intrinsic metric of the cuts of I + ; that is,the connection of the standard metric of S2 , σ. The remaining spin connection
coeﬃcient, σ = Γ01 00 , encodes the (non-trivial) dynamical degrees of freedom
in the set up. Its relation with the notion of gravitational radiation will be
brieﬂy explored in the next subsection.
10.3.2 The radiation ﬁeld and the news function
To explore the relation between the spin connection coeﬃcient σ and the notionof gravitational radiation it is convenient to expand the Ricci, Cotton and Bianchi
identities – that is, the conformal ﬁeld Equations (8.35b), (8.37a) and (8.37b) – interms of the gauge given by Proposition 10.1. An inspection of the componentsof the Ricci identity not used in the derivation of the NP gauge, taking into
account that ΨABCD 0, provides the relations
Φ00

−σσ̄,

−ð̄σ,

Φ01

Φ02

−σ̇,

where ˙ denotes diﬀerentiation with respect to the retarded time u. In addition,
one also ﬁnds
Φ11

δα − δ̄β + 4αβ.

As α and β describe the Levi-Civita connection of the standard metric of S2 , it
can be readily veriﬁed that Φ11 corresponds, essentially, to the curvature of S2 –recall that in two-dimensional manifolds the curvature is encoded in the Ricci
scalar.
The relation between σ and the components of the rescaled Weyl tensor canbe established by inspection of the Bianchi identity (8.37b) at I + . Choosing,for convenience Ξ so that e00 (Ξ) −1, one ﬁnds that
φ4

¨,
−σ̄

φ3

˙
−ðσ̄.

Moreover, one also obtains the constraint
φ2 + σ σ̄˙ + ð2 σ̄

φ̄2 + σ̄ σ̇ + ð̄2 σ.

In view of the Peeling theorem, Theorem 10.4, the component φ4 describes
the leading term of the gravitational ﬁeld – the so-called radiation ﬁeld or0,outgoing ﬁeld . In particular, if σ̄˙ is constant along I + one has that φ4
φ3 0, and one interprets this situation as describing the absence of gravitationalradiation – that is why σ̄˙ is sometimes called the news function. The
component φ2 is interpreted as describing the Coulomb part of the gravitational
ﬁeld while φ1 and φ0 are associated with incoming radiation; see Szekeres(1965).

238

Asymptotics
10.4 Other aspects of asymptotics

The present chapter provides a minimalistic account of the theory of asymptoticsof the gravitational ﬁeld. A detailed account would go beyond the scope of thisbook. It is, nevertheless, of interest to brieﬂy highlight certain topics.
10.4.1 The Bondi mass
The analysis of the asymptotics of the gravitational ﬁeld allows one to describe
in a rigorous manner the loss of energy of an isolated system due to gravitationalradiation. This physical process is described in terms of the so-called Bondi mass;
see Trautman (1958), Bondi et al. (1962), Sachs (1962b) and also Penrose (1965).
In terms of the notation introduced in this chapter, the Bondi mass mB overa cut C of I + is given by the surface integral
1φ2 + σ σ̄˙ dS.
mB ≡ −
2 C
A concise deduction of the above expression can be found in Stewart (1991).
Moreover, it can be shown that under suitable assumptions mB ≥ 0; see
Ludvigsen and Vickers (1981, 1982). A further calculation renders that1
|σ̇|2 dS ≤ 0.
ṁB = −2 C
The above inequality is called the Bondi mass-loss formula and encodes theloss of mass of an isolated system due to the energy that is carried away by(outgoing) gravitational radiation.
10.4.2 The Bondi-Metzner-Sachs groupAs already mentioned, one of the central objectives of the theory of asymptoticsof the gravitational ﬁeld is to identify universal structures in a wide class ofspacetimes and, in turn, use these to extract physical insight into the behaviourof isolated systems in general relativity. An example of this type of universal
structures is given by the so-called Bondi-Metzner-Sachs (BMS) group;
see Sachs (1962a), Bondi et al. (1962)and Newman and Penrose (1966).
In what follows let (u, r, θA ) denote a Bondi coordinate system deﬁned ina neighbourhood of the future null inﬁnity I + of an asymptotically simple
spacetime. The BMS group is deﬁned by the following transformations on the uand θ = (θA ) coordinates:(10.34a)u = K(θ) u − α(θ) ,θA = θA (θ2 , θ3 ),A

A(10.34b)where the map (θ ) → (θ ) is a conformal transformation of S onto itself, andK(θ) is the associated conformal factor so that2σ  = K 2 σ,10.4 Other aspects of asymptotics

239

and where α(θ) is an arbitrary smooth real function on S2 . The particular
BMS transformations for which θA = θA are called supertranslations. Undera supertranslation, the system of null hypersurfaces Nu• with u• constant istransformed into a diﬀerent system Nu• . Expanding the function α(θ) in termsof spherical harmonics Ylm – see the Appendix to this chapter – one ﬁnds that∞lα(θ) =

alm Ylm ,l=0 m=−lwith alm ∈ C. Thus, the supertranslations are an inﬁnite-dimensional subgroupof the BMS group. The particular (four-dimensional) case for which alm = 0 forl > 2 is called the translations subgroup.
Generic asymptotically simple spacetimes do not possess Killing vectors – inthe conformal picture Killing vectors of the physical spacetime correspond to
conformal Killing vectors. The BMS group arises from a notion of asymptotic
symmetries which ensures the existence of non-trivial solutions for genericspacetimes, that is, a diﬀeomorphism ϕ : I + → I + satisfying the conditionsϕ∗ q = ϑ2 q,

ϕ∗ N = ϑ−1 N ,(10.35)for some function ϑ > 0 and where the tensor ﬁelds q and N are as given
in Section 10.1.2. It can be veriﬁed that the BMS transformations (10.34a)and (10.34b) satisfy the conditions in (10.35) with K = ϑ. A particular typeof asymptotic symmetries corresponds to those generated by an asymptotic
Killing vector , that is, a ﬁeld ξ on I + satisfying the conditions£ξ q = 2ϑq,

£ξ N = −ϑN .

Given an asymptotically simple spacetime (M̃, g̃) endowed with a Killing
vector ξ̃, let (M, g, Ξ) denote a conformal extension thereof. Given that 0 =£ξ̃ g̃ = £ξ̃ (Ξ−2 g), it follows that£ξ̃ g = 2 Ξ−1 ξ̃(Ξ) g,(10.36)for Ξ = 0, so that ξ̃ is a conformal Killing vector of g on M̃. Since this vector
is determined by the smooth metric g, it extends smoothly to I + as a vector
ξ. Now, the left-hand side of Equation (10.36) extends smoothly to I + , and,
therefore, the right-hand side does so too. It follows that
ξ(Ξ) = α Ξ,(10.37)

with α a smooth function such that α = O(Ξ0 ) so that ξ is tangent to I + .
From Equation (10.36) one concludes that£ξ q = 2α q,£ξ N = −α N .

Accordingly, any Killing vector of (M̃, g̃) admits a unique extension to avector on I + and which deﬁnes an asymptotic Killing vector. The maximum

240

Asymptotics

number of linearly independent Killing vectors in a four-dimensional manifold
is 10. Accordingly, Killing vectors can give rise, at most, to 10 asymptotic
Killing vectors. By means of a direct calculation, it is possible to show thatthe function α in Equation (10.37) and the function α appearing in (10.34a)
are the same. Thus, the BMS transformations (10.34a) and (10.34b) are
asymptotic symmetries. In particular, the translations subgroup can be put incorrespondence with the asymptotic Killing vectors arising from translations in
the Minkowski spacetime.
For further details on the structure and properties of the BMS group, see, forexample, Penrose and Rindler (1986) and Schmidt et al. (1975). A discussionof the properties of Killing vectors in asymptotically simple spacetimes can befound in Ashtekar and Xanthopoulos (1978) and Ashtekar and Schmidt (1980).
10.4.3 Newman-Penrose constants
In Newman and Penrose (1965) – see also Newman and Penrose (1968) andPenrose and Rindler (1986) – it has been shown that in an asymptotically simple
spacetime (M̃, g̃) there exists a set of ten quantities deﬁned as integrals overcuts of null inﬁnity which are absolutely conserved in the sense that their value
is independent of the particular cut C on which they are evaluated – the socalled Newman-Penrose constants. In terms of the adapted frame {eAA }of Proposition 10.1 these constants are given by

Gm ≡

C

2 Ȳ2m e00 (φ0 )dS,m = −2, . . . , 2,

where 2 Y2m is a spin-weighted spherical harmonic; see the Appendix tothis chapter. A discussion of the relation between the above expression and theoriginal formula of Newman and Penrose can be found in Friedrich and Kánnár(2000a).
There exists no general consensus about the physical meaning or interpretationof the Newman-Penrose constants. An explicit computation for stationary
spacetimes shows that they are of the form
(mass) × (quadrupole) − (dipole)2 ;see, for example, Bäckdahl (2009). Evaluations of the Newman-Penrose spacetimes for dynamic spacetimes can be found in Friedrich and Schmidt (1987) andFriedrich and Kánnár (2000a). In particular, in the former reference it is shownthat for spacetimes possessing a conformal extension which includes the points
i+ and i− the Newman-Penrose constants correspond, essentially, to the value ofthe rescaled Weyl spinor φABCD at those points. Electrovacuum asymptoticallysimple spacetimes have a suitable generalisation of these absolutely conserved
constants; see Exton et al. (1969).

10.5 Further reading241

10.5 Further reading
An excellent introduction to the theory of asymptotics of the gravitational ﬁeld
is given in Stewart (1991) where the subject is called “asymptopia”. A related
account can be found in Penrose and Rindler (1986). A convenient entry point to
the extensive literature on the subject can be found in the review of Frauendiener(2004). A detailed discussion of the ideas and general philosophy behind thetreatment of the asymptotics of the gravitational ﬁeld by means of conformal
methods can be found in Geroch (1976). Accounts similar in spirit to the lattercan be found in Ashtekar (1980, 1987). A slightly diﬀerent perspective on thesubject can be found in Friedrich (1992); see also Friedrich (1998a, 1999). A
recent review on the subject of asymptotics is given in Ashtekar (2014).
Appendix: spin-weighted functions
Let {o, ι} denote a spinorial dyad deﬁned on a spacetime (M, g) and let{l, n, m, m̄} denote the associated null tetrad. As discussed in Section 3.1.10, thenull vectors m and m̄ span a spacelike subspace of T (M) which is orthogonal toboth l and n. Of particular interest is the case when this subspace corresponds
to the tangent bundle of a compact two-dimensional submanifold C of M. In
the following it is assumed that this is the case. From the expressiong = l ⊗ n + n ⊗ l − m ⊗ m̄ − m̄ ⊗ mof the metric g in terms of the null tetrad, it follows that the intrinsic metric σ
induced by g on C is given by
σ = −m ⊗ m̄ − m̄ ⊗ m.
There is a certain gauge freedom in the above expression since spin-boosts ofthe form1o → e 2 ic o,ι → e− 2 ic ι,1(10.38)with arbitrary c ∈ R which imply the transitionm → eic m,m̄ → e−ic m̄,leave the metric σ unchanged.
Given a spinor ηA1 ···An A1 ···Am of valence n + m, it is natural to consider the
behaviour of its components with respect to the dyad {o, ι} under the spin boost(10.38). For example, given p, q, r, t ∈ N such that p + q = n, r + t = m, thescalarη ≡ ηA1 ···Ap B1 ···Bq A1 ···Ar B1 ···Bt oA1 · · · oAp ιB1 · · · ιBq ōA1 · · · ōAr ῑB1 · · · ῑBt
(10.39)has a transformation given by1η → e 2 i(p+t−q−r)ϑ η.

242

Asymptotics

One says, then, that η has spin weight s = 12 (p+t−q−r). The spin weight of all
the possible components of ηA1 ···An A1 ···Am lies in the range −m − n ≤ s ≤ m + n.
In what follows, we adopt the standard Newman-Penrose conventions to denotethe directional covariant derivatives with respect to m and m̄ and let δ ≡ ma ∇a ,δ̄ ≡ m̄a ∇a . Generically, the directional derivatives δ and δ̄ acting on a spinweighted scalar do not give rise to scalars with a well-deﬁned spin weight. To
amend this deﬁciency it is convenient to deﬁne operators ð and ð̄ which, acting
on scalars with a given spin weight, give rise to new scalars with a well-deﬁned
spin weight. Given the spin-weighted scalar η of Equation (10.39), the action ofð and ð̄ is deﬁned to be








ðη ≡ oA1 · · · oAp ιB1 · · · ιBq ōA1 · · · ōAr ῑB1 · · · ῑBt× δ(η ιA1 · · · ιAp oB1 · · · oBq ῑA1 · · · ῑAr ōB1 · · · ōBt ),ð̄η ≡ oA1 · · · oAp ιB1 · · · ιBq ōA1· · · ō

Ar

ῑ

B1

· · · ῑ

(10.40a)Bt× δ̄(η ιA1 · · · ιAp oB1 · · · oBq ῑA1 · · · ῑAr ōB1 · · · ōBt ).

(10.40b)

The operators ð and ð̄ are complex conjugates of each other in the sense thatðη = ð̄η̄. If the scalar η has spin weight s, one can verify that ðη and ð̄η have,respectively, spin weights s + 1 and s − 1. Furthermore, ðη and ð̄η satisfy the
Leibnitz rule. In order to obtain alternative expressions for ðη and ð̄η let
α ≡ oA δ̄ιA = ιA δ̄oA ,β ≡ oA διA = ιA δoA ,consistent with standard Newman-Penrose notation. Expanding (10.40a) and(10.40b) and using the above deﬁnitions one obtains
ðη = (−1)p+r (δη + ((q − p)β + (t − r)ᾱ) η) , ð̄η = (−1)p+r δ̄η + (q − p)β̄ + (t − r)α η .
A computation with the above expressions shows that(ð̄ð − ðð̄)η = sη.
The above expressions are convenient for the discussion of spin-weighted
harmonics. In terms of standard spherical harmonics Ylm , these are given by
0 Ylm ≡ Ylm ,and for s = 0

⎧⎪2s (l − s)! s
⎪
⎪
(−1)s
ð Ylm
⎪
⎪
⎪
(l + s)!
⎨s Ylm ≡
(l + s)! −s⎪ð̄ Ylm⎪⎪⎪2s (l − s)!
⎪
⎪
⎩
0

0<s≤l
−l ≤ s < 0otherwise;see, for example, Stewart (1991) for further discussion.
Of special relevance for Theorem 10.3 is the following result:10.5 Further reading243

Lemma 10.1 Assume C to be diﬀeomorphic to S2 and let η denote a smoothscalar on C having spin weight s. If ðη = 0 and s < 0, then η = 0. Similarly, ifð̄η = 0 and s > 0, then η = 0.
Proofs of this result can be found in Penrose and Rindler (1984) and Stewart(1991). Remarkably, the result depends on the topology (genus) of C ; see
Frauendiener and Szabados (2001). For example, the above result is not valid
for surfaces diﬀeomorphic to the 2-torus S × S. It is of interest to point out
that Lemma 10.1 is equivalent to the statement that there exist no non-zerosymmetric trace-free, divergence-free, rank 2 tensor ﬁelds on S2 ; see Beig (1985)
and Frauendiener and Szabados (2001).

Part III
Methods of the theory of partial
diﬀerential equations

11
The conformal constraint equations

This chapter analyses the intrinsic equations implied by the conformal Einsteinﬁeld equations on non-null hypersurfaces. These equations are known as the conformal constraint equations. They play an essential role in the constructionof initial data sets for the conformal ﬁeld equations and in the identiﬁcation ofboundary conditions. Not surprisingly, these conformal constraint equations areclosely related to the standard Einstein constraint equations – consequently,
this chapter starts by considering the properties of the latter.
The solvability and behaviour of solutions to the conformal constraint
equations is closely related to the nature of the underlying three-dimensional
manifold on which the equations are imposed. As a consequence, this chapter
also provides a discussion of general properties of asymptotically Euclidean andasymptotically hyperboloidal 3-manifolds from a conformal point of view. The
systematic analysis of the constraint equations relies on methods of elliptic
partial diﬀerential equations. Hence, this chapter provides a discussion of someof the basic notions of this theory.
An important aspect of the conformal constraint equations – the so-called
propagation of the constraints – is discussed in Chapter 13. The analysis of theconstraint equations on null hypersurfaces is treated in Chapter 18.

11.1 General setting and basic formulae
Let (M̃, g̃) denote a spacetime satisfying the Einstein ﬁeld equations. In what
follows, it will be assumed that (M̃, g̃) can be conformally extended to an
unphysical spacetime (M, g). Accordingly, there exists an embedding φ : M̃ →M and a conformal factor Ξ such that φ∗ g = Ξ2 g̃. Now, let S̃ denote a threedimensional submanifold of M̃ and let ϕ : S̃ → M̃ denote the associatedembedding. As the composition φ ◦ ϕ : S̃ → M is also an embedding, thethree-dimensional manifold S̃ can be regarded, in turn, as a submanifold of M.
As discussed in Section 2.7.3, the spacetime metric g̃ induces a metric h̃ on S̃ via

248

The conformal constraint equationsh̃ = ϕ∗ g̃. Similarly, regarding S̃ as a hypersurface on M, the unphysical metricg also induces a metric h via the pull-back h = (φ ◦ ϕ)∗ g. A calculation showsthat
h = (φ ◦ ϕ)∗ g = (ϕ∗ ◦ φ∗ )g = ϕ∗ (Ξ2 |S̃ g̃) = Ω2 ϕ∗ g̃where Ω ≡ Ξ2 |S̃ is the restriction of Ξ to the hypersurface S̃. Following theconventions of previous chapters, h = Ω2 ϕ∗ g̃ will often be written as
h = Ω2 g̃.
Now, let ν̃ and ν denote, respectively, the g̃-unit and g-unit normals of S̃ anddeﬁne ≡ g̃(ν̃, ν̃) = g(ν, ν).
In accordance with the signature convention (+ − −−), the hypersurface S̃ isspacelike if  = 1 and timelike if  = −1. It follows thatν = Ξν̃,ν  = Ξ−1 ν̃ or, using index notation, νa = Ξν̃a and ν a = Ξ−1 ν̃ a . In what follows, the indicesof objects in M̃ are raised/lowered using the metric g̃, while the indices of objects
on M are moved using g.
11.1.1 The transformation formulae for the extrinsic curvature
Having discussed the relation between the 3-metrics and the unit normals to S̃,one is in the position to consider the relation between the extrinsic curvaturesK̃ and K. Given spatial vectors u, v ∈ T (S̃) – so that ν̃, u = ν̃, v = 0 – one
has that˜ u ν̃, v,K̃(u, v) = ∇K(u, v) = ∇u ν, v;˜ = S(Υ) one readily has thatsee Equation (2.43). Recalling that ∇ − ∇˜ u ν − S(Υ, ν; u);∇u ν = ∇the minus sign arises from the fact that ν is a covector. In abstract index notation
S(Υ, ν; u) is given by Sab cd Υc νd ub from where a short calculation gives that˜ c Ξν̃d ub ,Sab cd Υc νd ub = Sab cd ∇˜ c Ξν̃d˜ b Ξν̃a ub − g̃ab ub g̃ cd ∇=∇˜ c Ξ)ν̃a − ΞΣg̃ac uc ,= (uc ∇where
Σ ≡ g ab ∇a Ξνb = g  (dΞ, ν) = Ξ−1 g̃  (dΞ, ν̃)11.1 General setting and basic formulae249

is the derivative of Ξ in the direction of the g-unit normal to S̃. Accordingly, one
has thatS(Υ, ν; u) = u(Ξ)ν̃ − ΞΣg̃(u, ·),from where, recalling that u, v ∈ T (S̃) so that g̃(u, v) = h̃(u, v), it follows that˜ u ν, v − u(Ξ)ν̃, v + ΩΣg̃(u, ·), v
K(u, v) = ∇˜ u ν̃, v + ΩΣh̃(u, v)
= Ω∇


= Ω K̃(u, v) + Σh̃(u, v) ,where to pass from the ﬁrst to the second line it has been used that ν̃, v = 0
as T (S̃).
Summarising, the calculations in the previous paragraphs show thathij = Ω2 h̃ij ,
Kij = Ω K̃ij + Σh̃ij .

(11.1a)(11.1b)

These are the basic transformation formulae for the remainder of this chapter.
Taking the trace of the transformation formula for the extrinsic curvature,
Equation (11.1b), it follows thatΩK = K̃ + 3Σ,
where K̃ ≡ h̃ij K̃ij and K ≡ hij Kij – these scalars are sometimes called,respectively, the physical and unphysical mean curvature of S̃ . The scalars
Σ, K admit a geometric interpretation: if Σ = K = 0, then, necessarily, K̃ = 0
and the hypersurface S̃ is maximal in M̃ with respect to both the metrics g̃
and g – that is, it encloses a maximum volume for a given area.
11.1.2 Decompositions in electric and magnetic parts
A key ingredient in the analysis of the conformal constraint equations is thedecomposition in electric and magnetic parts of tensors with antisymmetric pairs of indices. Let S denote a hypersurface on a spacetime (M, g), and letν denote the unit normal to the hypersurface. The projector to S is the tensor
ha b given byha b ≡ δa b − νa ν b .
It follows thatha b νb = 0,

h a b hb c = ha c .

Furthermore, using the properties of the spacetime volume form abcd – see
Section 2.5.3 – one can deduce that1ha [c hb d] = − abe cde ,(11.2)
2

250

The conformal constraint equations

where abe ≡ f abe ν f is the three-dimensional volume form.
∗≡
Now, let Fab denote an antisymmetric tensor of rank 2 and let Fab1
cd− 2 ab Fcd denote its Hodge dual. Its electric and magnetic parts are deﬁned,
respectively, to beFa ≡ Fcb ν b ha c ,∗ bFa∗ ≡ Fcb
ν ha c .

It can be veriﬁed thatFa ν a = Fa∗ ν a = 0,

h a b F b = Fa ,ha b Fb∗ = Fa∗ ,so that the electric and magnetic parts are said to be spatial tensors. Together,Fa and Fa∗ encode the same information as the original tensor Fab . In order to
see this, one writes
Fab = Fcd δa c δb d = Fcd (ha c + νa ν c )(hb d + νb ν d )= Fcd ha c hb d + Fcd ha c νb ν d + Fcd hb d νa ν c= 2F[a νb] + Fcd ha c hb d .

(11.3)The term Fcd ha c hb d is, in turn, manipulated using the identity (11.2) as follows:1
Fcd ha c hb d = Fcd ha [c hb d] = − Fcd f cde νf abe
2
∗ f
= Fef
ν ab e = Fe∗ ab e .

(11.4)

Thus, combining Equations (11.3) and (11.4), one concludes that
Fab = 2F[a νb] + Fe∗ e ab .
The decomposition in electric and magnetic parts can be extended to tensors
Wabcd with the same symmetries as the Weyl tensor; such tensors are sometimes
known as Weyl candidates. By analogy to the rank-2 case one deﬁnes theν-electric and ν-magnetic parts of Wabcd to beWac ≡ Webf d ν b ν d ha e hc f ,

∗
∗
b d
e f
Wac
≡ Webfd ν ν ha hc ,∗with Wabcd≡ − 12 cd ef Wabef denoting the right Hodge dual of Wabcd . In thesubsequent discussion it is convenient to consider

Wabc ≡ Wef gh ν f ha e hb g hc h .
It can be veriﬁed that1∗
= − Wacd b cd .
Wab2

11.1 General setting and basic formulae251∗
As in the rank-2 case, the tensors Wab and Wab(or, alternatively, Waband Wabc ) encode the same information as Wabcd . The argument to show thisequivalence is similar to that of the rank-2 case:Wabcd = Wef gh δa e δb f δc g δd h
= Wef gh (ha e + νa ν e )(hb f + νb ν f )(hc g + νc ν g )(hd h + νd ν h )= Wef gh ha e hb f hc g hd h + Wcab νd − Wdeb ha e νc + Wacd νb+ Wac νb νd − Wad νb νc − Wbcd νa − Wbc νa νd + Wbd νa νc .

(11.5)∗
From the deﬁnition of the magnetic part Wab
it follows that∗
Wabc = e bc Wae.

(11.6)

abc def = −6δa [d δb e δc f ] ,(11.7)

Moreover, using that

it follows that1
Wef gh ef z abz ghx cdx4∗
= ∗ Wrzsx
ν r ν s ab z cd x = −Wzx ab z cd x

Wef gh ha e hb f hc g hd h == Wca hbd + Wdb hac − Wcb had − Wda hbc .

(11.8)

Combining Equations (11.5), (11.6) and (11.8) one obtains the desired decom∗:
position of Wabcd in terms of Wab and Wab∗ e∗ e
Wabcd = 2(lb[c Wd]a − la[c Wd]b ) − 2(ν[c Wd]e ab + ν[a Wb]e
 cd ),(11.9)

where lab ≡ hab − νa νb . A similar computation renders
∗
∗
∗ e
Wabcd
= 2ν[a Wb]e e cd − 4We[a b] e [c νd] − 4ν[a Wb][cνd] − Wef ab f cd .

(11.10)

Expressions in terms of an adapted frame
The decomposition discussed in the previous paragraphs acquires a particularly
simple form when supplemented with a frame {ea } adapted to the hypersurface
S. For such a frame, the projection of a particular index with respect to thenormal corresponds to replacement of the corresponding frame index with ⊥while the spatial part of a tensor is given by the replacement of the spacetime
frame indices a , b , c , . . . with the spatial frame indices i , j , k , . . . In particular,
the three-dimensional volume form satisﬁes ijk = ⊥ijk , and the electric andmagnetic parts of the antisymmetric tensor Fab are represented, respectively, by
Fi = Fi⊥ ,∗Fi∗ = Fi⊥.

252

The conformal constraint equations∗
In the case of the Weyl candidate Wabcd one has that the tensors Wab , Waband
Wabc correspond to

Wij = Wi⊥j⊥ ,∗∗Wij= Wi⊥j⊥,Wijk = Wi⊥jk .

11.2 Basic notions of elliptic equations
Elliptic diﬀerential operators arise naturally in the study of the constraint
equations of general relativity on spacelike hypersurfaces. In view of this,some basic properties of elliptic operators on Riemannian manifolds are brieﬂy
discussed.
Let (S, h) denote a Riemannian three-dimensional manifold with h a negative
deﬁnite metric. A linear diﬀerential operator of order M over S is a map
between tensor bundles
L : Ti1 ···iS (S) → Tk1 ···kN (S),S, N ∈ N,of the formM

aj1 ···jr i1 ···iS k1 ···kN Dj1 · · · Djr vi1 ···iS ,(Lv)k1 ···kN ≡(11.11)

r=0

for a smooth vi1 ···iS ∈ Ti1 ···iS (S) and where the coeﬃcients aj1 ···jr i1 ···iS k1 ···kN
are smooth functions over S. The principal part of L consists of the terms inEquation (11.11) with the highest order derivatives, that is,
aj1 ···jM i1 ···iS k1 ···kN Dj1 · · · DjM vi1 ···iS .
Closely related to the principal part is the symbol of L, σ L (ξ), deﬁned pointwise
on S, for ξ ∈ T ∗ |p (S) as the linear map
σ L (ξ) : Ti1 ···iS |p (S) → Tk1 ···kN |p (S),given by(σ L (ξ)v)k1 ···kN ≡ aj1 ···jM i1 ···iS k1 ···kN ξj1 · · · ξjM vi1 ···iS .
Observe that the symbol is obtained by the formal replacement of thederivatives Di → ξi in the principal part of the operator. The symbol σ L (ξ)
determines the nature of the diﬀerential operator. In particular, L is said tobe underdetermined elliptic at p ∈ S if σ L (ξ) is surjective for all ξ = 0;
L is overdetermined elliptic at p ∈ S if σ L (ξ) is injective. Finally, L iselliptic if σ L (ξ) is bijective, that is, if it is injective and surjective. If thecoeﬃcients aj1 ···jr i1 ···iS k1 ···kN in the operator (11.11) depend not only on the
point on S but also on the derivatives Dj1 · · · Djl , l < r, then L is said to11.3 The Hamiltonian and momentum constraints

253

be quasilinear. The deﬁnitions of (underdetermined, overdetermined) elliptic
diﬀerential operators extend in a natural way to the quasilinear case.
The paradigmatic example of an elliptic operator is the Laplace operator of
the metric h:Δh φ ≡ hij Di Dj φ,

φ ∈ X(S).

In this case the operator is equal to its principal part. Moreover, its symbol
is given by hij ξi ξj < 0 for ξi = 0 (as a consequence of negative-deﬁniteness),from where it follows that the symbol is a bijection and, hence, Δh is an elliptic
operator. Particular examples of overdetermined and underdetermined elliptic
operators are discussed in Section 11.3.3.
Associated to the diﬀerential operator L in (11.11) one has its formal adjointL∗ given byM(L∗ u)i1 ···iS ≡(−1)r Dj1 · · · Djr (aj1 ···jr i1 ···iS k1 ···kN uk1 ···kN ),r=0

for smooth uk1 ···kN ∈ Tk1 ···kN (S). The above expression comes from the identity
between inner products
k1 ···kN(Lv)k1 ···kN udμh =vi1 ···iS (L∗ u)i1 ···iS dμh ,(11.12)S

S

which is obtained by repeated integration by parts. In the previous expression,dμh denotes the volume element of h. For simplicity, in the identity (11.12)
it is assumed that S is a compact manifold so that the integrals are welldeﬁned. Important for the subsequent discussion is the fact (veriﬁable using
the deﬁnitions given in the previous paragraphs) that L is an underdetermined
elliptic operator if and only if L∗ is overdetermined elliptic. Moreover, if L isunderdetermined elliptic, then L ◦ L∗ is elliptic.
The interested reader is referred to appendix II in Choquet-Bruhat (2008) for
further details on the theory of elliptic equations. An alternative summary canbe found in the appendix of Besse (2008).
11.3 The Hamiltonian and momentum constraints
Before proceeding to analyse the conformal constraint equations, it is convenient
to discuss the intrinsic equations implied by the Einstein ﬁeld equations
1
R̃ab − R̃g̃ab + λg̃ab = T̃ab2
on a non-null hypersurface of a spacetime (M̃, g̃) – the so-called Einsteinconstraint equations.

254

The conformal constraint equations
11.3.1 Derivation of the Einstein constraint equations

Starting from the Gauss-Codazzi identity, Equation (2.47) and contracting withh̃ik one obtainsr̃jl + K̃ K̃jl − K̃ k j K̃kl = h̃ik R̃ijkl= η ab R̃ajbl − R̃⊥j⊥l= R̃jl − R̃⊥j⊥l .
Contracting this last equation with h̃jl one ﬁnally obtainsr̃ + K̃ 2 − K̃jl K̃ jl = h̃jl R̃jl − h̃jl R̃⊥j⊥l= η ab R̃ab − R̃⊥⊥ − η ab R̃⊥a⊥b= R̃ − 2R̃⊥⊥ .
Similarly, starting from the Codazzi-Mainardi identity, Equation (2.48), andcontracting with h̃ij one has thatD̃j K̃kj − D̃k K̃ = h̃ij R̃i⊥jk= η ij R̃i⊥jk = R̃⊥k ,where to pass from the ﬁrst to the second line one uses that R̃⊥⊥jk = 0.
Using the Einstein ﬁeld equations in the frame component form
1R̃ab − ηab R̃ + ληab = T̃ab2
one obtains the so-called Einstein constraint equations
),r̃ + K̃ 2 − K̃jl K̃ jl = 2(λ − ˜

(11.13a)D̃j K̃kj − D̃k K̃ = j̃k ,(11.13b)where˜ ≡ T̃⊥⊥ ,j̃k ≡ T̃⊥kare, respectively, the energy density and the components of the energy ﬂux
vector of the energy-momentum tensor in the direction of ν̃. Equations (11.13a)and (11.13b) are known, respectively, as the Hamiltonian constraint andthe momentum constraint. The tensorial version of Equations (11.13a) and(11.13b) is given by),r̃ + K̃ 2 − K̃jl K̃ jl = 2(λ − ˜D̃j K̃kj − D̃k K̃ = j̃k .

(11.14)

Finally, it is observed that in index-free notation the constraint equations canbe written as),r[h̃] + (trh̃ K̃)2 − |K̃|2h̃ = 2(λ − ˜divh̃ K̃ − grad trh̃ K̃ = j̃.

11.3 The Hamiltonian and momentum constraints

255

In what follows, a collection (S̃, h̃, K̃, ˜, j̃) such that the negative deﬁnite
metric h̃ and the symmetric rank-2 tensor K̃ satisfy the Einstein constraints(11.14) with  = 1 on the three-dimensional manifold S̃ will be known as an
initial data set for the Einstein ﬁeld equations. If ˜ = 0 and j̃ = 0, one speaksof a vacuum initial data set.
An important class of initial data sets is that for which K̃ = 0 and j̃ = 0, sothat one is left only with the Hamiltonian constraint in the formr[h̃] = 2(λ − ρ̃).
Such an initial data set is called time reﬂection symmetric (or timesymmetric for short); it follows from the properties of the Einstein reduced
equations that for this type of initial data one has ∂t hαβ = 0 on the initial
hypersurface S̃ so that the resulting solution to the Einstein ﬁeld equations isinvariant under the replacement t → −t.
11.3.2 The conformal Hamiltonian and momentum
constraint equations
Regarding, as in Section 11.1, the three-dimensional manifold S̃ as a hypersurface
on both (M̃, g̃) and (M, g), it follows from a computation using the transformation rules (11.1a) and (11.1b) together with the transformation rules for the
Ricci scalar, Equation (5.16c), that Equation (11.14) can be reexpressed in termsof unphysical quantities as:
1
2ΩDi Di Ω − 3Di ΩDi Ω + Ω2 r − 3Σ22

1 2 2+ Ω K − Kij K ij + 2ΩΣK = λ − Ω4 ,
2




Ω3 Di Ω−2 Kik − Ω Dk K − 2Ω−1 Dk Σ = Ω3 jk ,(11.15a)(11.15b)where ≡ Ω−4 ˜,

jk ≡ Ω−3 j̃k ,(11.16)denote, respectively, the unphysical energy density and the ﬂux vector.
11.3.3 The Hamiltonian and momentum constraint
as an elliptic system
The Einstein constraint Equations (11.14) on a spacelike manifold S̃ (i.e.  = 1)
have been studied extensively in the literature; see, for example, Bartnik andIsenberg (2004) for a review of the topic and see also Choquet-Bruhat (2008),
chapter 7, and Choquet-Bruhat and York (1980). In this section an adaptationof the so-called conformal method of Licnerowicz, Choquet-Bruhat and Yorkto analyse the conformal Hamiltonian and momentum constraints (11.15a) and(11.15b) will be discussed; see, for example, York (1971, 1972). This approach

256

The conformal constraint equations

works directly on a compact unphysical manifold S which is a conformal
extension of the physical manifold S̃. The key idea in this analysis is to showthat these constraint equations imply an elliptic system of equations for suitableconformal ﬁelds. Proceeding in this way, one also obtains an insight into thenature of the freely speciﬁable data in the Einstein constraints. The use of acompact manifold S simpliﬁes some of the technical aspects of the analysis. Thisapproach to the Einstein constraint equations has been advocated in Friedrich(1988, 1998c, 2004, 2013), Dain and Friedrich (2001) and Beig and O’Murchadha(1991, 1994).
Following the discussion of the previous paragraph, let (S, h) denote a compact
Riemannian manifold with h negative deﬁnite and set  = 1 so that S can be
regarded as a spacelike hypersurface of an unphysical spacetime (M, g). In what
follows, for simplicity, it is assumed that the matter ﬁelds  and j are known
on S.
The ﬁrst step to transform Equations (11.15a) and (11.15b) into an elliptic
system is given by the transformation law of the three-dimensional Ricci scalar,
Equation (5.17), which suggests introducing a conformal factor ϑ satisfyingΩ = ϑ−2 . By substituting this deﬁnition into Equation (11.15a) one ﬁnds that

1
1
1
3
1
Δh ϑ − r[h]ϑ = (Kij K ij − K 2 )ϑ + (ϑ−3  − ϑ5 λ) + Σ2 ϑ5 − ϑ3 ΣK,
8
8
4
4
2
(11.17)
where, as before, Δh ≡ hij Di Dj and the notation r[h] has been used to makeexplicit the dependence of the Ricci scalar on the metric h. Following thestandard use in the literature, this equation will be known as the Licnerowicz
equation. If the ﬁelds h (and hence r[h]), Kij , K,  and Σ are known, this lastequation can be read as a non-linear elliptic equation determining ϑ. For futureuse, it is convenient to deﬁne the Yamabe operator Lh : X(S) → X(S) as
1
Lh ϑ ≡ Δh ϑ − r[h]ϑ,8(11.18)so that Equation (11.17) can be rewritten as
Lh ϑ =


1
1
1
1(Kij K ij − K 2 )ϑ + ( − λ)ϑ−3 + Σϑ3 K − ϑ2 Σ .
84
2
6

The Yamabe operator has nice conformal transformation properties; see
Equation (11.23) below.
Equation (11.15b) suggests that the extrinsic curvature Kij should be split
into a trace-free part multiplied by a power of the conformal factor and a puretrace part. In this spirit one writes

11.3 The Hamiltonian and momentum constraints
1
Kij = ϑ−4 ψij + Khij ,3257

hij ψij = 0,

which, substituted into (11.15b), yields
Di ψij =2 6
ϑ Dj (ϑ−2 K) − 2ϑ−6 Dj Σ + jj .
3

In view of the latter, it is convenient to reintroduce the physical trace K̃ = ΩK =
ϑ−2 K so that one obtains
Di ψij =2 6
ϑ Dj K̃ − 2ϑ−4 Dj Σ + jj .
3

(11.19)This last equation is to be read as an equation for the trace-free tensor ψij . If
K̃ is a constant and Σ = 0, then Equations (11.17) and (11.19) decouple.
Following the discussion of Section 11.2 it can be veriﬁed that the principal partof Equation (11.19) is underdetermined elliptic. To transform Equation (11.19)
into an elliptic equation one makes use of a so-called York splitting ; see York(1973). One considers an ansatz for ψij of the form2ψij = Di ςj + Dj ςi − hij Dk ς k + ψij,
3(11.20)
where ςi is some covector on S and ψij
is a freely speciﬁable symmetric andtrace-free tensor. The operator (Lh ς)i deﬁned by2(Lh ς)i ≡ Di ςj + Dj ςi − hij Dk ς k ,3
is called the conformal Killing operator . It can be veriﬁed to be theformal adjoint of the divergence operator acting on symmetric trace-free tensors.
Substituting the ansatz (11.20) into Equation (11.19) one obtains
22.
Δh ςj + Di Dj ςi − Dj Dk ς k = ϑ6 Dj K̃ − 2ϑ−4 Dj Σ + jj − Di ψij
3
3(11.21)The symbol of this equation can be seen to be2(σ div◦L (ξ)ς)j = ξ i ξi ςj + ξ i ξj ςi − ξj ξ k ςk .
3
Contracting with ξ j one immediately ﬁnds that1
(σ div◦L (ξ)ς)j ξ j = (ξi ξ i )(ςk ς k ) + (ςi ξ i )2 > 0
3forξi , ςj = 0.

Thus, it follows that (11.21) is a linear elliptic equation for the covector ςi .
The freely speciﬁable data for this equation is the symmetric trace-free tensor
. As in the case of Equation (11.19) it decouples from the Licnerowicz
ψij
Equation (11.17) if K̃ is constant and Σ = 0. The analysis of the coupled system(11.17)–(11.19) is much more challenging; see, for example, Holst et al. (2008a,b).

258

The conformal constraint equations
Gauge freedom

The conformal method described in the previous paragraphs has a conformal
gauge freedom. More precisely, if φ is a positive function on S, then a directcomputation shows that the transitions
hij → φ4 hij ,ψij → φ−2 ψij ,Σ → φ Σ,2Ω → φ2 Ω, → φ−8,

Kij → φ2 Kij ,ji → φ−6

ji ,(11.22a)(11.22b)yield another solution to the conformal constraint Equations (11.15a) and(11.15b) with the same physical data (h̃, K̃). This gauge freedom can be
exploited to simplify certain speciﬁc computations. In particular, lettingh = φ4 h, a calculation using the transformation laws for conformal transformations shows that




1
1
(11.23)φ−5 Δh − r[h] ϑ = Δh − r[h ] (φ−1 ϑ),
8
8that is,φ−5 Lh [ϑ] = Lh (φ−1 ϑ).
11.3.4 The Yamabe problem
A classic question of Diﬀerential Geometry is the so-called Yamabe problem
which, given a compact three-dimensional Riemannian manifold (S, h), asks
whether it is possible to conformally rescale the (smooth) metric h to a metricwith constant Ricci scalar; see Yamabe (1960). This problem requires ﬁnding a
positive conformal factor ω and a constant r• satisfying the equation
1(r[h]ω − r• ω 5 ),(11.24)8
which follows from the transformation equation for the three-dimensional Ricci
scalar Equation (5.17). The Yamabe problem has been solved in the aﬃrmative;
see Trudinger (1968), Aubin (1976) and Schoen (1984). In particular, one has
the following (e.g. Lee and Parker (1987); O’Murchadha (1988)):Δh ω =

Theorem 11.1 (resolution of the Yamabe problem) Let h be a smooth
Riemannian metric on a compact manifold S. There exists a smooth, positivedeﬁnite function ω on S such that r[ω 4 h] is constant.
Theorem 11.1 allows the classiﬁcation of Riemannian metrics according towhether they can be rescaled to a metric with constant Ricci scalar which is
positive, negative or zero – a given metric h cannot be rescaled to two diﬀerent
metrics with constant curvature of diﬀerent signs. Thus, the resulting Yamabe
classes are conformal invariants. As will be seen in Section 11.5, this observation
plays a role in the construction of initial data sets on compact manifolds.
Remarkably, the analogous Yamabe problem on non-compact manifolds turns

11.4 The conformal constraint equations

259

out not to be true as shown by a number of counterexamples; see, for example,
Zhiren (1988).
11.4 The conformal constraint equations
Having analysed the standard Einstein constraint equations, focus is now on theconstraint equations implied by the conformal Einstein ﬁeld equations. Theseequations can be regarded as an extension of the conformal Hamiltonian andmomentum constraints (11.15a) and (11.15b).
11.4.1 The derivation of the equations
In this section the frame version of the conformal Einstein ﬁeld equations,
Equations (8.32a) and (8.32b), are considered. By making use of an orthonormal
frame adapted to the geometry of the hypersurface under consideration, as
described in Section 2.7.3, the split of the equations follows almost directly.
In what follows, let (M, g) denote an unphysical spacetime and let S denote a
hypersurface thereof. As in Section 11.1.1, let Σ denote the covariant derivativein the direction of the g-unit normal. The evaluation of a spacetime frame index
in the direction of the unit normal (i.e. the values 0 or 3 depending on the causal
character of S) will be indicated by the symbol ⊥ .
The constraints implied by Zab . Given
1
Zab ≡ ∇a ∇b Ξ + ΞLab − sηab − Ξ3 T{ab} ,2(11.25)the information of the conformal equation Zab = 0 which is intrinsic to thehypersurface S is encoded in the components
Zij = 0,Z⊥i = 0.

(11.26)

In order to obtain explicit intrinsic expressions for these equations it is observedthat∇a ∇b Ξ ≡ ea a eb b ∇a ∇b Ξ = ea (eb (Ξ)) − Γa c b ec (Ξ).
Hence, in particular, one has that∇i ∇j Ξ = ei (ej (Ξ)) − Γi c j ec (Ξ)= ei (ej (Ξ)) − Γi k j ek (Ξ) − Γi ⊥ j e⊥ (Ξ)= ei (Dj Ξ) − γi k j Dk Ξ + Kij Σ= Di Dj Ξ + Kij Σ,(11.27)

260

The conformal constraint equations

where, in the last term of the last line, Equation (2.45) for the extrinsic curvature
has been used. A similar computation shows that∇i ∇⊥ Ξ = ei (e⊥ (Ξ)) − Γi c ⊥ ec (Ξ)= ei (Σ) − Γi k ⊥ ek (Ξ) − Γi ⊥ ⊥ Σ= Di Σ − Ki k Dk Ξ,(11.28)where, in the third line, it has been used that Γa ⊥ ⊥ = 0 as a consequence of themetricity of the connection and the fact that Ki k = Γi k ⊥ .
Substituting the above expressions into Equation (11.26) and taking into
account deﬁnition (11.25) one obtains the constraint equations
1
1
Di Dj Ω = −Kij Σ − ΩLij + shij + Ω Tij − T hij ,2
41
Dj Σ = Kj k Dk Ω − ΩLj + Ω3 jj ,2
whereLi ≡ Li⊥ ,Ω ≡ Ξ|S .

The constraints implied by Za . Given
1
1
Za ≡ ∇a s + Lac ∇c Ξ − Ξ2 ∇c ΞT{ac} − Ξ3 ∇c T{ca} ,2
6(11.29)the intrinsic information of the equation Za = 0 is encoded in the components
Zi = 0.

(11.30)Now, the spatial components of the term Lab ∇b Ω in Equation (11.29) can be
expanded as
Lib ∇b Ω = Lib η ba ∇a Ω
= Li⊥ η ⊥⊥ ∇⊥ Ω + Lik η kl ∇l Ω= Li Σ + Lik Dk Ω.
By similar arguments one concludes that1∇c ΞT{ic} = Σji − Tik Dk Ω − Di T,
41∇c T{ic} = ∇⊥ ji + Dk Tki − Di T.
4
It is important to observe in ∇c T{ic} the presence of the term ∇⊥ ji which
requires further information about the matter model in order to be cast in aform intrinsic to the hypersurface S. In the case of trace-free matter, one hasthat ∇c T{ic} = 0, so that no further considerations are required.

11.4 The conformal constraint equations

261

From the discussion in the previous paragraphs it follows that Equation (11.30)
can be reexpressed as1 
1Di s = −Li Σ − Lik Dk Ω + Ω2 Σji − Tik Dk Ω − Di T
2
4

1 3
1
k
+ Ω ∇⊥ ji + D Tki − Di T .
6
4
The constraints implied by Δcdb . Given
Δcdb ≡ ∇c Ldb − ∇d Lcb − ∇a Ξda bcd − ΞTcdb ,(11.31)the information intrinsic to the hypersurface S of the conformal equation
Δcdb = 0 is encoded in the components
Δijk = 0,Δij⊥ = 0.

(11.32)A calculation similar to that leading to Equations (11.27) and (11.28) yields∇i Ljk = Di Ljk + Kik Lj ,∇i Lj = Di Lj + Ki k Lkj .
Given the components dabcd of the rescaled Weyl tensor with respect to theadapted frame {ea }, it is convenient to deﬁne
dij ≡ di⊥j⊥ ,dijk ≡ di⊥jk .

Following the discussion of Section 11.1.2, dij corresponds to the components ofthe electric part of the rescaled Weyl tensor, while dijk encodes the informationof the magnetic part. It can be veriﬁed that
dij = dji ,di i = 0,dijk = −dikj ,d[ijk] = 0,

dijkl = 2(hi[k dl]j + hj[l dk]i ).

(11.33a)(11.33b)

It follows from the latter expressions, together with (11.31), that theconstraints (11.32) can be reexpressed as
Di Ljk − Dj Lik = −Σdijk + Dl Ωdlkij − (Kik Lj − Kjk Li ) + ΩTijk ,Di Lj − Dj Li = Dl Ωdlij + Ki k Ljk − Kj k Lik + ΩJij ,where
Jjk ≡ Tjk⊥ .
The constraints implied by Λbcd . GivenΛbcd ≡ ∇a da bcd − Tcdb ,262

The conformal constraint equations

as a consequence of the decomposition of the Weyl tensor in electric and magnetic
parts, it follows that the information of the equation Λbcd = 0 which is intrinsic
to the hypersurface S is contained in the components
Λ⊥ij = 0,Λ⊥j⊥ = 0.

(11.34)

Observing that
∇a daijk = η ab ea (dbijk ) − Γa c b dcijk − Γa c i dbcjk − Γa c j dbick − Γa c k dbijc ,one concludes, by arguments similar to those used to obtain Equations (11.27)
and (11.28), that∇a da ⊥jk = Di dijk + (K i k dji − K i j dki ),∇a da ⊥j⊥ = Di dij − K ik dijk .
It follows from the previous discussion that the constraint Equations (11.34) canbe reexpressed as
Di dijk = (K i j dki − K i k dji ) + Jjk ,Di dij = K ik dijk + Jj ,where
Jjk ≡ Tjk⊥ ,Jj ≡ Tj⊥⊥ .

The explicit form of Jjk and Jj depends on the matter model under consideration. In the case of the electromagnetic ﬁeld, they can be expressed in terms of theelectric and magnetic parts of the Faraday tensor and their spatial derivatives.
The constraint Z = 0. Recall that1
Z ≡ 6Ξs − 3∇c Ξ∇c Ξ + Ξ4 T − λ.
4As discussed in Section 8.2.4 the equation Z = 0 is, in fact, a constraint
equation whose propagation is ensured by the other conformal ﬁeld equations;see Lemma 8.1. Following the procedure employed in the decomposition of theother conformal equations, it can be expressed in terms of quantities intrinsic to
the hypersurface S as1
λ = 6Ωs − 3Σ2 − 3Dk ΩDk Ω + Ω4 T.
4
11.4.2 The Gauss-Codazzi and Codazzi-Mainardi equations in termsof conformal ﬁelds
The intrinsic equations discussed in the previous section are supplemented by
the Gauss-Codazzi and Codazzi-Mainardi equations, Equations (2.47) and (2.48)

11.4 The conformal constraint equations

263

Rijkl = rijkl + Kik Kjl − Kil Kjk ,Ri⊥jk = Dj Kki − Dk Kji ,expressed in terms of conformal ﬁelds. As a consequence of the decomposition of
the four-dimensional Riemann tensor Rabcd in terms of the Weyl and Schouten
tensor, Equation (2.21b), one has that
Rabcd = Ξdabcd + ηac Ldb − ηad Lcb + Lac ηdb − Lad ηcb ,while the three-dimensional Riemann tensor rijkl can be expressed in terms ofthe three-dimensional Schouten tensor lij asrijkl = hik llj − hil lkj + hjl lki − hjk lli ,1
lij ≡ rij − rhij ;4

see Equation (2.40). A direct calculation using the above expressions yields thetwo additional constraint equations
Dj Kki − Dk Kji = Ωdijk + hij Lk − hik Lj ,11
lij = Ωdij + Lij − Kk k Kij − Khij + Kki Kj k − Kkl K kl hij .
44These equations provide the link between the spatial curvature tensor lij andthe spacetime curvature as described by dab , dabc , Lab and La .
11.4.3 Summary of the equations and basic properties of theconformal constraint equations
As a summary of the discussion of the previous sections, the conformal constraint
equations are collected:


1 3
1
Di Dj Ω = −ΣKij − ΩLij + shij + Ω Tij − T hij ,2
41
Di Σ = Ki k Dk Ω − ΩLi + Ω3 ji ,
2


1 2
1
k
k
Di s = −Li Σ − Lik D Ω + Ω Σji − Tik D Ω − Di T
2
4


1
1
+ Ω3 ∇⊥ ji + Dk Tki − Di T ,64(11.35a)(11.35b)(11.35c)

Di Ljk − Dj Lik = −Σdkij + Dl Ωdlkij− (Kik Lj − Kjk Li ) + ΩTijk ,Di Lj − Dj Li = D Ωdlij + Ki Ljk − Kj Lik + ΩTij⊥ ,lkk(11.35d)(11.35e)

264

The conformal constraint equations
Dk dkij =  K k i djk − K k j dik + Jij ,(11.35f)

Di dij = K ik dijk + Jj ,(11.35g)

1
λ = 6Ωs − 3Σ2 − 3Dk ΩDk Ω + Ω4 T,
4(11.35h)

Dj Kki − Dk Kji = Ωdijk + hij Lk − hik Lj ,(11.35i)1
1
lij = Ωdij + Lij − K(Kij − Khij ) + Kki Kj k − Kkl K kl hij .
4
4(11.35j)

Using the identity (11.7) and recalling that d∗ij = − 12 dikl j kl , Equations (11.35f)
and (11.35g) can be rewritten in the alternative form
1
Di d∗ij = −j kl K i k dli − j kl Jkl ,2
Di dij = l jk K ik d∗il + Jj .

(11.36a)(11.36b)

The conformal constraint Equations (11.35a)–(11.35j) are not independent
since integrability conditions have been used in their derivation. A list of various
relations between the vacuum constraint equations can be found in Friedrich(1983). In particular, it can be shown that


1 4
2
k
Di 6Ωs − 3Σ − 3Dk ΩD Ω + Ω T = 0,
4
consistent with the fact that the left-hand side of Equation (11.35h) equals thecosmological constant λ.
For future reference, it is observed that from Equation (11.35j) it follows thatrij = Ωdij + Lij + Lk k hij − KKij + Kik K k j ,(11.37a)r = 4Lk k − K 2 + Kij K ij .

(11.37b)

The vacuum version of the conformal constraint equations is obtained by
setting the matter ﬁelds Tij , T , ji , Tijk , Ji , Jij equal to zero. In the derivation of
the conformal constraint Equations (11.35a)–(11.35j), it has been assumed thatthe connection D is the Levi-Civita connection of the intrinsic metric h. Thus,by analogy to the full conformal ﬁeld equations one also has the relations
σi k j = 0,Πk lij = π k lij ,(11.38)where σi k j , Πk lij and π k lij denote, respectively, the components of the torsion,
the geometric curvature and the algebraic curvature of the connection D.
Explicitly, one has that
σi k j ek ≡ [ei , ej ] − (γi k j − γj k i )ek ,Πk lij ≡ ei (γj k l ) − ej (γi k l ) + γm k l (γj m i − γi m j ) + γj m l γi k m − γi m l γj k m ,πklij ≡ hik llj − hil lkj + hjl lki − hjk lli .

11.4 The conformal constraint equations

265

Given a collection of matter ﬁelds on S,m ≡ (Tij , T, , ji , ∇⊥ ji , Ji , Jij ),by a solution to the conformal constraint equations on S it will be
understood a collectionu ≡ (Ω, Σ, s, ei , γi k j , Kij , Lij , Li , dij , dijk )satisfying Equations (11.35a)–(11.35j) together with the supplementary conditions (11.38).
The relation between the conformal constraint Equations (11.35a)–(11.35j)
and the conformal Hamiltonian and momentum constraints (11.15a)–(11.15b) issummarised in the following lemma.
Lemma 11.1 (relation between the solutions to the Einstein constraints
and the conformal constraints) A solution to the conformal constraints(11.35a)–(11.35j) for a collection m of matter ﬁelds implies a solution to
the conformal Hamiltonian and momentum constraints (11.15a) and (11.15b).
Conversely, a solution of (11.15a) and (11.15b) together with a collection ofmatter ﬁelds m gives rise to a solution to (11.35a)–(11.35j) on the points of S
for which Ω = 0.
Proof Using Equations (11.35a) and (11.35h) to eliminate Lk k one readily
obtains the conformal Hamiltonian Equation (11.15a). Similarly, starting from
Equation (11.35i) and using Equation (11.35b) to eliminate Li one obtains theconformal momentum constraint (11.15b). Thus, any solution to the conformal
constraints (11.35a)–(11.35j) implies a solution to the conformal Hamiltonianand momentum constraints, Equations (11.15a) and (11.15b).
Assume now one has a collection (Ω, h, K, Σ, , ji ) satisfying Equations (11.15a) and (11.15b) together with a collection (Tij , T, Ji , Jij , ∇⊥ ji )consistent with the matter ﬁelds  and ji . Let now {ei } denote an h-orthonormal
frame. Using this frame one can compute the components lij and Kij of thethree-dimensional Schouten tensor and of the extrinsic curvature. If Ω = 0,
one can use the conformal constraint (11.35h) to compute the ﬁeld s. Next,
one makes use of Equations (11.35a) and (11.35b) to compute Lij and Li . A
computation using the commutator of the covariant derivative Di shows that
Equation (11.35c) is automatically satisﬁed. Once the components Lij and Li
are known, one can use Equations (11.35i) and (11.35j), respectively, to computedijk and dij – it can be veriﬁed that the resulting ﬁelds are trace free. A ﬁnal
computation using the three-dimensional Bianchi identity in the form
Di rij =

1
Dj r,
4

together with the irreducible decomposition of the three-dimensional Riemanntensor rijkl , the decomposition of dijkl into the electric and magnetic parts

266

The conformal constraint equations

and the commutator of Di shows that Equations (11.35d)–(11.35g) are also
automatically satisﬁed. Thus, the ﬁelds obtained constitute the required solution
to the conformal constraint equations; see Friedrich (1983) for further details.
Remark. In order to make assertions about the behaviour of solutions tothe conformal constraint equations at points where Ω = 0, the equations
need to be supplemented with boundary conditions. Several diﬀerent classes ofboundary conditions on three-dimensional manifolds will be considered: compact manifolds, asymptotically Euclidean manifolds and hyperboloidal
manifolds.
11.4.4 The conformal constraints at the conformal boundary
By construction, the conformal constraint equations can be evaluated in a regularmanner at a non-null hypersurface belonging to the conformal boundary of
spacetime. By deﬁnition such a hypersurface satisﬁes the conditionsdΩ = 0.

Ω = 0,

Following the convention introduced in Chapter 6 this hypersurface will bedenoted by I . The null case will be discussed in Chapter 18.
The deﬁning properties of the hypersurface I lead to a number of simpliﬁcations in the conformal constraint equations. In particular, dΩ is normal to I
so that, in terms of a tetrad adapted to the hypersurface, one has Di Ω = 0.
Assuming that the matter ﬁelds Tij , T and Tijk are smooth at I one ﬁnds
that on the hypersurface the conformal constraints (11.35a)–(11.35j) imply theequationsshij

ΣKij ,

(11.39a)Di Σ

0,(11.39b)

Di s−Li Σ,(11.39c)

Di Ljk − Dj Lik−Σdijk − (Kik Lj − Kjk Li ),Di Lj − Dj Li Ki Ljk − Kj Lik ,
Dk dkij  K k i djk − K k j dik + Jij ,
k

i

D dij
λ

k

ik

K dijk + Jj ,−3Σ2 ,

Dj Kki − Dk Kji hij Lk − hik Lj ,1
1
lij Lij − K Kij − Khij + Kki Kj k − Kkl K kl hij ,4
4(11.39d)(11.39e)(11.39f)(11.39g)(11.39h)(11.39i)(11.39j)where denotes equality at the conformal boundary. From Equations (11.39b)and (11.39h) it follows that Σ is a constant on I with a value given by11.4 The conformal constraint equations267Σ = −λ/3 – observe that if  = 1, then λ < 0, and if  = −1, then λ > 0,
for the previous expression to make sense. Moreover, from Equation (11.39a) the
extrinsic curvature of I is proportional to the intrinsic metric.
A procedure for constructing solutions to Equations (11.39a)–(11.39j) in thevacuum case (so that Jjk = 0, Jj = 0) has been given in Friedrich (1986a, 1995).
The fundamental idea is to identify the function s and the 3-metric h on I as
freely speciﬁable data. Instead of working directly with s, it is more convenient
to use a smooth function κ ∈ X(S) such thatsΣκ.

(11.40)

It follows directly from Equations (11.39a), (11.39c) and (11.39j) thatKij

κhij ,

Li

−Di κ,Lij

1
lij + κ 2 hij .
2(11.41)Substituting these expressions into Equation (11.39d) one obtains, after some
simpliﬁcation, thatdijk−Σ−1 yijk

(11.42)where yijk ≡ Di ljk − Dj lik denote the components of the Cotton tensor of themetric h; see Section 5.2.2. Alternatively, one can writed∗ij−Σ−1 yij ,with yij ≡ − 12 yklj i kl the components of the Bach tensor. It can be veriﬁedthat the integrability conditions (11.39e), (11.39f) and (11.39i) are automaticallysatisﬁed by (11.41) and (11.42). Finally, by substituting into Equation (11.39g)
one obtains that
Di dij0.

This is the only diﬀerential condition that has to be solved in this procedure.
This can be done by means of a York splitting so as to obtain an elliptic equation
for the components of a covector.
The discussion of the previous paragraph is summarised in the following:
Proposition 11.1 (solutions to the conformal constraint equations
at the conformal boundary ) Given a three-dimensional metric h, an
h-divergence-free and trace-free ﬁeld dij and a smooth function κ, the ﬁelds s,
Kij , Li , Lij , dijk as given by Equations (11.40), (11.41) and (11.42) constitutea solution to the vacuum conformal constraint equations with Ω = 0.
As will be seen in later chapters, a solution to Equations (11.39a)–(11.39j)
constitutes, in the case of  = 1 (i.e. I spacelike), initial data at, say, past nullinﬁnity for de Sitter-like spacetimes. In the case  = −1 (i.e. I timelike), thesolution gives boundary data for an anti-de Sitter-like spacetime.

268

The conformal constraint equations

Remark. The procedure indicated in the previous paragraphs can be extended
to the matter case if the ﬁeld Ji is known.
Exploiting the conformal freedom
The conformal freedom inherent to the conformal ﬁeld equations can be employed
to express the solution to the conformal constraint equations at the conformalboundary in an even simpler form. Recall the discussion in Section 8.2.5 on thetransformation properties of the various ﬁelds appearing in the conformal ﬁeld
equations. In particular, it follows from Equation (8.29b) that, under a rescalingof the form g  = ϑ2 g which implies a rescaling
h

ϑ2 h

of the intrinsic metric of I , the ﬁeld s on I transforms as
s

 −1

ϑ s + ϑ−2 ∇a ϑ∇a Ξ .

In particular, it is always possible to choose ϑ at I so that locallys

0.

Accordingly, in this particular conformal gauge one has that Equation (11.40)
implies κ  = 0 and, moreover,

Kij

0,

Li

0,

Lij

lij .

11.5 The constraints on compact manifolds
An important class of initial data sets for the Einstein ﬁeld equations involves
physical 3-manifolds S̃ which are compact. This type of initial data set is of
relevance in the discussion of cosmological models. In particular, in the vacuum
case with negative cosmological constant one expects these initial data sets togive rise to de Sitter-like spacetimes. Initial data sets on compact manifolds havebeen studied extensively in the literature, and there is a good understanding of
the required conditions on the free data in order to ensure existence of solutions
to the Einstein constraint equations; see, for example, Isenberg (1995).
For this type of initial data one can set, without loss of generality, Ω = 1 andΣ = 0 and let S = S̃. For simplicity of the presentation, in the remainder of thissection the discussion is restricted to the vacuum case. Furthermore, it is assumed
that the physical mean curvature K̃ is constant so that Equations (11.17) and(11.21) decouple from each other. The fundamental tool in the analysis of thesolvability of the constraint equations is given by the maximum principle forthe Laplacian of a Riemannian metric. A convenient formulation of this result isgiven by (see Isenberg (1995)):

11.5 The constraints on compact manifolds

269

Proposition 11.2 (maximum principle for compact manifolds) Let(S, h) denote a Riemannian manifold with S compact. Given a smooth ψ ∈ X(S)such that Δh ψ has the same sign on the whole of S, then ψ must be a constant.
As a consequence of the above principle, the equationΔh ψ = F (x, ψ),with ψ > 0 has no solution if F (x, ψ) does not change sign on S except
for the case where F (x, ψ) = 0. Using this observation it is easy to see thatcertain combinations of free data cannot give rise to solutions of the constraint
equations. As an example, consider time-symmetric data (i.e. K̃ = 0) withvanishing cosmological constant on a compact manifold S. As a consequenceof the conformal gauge freedom given in Equations (11.22a) and (11.22b) andof the Yamabe theorem, Theorem 11.1, one can assume that r[h] is a negative
constant on S – such a metric is said to be of positive Yamabe class. An
example of this situation is S3 with its standard metric. One is then left with a
Licnerowicz equation of the formΔh ϑ =

1r[h]ϑ.
8

If ϑ is required to be positive everywhere on S, it follows that Δh ϑ < 0 everywhere
so that no positive solution can exist since, as a consequence of the maximum
principle, ϑ must be a constant so that Δh ϑ = 0 which is a contradiction. To
get around this situation one can consider initial data with a negative (i.e. deSitter-like) cosmological constant. Keeping the time symmetry of the initial data
and the condition r[h] < 0, one obtains the Licnerowicz equationΔh ϑ =

1
1
r[h]ϑ − λϑ3 .
8
4(11.43)

The right-hand side of this equation has no deﬁnite sign for positive ϑ, so there
is no obstruction to the existence of solutions. In any case, a further argument(not discussed here) is required to show that Equation (11.43) does indeed havea solution.
The methods in Isenberg (1995) allow one to prove the following proposition:
Proposition 11.3 (solvability of the Einstein constraints with cosmological constant on a compact manifold ) Let (S, h) be a Riemannian manifoldwith S ≈ S3 and h conformal to a metric with constant negative Ricci scalar(positive Yamabe class). Then the vacuum Einstein constraints with de Sitterlike cosmological constant have a solution for an arbitrary choice of the seed
and constant physical mean curvature K̃.
metric h, trace-free tensor ψij
The initial data sets given by this proposition will be used to construct deSitter-like spacetimes in Chapter 15.

270

The conformal constraint equations
11.6 Asymptotically Euclidean manifolds

Spacetimes with λ = 0 can be thought of as describing isolated systems for which
the eﬀects of cosmological expansion are neglected. An important class of thesespacetimes consists of those solutions to the Einstein ﬁeld equations which are
asymptotically simple in the sense of Deﬁnition 7.1, that is, asymptotically simple
and empty. Proposition 14.3 shows that these spacetimes are globally hyperbolic,suggesting a systematic procedure for their construction through suitable initial
data prescribed on a Cauchy hypersurface.
In order to develop intuition, it is convenient to look at the Minkowski
spacetime (R4 , η̃). A foliation of this spacetime is given by the hypersurfacesof constant time t. These hypersurfaces are Riemannian manifolds of the form
(R3 , −δ). It can be veriﬁed that these hypersurfaces are extrinsically ﬂat; thatis, their extrinsic curvature K̃ = 0 vanishes. Of course, these are not the only
possible types of Cauchy hypersurfaces in this spacetime.
As a second example, consider the Schwarzschild spacetime. In terms of theso-called Schwarzschild isotropic radial coordinate1r̄ ≡ r − m + r(r − 2m) ,2the line element of the spacetime can be rewritten as2
1 − m/2r̄
m 4g̃ S =
dt ⊗ dt − 1 +(dr̄ ⊗ dr̄ + r̄2 σ).
1 + m/2r̄
2r̄
An example of a Cauchy hypersurface for this spacetime is given by the t = 0
hypersurface. One can verify that the intrinsic metric and the extrinsic curvatureof this hypersurface are given, respectively, by

m 4
δ,
K̃ S = 0.
(11.44)h̃S = − 1 +2r̄
The most general form of the above initial data set is obtained by performing atranslation of the radial coordinate to obtain

4
m
δ,
(11.45)h̃S = − 1 +2|y − y0 |with |y − y0 |2 ≡ (y 1 − y01 )2 + (y 2 − y02 )2 + (y 3 − y03 )2 where (y α ) = (y 1 , y 2 , y 3 )are standard Cartesian coordinates and (y0α ) ∈ R3 arbitrary. Observe that the
metric h̃S is, in fact, conformally ﬂat and that h̃S → −δ as r̄ → ∞. Moreover,
one has that
 

1
m 4
2m
+O
1+
=1+
.
(11.46)2r̄
r̄
r̄2
To understand the behaviour as r̄ → 0, it is observed that under the coordinate
inversion r̆ ≡ m2 /4r̄ one has thatm 4h̃S = − 1 +(dr̆ ⊗ dr̆ + r̆2 σ).
2r̆11.6 Asymptotically Euclidean manifolds

271

Figure 11.1 Embedding diagram of the Einstein-Rosen bridge in the standard
time-symmetric Schwarzschild hypersurface. √The diagram is obtained as thesurface of revolution of the curve z = ±ln(1+ r2 − 1); see Morris and Thorne(1988) for more details.

Thus, the behaviour of the metric h̃ is identical for both r̄, r̆ → ∞. Thereis a discrete reﬂexion symmetry with respect to the two-dimensional surface{r = m/2}. Thus, the topology of the hypersurface is S ≈ R × S2 . One says that
S has a non-trivial topology with two asymptotically ﬂat regions (see next
section) joined by a so-called Einstein-Rosen bridge. A representation of thisis given in Figure 11.1.
An example of an initial data set with non-vanishing extrinsic curvature isgiven by the family of conformally ﬂat initial data sets for the Schwarzschildspacetime with extrinsic curvature given byK̃ αβ =

A
(3y α y β + |y|2 δ αβ ),|y|3where |y|2 = δαβ y α y β and (y α ) are, again, standard Cartesian coordinates; see
Beig and O’Murchadha (1998), Estabrook et al. (1973) and Reinhart (1973). It
can be veriﬁed that K̃ = h̃αβ K̃αβ = 0 as hαβ y α y β = −|y|2 . This hypersurface
has the nontrivial topology of R×S2 . However, in contrast to the time-symmetriccase, the conformal factor ϑ cannot be written in a closed form. Nevertheless, theleading terms of its asymptotic expansion are the same as in Equation (11.46)
with |y| playing the role of the radial coordinate r̄.
11.6.1 Deﬁnition in terms of physical ﬁelds
The hypersurfaces discussed in the previous paragraphs are examples of asymptotically Euclidean manifolds. Given a three-dimensional manifold S̃, anasymptotic end is a subset Ẽ ⊂ S̃ which is diﬀeomorphic to the complementof a closed ball on R3 ; that is,Ẽ ≈ (y α ) ∈ R3 | |y| > r0 ,where r0 is some positive real number and |y|2 ≡ δαβ y α y β . By identifying Ẽ withthe complement of a ball, the triple y = (y α ) can be used as coordinates on
the asymptotic end – so-called asymptotically Cartesian coordinates. The

272

The conformal constraint equations

hypersurfaces of the Minkowski and Schwarzschild spacetimes have, respectively,
one and two asymptotic ends. More generally, a three-dimensional manifold S̃ issaid to have N asymptotically ﬂat ends if there exists a compact subset ofS̃ such that its complement is the union of disjoint subsets Ẽk , k = 1, 2, . . . , N ,each of which is an asymptotic end. In terms of the above concepts one can now
introduce the key deﬁnition of this section:
Deﬁnition 11.1 (asymptotically Euclidean manifolds) An initial data set
for the vacuum Einstein ﬁeld equations (S̃, h̃, K̃) is said to be asymptoticallyEuclidean if S̃ is a three-dimensional manifold with N asymptotically ﬂatends Ẽk , k = 1, . . . , N such that on each Ẽk the 3-metric and the extrinsic
curvature satisfy, in terms of asymptotically Cartesian coordinates on the end,
the asymptotic behaviour




1
2mk
,
(11.47a)h̃αβ = − 1 +δαβ + O2
|y|
|y|2


1
K̃αβ = O1,(11.47b)|y|2where mk , k = 1, . . . , N are constants.
The notation O1 and O2 in Equations (11.47a) and (11.47b) is explained in the
Appendix to this chapter. More general notions of asymptotic ﬂatness for threedimensional manifolds have been considered in the literature; see, for example,
Chaljub-Simon (1982), Chaljub-Simon and Choquet-Bruhat (1980), ChoquetBruhat and York (1980) and Christodoulou and O’Murchadha (1981). Their
precise formulation require the use of the notion of weighted Sobolev spaces; see,for example, appendix I of Choquet-Bruhat (2008) and Bartnik (1986). Thesedeﬁnitions are tailored for the analysis of the elliptic equations arising from theconstraint equations.
The asymptotic conditions in Deﬁnition 11.1 ensure the ﬁniteness of the
ADM-linear momentum and ADM-angular momentum 1 of each asymptotic end. These asymptotic quantities are given, respectively, by the surface
integrals
1lim(K̃αβ − K̃ h̃αβ )nβ dSh̃ ,Pα ≡8π r→∞ Sr
1lim˜αβγ y β (K̃ γδ − K̃ h̃γδ )nδ dSh̃Jα ≡8π r→∞ SrwithSr ≡ (y α ) ∈ S̃ | |y| = r ,1 ADM stands for Arnowitt-Deser-Miser, pioneers of the Hamiltonian formulation of generalrelativity; see Arnowitt et al. (1962) and Arnowitt et al. (2008) for a republication of thisclassical review.

11.6 Asymptotically Euclidean manifolds

273

nα its outward pointing normal and dSh̃ the surface element induced by h̃ on Sr .
The constants mk in Deﬁnition 11.1 correspond to the ADM mass of each of
the asymptotic ends. They are also computable as surface integrals of the sphereat inﬁnity via the expression
1limh̃αβ (∂α h̃βγ − ∂γ h̃αβ )nγ dSh̃ .
m=−
16π r→∞ SrStrictly speaking, m is the time component of a 4-vector, the ADM
4-momentum, whose spatial components are given by P α ; thus, it is moreaccurately described as an energy.
Deﬁnition 11.1 can be extended to initial data sets with matter. In thesesituations, decay conditions for the matter sources which are compatible withthe decay for h̃ and K̃ are given by (11.47a) and (11.47b). Direct inspection of
the constraint Equation (11.14) suggests that




1
1
=
O
,
j̃
.
˜ = Oα|y|3|y|3These conditions can be reﬁned via a more careful analysis of the constraint
equations.
It is possible to have an initial data set with several asymptotic ends, someof which are not asymptotically Euclidean. The simplest example is given by
the extremal Reissner-Nordström spacetime; see Equation (6.43). The intrinsic
metric of the hypersurface t = 0 is given, in terms of the extremal ReissnerNordström isotropic radial coordinate r̄ = r − m, by

m 2
δ.
h̃ = − 1 +r̄(11.48)

Clearly


m 2
2m
+O
1+
=1+
r̄
r̄



1
r̄2


as r̄ → ∞.

Thus, for large r̄, the extremal Reissner-Nordström 3-metric (11.48) has an
asymptotically Euclidean end. To discuss the behaviour as r̄ → 0, consider the
new radial coordinate ř = − ln r̄, so that ř → ∞ as r̄ → 0. It follows that interms of this coordinate the metric (11.48) can be rewritten ash̃ = − m + e−ř (dř ⊗ dř + σ).
This metric approaches a constant multiple of the standard metric of the cylinder
R+ ×S2 as ř → ∞. Accordingly, one speaks of a cylindrical asymptotic end . A
similar type of asymptotic behaviour can be found, for example, in hypersurfacesof the extremal Kerr spacetime; see, for example, Dain and Gabach-Clement(2011).

274

The conformal constraint equations
11.6.2 Deﬁnition using conformal rescalings

The notion of asymptotically Euclidean manifolds can be strengthened byrequiring the physical hypersurface S̃ to have a conformal extension which isa point compactiﬁcation. This approach provides a more geometrical setting
for the discussion of the asymptotic behaviour of the various ﬁelds, that is,
independent of the use of particular asymptotically Cartesian coordinates. This
point of view was ﬁrst introduced by Geroch (1972b).
Deﬁnition 11.2 (asymptotically Euclidean and regular manifolds) A
three-dimensional Riemannian manifold (S̃, h̃) will be said to be asymptoticallyEuclidean and regular if there exists a three-dimensional, orientable, compact
manifold (S, h) with points ik ∈ S, k = 1, . . . , N with N some integer, adiﬀeomorphism ϕ : S \ {i1 , . . . , iN } → S̃ and a function Ω ∈ C 2 such that:(i) Ω(ik ) = 0, dΩ(ik ) = 0, Hess Ω(ik ) = −2h(ik ).
(ii) Ω > 0 on S \ {i1 , . . . , iN }.
(iii) h = Ω2 ϕ∗ h̃ on S \ {i1 , . . . , iN } with h ∈ C 2 (S) ∩ C ∞ (S \ {i1 , . . . , iN }).
More generally, a function Λ1/2 such that Λ satisﬁes conditions (i) in the abovedeﬁnition is called an asymptotic distance function. The function Λ does notneed to be deﬁned globally on S.
When no confusion arises, condition (iii) in Deﬁnition 11.2 will simply be
written as h = Ω2 h̃ so that S \ {i1 , . . . , iN } and S̃ are identiﬁed. As will
be seen in the following, for asymptotically Euclidean and regular manifolds,
suitable neighbourhoods of the points ik – the points at inﬁnity – are mapped
to the asymptotic ends of S̃. Thus, one can use local diﬀerential geometry to
discuss the asymptotic properties of the initial data set (S̃, h̃). The questionof the diﬀerentiability of Ω and h at i1 , . . . , iN will be addressed later in thissubsection. Deﬁnition 11.2 is purely Riemannian; that is, it makes no reference
to the extrinsic curvature. The behaviour of the extrinsic curvature at the points
at inﬁnity will be discussed in the subsequent paragraphs.
There is some conformal gauge freedom in Deﬁnition 11.2. A replacementof the formh → φ4 h,Ω → φ2 Ω,(11.49)with φ(ik ) = 1 gives rise to the same physical metric h̃ = Ω−4 h and preserves
the boundary conditions in point (i) of the deﬁnition. This gauge freedom canbe used to select conformal metrics with special properties. For example, given
a particular point at inﬁnity i, and choosing φ such that1
Δh φ − r[h]φ = 08on Bε (i),(11.50)with Bε (i) the ball of radius ε centred at i for some ε > 0, it follows from
Equation (11.23) that11.6 Asymptotically Euclidean manifoldsr[h ] = 0275

on Bε (i).

A general property of elliptic equations with smooth coeﬃcients is that they
can always be solved locally; see, for example, Besse (2008) and Garabedian(1986). Thus, the requirement (11.50) can always be satisﬁed. In other words, theconformal metric h can always be chosen so that it vanishes in a neighbourhoodof one of the points at inﬁnity. In general, this statement is not true globally.
Normal coordinates around i
The consequences of Deﬁnition 11.2 are better analysed by means of normal
coordinates. Consider the set of h-geodesics γv ⊂ S starting at a particularpoint at inﬁnity i (i.e. γv (0) = i) with initial velocity v ∈ T |i (S). Moreover, let
T denote the subset of T |i (S) deﬁned byT ≡ v ∈ T |i (S) γv is deﬁned on an interval containing [0, 1] .
On the set T one can deﬁne the exponential map at i, expi : T → S, throughthe condition expi (v) = γv (1); that is, the exponential map sends the vector v tothe point at a unit parameter distance along the unique geodesic through i withinitial velocity v. It can be shown that there exists a neighbourhood Q ⊂ T |i (S)of the vector 0 such that the exponential map at i gives a diﬀeomorphism onto aneighbourhood U ⊂ S of i; see, for example, O’Neill (1983) for a proof. If v ∈ Q
implies that λv ∈ Q for all λ ∈ [0, 1], then one says that Q is star shaped
and U = expi (Q) is called a normal neighbourhood of i. In particular, if
U = Bε (i), the open ball of radius ε > 0 with respect to h, one has a geodesicball .
In what follows, assume one has a normal neighbourhood U around i and that
one is given an orthonormal basis {ei } for T |i (S). Given p ∈ U and v such thatp = exp(v), then writing v = xi ei one can use the components x = (xi ) ∈ R3
as coordinates for the point p – these are the normal coordinates determinedby the basis {ei }. For consistency, the normal coordinates will be written as(xα ) rather than (xi ). In terms of normal coordinates a geodesic through theorigin has the form x(s) = (s xα ) where s is an aﬃne parameter. As ẋ = (xα )and ẍ = 0, it follows from the geodesic equation that γβ α γ (i)xβ xγ = 0 with
γβ α γ being the Christoﬀel symbols of the metric h. As this has to hold for any
geodesic on U , one concludes that γβ α γ (i) = 0. It also follows that ∂α hβγ = 0,
so that one can write
hαβ = −δαβ + O(|x|2 )close to i,(11.51)

where |x|2 ≡ δαβ xα xβ . Moreover, from the above construction it follows thatxα hαβ = −δαβ xα .

(11.52)

276

The conformal constraint equations

For future use it is observed that, in terms of normal coordinates and suﬃciently
close to i, one has
dμh = |x|2 dσ + O(|x|3 ),(11.53)where dμh is the volume element of the metric h and dσ denotes the area elementof the unit 2-sphere S2 .
For later use, it is convenient to deﬁne the (square of the) geodesic
distance Γ2 ≡ |x|2 . One has that Γ2 is a smooth function of the normal
coordinates. It can be veriﬁed thathαβ Dα ΓDβ Γ = −4Γ

(11.54)and thatΓ(i) = 0,Dα Γ(i) = 0,Dα Dβ Γ(i) = −2hαβ ,Dα Dβ Dγ Γ(i) = 0.

Hence, Γ satisﬁes the boundary conditions (i) in Deﬁnition 11.2 so it is an
asymptotic distance function. Observe that, in general, Γ is not deﬁned globallyon S.
Remark. The results obtained using normal coordinates can be strengthenedby exploiting the conformal freedom in (11.49). In particular, a conformal factor
can always be found such that the Riemann curvature tensor of the resulting
rescaled metric vanishes at i. In order to see this, given the metric h, let Ω ≡ ef
with f ∈ X(S) such that
f=

1 α β
x x lαβ (i)2on Bε (i),where lαβ (i) denotes the components with respect to the normal coordinates of
the three-dimensional Schouten tensor at i. A calculation shows thatΩ (i) = 1,

Dα Ω (i) = 0,Dα Dβ Ω (i) = lαβ (i).

Hence, using the conformal transformation formula for the Schouten tensor
(i) = 0. As in dimension(5.16b) one ﬁnds that the metric h ≡ Ω2 h satisﬁes lαβ
3 the Riemann tensor is completely determined by the Schouten tensor one(i) = 0 as claimed. The metric h satisﬁes the improved
concludes that rαβγδ
expansionhαβ = −δαβ + O(|x|3 )close to i;compare with (11.51).
The construction described in the previous paragraph is not the only possible
way of exploiting the conformal gauge freedom. Depending on the particularanalysis, other choices may be more convenient – for example, the conformal
normal gauge introduced in Friedrich and Schmidt (1987) and Friedrich (1998c)
or the central harmonic gauge used in Friedrich (2013).

11.6 Asymptotically Euclidean manifolds

277

Asymptotically Euclidean data versus asymptotically Euclidean
and regular data
It is useful to compare the two deﬁnitions of asymptotic ﬂatness presented inthis Section: Deﬁnitions 11.1 and 11.2. Condition (i) in Deﬁnition 11.2 restricts
the form of the conformal factor Ω in a neighbourhood Ba (ik ) of a given point
at inﬁnity ik . More precisely, one has thatΩ = |x|2 f (x)near ik ,(11.55)where f is continuous with f (0) = 1. Given the normal coordinates (xα ) on
Ba (ik ), one can introduce inversion coordinates y α ≡ xα /|x|2 so thath̃αβ = Ω−2 hαβ = −δαβ + O(|y|−1 )as|y| → ∞.

Thus, to recover the mass term in the expansion (11.47a) one requires further
information about the ﬁelds Ω and h.
With regards to the second fundamental form, it follows from the transformation rules discussed in Section 11.1.1 thatK̃αβ = Ω−1 Kαβ = Ωψαβ .
Hence, if the physical ﬁeld K̃αβ satisﬁes the decay given by condition (11.47b),thenK̃αβ = O(|x|0 ),ψαβ = O(|x|−4 ),as |x| → 0.

Consequently, the decay conditions of Deﬁnition 11.1 imply a tensor ψαβ which
is singular at the points at inﬁnity. To have a regular ψαβ one needs the stronger
decay condition K̃αβ = O(1/|y|6 ). This decay excludes the possibility of a nonvanishing ADM linear momentum and ADM angular momentum.
The regularity at the points at inﬁnity
The regularity requirements on Ω and h of Deﬁnition 11.2 are given with respect
to some suitable coordinate system. A natural choice is the normal coordinates
x = (xα ) centred at the point at inﬁnity – intuitively, one expects the regularity
with respect to normal coordinates to be optimal. In these coordinates thefunction |x| is not smooth at i as its second derivative is not well deﬁned there.
More generally, even powers of |x| will be smooth, while odd ones will be only
C k , for some k.
Initial data sets for static vacuum spacetimes admit a conformal metric whichis, in fact, analytic at the point at inﬁnity; see Beig and Simon (1980b) and Beig
and Schmidt (2000). Remarkably, this is not the case for stationary solutions
which can be seen to be only C 2 at the point at inﬁnity; see Dain (2001b).
More precisely, any asymptotically Euclidean data set for a stationary spacetime

278

The conformal constraint equations(and in particular for the Kerr solution!) has a conformal metric of the form hwhich, in a suitable neighbourhood Ba (ik ) of inﬁnity, takes the formh = h + |x|3 h ,with h and h analytic tensors with respect to normal coordinates.
11.6.3 Fundamental solutions and punctures
Consider now, for simplicity, an asymptotically Euclidean and regular manifold(S, h) with a single point at inﬁnity i. Suppose, for ease of the presentation, thatthe conformal factor Ω = ϑ−2 satisﬁes the Yamabe equation1
Δh ϑ − r[h]ϑ = 0
8

on S \ {i}.

(11.56)

Condition (i) of Deﬁnition 11.2 implies a singular behaviour for the conformal
factor ϑ. Indeed, from Equation (11.55) it follows that
ϑ|x| → 1

as |x| → 0.

(11.57)

In order to develop a better understanding of the singular behaviour at i consider
the integral



1
Iε ≡
Δh ϑ − r[h]ϑ dμh
8Bε (i)
over an open ball Bε (i) of a suitably small radius ε > 0 centred at i. To simplify
the evaluation of the integral it is assumed that the metric h has been chosensuch that r[h] = 0 on Bε (i); as seen in Section 11.6.2 this is is always possible
locally. Using the divergence theorem (see the Appendix to this chapter), one hasthatΔh ϑdμh = −dϑ, ndSh ,Iε =Bε (i)∂Bε (i)where n is the outward-pointing unit normal to ∂Bε (i) and dσh is the surface
element of ∂Bε (i) implied by h. From the expansion (11.53) it follows for
suﬃciently small ε that dSh = ε2 dσ + o(ε2 ) with dσ the surface element of
S2 . Moreover, as a consequence of (11.57) one has
1dϑ, n = − 2 + o(ε−2 ).
εPutting everything together one concludes thatIε = −
dσ + o(ε) −→ 4π∂Bε (i)so thatSΔh ϑdμh = 4π.

as ε → 0,

11.6 Asymptotically Euclidean manifolds

279

The latter implies that one can writeΔh ϑ = 4πδ(i),where δ(i) denotes the Dirac’s delta distribution with support at the
point i; see the Appendix to this chapter for more details and references. To
obtain the expression for a generic metric with a non-vanishing Ricci scalar ina neighbourhood of i one makes use of the transformation law for the Yamabe
equation, Equation (11.23), to obtain
1
with h = φ4 h.
Δh − r[h ] (φ−1 ϑ) = 4πφ−5 δ(i)8As δ(i) has support only on i and φ(i) = 1 one ﬁnally concludes that


1


with ϑ = φ−1 ϑ.
Δh − r[h ] ϑ = 4πδ(i)8This expression provides an alternative description of the singular behaviour ofsolutions to the Yamabe equation which satisfy the boundary condition (i) of
Deﬁnition 11.2. The previous discussion can be generalised to manifolds (S, h)with several points at inﬁnity. For example, if S = S3 and h = −h̄ the standard
metric of S3 , then the Yamabe equation




1
Δ−h̄ − r[−h̄] ϑ = 4π δ(iN ) + δ(iS ) ,8
where δ(iN ) and δ(iS ) are supported, respectively, at the north and south
poles of S3 , describes the conformal factor ϑ for time-symmetric data for the
Schwarzschild spacetime. Letting φ ≡ 1 + m/2r, it follows from combining theﬁrst equation in (11.44) with the conformal factor ω compactifying R3 into S3
given in (6.5) thath̃ = −Ω2 h̄,Ω = ωφ−2 .

Setting α = 1 in Equation (6.5), one has that
2 sin2
Ω= 

ψ
2

ψ
m
tan
1+
2
2

2 .

One can verify that Ω and dΩ vanish at ψ = 0, π (the north and south poles of
S3 ). Moreover, one hasΩ = ψ 2 + O(ψ 3 ),

Ω = (ψ − π)2 + O((ψ − π)3 ),so that the fundamental solution ϑ = Ω−1/2 has the expected singular behaviour.

280

The conformal constraint equations
Conformal decompactiﬁcation of initial data sets on compact manifolds

From a geometric point of view, the purpose of introducing a conformal
factor ϑ which is singular at the point at inﬁnity is to produce a conformal
decompactiﬁcation of the manifold S. As an example, consider a vacuum
maximal initial data set (S̃, h̃, K̃) with S̃ compact. Under these assumptions
one has that the Einstein constraints (11.14) reduce tor[h̃] = −K̃ij K̃ ij ,D̃i K̃ij = 0,

K̃ = h̃ij K̃ij = 0.

If given a point i ∈ S̃ one can ﬁnd a solution ϑ̄ to the equation
1
1
Δh̃ ϑ̄ − r[h̃]ϑ̄ − r[h̃]ϑ̄−7 = 4πδ(i),88
it follows from a calculation involving the conformal transformation propertiesof the various ﬁelds thath̄ ≡ ϑ̄h̃,K̄ ≡ ϑ−2 K̃,gives rise to an asymptotically Euclidean and regular solution to the Einsteinconstraints
r[h̄] = −K̄ij K̄ ij ,D̄i K̄ij = 0,K̄ ≡ h̄ij K̄ij = 0.

As pointed out in O’Murchadha (1988), this construction can be used to argue
that, in a certain sense, there are more asymptotically ﬂat initial data sets thaninitial data sets on compact surfaces.
The Yamabe invariant
The possibility of conformally decompactifying a compact Riemannian manifold(S, h) to obtain a physical manifold (S̃, h̃) which is asymptotically Euclidean
and regular depends on being able to solve the equation
1Δh − r[h] ϑ = 4πδ(i).
(11.58)8The discussion in Section 11.5 suggests that this may not be possible for allcases. To explore this further, consider a test function φ ∈ X(S). A calculationshows that



 2

2
2
2
2
ϑ |Dφ| + φ |Dϑ| dμh +|D(ϑφ)| dμh =ϑDi ϑDi φ2 dμh
S
S
S


 2

ϑ |Dφ|2 + φ2 |Dϑ|2 dμh −=Di (ϑDi ϑ)φ2 dμh
S
S

2
2
2
=
ϑ |Dφ| dμh −ϑφ Δh ϑdμh ,SS

11.6 Asymptotically Euclidean manifolds

281

where the second equality follows by integration by parts on a compact manifoldof the last integral in the ﬁrst line. As |Dφ|2 = Di φDi φ < 0 and ϑ > 0 it follows
that


2
ϑφ2 Δh ϑdμh
− |D(ϑφ)| dμh >
S
S


1
> 4π
ϑφ2 δ(i)dμh +ϑ2 φ2 r[h]dμh
8 S
S

1
2
> 4πϑ(i)φ (i) +r[h]ϑ2 φ2 dμh ,
8 S

1
>
r[h]ϑ2 φ2 dμh ,8 S
where the last inequality follows from the fact that φ is an arbitrary test function.
Thus setting ζ ≡ ϑφ one concludes that

8|Dζ|2 + r[h]ζ 2 dμh > 0,
− infζ∈X(S)

S

where inf denotes the inﬁmum, that is, the biggest lower bound. The latter is anecessary condition for the existence of a solution to Equation (11.58). Under
some further technical assumptions, it can be shown to be a suﬃcient condition;see, for example, Friedrich (2011). The above expression can be reformulated
in a conformal way by adding a suitable normalisation factor. Accordingly, onedeﬁnes the Yamabe invariant (number) of h as ij
8h Di ζDj ζ + r[h]ζ 2 dμhS.
Y [h] ≡ − inf1/3ζ∈X(S)

S

ζ 6 dμh

The conformal invariance of the above expression follows from the transformation
properties of the three-dimensional Ricci scalar and of the volume element.
Accordingly, the Yamabe number is, in fact, a property of the conformal class[h]. In particular, if (S, h) is such that Y [h] > 0, then there exists h̄ ∈ [h] suchthat r[h̄] < 0 on S; see Lee and Parker (1987).
11.6.4 Constructing solutions to the constraint equations using
fundamental solutions
As already mentioned, fundamental functions of the Yamabe equation oncompact manifolds allows one to obtain solutions to the Hamiltonian andmomentum constraints by means of a procedure of conformal decompactiﬁcation.
In this section an overview of some of the technical details of this construction
is provided.
In the ﬁrst instance, attention is restricted to the time-symmetric case.
Furthermore, it is assumed that there is only one point at inﬁnity. Given a

282

The conformal constraint equations

compact Riemannian manifold (S, h) and a point at inﬁnity i, the constructionof a time-symmetric initial data set (S̃, h̃) requires a global solution to
Equation (11.58). As already observed, the function Γ = |x|, deﬁned only ina neighbourhood Ba (i) with a > 0, satisﬁes the required boundary conditions
for a solution to Equation (11.58). Indeed, it can be shown that the solution ϑ
satisﬁesϑ = Γ−1 +m+ O(Γ),
2

near i,where m is a constant; see Lee and Parker (1987). The above expansion is also
valid for any other choice of asymptotic distance function – the constant m is
independent of the particular choice. As Γ2 is a smooth function on its domainof deﬁnition, it can be extended to a smooth function (to be denoted again byΓ2 ) on the whole of the compact manifold S; see the Appendix to this chapter
for further discussion. To obtain the global solution to Equation (11.58), one
considers the ansatz
ϑ = Γ−1 +

m
+ W,
2

(11.59)with W some smooth function on S. To make eﬀective use of this ansatz it
is assumed, without loss of generality, that the conformal metric h satisﬁes
rαβγδ (i) = 0 so that one can write
hαβ = −δαβ + h̄αβ ,h̄αβ = O(|x|3 ).

Hence, using the identity
Δh ϑ = √√1− det h hαβ ∂β ϑ ,∂α− det hit follows that1
Lh = Δh − r[h]8= Δ−δ + L̄ + r[h],withL̄ ≡ h̄αβ ∂α ∂β + bα ∂α ,h̄αβ = O(|x|3 ),bα = O(|x|2 )and r[h] = O(|x|). Using the above expressions one can compute that
 
 
1
1
Lh
= Δ−δ
+ f¯
with f¯ = O(|x|0 ).
ΓΓNow, a calculation similar to the one discussed in Section 11.6.3 shows that
 
1
Δ−δ
= 4πδ(i),Γ11.6 Asymptotically Euclidean manifolds

283

so that substitution of ansatz (11.59) into Lh ϑ = 4πδ(i) leads to the regularequation
1
Δh W − r[h]W = f
8with f = O(|x|0 ),(11.60)for which a suitable existence theory is readily available. A unique smooth
solution to Equation (11.60) exists if the Yamabe number of the metric hsatisﬁes Y [h] > 0; see Beig and O’Murchadha (1991) and Friedrich (1998c).
A further argument using the maximum principle shows that ϑ – as given by
Equation (11.59) – with W solving Equation (11.60) is positive on S \ {i} andgives the unique global solution to Equation (11.58). It follows that (S̃, ϑ4 h) is
an asymptotically Euclidean and regular time-symmetric initial data set.
Data with a non-vanishing extrinsic curvature
The procedure to solve the Yamabe equation described in the previous section
can be extended to the case of an initial data set with a trace-free extrinsic
curvature. One ﬁrst needs a solution to the momentum constraint. Several
procedures to construct solutions to the maximal momentum constraint (and inparticular of the elliptic Equation (11.21)) have been considered in the literature;see, for example, Beig and O’Murchadha (1996), Chaljub-Simon (1982) and Dainand Friedrich (2001). In particular, it is well understood how to specify thein Equation (11.21) so as to ensure non-vanishing ADM linear
free datum ψij
momentum and ADM angular momentum.
In what follows, assume that Equation (11.21) has been solved for a particular. Substituting the transverse and trace-free tensor ψijchoice of the free datum ψij
obtained from the York splitting (11.20) into the Licnerowicz Equation (11.17)
yields the equation
1
1
Δh ϑ − r[h]ϑ = ψij ψ ij ϑ−7 .
8
8As in the case of the Yamabe equation, one can incorporate the singularbehaviour of the conformal factor required to decompactify the compact manifold
S via a Dirac’s delta. This leads to the equation
1
1
Δh ϑ − r[h]ϑ = 4πδ(i) + ψij ψ ij ϑ−7 .
8
8(11.61)

To construct a solution to this equation one ﬁrst considers a solution ϑ• toEquation (11.58) – such solution exists if Y [h] > 0. One uses ϑ• to write theansatzϑ = ϑ• + V
with V a smooth function to be determined. Equation (11.61) yields
1
1
−1
−7
.
Δh V − r[h]V = ψij ψ ij ϑ−7• (1 + ϑ• V )8
8(11.62)

284

The conformal constraint equations

Observe that if ψij = O(|x|−4 ), then, in principle, 18 ψij ψ ij ϑ−7= O(|x|−1 ) so•that the right-hand side of Equation (11.62) is still singular. This singularity is,nevertheless, mild, and suitable existence results are available; see theorem 12
in Dain and Friedrich (2001) and also the appendix in Beig and O’Murchadha(1994). The solution ϑ is smooth, and, again, it can be veriﬁed that it satisﬁes
ϑ > 0 on S \ {i}.
11.7 Hyperboloidal manifolds
In certain applications of the conformal ﬁeld equations it is convenient to consider
initial data sets prescribed on hypersurfaces similar to the hyperboloids ofthe Minkowski spacetime discussed in Section 6.2.4. Hyperboloidal 3-manifolds
arise in the construction of asymptotically simple spacetimes with vanishingcosmological constant and in the construction of anti-de Sitter like spacetimes.
11.7.1 Hyperboloidal initial data sets
For the sake of the presentation, the discussion in this section is restricted to
the vacuum case with vanishing cosmological constant. Based on the intuition
gained through the analysis of hyperboloids in the Minkowski spacetime one has
the following deﬁnition (see Friedrich (1983) and Kánnár (1996a)):Deﬁnition 11.3 (hyperboloidal initial data sets) A triple (S̃, h̃, K̃) satisfying the vacuum Einstein constraint equations is called a hyperboloidal initial
data set if:(i) There exists a conformal compactiﬁcation whereby S̃ is diﬀeomorphically
identiﬁed with the interior of a manifold S with boundary ∂S such that S
is diﬀeomorphic to the closed unit ball in R3 (whence ∂S is diﬀeomorphic
to S2 ).
(ii) There exist functions Ω and Σ on S such that Ω > 0 on S̃ and Ω = 0 andΣ > 0 on ∂S.
(iii) The conformal ﬁelds
h = Ω2 h̃,

K = Ω(K̃ + Σh̃),extend smoothly to S. Moreover, one has that h (dΩ, dΩ) = Σ2 on ∂S.
The simplest type of hyperboloidal initial data sets consists of the case where
the physical extrinsic curvature is pure trace; that is, one has
K̃ =1K̃ h̃.
3(11.63)As a consequence of the momentum constraint and assuming (11.63) it follows
that K̃ must be a constant. From the conformal Hamiltonian constraint (11.15a)11.7 Hyperboloidal manifolds

285

one concludes that
4ΩDi Di Ω − 6Di ΩDi Ω + 2Ω2 r = K̃ 2 .

(11.64)In order to encode the right behaviour of the conformal factor Ω at ∂S one
introduces a so-called boundary deﬁning function ρ, that is, a real function
over S satisfying
ρ|∂S = 0,dρ|∂S = 0.

Given a Riemannian manifold (S, h), such a function can always be constructed.
Making use of the ansatz Ω = ρϑ−2 with ϑ > 0 on S, it follows from (11.64) that


3
1
1
1
Di ρDi ρ − r[h]ϑρ2 θ − ρϑ2 Δh ρ ϑ = − K̃ 2 ϑ−5 .
ρ2 Δh ϑ − ρDi ρDi ϑ +282
8(11.65)The latter is an elliptic equation for ϑ which becomes singular at ∂S as its
principal part vanishes at this set.
The properties of solutions to Equation (11.65) have been analysed inAndersson et al. (1992). One has the following:
Theorem 11.2 (existence of hyperboloidal initial data sets) Let (S, h) bea smooth Riemannian manifold with boundary ∂S. Then, there exists a uniquepositive solution ϑ to Equation (11.65). Moreover, the following are equivalent:(i) The function ϑ and the tensors


1
1
2
Lij ≡ − D{i Dj} Ω +r + K 2 hij ,
Ω
12
3
dij ≡

1
1
D{i Dj} Ω + r{ij} ,Ω2
Ω

(11.66a)(11.66b)determined on S̃ by h and Ω = ρϑ−2 extend smoothly to all of S.
(ii) The Weyl tensor C a bcd computed from the data on S vanishes on ∂S.
(iii) The conformal class [h] is such that the extrinsic curvature of ∂S withrespect to its embedding in (S, h) is pure trace.
The expressions for the ﬁelds Lij and dij correspond to the spatial partof the Schouten tensor and the electric part of the rescaled Weyl tensor as
determined by the conformal constraint equations of Section 11.4.3. Observethat the expressions for the ﬁelds are formally singular at Ω = 0, so that theconclusion of the theorem is non-trivial and ensures the existence of regular
hyperboloidal data for the conformal ﬁeld equations. Extensions of Theorem
11.2 to more general forms of the extrinsic curvature have been analysed inAndersson and Chruściel (1993, 1994).

286

The conformal constraint equations
Initial data for anti-de Sitter-like spacetimes

By making the identiﬁcation K̃ 2 → λ with λ > 0 in Equation (11.64), hyperboloidal initial data sets can be interpreted as initial data sets for anti-deSitter-like spacetimes. Thus, all available knowledge about the existence ofhyperboloidal initial data sets can be transferred to this setting. This idea has
been investigated for a larger class of data than the one considered in this sectionin Kánnár (1996a).
11.8 Other methods for solving the constraint equations
The analysis of the Einstein constraint equations carried out in the previous
sections relies on a systematic use of the conformal method of Licnerowicz,
Choquet-Bruhat and York. There are, however, other alternative procedures,
each providing a diﬀerent insight into the properties of the solutions to theconstraint equations; see, for example, Bartnik and Isenberg (2004). In thissection, methods of particular relevance for the analysis of the conformal ﬁeld
equations are brieﬂy discussed: the ﬁrst one based on the so-called extended
constraint equations, and the second one being the so-called exterior gluing
procedure.
11.8.1 The extended constraint equations
Given a solution to the conformal constraint equations, Lemma 11.1 shows
how to construct initial data for the conformal Einstein ﬁeld equations. It
is, nevertheless, of interest to directly obtain a solution to the conformal
constraint equations without having to solve the Einstein constraint equations. A
construction of this type is of importance as the expressions for the rescaled Weyltensor and the Schouten tensor in terms of the conformal factor and the intrinsic
3-geometry of the hypersurface are formally singular at the points where Ω = 0;see, for example, Equations (11.66a) and (11.66b) in Theorem 11.2. Currently
available results in this direction are restricted to the case where the Ω = 1;
see Butscher (2002, 2007). Despite this limitation, they provide insight into theproperties and structure of the conformal constraint equations and lead to aprocedure for the construction of initial data sets by perturbative methods.
Assuming that the matter ﬁelds vanish, and setting Ω = 1, Σ = 0 in theconformal constraint equations (11.35a)–(11.35j) one ﬁnds that the essential
equations of the system can be rewritten in tensorial form as
Dj Kki − Dk Kji = l jk d∗il ,k

D dki = K ki d∗jl ,jk l(11.67a)(11.67b)

Dk d∗ki = −i jl Kj k rkl ,(11.67c)

rij = dij + KKij − Ki Kkj .
k(11.67d)11.8 Other methods for solving the constraint equations

287

These equations are known as the extended Einstein constraints since asolution thereof implies a solution to the Einstein vacuum constraints; see
Lemma 11.1 in this chapter and lemma 1 in Butscher (2007) for a more detailed
discussion. The ﬁrst three equations constitute an underdetermined elliptic
system for the ﬁelds Kij , dij and d∗ij .
A direct computation shows that the formal adjoint of the operator in theprincipal part of Equation (11.67a) is the divergence with respect to the index
j . Applying this divergence to the equation and writing
1
Kij = ψij − Khij3with ψij hij = 0

one obtains the equation
Dj Dj ψki − Dj Dk ψji = l jk Dj d∗il +
1
hki Dj Dj K − Di Dk K .
3

If the ﬁelds d∗ij and K are known, this equation can be veriﬁed to be an elliptic
equation for the trace-free part ψij .
Equations (11.67b) and (11.67c) can be transformed into fully elliptic equations by means of a York splitting of the ﬁelds dij and d∗ij ; see Section 11.3.3.
Hence, writing2
dij = Di υj + Dj υi − Dk υ k hij + dij ,3
2
d∗ij = Di uj + Dj ui − Dk uk hij + d∗ij ,3where dij and d∗ij are freely speciﬁable symmetric trace-free tensors, one obtains
elliptic equations for the ﬁelds υi and ui whose principal part is identicalto that of Equation (11.21). Finally, Equation (11.67d) can be transformed
into an elliptic equation for the components of the 3-metric h by introducing
harmonic coordinates x = (xα ), Δh xα = 0; compare the analogous useof wave coordinates in the case of a Lorentzian metric to obtain the reduced
Einstein ﬁeld equation discussed in the Appendix to Chapter 13.
The system of elliptic equations for the ﬁelds Kij , υi , ui , hij discussed in theprevious paragraphs is called the auxiliary system. Solutions to the auxiliary
system could be obtained, in principle, by means of perturbative methods relying
on the use of the implicit function theorem – see, for example, Ambrosetti and
Prodi (1995) – if some background solution is known. The solutions thus obtained
are not a priori solutions to the original Equations (11.67a)–(11.67d). Hence, ina second step, one needs to investigate the conditions under which a solution to
the auxiliary system implies a solution to the extended Einstein constraints and,
consequently, a solution to the vacuum Einstein constraints. This strategy has
been investigated in Butscher (2002, 2007) to obtain asymptotically Euclidean
solutions to the extended constraints which are close to data for the Minkowski
spacetime. The particular details require the use of weighted Sobolev spaces

288

The conformal constraint equations

to control the decay of the various ﬁelds. These methods can be adapted, inprinciple, to obtain data on S3 corresponding to perturbations of de Sitter initialdata.
11.8.2 Exterior asymptotic gluing
The exterior asymptotic gluing is a method to construct solutions to the
Einstein constraint equations by gluing the interior region of an asymptoticallyEuclidean solution to the Einstein vacuum constraints to an asymptotic end of
initial data for the Kerr spacetime or, in fact, of any stationary solution; seeCorvino (2000), Chruściel and Delay (2003), Corvino and Schoen (2006) andCorvino (2007). More precisely, given a smooth asymptotically Euclidean initial
data set for the vacuum Einstein ﬁeld equations (S̃, h̃, K̃) and a given compact
subset U ⊂ S̃ such that S̃ \ U is an asymptotic end, it is possible to show that
there exists another smooth asymptotically Euclidean solution to the vacuum
Einstein constraints (S̃, h̄, K̄) which is identical to the original solution on U
and coincides with initial data for the Kerr spacetime on S̃ \ Ū for some Ū ⊂ S̃.
In addition, the initial data set (S̃, h̄, K̄) contains an annular transition regionin which the initial data can be controlled. In the case of time-symmetric initial
data sets this method glues any interior region to an exterior region of a slice of
the Schwarzschild spacetime.
The underlying idea in the asymptotic exterior gluing method is to exploit theunderdetermined character of the Einstein constraints as a system of partial
diﬀerential equations for the ﬁelds (h̃, K̃). Prior to the development of theasymptotic exterior gluing methods Cutler and Wald (1989) have shown that it ispossible to make use of the standard conformal method to construct solutions to
the time symmetric constraints containing a Minkowskian interior region and a
Schwarzschildean exterior region joined together by an annular region containing
a purely magnetic solution to the Einstein-Maxwell constraints.
As will be discussed in Chapter 20, initial data sets obtained by means ofasymptotic exterior gluing play a key role in the construction of Minkowski-like
asymptotically simple spacetimes. For simplicity, in the remainder of this section
attention is restricted to the time-symmetric case for which the Einstein vacuum
constraints reduce to r[h̃] = 0. In the present context, one regards the Ricci
scalar as a map between the space of Riemannian metrics over S̃ and X(S̃). Undercertain circumstances this mapping is an isomorphism; that is, given a metric hand f ∈ X(S̃) such that r[h] = f and given a further g ∈ X(S̃) close enough to f ,then there exists another metric h̄ close to h such that r[h̄] = g. This propertyof the Ricci scalar operator is the essential ingredient in the gluing procedure.
As part of the gluing construction, one connects the inner region (U , h̃) and an
asymptotic region (E, h̃S ) with h̃S as given in Equation (11.45) for some choice(so far undetermined) of the constants m and (xα0 ) through an annular region.
A positive deﬁnite symmetric tensor h̆ is deﬁned on S̃ by requiring it to be
identical to h̃ on U and to h̃S on E, while on the asymptotic region it is chosen

11.8 Other methods for solving the constraint equations289

so that it interpolates smoothly between h̃ and h̃S . By construction r[h̆] = 0 in
both U and E, while r[h̆] = 0 in the transitional annular region. Nevertheless, bymoving U suitably into the asymptotic region, one can make r[h̆] small enoughso that the isomorphism properties of the Ricci scalar operator can be used toensure the existence of a tensor k with support on an annular region such thath̄ ≡ h̆ + k is a Riemannian metric with r[h̄] = 0 on S̃.
The asymptotic exterior gluing construction requires a careful analysis of theproperties of the linearised Ricci operator
Rh [h̄] ≡ −Δh (trh (h̄)) + divh (divh (h̄)) − h(h̄, Ric[h̄]).
For a ﬁxed metric h, the latter is an underdetermined elliptic operator. It canbe transformed into an elliptic system by composition with its formal adjointRh∗ (f ) ≡ −(Δh f )h + Hess(f ) − f Ric[h].
The composite elliptic operator Rh∗ ◦ Rh is a fourth-order partial diﬀerential
operator. Once the linearised problem is controlled, the non-linear problem isthen solved by means of an iteration. To conclude, one needs to show that the
metric h̄ is indeed a solution to r[h̄] = 0. It is in this part of the constructionthat the value of the constants m and (xα0 ) are ﬁxed. A reﬁned version of theoriginal construction in Corvino (2000) has been given in Corvino (2007), fromwhich the following result has been adapted:
Theorem 11.3 (exterior asymptotic gluing construction) Let (S̃, h̃)denote an asymptotically Euclidean initial data set for the Einstein vacuum
equations. Let E ⊂ S̃ be any asymptotically ﬂat end of S̃. Given r0 > 0 let Er0 ⊂ E
be an exterior region in E expressed in asymptotically Cartesian coordinates by
Er0 = {(xα ) ∈ R3 | |x| > r > r0 }. Suppose, furthermore, that in these coordinates
the metric h̃ has the form2mh̃αβ = − 1 +δαβ + O3 (|x|−2 ).
|x|
Let k be a non-negative integer. Then for any ε > 0 there exists r∗ > 0 and asmooth metric h̄ satisfying r[h̄] = 0 and ||hαβ − h̄αβ ||C k (E) < ε so that h̄ is equal
to h̃ on U = S̃ \ Er∗ and identical to an asymptotically ﬂat end of a standard
Schwarzschild slice on E2r∗ .
The precise deﬁnition of the supremum norm || ||C k (E) is discussed in the
Appendix to this chapter. A schematic depiction of the construction of Theorem
11.3 is given in Figure 11.2. In the applications of this result to the existence ofasymptotically simple spacetimes, it is important to control the location of theexterior region Er∗ and to ensure that r∗ → ∞ as one moves along a family of
initial data sets tending, say, to data for the Minkowski spacetime. This possible
degeneracy has been dealt with by imposing some reﬂexion symmetry properties

290

The conformal constraint equations
Ũ

E

Figure 11.2 Schematic depiction of the exterior gluing construction given by
Theorem 11.3. It contains an inner region Ũ where the 3-metric has a ﬁxed
arbitrary value h̃, an annular transition region between Er∗ and E2r∗ and an
exterior region E where it is equal to data for a member of the Schwarzschildfamily of solutions.

on the metric h̃; see Chruściel and Delay (2003). An alternative solution has
been provided in Corvino (2007). This result makes use of symmetric (0, 2)tensors k satisfying the condition R−δ (k) = 0. Making use of a York splitting
the tensor k can be decomposed in a unique way into a traceless term withvanishing divergence, a trace part and a part which is the conformal Killing
operator of a covector; see Chaljub-Simon (1982). The tensor k is said to be nondegenerate if its transverse-traceless part is non-zero. Using this terminology
one has the following stability result (see Corvino (2007) for further details and its
proof):
Theorem 11.4 (stability of the exterior gluing construction) Let k be anysmooth, compactly supported symmetric (0, 2)-tensor on R3 with R−δ (k) = 0.
Moreover, for suﬃciently small ε > 0 leth̃ = −ϑ4 (δ + εk)
be asymptotically ﬂat and satisfy r[h̃] = 0. If k is non-degenerate, there existsr∗ > 0 so that for all ε small enough there is a metric h̄ with r[h̄] = 0 which
agrees with h̃ in the closed ball B r∗ (0) and is exactly Schwarzschild on E2r∗ .
Consequently, the Riemannian manifold (R3 , h̄) admits a smooth conformal pointcompactiﬁcation in the sense of Deﬁnition 11.2.
This theorem guarantees the existence of time-symmetric solutions to thevacuum Einstein constraint equations which are both close to data for the
Minkowski spacetime and exactly Schwarzschildean in a non-trivial exterior
region; see Section 20.5.
Versions of the asymptotic exterior gluing construction for initial data setswith non-vanishing extrinsic curvature can be found in Chruściel and Delay(2003) and Corvino and Schoen (2006). There are adaptations of the exterior
gluing method to the case of hyperboloidal initial data sets with constant scalar
curvature; see Chruściel and Delay (2009).

11.9 Further reading

291

11.9 Further reading
The best point of entry to the extensive literature on the Einstein constraint
equations is through reviews such as those of Bartnik and Isenberg (2004) or
Isenberg (2013). An older, classical review on the topic is given in ChoquetBruhat and York (1980). An alternative review aimed at applications innumerical relativity is Cook (2000). A detailed account of the conformal method
to solve the constraint equations, as seen by one of the main contributors of
the topic, can be found in Choquet-Bruhat (2008) – this reference contains, inaddition, a discussion of the basic aspects of weighted Sobolev spaces. Closely
related to the latter is the reference Choquet-Bruhat et al. (2000). A discussionof basic aspects of the theory of elliptic diﬀerential equations and its application
to the analysis of the Einstein constraints can be found in Rendall (2008). An
alternative account of the basic aspects of the analysis of elliptic equations witha number of worked-out examples is Dain (2006). Finally, a detailed account of
the conformal equations under the assumption of spherical symmetry is given in
Guven and O’Murchadha (1995).
By contrast, the accounts on the conformal Einstein constraints are much more
restricted in number. The original references are Friedrich (1983, 1984, 1986a,
1995, 2004); see also the discussion in Frauendiener (2004). A systematic analysis
of hyperboloidal initial data sets can be found in Andersson et al. (1992) andAndersson and Chruściel (1993, 1994).
The notion of asymptotically Euclidean and regular manifolds can be traced
back to the discussion in Geroch (1972b). These ideas have been further
elaborated in Friedrich (1988, 1998c). Accounts of the use of Dirac’s deltas torepresent the points at inﬁnity can be found in Beig and O’Murchadha (1991,
1994). A neat application of this approach to the construction of initial data
sets with a conformal toroidal symmetry is given in Beig and Husa (1994).
Applications of the method to the construction of initial data for the collisionof Kerr-like black holes can be found in Dain (2001a,c). Finally, a detailed
construction of initial data sets admitting expansions in powers of the geodesic
distance is given in Dain and Friedrich (2001).
Appendix: some results of analysis
As in the main text of this chapter, let (S, h) denote a Riemannian manifold.
Moreover, let p ∈ S denote a point and consider normal coordinates x = (xα )centred at p; that is, xα (0) = 0.
Order symbols. The behaviour of functions f : S → R near p can be conveniently described by means of the big O and small o notations. More precisely,
given f , g : S → R, if for some x = (xα ) suﬃciently close to 0 there exists a
positive constant M such that|f (x)| ≤ M |g(x)|,292

The conformal constraint equations

one writes f (x) = O(g(x)), and one says that f is at most of the order of g. If,
in addition, one has that∂α f (x) = O(∂α g(x)),···∂α1 . . . ∂αk f (x) = O(∂α1 · · · ∂αk g(x)),

for some integer k one writes f (x) = Ok (g(x)).
If given f, g one has f (x)/g(x) → 0 as xα → 0, then one writes
f (x) = o(g(x)),
and one says that the order of f is bigger than that of g. Again, if∂α f (x) = o(∂α g(x)),∂α1 · · · ∂αk f (x) = o(∂α1 · · · ∂αk g(x)),...

one writes f (x) = ok (g(x)). For further discussion, see, for example, Courant
and John (1989).
Taylor expansions. If a function f : Rn → R is of class C k on the open ballBa (0) ⊂ Rn one has thatf (x) = f (0) + ∂α f (0)xα ++ ··· +

1∂α ∂α f (0)xα1 xα2
2! 1 2

1∂α · · · ∂αk−1 f (0)xα1 · · · xαk−1 + O(|x|k ).
(k − 1)! 1

For further discussion, see, for example, Courant and John (1989).
Supremum norm. Given U ⊂ Rn and f ∈ C k (U ), one deﬁnes the supremum
norm assup{|∂α1 · · · ∂αl f (x)| , x ∈ U }||f ||C k (U ) =0≤l≤kwhere U denotes the closure of U . For further discussion on this and other related
norms, see, for example, Ambrosetti and Prodi (1995).
Extension of smooth functions. Let U ⊂ S denote a closed subset and f :U → Rk a smooth function. There exists a smooth function f˜ : S → Rk suchthat f˜|U = f and whose support is contained in S \ U ; in other words, f˜ isnon-vanishing in S \ U . In a slight abuse of notation f˜ will be denoted, again, by
f . For more details on this result, see Lee (2002).
Dirac’s delta. Let now S denote a compact manifold and p ∈ S a ﬁxed point
within. The Dirac’s delta δ(p) with support on p is the distribution (i.e. a linear
functional C 0 (S) → R) satisfyingf (x)δ(p)dμh = f (p),for all f ∈ C 0 (S).
S

In particular, one has that

Sδ(p) dμh = 1.

11.9 Further reading

293

If f (p) = 0, one has the distributional equality
f (x)δ(p) = 0.
For further details, the reader is referred to Appel (2007).
Divergence theorem. Given (M, g) a manifold with metric (Riemannian orLorentzian) and, within, U ⊂ M a compact subset and a smooth covector ω,
one hasdivω dμh =ω, ν dSh ,U∂Uwith ν the outward pointing unit normal to ∂U ; see, for example, Frankel (2003)
for further details.

12
Methods of the theory of hyperbolic
diﬀerential equations

This chapter discusses the notions of the theory of hyperbolic diﬀerential
equations and the existence theorems employed to construct solutions to theconformal Einstein ﬁeld equations. Conformal methods allow, under suitable
circumstances, the use of very general theorems of the theory of partial
diﬀerential equations (PDEs) to obtain conclusions of a global nature about
solutions to the Einstein ﬁeld equations. The results presented in this chapter
have been tailored to ﬁt the general discussion of this book.
The basic result of the theory of hyperbolic PDEs that will be used in thisbook is Kato’s existence, uniqueness and stability result for symmetric hyperbolic
systems; see Theorem 12.4. In view of applications to the construction of antide Sitter-like spacetimes a basic existence and uniqueness result of the initialboundary value problem of symmetric hyperbolic equations is also discussed;
see Theorem 12.6. The chapter concludes with an overview of the basic theory
behind characteristic initial value problems; see Theorem 12.7.
12.1 Basic notions
As will be seen in Chapter 13, the conformal Einstein equations give rise toquasilinear evolution equations which, in local coordinates x ≡ (xμ ) on an
open set U ⊂ M of the spacetime manifold, take the form
Aμ (x, u)∂μ u = B(x, u)(12.1)

where u is a CN -valued unknown for some positive integer N and Aμ , μ = 0, . . . 3,
are (N × N ) matrix-valued functions of the coordinates and of the vector-valued
unknown u; thus, one has as many equations as components in the vector u.
Finally, B(x, u) is a vector-valued function of x and u. In what follows, it will
be assumed that the components of u are scalars. The functions Aμ (x, u) andB(x, u) are, in principle, non-linear functions of the entries of u. If the matrices

12.1 Basic notions

295

Aμ do not depend on u, one has a semilinear system. Without loss of generality,
U can be regarded as some suitable subset of R4 .
Following the terminology of Section 11.2 the termAμ (x, u)∂μ u
is known as the principal part of Equation (12.1). The symbol with respect
to the unknown u at the point p ∈ U with coordinates x = x(p) for a covectorξ ∈ T ∗ |p (U ) is given by the matrix
σ(x, u, ξ) ≡ Aμ (x, u)ξμ .
Under a coordinate transformation x = x (x), it follows from Equation (12.1)
thatAμ (x , u)∂μ u = B(x , u),
withAμ (x , u) =∂xμ νA (x(x ), u).
∂xν(12.2)

It then follows from the transformation law of covectors under coordinatetransformations and Equation (12.2) that the symbol of the diﬀerential
Equation (12.1) is an invariant.
12.1.1 Symmetric hyperbolic systems
The basic properties of the PDE (12.1) and of its solutions depend on the
structure of its principal part. Given a matrix A, the operation of taking thetranspose of its complex conjugate will be denoted by A∗ . One has the following
deﬁnition:
Deﬁnition 12.1 (symmetric hyperbolic systems) Given a solution u(x),the system (12.1) is said to be symmetric hyperbolic at (x, u) if:(i) the matrices Aμ (x, u) are Hermitian; that is (Aμ )∗ = Aμ(ii) there exists a covector ξ such that σ(x, u, ξ) = Aμ (x, u)ξμ is a positivedeﬁnite matrix.
Given two vectors u, v ∈ CN , their inner product is deﬁned by
u, v ≡ u∗ v.
It follows then that u, v = v, u with the overbar denoting the usual complex
conjugation of scalars. Moreover, if A is a Hermitian N × N matrix, thenu, Av = Au, v,u, Au = u, Au.

296

Methods of the theory of hyperbolic diﬀerential equations
Spacelike and timelike hypersurfaces with respect to asymmetric hyperbolic system

In what follows, let S denote a hypersurface on U ⊂ M deﬁned in terms of asmooth scalar φ ∈ X (M) as
S ≡ {p ∈ U | φ(p) = 0},(12.3)

where it is assumed that dφ = 0 so that S has everywhere a well-deﬁned normal.
The positivity condition (i) in Deﬁnition 12.1 allows one to deﬁne the causal
nature of the hypersurface S with respect to solutions of Equation (12.1).
More precisely, the hypersurface S is said to be spacelike with respect toa solution u to the symmetric hyperbolic system (12.1) if σ(x, u, dφ)
is positive deﬁnite for p ∈ S. If σ(x, u, dφ) has a non-vanishing determinant
and is not positive deﬁnite, one says that S is timelike for the solution u.
Finally, if σ(x, u, dφ) has a vanishing determinant, one says that S is null – thiscase is tied to the notion of characteristics to be discussed in the next section.
These causal deﬁnitions are, in principle, independent of the homonymous notion
deﬁned in terms of a metric g on M. However, as discussed in Chapter 14, forevolution equations arising from the Einstein ﬁeld equations, the geometric and
PDE notions agree; see Theorem 14.1.
12.1.2 Initial value problems and characteristics
Of particular relevance for a symmetric hyperbolic system of the form (12.1) isthe so-called initial value problem whereby some initial data on a hypersurface
S is prescribed and one purports to obtain the solution to the equation awayfrom the initial hypersurface.
An initial data set for Equation (12.1) on a hypersurface S which is spacelike
with respect to Equation (12.1) consists of a CN -valued function u on S whichis interpreted as the value of the solution u to Equation (12.1) on S. A question
which arises naturally in this context is whether all the components of the vectoru can be speciﬁed freely on S.
It is convenient to introduce on U coordinates x = (x0 , x) = (x0 , x1 , x2 , x3 )adapted to S so that the hypersurface is represented by the condition x0 = 0.
Using these adapted coordinates and the initial data u one can compute thespatial derivatives ∂α u of u on S. In order to determine the time derivatives∂0 u on S one substitutes the above into Equation (12.1) to obtain
A0 (0, x; u )(∂0 u) + Aα (0, x; u )∂α u = B(0, x; u ),(12.4)

where it is observed that (∂α u) = ∂α u . This equation can be read as a linear
algebraic system for (∂0 u) ≡ ∂0 u|S which can be solved if A0 (0, x, u ) can beinverted, that is, if
det A0 (0, x; u ) = 0.

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297

If det A0 (0, x; u ) = 0, then M ≡ rank A0 (0, x, u ) < N , and one can makelinear combinations of the equations in (12.4) to obtain a new system on S of
the formĀ0 (0, x; u )(∂0 u) + Āα (0, x; u )∂α u = B(0, x; u ),
where

⎛

a011 (0, x; u )
⎜
..
⎜
.
⎜
⎜ 0
⎜ aM 1 (0, x; u )0Ā (0, x, u ) = ⎜
⎜
0
⎜
..
⎜
⎝
.
0

···
..
.
···
···
..
.
···

⎞
a01N (0, x; u )
⎟
..
⎟
.
⎟
⎟
0
aM N (0, x; u ) ⎟
⎟
⎟
0
⎟
..
⎟
⎠
.
0

has N − M rows consisting of zeros. Hence, not all the derivatives (∂0 u) aredetermined by the initial data, and one has N −M constraint equations whichhave to be satisﬁed by the initial data u .
The discussion of the previous paragraphs leads to the following generaldeﬁnition which also applies to evolution systems of the form (12.1) which are
not necessarily symmetric hyperbolic:Deﬁnition 12.2 (characteristic surfaces of a ﬁrst order PDE ) A
hypersurface S deﬁned by a condition of the form (12.3) is said to be acharacteristic of a solution u of Equation (12.1) if
det σ(x, u, dφ) = 0 for p ∈ S.
(12.5)If
det σ(x, u, dφ) = 0

for p ∈ S,

then S is said to be nowhere characteristic for the solution u ofEquation (12.1).
On a characteristic, the system (12.1) implies M transversal equations andN − M interior equations on S. If M = 0, so that the full system (12.1)
reduces to interior equations, one says that S is a total characteristic of thesystem. More generally, given a point p ∈ U , one deﬁnes its characteristic set(or Monge cone) with respect to a solution u of Equation (12.1) as the subset
Cp∗ ⊂ T ∗ |p (U ) deﬁned byCp∗ ≡ ξ ∈ T ∗ |p (U ) det(σ(x, u, ξ)) = 0, ξ = 0 .
That is, the elements of Cp∗ are in the kernel of the symbol. The covectors ξare sometimes called the null directions at p. The quantity det(σ(x, u, ξ)) canbe read as a polynomial for the components of the covector ξ – the so-called
characteristic polynomial.

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Methods of the theory of hyperbolic diﬀerential equations
Well-posedness

An initial value problem for a system of the form (12.1) (not necessarily
symmetric hyperbolic) with data prescribed on a hypersurface S which is nowherecharacteristic and timelike with respect to the evolution system at the prescribeddata u will be called a Cauchy initial value problem. If the initial data
is prescribed on a hypersurface N which is characteristic, one speaks of acharacteristic initial value problem.
The deﬁnitions given in the previous paragraph are motivated by the notionof well-posedness. In broad terms, an initial value problem is well posed if:(i) There exist solutions to all initial data.
(ii) The solutions are uniquely determined by the initial data.
(iii) The solutions depend continuously on the initial data.
The ﬁrst step in the analysis of the well-posedness of an initial value problem fora given class of PDEs is the formulation of the above requirements in a precise
manner; see, for example, Rendall (2008) for further discussion on this. Initialvalue problems which are not well posed are said to be ill-posed.
The Cauchy problem for a symmetric hyperbolic system of the form (12.1)
is well-posed. By contrast, an initial value problem with data prescribed on atimelike hypersurface is ill-posed. A further example of an ill-posed problem is
the Cauchy problem for elliptic equations. In the case of characteristic initialvalue problems the well-posedness of the problem depends on the causal relation
between the region where one wants to obtain the solution and the initialcharacteristic surfaces; see Section 12.5.1. Although well-posed initial value
problems are of natural importance in general relativity, ill-posed problems also
arise in applications such as the uniqueness of stationary black holes; see, forexample, Ionescu and Klainerman (2009a,b).
12.1.3 Some examples
The discussion of the previous paragraphs is best illuminated by a couple of
explicit examples. Many of the features of these examples are generic and arisein the analysis of the evolution equations implied by the (conformal) Einsteinﬁeld equations.
In what follows, let (M, g) denote a spacetime. On U ⊂ M consider somelocal coordinates x = (xμ ) and a null frame {eAA } with associated cobasis
{ω AA }. In terms of the local coordinates one writes eAA = eAA μ ∂ μ andω AA = ω AA μ dxμ . Moreover, let {A A } be a spinorial frame giving rise to thevector frame {eAA }; see the discussion in Section 3.1.9.
A spinorial curl equation
As a ﬁrst example consider on U ⊂ M a spinorial equation of the form
∇Q A ϕQA···D = FA A···D ,(12.6)

12.1 Basic notions

299

for the components ϕQA···D of a spinor ϕQA···D with respect to the spin frame{A A }. The spinor ϕQA···D is not assumed to have any particular symmetries
and the ﬁeld FA A···D may depend on the coordinates or any other ﬁeld. Noticethat the unknowns of Equation (12.6) are scalars.
It is claimed that the combination∇Q 1 ϕQA···D = F1 A···D ,−∇Q

0 ϕQA···D = −F0 A···D ,(12.7a)(12.7b)is a symmetric hyperbolic system. In order to see this, observe that
∇Q A ϕQA···D = ∇1A ϕ0A···D − ∇0A ϕ1A···D .
Thus, the principal part of the system (12.7a) and (12.7b) can be written in
matricial form as e11 μ −e01 μ
ϕ0A···DAμ ∂ μ ϕ ≡∂.
μ−e10 μ e00 μ
ϕ1A···DThe matrices Aμ are Hermitian as e00 and e11 are real vectors and e01 = e10 .
Letting ξμ ≡ ω 00 μ + ω 11 μ , a calculation shows that
#
$




e11 μ ω 00 μ + e11 μ ω 11 μ −e01 μ ω 00 μ − e01 μ ω 11 μμ
A ξμ =.
−e10 μ ω 00 μ − e10 μ ωμ11e00 μ ω 00 μ + e00 ω 11 μUsing eAA μ ω BB μ = A B A B it follows thatμ

A ξμ =

1 0
0 1


,

which is clearly positive deﬁnite. Thus, the system (12.7a) and (12.7b) issymmetric hyperbolic as claimed. Given a generic covector ξ, the characteristic
polynomial is given bye11 μ ξμ −e01 μ ξμμdet(A ξμ ) = det−e10 μ ξμ e00 μ ξμ= eμ11 e00 ν − e01 μ e10 ν ξμ ξν
1= g μν ξμ ξν ,2where, in the last equality, Equation (3.30) relating the null frame and themetric has been used. Thus, the characteristics of Equation (12.6) are given
by null hypersurfaces with respect to the metric g. Furthermore, spacelike
hypersurfaces with respect to solutions to the equation coincide with theg-spacelike hypersurfaces so that the causal notions given by Equation (12.6)
and the background metric g coincide.

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Methods of the theory of hyperbolic diﬀerential equations
The wave equation as a symmetric hyperbolic system

As a second example consider the wave equation∇a ∇a φ = 0(12.8)

on a region U ⊂ M. In contrast to the previous example, this equation is second
order, and thus, it does not ﬁt into the scheme discussed so far. Nevertheless,the wave equation can be recast as a symmetric hyperbolic system for the scalarﬁeld φ and some further auxiliary ﬁelds.
The spinorial version of Equation (12.8) is given by∇AA ∇AA φ = 0.

(12.9)As a ﬁrst step one introduces the auxiliary variable φAA ≡ ∇AA φ. Reading
this deﬁnition as an equation for the scalar ﬁeld φ and contracting with a spinorτ AA representing a timelike vector τ a , one obtains the evolution equation
Pφ = ϕ,(12.10)

where ϕ ≡ τ AA φAA and P ≡ τ AA ∇AA is the directional derivative
along τ a ; see Section 4.3.1. Now, deﬁning ϕAB ≡ τ(B A φA)A one obtains thedecomposition
φAA =

1ϕτAA − τ Q A ϕAQ .
2(12.11)

Having introduced the auxiliary variable φAA one needs to consider a suitableﬁeld equation for it. A convenient choice is given by the no torsion condition∇AA ∇BB  φ − ∇BB  ∇AA φ = 0,
which, in view of the deﬁnition of φAA , can be rewritten as∇AA φBB  − ∇BB  φAA = 0.

(12.12)Contracting the indices A and B  and using the see-saw rule one obtains
∇(A Q φB)Q = 0,
which, as a result of the hermicity of φAA is completely equivalent toEquation (12.12). Finally, using the identity1∇A Q φBQ = ∇(A Q φB)Q − AB ∇QQ φQQ2and observing that from Equation (12.9) it follows that ∇QQ φQQ = 0, one
concludes that

∇A Q φBQ = 0.

12.2 Uniqueness and domains of dependence

301

Using Equation (12.11) one can perform a space spinor split of this equation.
After some calculations one obtains the pair of equations
Pϕ + 2DAB ϕAB = 0,

(12.13a)PϕAB − DAB ϕ + 2D(A ϕB)Q = 0,
Q

(12.13b)where DAB denotes the directional derivative associated to the Sen connection
relative to τ AA ; see Section 4.3.1. Equations (12.10), (12.13a) and (12.13b)
are the basic evolution equations. For simplicity of presentation in Equations
(12.13a) and (12.13b) the covariant derivatives of τ AA have been assumed tovanish. To obtain a system which is symmetric hyperbolic, some normalisation
factors have to be added. Some experimentation renders
Pφ = ϕ,Pϕ + 2DAB ϕAB = 0,
4
PϕAB − DAB ϕ + 2D(A Q ϕB)Q = 0,
(A + B)!(2 − A − B)!
which is claimed to be symmetric hyperbolic. From these equations, a calculation
similar to the one carried out for the Maxwell equations yields the following
matricial expression for the principal part:
⎛

τμ
⎜ 0
⎜
μ
A ∂μ φ ≡ ⎜
⎜ 0
⎝ 0
0

0
τμ
−2e00 μ−4e01 μ−2e11 μ02e11 μ2τ − 4e01 μ−4e11 μ0
μ0−4e01 μ4e00 μ4τ μ−4e11 μ⎞φ
⎟ ⎜ ϕ ⎟
2e00 μ⎟ ⎜⎟⎟ ∂μ ⎜ ϕ0 ⎟ ,0⎟ ⎜⎟⎠ ⎝ ϕ1 ⎠4e00 μ2τ μ + 4e01 μ
ϕ2
0

⎞

⎛

where ϕ0 ≡ ϕ00 , ϕ1 ≡ ϕ01 and ϕ2 ≡ ϕ11 . Taking into account the reality
conditions satisﬁed by the various frame coeﬃcients one concludes that thematrices are Hermitian. Moreover, a short computation shows that Aμ τμ is
positive deﬁnite so that, indeed, one has obtained a symmetric hyperbolic
system for the wave equation. Finally, a further computation shows that thecharacteristic polynomial of the system is given bydet(Aμ ξμ ) = 8(τ μ ξμ )(g νλ ξν ξλ )2 .
Accordingly, g-null hypersurfaces are characteristics of the system.

12.2 Uniqueness and domains of dependence
An important property of the Cauchy initial value problem for symmetric
hyperbolic systems is the uniqueness of solutions for a given prescriptionof initial data. The discussion of the uniqueness of solutions is naturally carriedout in subsets of R4 known as lens-shaped domains. A lens-shaped domain

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Methods of the theory of hyperbolic diﬀerential equations
S1
G
S0

Figure 12.1 Schematic depiction of a lens-shaped domain G. The hypersurfaces
S0 and S1 are spacelike with respect to a solution u of a symmetric hyperbolic
system of the form (12.1).

with respect to a solution u to a symmetric hyperbolic system of the form (12.1)
is an open subset G ⊂ R4 with compact closure and whose boundary is givenby the union of two subsets S0 and S1 of hypersurfaces which are spacelike withrespect to u; see Figure 12.1. In terms of these domains one has the following
result which exploits all the algebraic conditions in Deﬁnition 12.1:
Theorem 12.1 (uniqueness of solutions of symmetric hyperbolic systems) Let G be a lens-shaped domain. If u1 and u2 are two solutions to theinitial value problem for the symmetric hyperbolic system
Aμ (x, u)∂μ u = B(x, u),u|S0 = uthen u1 = u2 on G.
Proof This proof follows closely the discussion in Friedrich and Rendall (2000).
Assume one has a symmetric hyperbolic system of the form (12.1) such that thematrices Aμ and B are C 1 functions of their arguments. Moreover, let u1 andu2 denote two C 1 solutions. Let G denote a lens-shaped region with respect to
u1 and u2 whose boundary is given by the union of two hypersurfaces S0 and S1 .
Using a reﬁned version of the mean value theorem (see the Appendix to thischapter for further discussion) it follows that there exist continuous functions
Mμ and N such thatAμ (x, u1 ) − Aμ (x, u2 ) = Mμ (x, u1 , u2 )(u1 − u2 ),B(x, u1 ) − B(x, u2 ) = N(x, u1 , u2 )(u1 − u2 ).
It follows then from Equation (12.1) thatAμ (x, u1 )∂μ (u1 − u2 ) + Mμ (x, u1 , u2 )∂μ u2 + N(x, u1 , u2 ) (u1 − u2 ) = 0.
This equation can be written in a more compact form as
Aμ (x, u1 )∂μ (u1 − u2 ) = Q(x, u1 , u2 , ∂u2 )(u1 − u2 )with Q(x, u1 , u2 , ∂u2 ) a continuous function of its arguments.

12.2 Uniqueness and domains of dependence

303

Now, choosing coordinates such that x = (t, x) and using the evolution
Equation (12.1) one can verify the identity


−kt
μ
∂μ e u1 − u2 , A (x, u1 )(u1 − u2 )
= e−kt u1 − u2 , P(x, u1 , u2 , ∂u2 )(u1 − u2 ),(12.14)

whereP(x, u1 , u2 , ∂u2 ) ≡ −kA0 (x, u1 ) + ∂μ Aμ (x, u1 )+ Q(x, u1 , u2 , ∂u2 ) + Q∗ (x, u1 , u2 , ∂u2 ).
Integrating the identity (12.14) over the lens-shaped region G and using theGauss theorem one has that∂μ e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 ) d4 xG=e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS
S1

e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS,(12.15)−S0

where d x is the standard volume element in R4 and νμ denotes the outward
pointing unit normal to ∂G.
As S1 is spatial with respect to the symmetric hyperbolic system, it follows thatAμ (x, u1 )|S1 is positive deﬁnite. Hence, the integral over S1 in Equation (12.15)
is non-negative. By assumption one has that (u1 −u2 )|S0 = 0 so that the integral
over S0 in (12.15) vanishes. Hence, one concludes thate−kt u1 − u2 , P(x, u1 , u2 , ∂u2 )(u1 − u2 )d4 x ≥ 0.
(12.16)
4

G

Finally, as the matrix A0 (x, u1 ) is positive deﬁnite and G is compact, it follows
that the constant k > 0 can be chosen so that P(x, u1 , u2 , ∂u2 ) is negative
deﬁnite uniformly on G. In other words, there exists a positive constant C suchthat
0 > −Cu1 − u2 , u1 − u2  ≥ u1 − u2 , P(x, u1 , u2 , ∂u2 )(u1 − u2 ).
Accordingly, the integral over G in inequality (12.16) can be made negative by asuitable choice of k. This is a contradiction unless u1 = u2 in G.
A corollary of the above theorem is the following:
Corollary 12.1 If u|S0 = 0 and B(x, u) is homogeneous in u, then u = 0
in G.
Proof The result follows directly from the previous theorem, observing thatu = 0 is a solution.

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Methods of the theory of hyperbolic diﬀerential equations

The uniqueness Theorem 12.1 shows that, in a neighbourhood of an initial
hypersurface S, the solution of a symmetric hyperbolic system is determinedby initial data on a compact subset of S as any point suﬃciently close to S
is contained in a lens-shaped region. This consideration leads to the notion of
domain of dependence.
Deﬁnition 12.3 (domain of dependence) Let R ⊂ S. The domain of
dependence D(R) of R is the set of all points p ∈ U ⊂ R4 such that thevalue of a solution u to Equation (12.1) at p is determined (uniquely) by the
restriction of the initial data to R.
Remark. The term “domain of dependence” is sometimes used in the PDE
literature to denote the set of points determining the value of a solution u at agiven point. The notion of domain of dependence used in this book is then calleddomain of inﬂuence; see Rendall (2008) for further discussion.
The main property singled out by Deﬁnition 12.3 is that the solution ofa symmetric hyperbolic system is determined at a given point by data on
a proper subset of the initial hypersurface. Thus, the process of solving the
Cauchy problem for the symmetric hyperbolic system (12.1) can be localised inspace. This is a particular property of hyperbolic diﬀerential equations which
distinguishes them from other types of PDEs. More precisely, if two initial data
sets u and ū coincide on an open subset R ⊂ S, then the corresponding
domains of inﬂuence and the solutions u and ū coincide as well. In other words,
in the domain of inﬂuence D(R) a solution u is independent of the behaviourof the data u outside R. In particular, there is no need to impose boundaryor fall-oﬀ conditions away from R. This observation is usually known as the
localisability property of symmetric hyperbolic systems; that is, the theory
does not depend on the global knowledge of the initial data in space. A related
observation is that if on S one has two diﬀerent intersecting coordinate patches
R and R such that on R ∩ R one has x = x (x), then, as a consequence thetransformation rule of Equation (12.1) and the uniqueness of the solution on
D(R ∩ R ), one has that u (x ) = u(x(x )).
Finite speed of propagation of solutions
A consequence of the existence of a domain of dependence for symmetric
hyperbolic systems is the so-called ﬁnite speed of propagation of theirsolutions. A rough estimate of this phenomenon can be constructed using anargument given in Rendall (2008).
As in previous sections, let u denote a solution to a symmetric hyperbolic
system of the form (12.1) with initial data u prescribed on the hypersurface
S0 ≡ {p ∈ U | t(p) = 0}.
In what follows, assume that the support of u is contained on a ball of radiusr around the origin.

12.2 Uniqueness and domains of dependence

305

Now, given a point p ∈ U with coordinates (t• , x• ) ≡ (t, xα• ) and a constantβ > 0 consider the paraboloidal hypersurface

 β


x (p) − xβ• .
Sβ;(t• ,x• ) ≡ p ∈ U t(p) = t• − β δαβ xα (p) − xα•
The normal to these hypersurfaces is given byν = dt − 2βδαβ xα dxβ .
Hence, assuming that A0 (x, u) is positive deﬁnite on S0 it follows thatAμ (x, u)νμ = A0 (x, u) + 2βδαβ xα Aβ (x, u)
can be made positive deﬁnite by choosing β suﬃciently small, say, β < β0 , sothat Sβ;(t• ,x• ) is spacelike with respect to Equation (12.1). For this choice of β
the region G bounded by S0 and Sβ;(t• ,x• ) is a lens-shaped domain. Now, it canbe veriﬁed that the intersection of Sβ;(t• ,x• ) with S0 lies outside a ball of radiust•β,|x• |2 ≡ δαβ xαr ≡ |x• | −• x• .
βThus, if|x• | −t•> r ,
βthen the solution satisﬁes u(t• , x• ) = 0 as (t• , x• ) lies on the boundary of a lensshaped region with trivial data. Accordingly, the support of u on the hypersurfaceSt• = p ∈ U | t(p) = t•must lie within a ball of radius r + t• /β; see Figure 12.2 for further details.
Thus, the support of the solution gradually spreads in space at ﬁnite speed.
t

r +

t/β
(t•, x•)supp u•Sβ;(t•,x•)

supp uS0

x

Figure 12.2 Schematic depiction of the rough estimate of the spread of the
support of a solution to a symmetric hyperbolic system. The solution at (t• , x• )is determined by trivial data at the initial hypersurface S0 ; see the main text
for further details.

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Methods of the theory of hyperbolic diﬀerential equations
12.3 Local existence results for symmetric hyperbolic systems

The purpose of this section is to analyse the basic existence and stability results
for symmetric hyperbolic systems of the form (12.1). The precise formulationof existence results is more technical than the one for uniqueness and requires
a certain number of notions from the theory of functional analysis. These are
discussed in the following subsection.
12.3.1 Sobolev spaces
The precise discussion of existence results for symmetric hyperbolic systems iscarried out in terms of Sobolev spaces. The purpose of this section is to introduce
some of the basic ideas concerning these function spaces. In a ﬁrst step, thediscussion will consider Sobolev spaces of functions over R3 . These notions canbe suitably extended to three-dimensional manifolds with a diﬀerent topology.
In what follows, let x ≡ (xα ) denote some particular choice of Cartesian
coordinates and let d3 x be the standard volume element of R3 . The discussionof solutions of symmetric hyperbolic systems of the form (12.1) leads to consider
CN -valued functions on R3 ; that is, w : R3 → CN . The space of smoothfunctions of this type will be denoted by C ∞ (R3 , CN ). On C ∞ (R3 , CN ) one canintroduce, for m ∈ N, a Sobolev norm via#m3||w||R3 ,m ≡$1/2

|∂αk · · · ∂α1 w| d x
2 3

k=0 α1 ,...αk =1R3,(12.17)for w ≡ (w1 , . . . , wN ) ∈ C ∞ (R3 , CN ) where |w|2 = w, w is the standard normin CN . For example, if u = (u) is a C-valued function, one has that||u||2R3 ,1 =(u u + ∂1 u ∂1 u + ∂2 u ∂2 u + ∂3 u ∂3 u)d3 x.
R3

Not all functions w ∈ C ∞ (R3 , CN ) satisfy ||w||R3 ,m < ∞. For example, aconstant function from R3 to CN will have inﬁnite Sobolev norm. In order for afunction to have ﬁnite Sobolev norm, it must decay suitably at inﬁnity. In viewof the localisability property of hyperbolic equations discussed in Section 12.2
this restriction does not pose a problem in the subsequent considerations. Thus,in what follows, attention is restricted, for given m ∈ N, to the spacew ∈ C ∞ (R3 , CN ) ||w||R3 ,m < ∞of CN -valued functions over R3 with ﬁnite Sobolev norm of order m. This set
is clearly a vector space, but not a Banach space; that is, not all Cauchy
sequences of functions in the set have a limit in the space. To obtain a
Banach space one needs to complete the space by including the limit pointsof its Cauchy sequences. The completion of the space under the norm || ||R3 ,mdeﬁned by Equation (12.17) is called the Sobolev space H m (R3 , CN ). Given12.3 Local existence results for symmetric hyperbolic systems

307w• ∈ H m (R3 , CN ), the (open) ball of radius ε centred at w• with respect to thenorm || ||R3 ,m is deﬁned as the setBε (w• ) ≡ w ∈ H m (R3 , CN ) ||w − w• ||R3 ,m < ε .
When discussing symmetric hyperbolic systems of the form (12.1), it isconvenient to consider their solutions u as H m (R3 , CN )-valued functions of thetime coordinate t. This point of view is expressed by writingu(t, ·) : [0, T ] −→ H m (R3 , CN ).
If a CN -valued function u is such that for every t ∈ [0, T ], u(t, ·) ∈ H m (R3 , CN )with C k -dependence on t, one writes
w ∈ C k [0, T ]; H m (R3 , CN ) .
For further details on Sobolev spaces, the reader is referred to Evans (1998).
Embedding theorems
Functions in the Sobolev space H m (R3 , CN ) are not necessarily smooth. The
reason for this is that by completing the space one has included functions withlower regularity. There is, nevertheless, a relation between functions in H m andC k spaces. This relation is expressed in terms of so-called embedding theorems.
For the particular case under consideration one has the following:
Proposition 12.1 (Sobolev embedding theorem) If m ≥ 2 + k, thenH m (R3 , CN ) ⊂ C k (R3 , CN ).
In other words, if a function belongs to the H m space, then it has at leastm − 2 continuous derivatives. A proof of this result can be found in Taylor(1996a), chapter 4, section 1. It follows from Proposition 12.1 that a function
over R3 is smooth (i.e. C ∞ ) if it belongs to H m for every m.
Extensions of functions
To exploit the localisability property of hyperbolic equations it is often convenient to extend functions which are deﬁned only on bounded subsets R ⊂ R3 tofunctions with domain on the whole of R3 . Deﬁning in a natural way the norm|| ||R,m and the Sobolev space H m (R, CN ) one has the following result:
Proposition 12.2 (extension of functions on a compact domain)
Assume that R ⊂ R3 is bounded with smooth boundary ∂R. Then there exists a
linear operator
E : H m (R, CN ) −→ H m (R3 , CN )such that for each u ∈ H m (R, CN ):308

Methods of the theory of hyperbolic diﬀerential equations(i) E u = u almost everywhere in R.
(ii) E u has support in an open bounded set R ⊃ R.
(iii) There exists a constant C depending only on U and R such that||E u||R3 ,m ≤ C||u||R,m .
The CN -valued function E u is called an extension of u to R3 .
A discussion on how to prove this result can be found in Evans (1998).
12.3.2 Kato’s existence and stability theorems
Using the terminology introduced in the previous subsections, it is now possible
to discuss the basic existence and stability result for quasilinear symmetric
hyperbolic systems of the form
A0 (t, x, u)∂t u + Aα (t, x, u)∂α u = B(t, x, u).

(12.18)In what follows, it will always be assumed that the matrices Aμ are smoothfunctions of their arguments.
The basic local existence theorem
As it can be seen from the proof of the Uniqueness Theorem 12.1, the positivedeﬁniteness of the matrix A0 (t, x, u) plays a key role in determining theproperties of solutions to the equation. On an initial hypersurface S, thispositivity can be set by ﬁat by choosing suitable initial data. However, in viewof the quasilinearity of the equation, the positive-deﬁniteness could be violated
at some time as the solution evolves. Intuitively, one would expect this to leadto some sort of problems in the solution. For ﬁxed (t, x), and given a CN -valued
function w, one says that A0 (t, x, w) is positive deﬁnite and bounded awayfrom zero by δ > 0 if
z, A0 (t, x, w)z > δz, z
for all z ∈ CN .
The basic local existence result for the Cauchy problem of symmetric
hyperbolic systems to be considered in this book is the following:
Theorem 12.2 (local existence of solutions to symmetric hyperbolic
systems) Consider the Cauchy problem
A0 (t, x, u)∂t u + Aα (t, x, u)∂α u = B(t, x, u),u(0, x) = u (x) ∈ H m (R3 , CN )m ≥ 4,

for a quasilinear symmetric hyperbolic system. If δ > 0 can be found such that
A0 (0, x, u ) is positive deﬁnite with lower bound δ for all p ∈ R3 , then there12.3 Local existence results for symmetric hyperbolic systems

309

exists T > 0 and a unique solution u to the Cauchy problem deﬁned on [0, T ]×R3
such thatu ∈ C m−2 [0, T ] × R3 , CN .
Moreover, A0 (t, x, u) is positive deﬁnite with lower bound δ for (t, x) ∈[0, T ] × R3 .
This theorem is an adaptation of similar theorems given in Kato (1975a) andFriedrich (1986b). A proof of this result falls beyond the scope of this book. Theinterested reader is referred to references given above.
Remarks(a) For convenience, the regularity of the solution has been stated in terms of
C k spaces. However, the conclusions of the theorem can be expressed in amore detailed manner. In particular, one has that the solution satisﬁesu ∈ C 1 [0, T ], H m−1 (R3 , CN ) .
The latter can be shown to imply u ∈ H m ([0, T ] × R3 , CN ) which, in turn,using a Sobolev embedding theorem in four-dimensions gives the regularity
stated in the theorem.
(b) In most of the applications given in this book, the initial data u will beassumed to be smooth, so that u ∈ H m (R3 , CN ) for all m. However, as
R3 is an unbounded set, one cannot simply assume that u ∈ C ∞ (R3 , CN );compare the remark after Equation (12.17).
(c) As A0 is a smooth function of its arguments, it follows from the regularityof the solution u that z, A0 (t, x, u)z for z ∈ CN depends continuously on(t, x).
(d) As A0 (t, x, u) is positive deﬁnite for (t, x) ∈ [0, T ] × R3 , it follows thatthe hypersurfaces of constant t are spacelike with respect to the symmetric
hyperbolic system (and the solution).
(e) The value of the lower bound δ can often be determined by inspection.
The basic stability result
Of great relevance is the notion of Cauchy stability – namely, the idea that,given a symmetric hyperbolic system, initial data which are close to each othershould lead to solutions which are close in some sense and have a common
existence time interval. In view of the inherent error in the physical process ofmeasurement, Cauchy stability is fundamental for the applicability of diﬀerential
equations to describe physical phenomena. In mathematical terms, the precise
formulation of the closeness of initial data and solutions is expressed in termsof Sobolev norms.
In the remainder of this section let D be a bounded open subset of H m (R3 , CN )such that for w ∈ D the matrix A0 (0, x, w) is positive deﬁnite bounded away

310

Methods of the theory of hyperbolic diﬀerential equations

from zero by δ > 0 for all p ∈ R4 . The basic result describing the Cauchy
stability of the symmetric hyperbolic system (12.18) is the following theorem,
adapted from Kato (1975a):
Theorem 12.3 (basic Cauchy stability for symmetric hyperbolic systems) Let m ≥ 4. If u ∈ D is given as an initial condition for the system(12.18), then:(i) There exists ε > 0 such that a common existence time T can be chosen for
all initial conditions in the open ball Bε (u ) ⊂ D.
(ii) If the solution u with initial data u exists on [0, T ] for some T > 0, thenthe solutions to all initial conditions in Bε (u ) exist on [0, T ] if ε > 0 is
suﬃciently small.
(iii) If ε and T are chosen as in (i) and one has a sequence un ∈ Bε (u ) suchthat||un − u ||R3 ,m → 0

as n → ∞,then for the solutions un (t, ·) with un (0, ·) = un it holds that||un (t, ·) − u(t, ·)||R3 ,m → 0,

as n → ∞,uniformly for t ∈ [0, T ].
Remarks(a) Point (i) in the previous theorem essentially states that, given a suﬃciently
small ball in the space of data on which the Existence Theorem 12.2 can beapplied, then a common existence time for the solutions arising from thisdata can be found. Observe, however, that one has no control over the sizeof the common existence time; one only knows there is one.
(b) If the existence of a particular solution is known, then point (ii) states that,by shrinking the ball on the space of data, one can choose the known existence
time as the common existence time.
(c) Point (iii) states that data close to certain reference data give rise todevelopments which are also close to the reference solution; this is thestatement of Cauchy stability.
(d) The convergence stated in (iii) is uniform on [0, T ] × R3 .
12.3.3 Localising solutions
The localisability property of hyperbolic equations allows one to apply theexistence and stability results discussed in the previous sections to the case of
an initial data problem where data are prescribed only on a compact region R.
Given smooth initial data u for a symmetric hyperbolic equation of the form
(12.1) on a region R ⊂ R3 , one can make use of Proposition 12.2 to extend the12.3 Local existence results for symmetric hyperbolic systems

311

D(R)
R3
R

Figure 12.3 Localised solution arising from data prescribed on an open set
R ⊂ R3 . The associated domain of dependence is denoted by D(R).

initial data u to the whole of R3 in a controlled manner. Denoting this extension
by E u , one has that by point (iii) of Proposition 12.2, E u ∈ H m (R3 , CN ).
In order to make use of Theorems 12.2 and 12.3 it is necessary to assume that
A0 (0, x, E u ) is positive deﬁnite with some non-zero lower bound uniform on R3 .
Thus, one obtains a solution to Equation (12.1) with initial data on R3 givenby u(0, x) = E u (x). As a consequence of the uniqueness of solutions on thedomain of dependence, the solution u on D(R) is independent of the particularextension of the initial data u on R to R3 ; see Figure 12.3.
12.3.4 Existence and stability result on manifolds withcompact spatial sections
The existence and stability Theorems 12.2 and 12.3 can be modiﬁed so as toapply to Cauchy problems where data is prescribed on compact, orientable threedimensional manifolds. In what follows, the main ideas behind this construction
are discussed.
Patching together solutions
In the remainder of this section let S denote an orientable, compact threedimensional manifold – in most of the applications to be considered in this book
one has S ≈ S3 ; however, any other compact, orientable topology will work as
well. As a result of compactness, there exists a ﬁnite cover consisting of open
sets R1 , . . . , RM ⊂ S; that is, one has ∪M
i=1 Ri = S. On each of the open sets
Ri , i = 1, . . . , M , one can introduce local coordinates xi ≡ (xαi ) which allow
one to identify Ri with open subsets Bi ⊂ R3 . As S is assumed to be a smoothmanifold, the coordinate patches can be chosen so that the change of coordinates
on intersecting sets is smooth.
Now, assume that a smooth function u : S → CN has been prescribed on S.
In what follows, the restriction of u to a particular open set Ri will be denotedby ui . Using the local coordinates xi , the function ui can be regarded as afunction ui : Bi → CN .

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Methods of the theory of hyperbolic diﬀerential equations

D2

D3

D1

R3

Figure 12.4 Construction of a solution by patching localised solutions to data
prescribed on open sets D1 , D2 , D3 ⊂ R3 .

The strategy is now to use the same procedure as described in Section 12.3.3
to ensure the existence of a solution on the domain of dependence of Bi .
Accordingly, one makes use of Proposition 12.2 to extend ui to a function E ui
deﬁned on the whole of R3 . Using the extended functions E ui one deﬁnes thenormM||u ||S,m ≡||ui ||R3 ,m .

(12.19)i=1Assuming, as in Section 12.3.3, that A0 (0, x, E ui ) is positive deﬁnite withlower bound δi > 0, one obtains a unique solution ui of Equation (12.1) withinitial data u(0, x) = E ui (x) with existence interval [0, Ti ]. The solution on
D(Bi ) is independent of the particular extension E ui being used, so that one
can speak of a solution ui on a domain Di ⊂ [0, Ti ] × Ri ; see Figure 12.4.
Now, given two solutions ui and uj deﬁned, respectively, on intersecting
domains Di and Dj one has – following the discussion on the change of
coordinates given in Section 12.1 and as a consequence of uniqueness – thatui and uj must coincide on Di ∩ Dj . Proceeding in the same manner over thewhole ﬁnite cover of S, one obtains a unique solution u on [0, T ] × S withT ≡ mini=1,...M {Ti } which is constructed by patching together the localised
solutions u1 , . . . , uM deﬁned, respectively on the domains Di , . . . , DM . Observethat the compactness of S ensures the existence of a minimum non-zero existence
time for the whole of the domains Di .
A general existence and stability result
Using the ideas of the localisation of solutions discussed in the previous
subsection, one can formulate a quite general existence and stability result forsymmetric hyperbolic systems on manifolds whose spatial sections are given byorientable, compact three-dimensional manifolds. The hypotheses of this theorem
are very similar to the ones in Theorems 12.2 and 12.3.

12.4 Local existence for boundary value problems

313

Theorem 12.4 (existence and stability result for symmetric hyperbolic
systems on compact spatial sections) Given an orientable, compact, threedimensional manifold S, consider the Cauchy problem
A0 (t, x, u)∂t u + Aα (t, x, u)∂α u = B(t, x, u),u(0, x) = u (x) ∈ H m (S, CN )for m ≥ 4,

for a quasilinear symmetric hyperbolic system. If δ > 0 can be found such that
A0 (0, x, u ) is positive deﬁnite with lower bound δ for all x ∈ S, then:(i) There exists T > 0 and a unique solution u to the Cauchy problem deﬁned
on [0, T ] × S such thatu ∈ C m−2 [0, T ] × S, CN .
Moreover, A0 (t, x, u) is positive deﬁnite with lower bound δ for (t, x) ∈[0, T ] × S.
(ii) There exists ε > 0 such that one common existence time T can be chosen
for all initial conditions in the open ball Bε (u ) and such that Bε (u ) ⊂ D.
(iii) If the solution u with initial data u exists on [0, T ] for some T > 0, thenthe solutions to all initial conditions in Bε (u ) exist on [0, T ] if ε > 0 is
suﬃciently small.
(iv) If ε and T are chosen as in (ii) and one has a sequence un ∈ Bε (u ) suchthat||un − u ||S,m → 0,

as n → ∞,then for the solutions un (t, ·) with un (0, ·) ≡ un it holds that||un (t, ·) − u(t, ·)||S,m → 0,

as n → ∞uniformly in t ∈ [0, T ].
Remarks similar to the ones after Theorems 12.2 and 12.3 apply to this result.
Further discussion and details can be found in Friedrich (1991).
12.4 Local existence for boundary value problems
As will be seen in Chapter 17, the construction of anti-de Sitter-like spacetimes
leads one to consider initial boundary value problems for symmetric hyperbolic
systems of the form (12.1). In this type of problem one prescribes initial dataon a spacelike hypersurface S and boundary data on a timelike hypersurface T .
These two hypersurfaces intersect on a two-dimensional hypersurface E ≡ S ∩ T ,the edge, on which the initial and the boundary conditions need to satisfy somecompatibility conditions; see Figure 12.5. In view of the localisation property of314

Methods of the theory of hyperbolic diﬀerential equations

T

S

E=S∩T

Figure 12.5 Geometric setting of the initial boundary value problem forsymmetric hyperbolic systems. The initial data are prescribed on the threedimensional spacelike hypersurface S; boundary data are prescribed on thethree-dimensional timelike hypersurface T . The initial and boundary data
must satisfy certain compatibility conditions (corner conditions) on the edge
E =S ∩T.

symmetric hyperbolic systems, it is suﬃcient to analyse the problem close to theedge. The solution away from the boundary is obtained by patching domains ofdependence.
12.4.1 Basic setting
In a neighbourhood of a point p ∈ E, one can introduce coordinates x = (xμ )such that the domain U in which the solution to the boundary value problem
takes the form
U = {x ∈ R4 | x0 ≥ 0, x3 ≥ 0},
while the initial hypersurface and the boundary are given, respectively, byS ≡ {x ∈ U | x0 = 0},
T ≡ {x ∈ U | x3 = 0}.
The normal matrix A3 (x, u) in a symmetric hyperbolic system of the form
(12.1) plays a crucial role in the speciﬁcation of admissible boundary conditions
leading to a well-posed initial boundary value problem. Due to the use ofcoordinates adapted to the boundary, the properties of the matrix A3 determine
the relation between the timelike boundary T and the characteristics of thehyperbolic evolution equation.
In what follows let T(x) denote a smooth map from T to the vector subspacesof CN and require as boundary condition thatu(x) ∈ T(x),x∈T.

12.4 Local existence for boundary value problems

315

The map T is restricted by the requirements:(i) The set T is a characteristic of (12.1) of constant multiplicity; that is, 
dim Ker A3 = constant > 0,x∈T.
(ii) The map T satisﬁes the non-positivity condition
u, A3 (x, u)u ≤ 0,u ∈ T(x), x ∈ T .

(iii) The dimension of the subspace T(x), x ∈ T , is equal to the number of
non-positive eigenvalues of A3 (x, u) counting multiplicities.
An important property of Hermitian matrices is that they can be diagonalisedby unitary matrices and that all their eigenvalues are real. Accordingly, after a
redeﬁnition of the dependent variables one can assume that, at a given pointx ∈ T , the normal matrix A3 (x, u) has the form
⎞
⎛
0
0
−Ij×j
A3 (x, u) = κ ⎝κ > 0,
0 ⎠,
0
0k×k
0
0
Il×l
where Ij×j and Il×l are, respectively, j × j and l × l unit matrices and 0k×k is
the k × k zero matrix. Moreover, one has that j + k + l = N . Writing⎛⎞a(x)u(x) = ⎝ b(x) ⎠ ∈ Cj × Ck × Cl ,c(x)one ﬁnds that the linear subspaces admitted by condition (ii) are of the formc − Ha = 0
with H = H(x) an l × j matrix satisfying−a, a + Ha, Ha ≤ 0,a ∈ Cj .

This condition can be reexpressed, alternatively, as H∗ H ≤ Ij×j . The key
observation is that the above procedure gives no freedom to prescribe data forthe component b of u associated with the kernel of the normal matrix A3 (x, u).
In particular, if A3 (x, u) = 0, one has that the boundary is a total characteristic(see Section 12.1.2) and no boundary conditions can be speciﬁed on T – thesolution u on T is directly determined by the initial conditions on the edge E.
More generally, by a further redeﬁnition of the dependent variables one obtains
the inhomogeneous maximally dissipative boundary conditions
q(x) = c(x) − H(x)a(x),x∈T,with q(x) a Cl -valued function representing the free boundary data on T .

316Methods of the theory of hyperbolic diﬀerential equations
Corner conditions

To obtain a smooth solution to an initial boundary value problem for a symmetric
hyperbolic system of the form (12.18), the initial data prescribed on S and theboundary data at T must satisfy certain compatibility conditions at the edge
E = ∂S = S ∩ T – the so-called corner conditions. More precisely, if one has
initial data of the formu(0, x) = u (x)on S,

with u smooth and maximally dissipative boundary conditions of the form
T(t, x)u(t, x) = q(t, x)

on T ,(12.20)then one requires that
T(0, x)u |E = q(0, x).
Higher order corner conditions can be obtained by considering the system (12.18).
Evaluating at E one obtains
A0 (u )|E (∂t u)|E + Aα (u )|E (∂α u )|E = B(u )|E .
As A0 (u )|E is positive deﬁnite, the above equation can be used to solve for(∂t u)|E . The result should be consistent, upon substitution, with what is obtained
from diﬀerentiating the boundary condition (12.20). Namely,(∂t T)|E u|E + T|E (∂t u)|E = (∂t q)|E .
Further higher order boundary conditions are obtained in an analogous mannerby diﬀerentiating (12.18) successively with respect to t.
12.4.2 Uniqueness of the solutions to the boundary value problem
Insight into the role of the maximally dissipative boundary conditions can beobtained from the analysis of the uniqueness of solutions to the boundary
value problem. The argument follows a strategy similar to the one employed in
Theorem 12.1 with a domain G whose boundary consists of portions of the initial
hypersurface S0 , the boundary T and a hypersurface S1 which is spacelike withrespect to the symmetric hyperbolic system; see Figure 12.6. Set M = [0, ∞)× S
such that S and T can be identiﬁed, in a natural way, as the boundary of M.
Deﬁne the coordinate x0 ≡ t in such a way that S0 = {p ∈ M | t = 0}.
Theorem 12.5 (uniqueness of solutions of the initial boundary problem
with maximally dissipative boundary conditions) Let G be a domain asgiven above. If u1 and u2 are two solutions to the initial value problem for thesymmetric hyperbolic system
Aμ (x, u)∂μ u = B(x, u),u|S0 = u ,with the same maximally dissipative boundary conditions, then u1 = u2 on G.

12.4 Local existence for boundary value problems

317

S1
G

T

S0

Figure 12.6 Integration domain for the uniqueness argument for initialboundary value problems. The boundary ∂G consists of portions of the initial
hypersurface S0 , the timelike boundary T and a spacelike hypersurface S1 .

Proof Starting from the identity (12.14) one integrates over a domain G as
depicted in Figure 12.6, where S1 is spacelike with respect to the symmetric
hyperbolic system. Applying the Gauss identity one obtains
e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS
S1

e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS−
 S0
e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS−T
e−kt u1 − u2 , P(x, u1 , u2 , ∂u2 )(u1 − u2 )d4 x,
=
G

with
P(x, u1 , u2 ) ≡ −kA0 (x, u1 ) + ∂μ Aμ (x, u1 )+ Q(x, u1 , u2 , ∂u2 ) + Q∗ (x, u1 , u2 , ∂u2 )and Q obtained as in Theorem 12.1 using the mean value theorem. Exploiting
the positive deﬁniteness of A0 (x, u1 ), one can make the volume integral over G
negative. Moreover, as u1 and u2 coincide on S0 one obtains
e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS
S1

e−kt u1 − u2 , Aμ (x, u1 )(u1 − u2 )νμ dS ≤ 0,≤
T

where the last inequality follows from the negative deﬁniteness of the maximallydissipative boundary conditions. Thus, one obtains a contradiction with the factthat the surface integral over the spacelike hypersurface S1 is positive unless
u1 = u2 .

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Methods of the theory of hyperbolic diﬀerential equations
12.4.3 The basic existence result for the initial boundary value
problem of symmetric hyperbolic systems

One has the following basic local existence theorem for the initial boundary value
problem with maximally dissipative boundary conditions:
Theorem 12.6 (local existence for initial boundary value problems)Given the initial boundary value problem
A0 (t, x, u)∂t u + Aα (t, x, u)∂α u = B(t, x, u),(12.21a)T(t, x)u = q(t, x)

on T ,(12.21b)u(0, x) = u (x),on S,

(12.21c)with (12.21a) symmetric hyperbolic, A0 (0, x, u ) positive deﬁnite and q, u
smooth, assume that the boundary condition (12.21b) is maximally dissipativewith respect to the normal matrix A3 (t, x, u) and that the boundary data satisfy
corner conditions at E = S ∩ T to all orders. Then, the initial boundary value
problem has a unique smooth solution u(t, x) deﬁned on
MT = {p ∈ [0, ∞) × S | 0 ≤ t(p) < T },for some T > 0.
The reader is refereed to Guès (1990), Friedrich (1995) and Friedrich andNagy (1999) for details and remarks concerning the proof. As a consequenceof the localisability property of hyperbolic equations, the problem can be split
into two parts: an interior one away from the boundary in which the standard

D1

D2

T

T
D3
S

Figure 12.7 Construction of a solution to an initial boundary value problem
which is global in space by patching domains. The solution patch D1 near theboundary T is obtained using Theorem 12.6. The existence on the domains D2
and D3 away from the boundary are obtained by means of Theorem 12.3. The
uniqueness of solutions ensures that the solution on the intersections “match
together”. Due to the compactness of the initial hypersurface it is possible toobtain an existence time T common to all domains.

12.5 Local existence for characteristic initial value problems

319

local existence for the Cauchy problem (as described in Section 12.3) is used,
and a boundary part in which the boundary and edge conditions play a role; see
Figure 12.7. The local solutions are then patched together to obtain the solution
on the whole of MT .
Remark. The question of the stability of solutions to the initial boundary value
problem will not be analysed here. Stability questions for initial boundary value
problems are much more complicated than for the Cauchy case. At the time ofwriting, there are no applications of stability results for boundary value problems
involving the conformal ﬁeld equations.
12.5 Local existence for characteristic initial value problems
Characteristic initial value problems arise naturally in applications to generalrelativity; see Chapter 18. The purpose of this section is to discuss a method toanalyse the local existence of solutions to the characteristic initial value problem
for symmetric hyperbolic equations due to Rendall (1990). The idea behind thismethod is to reduce the characteristic problem to a standard Cauchy problem
where the standard theory of Section 12.3 can be applied.
12.5.1 General remarks on the characteristic problem
In what follows, consider a quasilinear symmetric hyperbolic system of the form
given by Equation (12.1) on R4 . In contrast to the analysis of the Cauchy problem
where it is convenient to single out one of the coordinates as a time coordinate,in the characteristic problem it is convenient to make use of coordinates adapted
to the characteristic hypersurfaces.
As discussed in Section 12.1.2, for quasilinear equations like (12.1), the notionof characteristic hypersurfaces depends on the solution u. Thus, it is, in principle,
unclear on which hypersurfaces one can prescribe the characteristic initial data.
There are two approaches to get around this diﬃculty:(i) Fix the data ﬁrst, then look for the hypersurface. Choose a smoothfunction v on U ⊂ R4 such that the matrices Aμ (x, v) are deﬁned at eachpoint of R4 , and choose a smooth function φ ∈ X (U ), dφ = 0 in U , suchthat the hypersurface
N ≡ {x ∈ U | φ(x) = 0}(12.22)is characteristic with respect to Aμ (x, v), that is, such thatdet(Aμ (x, v)∂μ φ) = 0.
The characteristic data on N is then given as the restriction of v to N ; thatis, u = v|N .

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Methods of the theory of hyperbolic diﬀerential equations
N

N

Z

Figure 12.8 Initial hypersurfaces N and N  on a characteristic initial value
problem. The set Z = Ø is the intersection of N and N  .

(ii) Choose the hypersurface ﬁrst, then look for suitable data. Alternatively, one can choose some hypersurface N in U ⊂ R4 deﬁned as in (12.22),
and then consider only those smooth functions u such thatdet(Aμ (x, u )∂μ φ) = 0.
Approach (ii) is more natural in applications where geometric information of
the initial hypersurface is available. This point of view will be adopted in the restof this section.
A peculiarity of characteristic initial value problems for the system (12.1) isthat data need to be prescribed on two intersecting characteristic hypersurfaces
N and N  ; see Figure 12.8. Intuitively, this is a consequence of the existence of asubsystem of equations in (12.1) which is intrinsic to the hypersurface N , so that
one does not have enough evolution equations transverse to the hypersurface forall the components of u. Alternatively, one can formulate characteristic initialvalue problems by prescribing initial data on a cone. This is a more technicallyinvolved problem and will not be discussed here. The interested reader is referred
to Cagnac (1981) and Dossa (1997) for further details.
Well- and ill-posed characteristic problems
In what follows, let N and N  denote two hypersurfaces on U ⊂ R4 with nonempty intersection Z ≡ N ∩ N  . One can introduce coordinates u and v suchthat, at least in a neighbourhood of N ∩ N  , one can writeN ≡ {p ∈ U | u(p) = 0},

N  ≡ {p ∈ U | v(p) = 0}.

(12.23)Given suitable initial data on N ∪ N  one would like to make some statement
about the existence and the uniqueness of solutions to Equation (12.1) on someopen setV ⊂ {p ∈ U | u(p) ≥ 0, v(p) ≥ 0}.
By symmetry, one could also look for a solution in the region{p ∈ U | u(p) ≤ 0, v(p) ≤ 0};
see Figure 12.9 (a).

12.5 Local existence for characteristic initial value problems
(a)

321

(b)
u ≥ 0, v ≥ 0

v=0

u=0

v=0

u=0

u ≥ 0, v ≤ 0

u ≤ 0, v ≥ 0

u ≤ 0, v ≤ 0
well-posed

ill-posed

Figure 12.9 Schematic representation of well-posed (a) and ill-posed (b)characteristic initial value problems.

The problem of looking for solutions in domains of the formV̄ ⊂ {p ∈ U | u(p) ≤ 0, v(p) ≥ 0}orV̄¯ ⊂ {p ∈ U | u(p) ≥ 0, v(p) ≤ 0}
is ill-posed – the reason will become clear once the Rendall’s reduction procedure
to a Cauchy problem is discussed in Section 12.5.3. Under suitable circumstances,it may be possible to establish uniqueness of a solution – but not existence – forthis ill-posed problem. These ideas have been used by Ionescu and Klainerman(2009a,b) to obtain a new strategy to prove the uniqueness of stationary black
holes.
12.5.2 Interior equations on the characteristic hypersurfaces
As seen in Section 12.1.2, on a characteristic surface, a system of the form (12.1)
implies a subsystem of interior equations on the hypersurface. Assuming that thefreely speciﬁable part of u is smooth on the characteristic hypersurface, theseinterior equations can be used to compute the remaining components of u andtheir derivatives to any arbitrary order. For conciseness, the subsequent analysis
is restricted to the characteristic N as given by (12.23). The situation on N  iscompletely analogous. Letting x ≡ (u, y) with y ≡ (y α ) = (v, x2 , x3 ) and using
the chain rule, Equation (12.1) can be rewritten asσ(u, y; u, du)∂u u + Aα (u, y; u)∂α u = B(u, y; u)withσ(u, y; u, du) ≡ Aμ (u, y; u)

∂u
.
∂xμ

(12.24)

322

Methods of the theory of hyperbolic diﬀerential equations

If N is a characteristic hypersurface for some function u , then one has that
det σ(0, y; u , du) = 0.
Let m = dim Ker σ(0, y; u , du). It follows that there exist m vectors k(i) , i =1, . . . , m such that
σ(0, y; u , du)k(i) = 0.
That is, the k(i) are eigenvectors of σ(0, y; u , du) with zero eigenvalue. Thus,
one has that
k(i) , σ(0, y; u , du)∂u u = σ(0, y; u , du)k(i) , ∂u u = 0,
since σ(u, y; u, du) is Hermitian as a consequence of the symmetric hyperbolicityof (12.1). Thus, from Equation (12.24) one obtains
k(i) , Aα (0, y; u )∂α u  = k(i) , B(0, y; u ),i = 1, . . . m,

(12.25)a system of m (scalar) interior equations for the components of u. In what
follows, it will be assumed that the components of u have been chosen such thatthe free data on N consist of N − m variables ŭ – the so-called u-data. The
remaining m variables, ū , constrained by the equations in (12.25), are called
the u-variables. Thus, one obtains the splitu = (ŭ , ū )on N .

(12.26)In terms of this split, the scalar intrinsic equations (12.25) can be rewritten in
matricial form asĀα (y, ŭ , ū )∂α ū = B̄(y, ŭ , ū ),(12.27)for some (m × m)-matrix valued smooth functions Āα and an m-vector valued
function B̄. For simplicity, it will assumed that the system (12.27) is a symmetric
hyperbolic system on N which can be solved, at least locally, in a neighbourhood
W ⊂ N of the two-dimensional surface Z where initial data for the u-variablesū is prescribed. In this way, one obtains the value of the whole components ofu on W. Assuming that ŭ is smooth on N , higher intrinsic derivatives can beobtained in a similar manner by formally diﬀerentiating Equation (12.27) withrespect to ∂α an arbitrary number of times, say, M . In this manner, one obtains
a system of the formĀα (y, ∂α ŭ , ∂α ū )∂α ūα = B̄(y, ∂α ŭ , ∂α ū , ūα ),where multi-index notation has been used so that∂α ŭ ≡ ŭ , ∂α ŭ , ∂α1 ∂α2 ŭ , . . . , ∂α1 · · · ∂αM −1 ŭ ,∂α ū ≡ ū , ∂α ū , ∂α1 ∂α2 ū , . . . , ∂α1 · · · ∂αM −1 ū(12.28)

12.5 Local existence for characteristic initial value problems

323

and
ūα ≡ ∂α1 · · · ∂αM ū .
By assumption, Equation (12.28) is a symmetric hyperbolic system on N
so that by prescribing initial data for ūα on Z and assuming that the lower
order intrinsic derivatives ∂α ū have been solved for, one obtains a solution in aneighbourhood of Z on N . Thus, one can obtain, recursively, the interior partial
derivativesu  , ∂ α u  , ∂ α1 ∂ α2 u  , . . . , ∂ α1 · · · ∂ αM u 

on W ⊂ N ,with W ⊃ Z.
Now, not only the interior derivatives on N can be computed. Using the split(12.26), the subset of N − m equations in (12.1) which are transversal to N canbe written as
Cu (0, y, ŭ , ū )∂u ŭ + Cα (0, y, ŭ , ū )∂α ŭ = D(0, y, ū , ū),(12.29)with Cμ smooth (N − m) × (N − m)-matrix valued functions and D an (N − m)vector valued function of their arguments. For clarity of the presentation it isconvenient to write ∂u ŭ ≡ (∂u ŭ) . By construction, the matrix Cμ is invertible,
so that Equation (12.29) can be regarded as an algebraic linear system of
equations determining the transversal derivatives ∂u ŭ on N in terms of uand ∂α u . To compute the transversal derivatives of the u-variables ū , one
diﬀerentiates the interior system (12.27) to obtain a system of the formĀα (y, ŭ , ū )∂α (∂u ū ) = B̄(y, ŭ , ∂u ŭ , ū , ∂u ū ).
As in the case of the system (12.27), the above system can be solved in someneighbourhood of Z on N if initial data for ∂u ū are given on Z. This procedurecan be repeated to obtain higher order transversal derivatives.
The procedure described in the previous paragraphs can also be implemented
on the characteristic hypersurface N  . By analogy to the case of N , one can split
the unknown u as
u = (ŭ , ū ),on N  ,where ŭ are v-data which can be speciﬁed freely on N  and ū are v-variables
constrained by interior equations analogous to (12.28). In what follows, theseinterior equations are assumed to be symmetric hyperbolic on N  . Applying aprocedure similar to that used on N , all the derivatives of u on N  to any desiredorder can be computed if ŭ is suitably smooth, and the required initial data aresupplied on Z.
The discussion described in the previous paragraphs is summarised in thefollowing proposition:

324

Methods of the theory of hyperbolic diﬀerential equations

Proposition 12.3 (evaluation of derivatives on the initial characteristic
surface) Let N and N  denote two characteristic hypersurfaces for the symmetric hyperbolic system (12.1) having a non-empty two-dimensional intersection
Z = N ∩ N  . If smooth u-data and v-data are prescribed, respectively, on N andN  and the values of the u-variables and v-variables are prescribed on Z in sucha way that the freely speciﬁable data are smooth on N ∪ N  , then all derivativesof u on N ∪ N  to any desired order can be computed in a neighbourhood W ⊂ N
of Z.
12.5.3 Reduction to a standard Cauchy problem
The observations summarised in Proposition 12.3 are the cornerstone of areduction procedure of the characteristic problem on N ∪ N  to a standard
Cauchy problem for which the theory discussed in Section 12.3 is applicable.
This approach to analysing the characteristic initial value problem for hyperbolic
equations was originally introduced by Rendall (1990).
In what follows, suppose that characteristic initial data have been prescribedon N ∪ N  in a manner consistent with Proposition 12.3 so that the values ofu and its derivatives to any order are known in a neighbourhood W of Z on
N ∪ N  . Rendall’s reduction proceeds ﬁrst by constructing an extension of u toa neighbourhood U of Z in R4 . This type of extension of functions is diﬀerent
from the one discussed in Section 12.3.1 where functions deﬁned on open subsets
of a certain space are extended to functions on the whole space. In the present
case one needs to extend a function deﬁned on a closed set of R4 . There exists a
general result, Whitney’s extension theorem, which allows one to obtain therequired extension; see the Appendix to this chapter for more details.
To apply Whitney’s extension theorem to the collection of ﬁelds{u , (∂μ u) , (∂μ1 ∂μ2 u) , . . . , (∂μ1 · · · ∂μM u) }(12.30)on W ⊂ N ∪ N  for some non-negative integer M , one has to verify that thevarious ﬁelds in this collection are related to each other in the same way as thederivatives of a function are related to each other in a Taylor expansion. The key
condition on these Taylor-like expansions ensuring the existence of an extension
is a requirement on the vanishing rate of the remainder of the expansions. Given
two points on N away from Z this vanishing of the remainder follows fromsmoothness of the free data on the characteristic hypersurface, and the fact thatthe derivative candidates in (12.30) have been obtained solving hyperbolic
diﬀerential equations on N and algebraic equations. The functions thus obtained
are smooth on N and admit a standard Taylor expansion on the characteristic
hypersurface. This also holds for two points on N  away from Z. Thus, thediﬃculty is to verify Whitney’s condition for two points, respectively, on N andN  , so that one writes
x = (0, v, x2 , x3 ),x = (u, 0, x2 , x3 ).

12.5 Local existence for characteristic initial value problems

325

The complication arises from the fact that the characteristic initial hypersurface
N ∪ N  is continuous only at Z. It is convenient to deﬁne a point x∗ ∈ Z asx∗ ≡ 0, 0, 12 (x2 + x2 ), 12 (x3 + x3 ) .
Using the cosines law, it follows that there exists a constant C > 0 such that|x − x∗ |2 + |x − x∗ |2 ≤ C|x − x |2 .
To apply Whitney’s extension theorem it is necessary to establish that theremainder of the Taylor-like expansion about x vanishes at x as fast as it would
do if the function had an extension as x and x tend to a common point, say, x∗ .
The inequality above shows that the points x and x cannot get closer to eachother without getting close to a point on Z. This idea, together with the Cauchy
stability of solutions to the interior equations which determine the constrained
components of the data on N ∪ N  yields the required vanishing rate.
Applying Whitney’s extension theorem to the collection of derivative candidates (12.30) one obtains a smooth function û in a neighbourhood U of Z on R4 .
The function û satisﬁesû = u ,∂μ û = (∂μ u) ,∂μ2 ∂μ1 û = (∂μ2 ∂μ1 u) , . . . ,on W ⊂ N ∪ N  . In general, û is not a solution to Equation (12.1) away fromN ∪ N  . Nevertheless,Δ ≡ Aμ (x, û)∂μ û − B(x, û)vanishes to all orders on W ⊂ N ∪ N  and
0 u > 0, v > 0,δ≡
Δ elsewhere,is smooth in a neighbourhood of N ∩ N  where û exists.
The desired reduction to a Cauchy problem is now obtained by considering
the equation
Aμ (x, û + v)∂μ (û + v) − B(x, û + v) = δ,

(12.31)for the unknown v together with the initial datav = 0,on

S ≡ {p ∈ R4 | u(p) + v(p) = 0}.

(12.32)

By assumption, the hypersurface S has a neighbourhood around N ∩ N  whichis spacelike with respect to Equation (12.31) so that the Cauchy problem givenby (12.31) together with the initial data (12.32) is well posed and the theory ofSection 12.3 is readily applicable. In particular, one obtains a unique solution v
in a neighbourhood V of Z on R4 ; see Figure 12.10.
Outside the intersection of V with the quadrant{p ∈ R4 | u(p) ≥ 0, v(p) ≥ 0},326

Methods of the theory of hyperbolic diﬀerential equations

N

N

t

U

S

Figure 12.10 Schematic reduction of a characteristic initial value problem toa Cauchy problem. The data on N ∪ N  (thick line) are extended by meansof Whitney’s extension theorem to a neighbourhood U (light gray region) of
Z ≡ N ∩ N  . This extension implies data for an auxiliary initial value problem
on the spacelike hypersurface S. The solution to the characteristic problem isfound in the upper and lower quadrants (dark gray areas).

Equation (12.31) takes the form
Aμ (x, û + v)∂μ (û + v) − B(x, û + v) = Aμ (x, û)∂μ û − B(x, û)
so that v = 0 is clearly a solution – by uniqueness, it is the only solution. In
contrast, on V ∩ {p ∈ R4 | u(p) ≥ 0, v(p) ≥ 0} one has the equation
Aμ (x, û + v)∂μ (û + v) = B(x, û + v).
As û + v coincides with u on N ∪ N  one concludes, again by uniqueness of thesolution of the reduced Cauchy problem, thatu ≡ û + v
is the required solution to the posed characteristic initial value problem.
The discussion in this section is summarised in the following theorem:
Theorem 12.7 (local existence for the standard characteristic problem)
Let N and N  denote two characteristic hypersurfaces for the symmetric
hyperbolic system (12.1) with smooth, freely speciﬁable data on N and N  as given
in Proposition 12.3. Then there exists a unique solution u to the characteristic
initial value problem in a neighbourhood V of Z with u ≥ 0, v ≥ 0.
Remark. If one were to attempt a similar reduction procedure to constructa solution on the regions for which either u ≥ 0, v ≤ 0 or u ≤ 0, v ≥ 0,

12.5 Local existence for characteristic initial value problems

327

N
N

Z

Figure 12.11 Characteristic cones N and N  intersecting on a two-dimensional
hypersurface Z which is diﬀeomorphic to S2 .

one would end up with an initial value problem with data prescribed on atimelike hypersurface. This is an ill-posed problem. Accordingly, the original
characteristic problems are, themselves, also ill-posed.
The case Z ≈ S2
A case that occurs naturally in applications of conformal methods in generalrelativity is an initial characteristic problem where the intersection Z = N ∩N  isdiﬀeomorphic to the 2-sphere S2 . This is the case, for example, of the intersectionof two light cones; see Figure 12.11. The method discussed in the previous section
can be adapted to this case; see Kánnár (1996b).
Assuming in what follows that Z ≈ S2 , consider an atlas {(U1 , φ1 ), (U2 , φ2 )} of
Z and closed sets V1 ⊂ U1 and V2 ⊂ U2 which also cover Z; that is, Z = V1 ∪ V2 .
Furthermore, deﬁne two smooth functions η1 and η2 with compact support on
R2 byη1 (x) ≡1x ∈ φ1 (V1 )0 x ∈ R2 \ φ1 (U1 ),η2 (x) ≡10x ∈ φ2 (V2 )x ∈ R2 \ φ2 (U2 ).

In what follows, denote by ŭ1 and ŭ2 , respectively, the restriction to U1 andU2 of the freely speciﬁable data on N ∪ N  . It follows that the functions η1 ŭ1and η2 ŭ2 deﬁne a smooth initial value data set on the initial hypersurfaces
N1 ≡ R+ × {0} × R2 and N2 ≡ {0} × R+ × R2 which coincides with u1 and u2
on R+ × {0} × φ1 (V1 ) and {0} × R+ × φ2 (V2 ).
The intrinsic equations on N1 and N2 can now be solved in a manner similar
to what was done in Section 12.5.2. In this way, one obtains two completecharacteristic initial data sets u1 and u2 on N1 ∪ N2 . The (interior andtransversal) derivatives of these data can be computed to any desired order.
Using Theorem 12.7 one obtains two solutions u1 and u2 in a neighbourhood V
of N1 ∩ N2 . Their restrictions to the Cauchy development of R+ × R+ × φ1 (V1 )and R+ × R+ × φ2 (V2 ) are local solutions to the original problem. The solutions
u1 and u2 can be glued together to obtain a global solution on Z ≈ R2 . As a

328

Methods of the theory of hyperbolic diﬀerential equations

consequence of the uniqueness of the local solutions u1 and u2 , it follows that
their restriction to a part of the Cauchy development of R+ × R+ × φ1 (V1 ∩ V2 )and R+ × R+ × φ2 (V1 ∩ V2 ) – where both solutions exist – must be related by acoordinate transformation. In this manner, one obtains a smooth function in aneighbourhood of Z.
12.6 Concluding remarks
This chapter has provided a succinct discussion of the theory of the local existence and uniqueness of quasilinear symmetric hyperbolic evolution equations.
Of course, this is not the only way the subject can be approached. Nor are theissues raised the only relevant ones in the analysis of the evolution problem ingeneral relativity. Thus, it is important to make some remarks concerning someideas and approaches which have been omitted.
12.6.1 Wave equations
The analysis of Section 12.1.3 gives a hint on how the theory of secondorder hyperbolic equations (wave equations) can be reduced to the analysis ofsymmetric hyperbolic equations. There is, however, a well-developed theory forthe local existence and stability of systems of quasilinear equations of the formg μν (x, u)∂μ ∂ν u = B(x, u, ∂u),(12.33)

which does not rely on the reduction to a ﬁrst order system; see Hughes et al.
(1977). Equation (12.33) is quasilinear in the sense that g μν is the contravariant
version of a Lorentzian metric which is allowed to depend not only on thecoordinates but also on the unknowns. This “stand alone” theory relaxes thediﬀerentiability assumptions made on the equation and data; see, for example,
Rendall (2008) for more details.
The results of Hughes et al. (1977) are similar, in spirit, to the results given in
Theorems 12.2 and 12.3: given suitably smooth initial data for Equation (12.33)
one obtains a unique solution for some existence time T ; moreover one also
has a Cauchy stability result. It should be pointed out that this theory applies,in fact, to a class of second-order equations more general than that given in
Equation (12.33).
Systems of quasilinear wave equations of the form (12.33) arise naturally inthe reduction procedure for the Einstein ﬁeld equations based on the use of wave
coordinates; see the Appendix to Chapter 14. Historically, this was the ﬁrst
approach to the Cauchy problem in general relativity; see Fourès-Bruhat (1952).
12.6.2 Global existence of solutions
Conformal methods allow the reformulation of several questions on the global
existence of solutions to the Einstein ﬁeld equations as a local existence question

12.7 Further reading329

for symmetric hyperbolic systems. Accordingly, the issue of global existenceof solutions to symmetric hyperbolic equations has not been addressed in thischapter. Nevertheless, this question is at the heart of current research work in
the area; see, for example, Klainerman (2008) for a discussion.
As already pointed out, the local theory of solutions to hyperbolic equations
depends solely on the properties of the principal part of the equations. To
construct a theory of global existence one has to include the lower order terms of
the equations into the analysis. Under certain circumstances, the analysis of theeigenvalues of the matrix arising from the linearisation of the lower order termsof a quasilinear system gives a strong indication of whether one can expect global
existence and stability of solutions; see, for example, Kreiss and Lorenz (1998).
More generally, one can identify structures in the evolution equations whichallow one to prove global existence. One of these structures is the so-called nullcondition; see, for example, Klainerman (1984).

12.7 Further reading
The theory of hyperbolic diﬀerential equations, in general, and their application
to the analysis of solutions of the Einstein ﬁeld equations, in particular, is anextensive area of research so that any list of references can provide only a partial
impression of the ﬁeld. For an overview of the whole ﬁeld of the theory of PDEs
and the interconnection between the various types of equations the reader isreferred to Klainerman (2008).
Readers interested in further details of the basic aspects of the theory of PDEs
are referred to the classical references by Garabedian (1986) and Courant andHilbert (1962). A modern introduction to the subject is given in Evans (1998).
A comprehensive exposition of the subject is given in the three-volume treatiseof Taylor (1996a,b,c). Detailed accounts of the theory of the Cauchy problem
for symmetric hyperbolic systems are discussed in the original references by
Kato (1975a); Fischer and Marsden (1972) for ﬁrst-order symmetric hyperbolic
systems and Hughes et al. (1977) for second-order equations. A review of theideas contained in these works can be found in Kato (1975b).
A comprehensive discussion of the role of PDEs in general relativity is given
in Rendall (2008). A more compact review is Friedrich and Rendall (2000).
Complementary discussions on the topics covered in these references can befound in Rendall (2006) and Reula (1998).

Appendix
A generalised mean value theorem
In the proofs of the uniqueness of solutions for symmetric hyperbolic systems,
Theorems 12.2 and 12.5, the following generalisation of the mean value theorem

330

Methods of the theory of hyperbolic diﬀerential equations

has been used; see Hamilton (1982). In the following result, M (N ×N, C) denotes
the set of N × N matrices with complex entries.
Lemma 12.1 Let U ⊂ RN , and let F : U → CN be a C 1 map. Then there exists
a continuous map M : U × U → M (N × N, C) such that
F(u) − F(v) = M(u, v)(u − v).
The proof is an application of the fundamental theorem of calculus.
Whitney’s extension theorem
In what follows, let α = (α1 , . . . , αn ), β = (β1 , . . . , βn ) denote multi-indices. The
factorial α! is deﬁned as α! ≡ α1 ! · · · αn !. Moreover, let|β| ≡ β1 + · · · + βn .
In terms of this notation one has the following:
Theorem 12.8 (Whitney’s extension theorem) Given a non-negative
integer k, suppose {fα }, |α| < k is a collection of real valued functions deﬁned
on a closed set A ⊂ Rn satisfying
fα (x) =|β|≤k−|α|1
fα+β (x )(x − x )β + Rα (x, x ),β!

for every x, x ∈ A and each multi-index α with |α| ≤ k such that for every
x0 ∈ A
asx, x → x0 .
Rα = o |x − x |k−|α| ,Then there exists a C k function g : Rn → R such thatg = f0 ,∂ α g = fα

on A.

In other words, for a closed set A, if one is given a function f and candidates
fα for its partial derivaties on A, then f can be extended to all of Rn in such a
way that the candidates are indeed the derivatives of f as long as the remainder
has a suitable behaviour. A priori, it is not possible to identify the functions fα
with the derivatives of f as A is a closed set and transversal derivatives f to ∂A
may not be deﬁned. For further details on Theorem 12.8 and its proof, see, forexample, Abraham and Robbin (1967).

13
Hyperbolic reductions

This chapter discusses several methods for the construction of symmetric
hyperbolic evolution systems out of the conformal Einstein ﬁeld equations. Oncesuitable evolution systems have been obtained, the methods of Chapter 12
allow, in turn, one to make statements about the existence of solutions to
the equations. Direct inspection of the conformal ﬁeld equations reveals thatthese are overdetermined – there are more equations than unknowns, even if thesymmetries of the various tensorial and spinorial ﬁelds are taken into account.
Thus, the process of hyperbolic reduction for the conformal ﬁeld equations
necessarily requires discarding some of the equations. The discarded equations
are then treated as constraints. It is a remarkable structural property of theconformal ﬁeld equations that these constraints satisfy a system of evolution
equations – a so-called subsidiary evolution system – from where it can beconcluded that the constraint equations will be satisﬁed if they hold at some
initial hypersurface and the evolution equations are imposed. This construction
is called the propagation of the constraints. The solution of the evolution
system together with the propagation of the constraints yields the required
solution of the conformal Einstein ﬁeld equations.
In this chapter, two diﬀerent procedures for the hyperbolic reduction of theconformal Einstein ﬁeld equations are considered. The ﬁrst method, based on thenotion of gauge source functions, exploits the fact that certain derivatives of
the conformal ﬁelds are not directly determined by the equations and, thus, canbe freely speciﬁed. In the spinorial formulation of the equations, once the requiredgauge source functions have been speciﬁed, the irreducible decomposition of thevarious zero quantities leads to the required evolution equations. The equations
obtained by this procedure include the conformal factor as an unknown.
The second hyperbolic reduction procedure presented in this chapter exploits
the properties of congruences of conformal geodesics to construct conformal
Gaussian gauge systems. As discussed in Chapter 5, the connection coeﬃcients and components of the Schouten tensor with respect to a frame which is332Hyperbolic reductions

Weyl propagated along the congruence satisfy certain relations which lead to aparticularly simple system of equations in which the evolution of all the geometricunknowns, save for the components of the rescaled Weyl spinor, are either ﬁxed
by the gauge or given by transport equations along the congruence. Moreover,as a consequence of the properties of the conformal geodesics one gains an apriori knowledge of the location of the conformal boundary; see Proposition 5.1.
Despite these attractive features, this method is less ﬂexible than the one basedon the use of gauge functions and may not be readily extended to non-vacuum
situations.
13.1 A model problem: the Maxwell equations on a ﬁxed backgroundTo illustrate the various aspects of the construction of evolution equations forthe conformal Einstein ﬁeld equations, it is convenient to analyse the analogous
problem for the Maxwell equations on a ﬁxed background.
In the remainder of this section, let U denote an open region of a spacetime(M, g). It will be assumed that U is covered by a non-singular congruence of
curves with tangent vector τ satisfying the normalisation condition g(τ , τ ) = 2.
The vector τ does not need to be hypersurface orthogonal. Let τ AA denote thespinorial counterpart of τ a . As discussed in Section 4.2.5, the spinor τ AA gives
rise to a Hermitian structure, and, accordingly, one can introduce a space spinor
formalism. Let {A A } denote a spin basis such that

τ AA = 0 A 0 A + 1 A 1 A ,(13.1)

and with {eAA } its associated null frame. At every point p ∈ U a basis of the
subspace τ ⊥ |p ⊂ T |p (U ) orthogonal to τ is given by eAB = τ(B A eA)A . In
terms of local coordinates x = (xμ ) in U one writes
eAB = eAB μ ∂ μ .

(13.2)

In principle, it is possible for the frame vectors eAB to have components withrespect to the time coordinate. The frame components eAB μ satisfy the reality
conditions
e01 μ = e01 μ ,e00 μ = −e11 μ .

(13.3)All spinorial objects will be expressed with respect to the spin basis {A A }.
In particular, the spinorial Maxwell Equation (9.15) is written as∇Q A ϕBQ = 0.

(13.4)In what follows, it will be convenient to introduce the zero quantity
ωA B ≡ ∇Q A ϕBQ ,so that (13.4) can be expressed as ωA B = 0. Here and in the remainder of thischapter, zero quantities such as ωA B serve as convenient bookkeeping devices
to denote the various ﬁeld equations.

13.1 A model problem: the Maxwell equations on a ﬁxed background 333
13.1.1 Space spinor description of the Maxwell equations
and hyperbolic reductions
The space spinor version of Equation (13.4) leads to a decomposition into
evolution and constraint equations. Following the discussion of Chapter 4 oneconsiders the unprimed zero quantity ωBA ≡ τB A ωA A . One then has thatωBA = ∇Q B ϕAQ =1 Q
 A PϕBQ + DQ A ϕBQ
2

1= − PϕAB + DQ A ϕBQ ,2
where P is the covariant directional derivative along τ , DAB is the Sen covariantderivative implied by ∇AA and ∇AB ≡ τB A ∇AA . In the above expressions,the decomposition∇AB =1
AB P + DAB
2(13.5)

has been used; see Section 4.3.1. The spinor ωBA can, in turn, be decomposedin irreducible parts as
ωBA =1
BA ω + ω(AB) ,2withω ≡ ω Q Q = D P Q ϕP Q ,1
ω(AB) = − PϕAB + DQ (A ϕB)Q .
2

Thus, the Maxwell Equations (13.4) imply the equations
ω = DP Q ϕP Q = 0,−2ω(AB) = PϕAB − D

Q(A ϕB)Q = 0.

(13.6a)(13.6b)The decomposition of the spinorial Maxwell equation given by (13.6a) and(13.6b) shows that Equation (13.4) is overdetermined. Equation (13.6a) will
be interpreted as a constraint equation on the orthogonal subspaces of the
distribution generated by the vector ﬁeld τ , while (13.6b) will be regarded as
suitable evolution equations for the symmetric spinorial ﬁeld ϕAB .
13.1.2 The symmetric hyperbolicity of the Maxwell
evolution equations
To apply the theory of Chapter 12 one needs to verify that the evolution
Equations (13.6b) give rise to a symmetric hyperbolic system for the independent
components of ϕAB . One considers the slightly modiﬁed version




2
PϕAB − DQ (A ϕB)Q = 0,(13.7)A+B

334

Hyperbolic reductions

where the binomial coeﬃcient in front of the equation has been included tomake the expression manifestly symmetric hyperbolic. The principal part ofEquation (13.7) can be written as




 μ
2
τ ∂μ ϕAB − eQ (A μ ∂|μ| ϕB)Q .
A+B

As a result of the symmetry of ϕAB , the above principal part contains three
independent expressions. These can be arranged in the matricial expression⎛

τ μ + e10 μμ⎝
A ∂μ ϕ ≡e11 μ
0−e00 μ
2τ μ
e11 μ

⎞

⎞
ϕ0
⎠ ∂ μ ⎝ ϕ1 ⎠ ,
e00 μ
μ
μ
τ − e01
ϕ2
0

⎛

with
ϕ0 ≡ ϕ00 ,ϕ1 ≡ ϕ01 ,ϕ2 ≡ ϕ11 .

Thus, making use of the reality conditions (13.3), it follows that the matrices
Aμ are Hermitian. Moreover, the matrix⎛

2
Aμ τ μ = ⎝ 0
0

0
4
0

⎞
0
0 ⎠
2

clearly is positive deﬁnite. Thus, Equation (13.7) implies a symmetric hyperbolic
system for the independent components of ϕAB . Finally, a direct computationshows that given an arbitrary covector ξμ ,det(Aμ ξμ ) = 2(τ μ ξμ ) τ ν τ λ + e00 ν e11 λ − e01 ν e10 λ ξν ξλ= 4(τ μ ξμ )(g νλ ξν ξλ ),where in the last line Equation (4.14) for the 1+3 decomposition of thespacetime metric has been used. Thus, g-null hypersurfaces are characteristics
of Equation (13.7) – these types of characteristics are often called physical characteristics. By contrast, the factor (τ μ ξμ ) is associated to gauge
characteristics.
For completeness, it is observed that the principal part of the constraint
equation is given, explicitly, by
e00 μ ∂μ ϕ0 + e01 μ ∂μ ϕ1 + e11 μ ∂μ ϕ2 ,so that, in general, it will contain derivatives in the time direction. Moregenerally, if the vector τ is not hypersurface orthogonal, then the constraint
equation ω = 0 will not be intrinsic to the leaves of a foliation.

13.1 A model problem: the Maxwell equations on a ﬁxed background 335
13.1.3 The subsidiary system for the spinorial Maxwell equations
The hyperbolic reduction for the Maxwell equations discussed in Section 13.1.1
splits Equation (13.4) into three evolution equations and one constraint equation.
Thus, if one wants to obtain a solution to Equation (13.4) through a Cauchy
initial value problem, one uses, in ﬁrst instance, the theory of Chapter 12 toshow the existence of a unique solution to the evolution equations. In a second
stage, one has to show that if the constraint equation is satisﬁed initially, then,by virtue of the evolution equations, it must be satisﬁed also at later times.
This last argument requires the construction of a suitable hyperbolic evolution
equation for ω.
To obtain an equation for the zero quantity ω one considers the expression
AAωA A . Using that ωA A = −τ Q A ωQA one has that∇∇AA ωA A = −∇AA τ Q A ωQA
= ∇AQ ωQA − (∇AA τ Q A )ωQA .
Now, using Equation (4.17), a calculation yields√
∇AA τ Q A = − 2χA P P Q ,(13.8)

so that∇AA ωA A = ∇AQ ωQA +√ A PQ
2χ PωQA .

Thus, the split (13.5) leads to the expression√
Pω + 2DAB ω(AB) + 2 2χA P P Q ωQA = 2∇AA ωA A .
If the evolution equations hold – that is, ω(AB) = 0 – then ωAB = 12 AB ω andone obtains√
Pω + 2χAB AB ω = 2∇AA ωA A .


The next step is to evaluate ∇AA ωA A in an alternative manner. Using thedeﬁnition of the zero quantity one has that∇AA ωA A = ∇AA ∇Q A ϕAQ .
From the commutator∇AA ∇BB  ϕCD − ∇BB  ∇AA ϕCD = −RP CAA BB  ϕP D − RP DAA BB  ϕCP ,suitably contracting indices one obtains∇AA ∇Q A ϕAQ = −2RP A AA Q A ϕP Q − 2RP Q AA Q A ϕAP .

336Hyperbolic reductions

Thus, combining the above equation with the decomposition
RABCC  DD = ΨABCD C  D + LBC  DD CA − LBD CC  DA ,(13.9)

where ΨABCD and LBC  DD denote, respectively, the spinorial counterparts of
the Weyl and Schouten tensors, one concludes that ∇AA ∇Q A ϕAQ = 0. Hence,the evolution equation for ω takes the form√
if ω(AB) = 0.
Pω + 2χAB AB ω = 0
The form of this equation implies, in together with Corollary 12.1, that if ω = 0
on some spacelike hypersurface S in U , then ω = 0 on lens-shaped domains
having S as base.
13.2 Hyperbolic reductions using gauge source functions
In this section hyperbolic reduction procedures for the conformal Einstein ﬁeld
equations based on the notion of gauge source functions are considered.
Gauge source functions naturally arise in the analysis of frame formulationsof the conformal Einstein ﬁeld equations written in terms of the Levi-Civita
connection ∇ of an unphysical metric g. The present analysis will be restricted
to the spinorial version of the conformal ﬁeld equations: Equations (8.36a) and(8.36b) or, alternatively, Equations (8.38a) and (8.38b).
Basic set up and assumptions
As in the analysis of the Maxwell equations in Section 13.1, all the calculations
will be performed in an open subset U ⊂ M of an unphysical spacetime (M, g)which is conformally related to a spacetime (M̃, g̃) satisfying the Einstein ﬁeld
equations. On U one considers some local coordinates x = (xμ ) and an arbitrary
frame {ca } which may or may not be a coordinate frame. Let {αa } denotethe dual coframe so that αa , cb  = δb a . In what follows, let ∇ denote theLevi-Civita covariant derivative of the metric g.
It will be assumed that U is covered by a non-singular congruence of curves
with tangent vector τ satisfying the normalisation condition g(τ , τ ) = 2. Thevector τ does not need to be hypersurface orthogonal. Let τ AA denote thespinorial counterpart of τ a . In what follows, only spin bases {A A } satisfying
condition (13.1) will be considered. All spinors will be expressed in components
with respect to this spin basis.

Let {eAA } and {ω AA } denote, respectively, the null frame and coframeassociated to the spin basis {A A }. By deﬁnition, one has that ω AA , eBB   =B A B  A . At every point p ∈ U a basis of τ ⊥ |p , the subspace of T |p (U )orthogonal to τ is given by eAB = τ(B A eA)A . The spatial frame can be
expanded in terms of the vectors ca as eAB = eAB a ca . If the basis {ca } is acoordinate basis, the last expression reduces to the one given in Equation (13.2).

13.2 Hyperbolic reductions using gauge source functions

337

A model equation
The general strategy behind the procedure of hyperbolic reduction using gauge
functions is best understood through a model equation.
In Section 12.1.3 it has been shown that spinorial equations of the form
∇Q A ϕQB  C···D = FA B  C···D(13.10)

imply a symmetric hyperbolic system for the components of the ﬁeld ϕQB  C···D
which is not assumed to have any special symmetries. This equation is now
contrasted with the equation∇AA ϕBB  C···D − ∇BB  ϕAA C···D = FAA BB  C···D .

(13.11)

Exploiting the antisymmetry in the pairs AA and BB  it follows that
∇Q (A ϕ|Q|B  )C···D =1 Q
F A QB  C···D .
2(13.12)Thus, while Equation (13.10) determines the full derivative ∇Q A ϕQB  C···D ,Equation (13.12) determines only its symmetric part. More precisely, writing
1∇Q A ϕQB  C···D = ∇Q (A ϕ|Q|B  )C···D − A B  ∇QQ ϕQQ C···D ,2(13.13)one has that the ﬁrst term in the right-hand side is determined by Equation (13.12), while the divergence ∇QQ ϕQQ C···D remains unspeciﬁed. Thus,in the absence of other equations providing information about this term, thelatter observation suggests completing Equation (13.13) by setting∇QQ ϕQQ C···D = fC···D (x),where fC···D ∈ X (M) are smooth freely speciﬁable functions of the coordinates.
In what follows, functions of this type will be known as gauge source
functions. Thus, from (13.13) one obtains the equation∇Q A ϕQAC···D =1 Q
1F A QB  AC···D − A B  fAC···D (x),22for which one can extract a symmetric hyperbolic evolution system for thecomponents of ϕAA C···D ; see the discussion of Section 12.1.3. In particular, thecharacteristics of this evolution system are null hypersurfaces of the spacetime
metric g.
As will be seen in the following subsections, several of the conformal Einsteinﬁeld equations admit an analysis similar to that of Equation (13.11). A detailed
discussion of the resulting evolution equations exploits the particular symmetriesof the ﬁeld appearing in the principal part.

338

Hyperbolic reductions
13.2.1 Coordinate gauge source functions

The purpose of this subsection is to analyse the evolution equations arising fromthe no-torsion condition in the frame and spinor formulations of the conformal
ﬁeld equations; see Equations (8.31a), (8.35a), (8.44a) and (8.53a). This leads tothe ﬁrst class of gauge source functions that will be considered in this chapter:the coordinate gauge source functions. Following the general discussion ofChapter 8, the no-torsion condition will be regarded as a diﬀerential condition on
the coeﬃcients of the frame {eAA }. Thus, the ultimate purpose of this section
is to derive a symmetric hyperbolic subsystem for these quantities.
In Section 8.3.2 an expression for the spinorial counterpart of the torsion tensor
ΣAA CC BB  in terms of the spinorial connection coeﬃcients ΓAA CC BB  hasbeen given; see Equation (8.35a). In what follows, it is more convenient to makeuse of an expression involving the reduced spin connection coeﬃcients. Using therelationΓAA CC BB  = ΓAA C B B  C + Γ̄A A C B  B C ,– compare Equation (3.33) – it can be seen thatΣAA QQ BB  eQQ = [eBB  , eAA ] − ΓBB  Q A eQA − Γ̄BB  Q A eAQ+ ΓAA Q B eQB  + Γ̄AA Q B  eBQ .

(13.14)Using the frame {ca } one can write
eAA = eAA a ca ,so that for ﬁxed frame spinorial indices AA , the coeﬃcients eAA a have thenatural interpretation of the components of eAA with respect to ca . However,there is an alternative interpretation: for ﬁxed frame index a , the coeﬃcients
eAA a correspond to the components of the covectors αa with respect to the
coframe ω AA . That is, one has

αa = eAA a ω AA ,
from where it follows that eAA a ω AA b = δb a . In view of this interpretation, it
is convenient to deﬁne∇CC  eBB  a ≡ eCC  b cb (eBB  a ) − ΓCC  Q B eQB  a − Γ̄CC  Q B  eBQ a ,(13.15)a

so that ∇CC  αa = ∇CC  eBB  ω BB . Expression (13.15) corresponds to theformula one would use to compute the covariant derivative of eBB  a if it werethe components of a tensor – which, of course, it is not.
Intuition into this general discussion is gained by considering the particularcase of a coordinate frame for which eAA = eAA μ ∂ μ so that
eAA (xν ) = eAA μ ∂μ (xν ) = eAA μ δμ ν = eAA ν .

13.2 Hyperbolic reductions using gauge source functions


339



Moreover, writing ω AA = ω AA μ dxμ one has thatdxμ = eAA μ ω AA .
That is, for ﬁxed coordinate index μ , the coeﬃcients eAA μ are the components
of the coordinate diﬀerential dxμ with respect to the coframe ω AA .
Returning to the general discussion, using the identity[f v, u] = f [v, u] − u(f )v
for v, u ∈ T (M) and f ∈ X (M), together with expression (13.15) one canrewrite Equation (13.14) asΣAA QQ BB  eQQ c = ∇BB  eAA c − ∇AA eBB  c − eAA a eBB  b Ca c b , (13.16)
where Ca c b are the commutation coeﬃcients deﬁned by[ca , cb ] = Ca c b cc .
In the case of a coordinate frame one obtains the simpler expression
ΣAA QQ BB  eQQ μ = ∇BB  eAA μ − ∇AA eBB  μ ,as [∂ μ , ∂ ν ] = 0.
A ﬁnal simpliﬁcation is obtained by exploiting the antisymmetry of Equation (13.16). Contracting the indices A and B  and symmetrising in AB one
concludes that1
(13.17)∇(A Q eB)Q a + eA Q b eBQ c Cb a c = ΣAB a ,
2
with


1
ΣAB a ≡ ΣA Q CC BQ eCC  a .
2
As the frame eAA is Hermitian, that is, eAA = eAA , one has that (13.17)
is completely equivalent to Equation (13.16). Moreover, if ΣAB a = 0, thenΣAA CC BB  = 0 and the connection is torsion free.
The structure of Equation (13.17) is similar to that of the model Equation(13.12), suggesting that by introducing a gauge source function one willobtain a symmetric hyperbolic system for the frame coeﬃcients eAA a . Now,
Equation (13.17) does not impose restrictions on the divergences ∇QQ eQQ a sothat one can set∇QQ eQQ a = F a (x),(13.18)where the coordinate gauge source functions F a (x) are smooth functions of
the coordinates x = (xμ ). In the case of a coordinate frame the above expression
reduces to∇QQ ∇QQ xμ = F μ (x),the so-called generalised wave coordinates condition.

(13.19)

340

Hyperbolic reductions

Combining the identity1∇(A Q eB)Q a = ∇A Q eBQ a + AB ∇P P eP P  a2with Equations (13.17) and (13.18) one ﬁnally obtains, for ΣAB a = 0, theequation
1
1∇A Q eBQ a + AB F a (x) + eA Q b eBQ c Cb a c = 0,
22from which a symmetric hyperbolic system for the frame components of eBQ a
can be deduced.

Geometric interpretation
The generalised wave coordinate condition (13.19) shows that a particular choiceof coordinate gauge is, implicitly, a choice of coordinates. Equation (13.19) canalways be solved locally by choosing some coordinates x = (x0 , xα ) on someﬁduciary surface S . If this surface is described by the condition x0 = 0, then it
is also natural to require that∂xα
= 0,
on S .
∂x0
Moreover, one needs the coordinate diﬀerentials dxμ to be linearly independenton S . These conditions ensure the existence of a solution to Equation (13.19)
close to S .
Conversely, given a particular coordinate choice on a spacetime (M, g), one
can use Equation (13.19) to compute the coordinate gauge source function F μ (x)associated with the coordinates. Thus, local coordinates and coordinate gaugesource functions are in a one-to-one correspondence.
Construction of coordinates in perturbations of spacetimes
The discussion of the previous subsection can be applied to the constructionof coordinates in spacetimes (M, g) which are perturbations of a certain exact
background spacetime (M̊,g̊). In this situation, one would expect the spacetime
manifolds M and M̊ to be diﬀeomorphic to each other so that coordinates in the
background spacetime could be used as coordinates in the perturbed spacetime.
This does not mean that the spacetimes (M, g) and (M̊,g̊) are isometric! The
intuition expressed in this paragraph will now be formalised.
In what follows, assume that one has two spacetimes (M, g) and (M̊,g̊) suchthat the manifolds M and M̊ are diﬀeomorphic. Let ϕ : M → M̊ denote a
diﬀeomorphism between them. This choice is clearly not unique. The subsequentdiscussion will single out a particular type of diﬀeomorphism between M and M̊.
Let x = (xμ ) and x̊ = (x̊μ ) denote, respectively, local coordinates on M
and M̊. In terms of these local coordinates the diﬀeomorphism ϕ is given byx̊μ = x̊μ (x) and its inverse by xμ = xμ (x̊). On M̊ consider a frame {c̊a } and its

13.2 Hyperbolic reductions using gauge source functions

341

dual coframe {α̊a }. The frame is not necessarily assumed to be g̊-orthonormal.
From this frame and coframe one can introduce a frame {ca } and a coframe{αa } on M using, respectively, the push-forward and the pull-back implied by
ϕ : M → M̊. More precisely,c̊a = (ϕ)∗ ca ,α̊a = (ϕ−1 )∗ αa .

αa = αa μ dxμ ,α̊a = α̊a μ dx̊μ ,Thus, writing

one concludes thatαa μ = α̊a ν

∂x̊ν
.
∂xμ

Now, observing that αa , eb  = δb a , it follows that ∇c αa = (∇c eb a )αb and,
consequently,∇b eb a = η cd ∇c αa , ed  = eb μ ∇b αa μ = ∇μ αa μ .
The above expression can be used to write the divergence ∇QQ eQQ a appearing
in Equation (13.18) in terms of quantities associated to the diﬀeomorphism ϕ :M → M̊.
Treating the coordinates x̊ = (x̊μ ) as scalars and recalling that α̊a ν =α̊a , ∂/∂x̊ν  so that the coeﬃcients α̊a ν are also scalars, one ﬁnds that λ∂x̊∂α̊a λ ∂x̊λ+∇ν αa μ = α̊a λ ∇νμ∂x∂xν ∂xμ˚ ρ α̊a λ= α̊a λ ∇ν ∇μ x̊λ + ∇∂x̊ρ ∂x̊λ,∂xν ∂xμwhere in the last equality the chain rule has been used. Consequently, one has
˚ ρ α̊a λ∇μ αa μ = α̊a λ ∇μ ∇μ x̊λ + g μν ∇∂x̊ρ ∂x̊λ= F a (x),∂xν ∂xμ

or, more suggestively,ρλ˚ ρ α̊a λ ∂x̊ ∂x̊ − F a (x) = 0.
∇μ ∇μ x̊σ + c̊a σ g μν ∇∂xν ∂xμ
So far, the diﬀeomorphism ϕ : M → M̊ has been kept completely general.
However, if one sets˚ ρ α̊a λg μν ∇∂x̊ρ ∂x̊λ= F a (x),∂xν ∂xμ

one ﬁnds that∇μ ∇μ x̊σ = 0.

(13.20)342

Hyperbolic reductions

That is, under condition (13.20), the diﬀeomorphism ϕ : M → M̊ given byx̊μ = x̊μ (x) is a wave map. Wave maps can be regarded as a generalisation of thegeodesic equation. Further discussion on this notion, which plays an importantrole in modern research in PDE theory and geometric analysis, can be found in
the review by Tataru (2004).
Now, it is convenient to regard the manifolds M and M̊ as being the sameand let x̊μ = x̊μ (x) be the identity map so that ∂x̊ρ /∂xν = δν ρ . This amounts
to saying that the coordinates x̊ = (x̊μ ) are used as coordinates of the perturbed
spacetime (M, g). In this case condition (13.20) reduces to˚ b α̊a b = F a (x).
∇If in the reference spacetime one has ω̊ a = α̊a so that α̊a b ≡ α̊a ,c̊b  = δb a ,then
˚ b α̊a b = −η bc Γ̊b a c .
∇Accordingly, the coordinate gauge source function F a (x) can be expressed interms of the connection of the background spacetime via
F a (x) = −η bc Γ̊b a c ,or, in spinorial terms

F a (x) = −AB A B eAA b eBB  c Γ̊b a c .
Space spinor decomposition of the equation for the frame coeﬃcients
The space spinor decomposition of Equation (13.17) provides a systematic
approach to the extraction of the required symmetric hyperbolic system.
Accordingly, one considers the space spinor split of the frame ﬁelds given by
eAA a =1
τAA ea − τ Q A eAQ a
2

with


ea ≡ τ AA eAA a ,

eAB a ≡ τ(A A eB)A a .

Alternatively, one can writeτB Q eAQ a =1
AB ea + eAB a .
2

Using
√
∇AB τCD = − 2τ D D χABCD ,– compare Equation (4.17) – together with the decomposition of ∇AB given in
Equation (13.5), it follows from Equation (13.18) that√√
Pea + 2DP Q eP Q a + 2ea χP Q P Q + 2 2eP Q a χP C CQ − 2F a (x) = 0.
(13.21)

13.2 Hyperbolic reductions using gauge source functions

343

A similar computation for Equation (13.17) yields
1
1
1
PeAB a − DAB ea + D(A Q eB)Q a − √ ea χ(A|Q| Q B)
2
2
2
√
1
+ 2eP (A a χB)Q QP − (eb eAB c + eAQ b eB Qc )Cb a c .
2
A further independent equation can be obtained from the Hermitian conjugateΣAB a =



aA
τB B Σ̄A B  a .
Σ+AB ≡ τA

Exploiting the identity







τA A τB B ∇Q A eQB  a = ∇Q A τB B eQB  a − eQB  a ∇Q A τB B ,one arrives at
1
1
1 a Q
a
Q
a
a
a
Σ+
AB = − PeAB + DAB e + D(A eB)Q + √ e χ (AB)Q
2
2
2
√
1
− 2eP Q a χP (AB) Q − (eb eAB c + eAQ b eB Qc )Cb a c .
2
The required evolution equation complementing (13.21) is then obtained froma
ΣAB a − Σ+AB = 0,

where

1 a
a
Q
Q
a
a
ΣAB a − Σ+AB = PeAB − DAB e − √ e χ(A|Q| B) + χQ(AB)
2
√
√
a
QP
+ 2eP Q a χP (AB) Q+ 2eP (A χB)Q
− ec eAB b Cb a c .

(13.22)A direct inspection shows that Equations (13.21) and (13.22) imply, for ﬁxed
frame index a , a symmetric hyperbolic system of four equations for ea andthe independent components of eAB a . A further computation shows that thecharacteristic polynomial of the system is given by−4(τ μ ξμ )2 (g λρ ξλ ξρ ).
As a by-product of the analysis one obtains the constraint equations impliedby (13.17) froma
ΣAB a + Σ+
AB = 0,

where

1 a
a
a
Q
Q
Q
ΣAB a + Σ+
AB = 2D (A eB)Q + √ e χ(A|Q| B) + χ (AB)Q
2√√+ 2eP (A a χB)Q QP − 2eP Q a χP (AB) Q− eb eAB c + eAQ b eB Qc Cb a c .
Expanding the principal part of this constraint equation, one ﬁnds it contains
derivatives in the time direction.

344

Hyperbolic reductions
13.2.2 Frame gauge source functions

After having analysed the gauge source conditions arising from the no-torsion
condition, one can now consider the gauge source functions associated to the Ricci
identity – that is, the condition requiring that the geometric and the algebraic
curvatures coincide. As with the no-torsion condition, the equality between thetwo expressions for the curvature is part of the frame and spinorial formulationsof the conformal ﬁeld equations; compare Equations (8.31b), (8.35b), (8.44b)and (8.53b).
Rather than working with the full expressions for the curvature spinors, inthe subsequent discussion it will be convenient to make use of the reduced
spinorial counterpart of the Riemann tensor in terms of the reduced connection
coeﬃcients:


RABCC  DD + ΣCC  QQ DD ΓQQ AB= ∇DD ΓCC  AB − ∇CC  ΓDD AB− ΓDD Q A ΓCC  QB − ΓCC  Q A ΓDD QB ,(13.23)where the deﬁnition∇DD ΓCC  AB ≡ eDD (ΓCC  AB ) − ΓDD Q C ΓQC  AB
− Γ̄DD Q C ΓQC  AB − ΓDD Q B ΓCC  AQ
has been used in order to obtain a more concise expression; see Section 8.3.2
for further details. This last expression is formally the same as the one for thecovariant derivative of a spinor ﬁeld with the same index structure as ΓCC  AB .
Equation (13.23) is encoded in the zero quantity
ΞABCC  DD ≡ RABCC  DD − ρABCC  DD ,where RABCC  DD and ρABCC  DD denote, respectively, the geometric andalgebraic curvatures. One has the symmetries
ΞABCC  DD = Ξ(AB)CC  DD = −ΞABDD CC  .
Exploiting the antisymmetry of Equation (13.23) one obtains the pair ofequations∇(C Q ΓD)Q AB + Γ(C Q Q |A| ΓD)Q QB = RABCD + ΣC QQ D ΓQQ AB ,(13.24a)∇P (C  Γ|P |D )AB + ΓP (C  Q |A ΓP |D )QB = RABC  D + ΣC  QQ D ΓQQ AB ,(13.24b)where
1
RABCQ D Q ,
2



1
ΣC QQ D ≡ ΣCP  QQ D P ,2

RABCD ≡1
RABQC  Q D ,21
ΣC  QQ D ≡ ΣP C  QQ P D .
2

RABC  D ≡

13.2 Hyperbolic reductions using gauge source functions

345

As the ﬁeld ΓAA BC is not Hermitian, two reduced equations are necessary to
encode the content of (13.23) – contrast this with the analysis of the no-torsion
Equation (13.14).
From the structure of Equations (13.24a) and (13.24b) one concludes that thederivative ∇QQ ΓQQ AB is not determined by the equations. Accordingly, one
can set∇QQ ΓQQ AB = FAB (x),(13.25)where FAB = F(AB) are smooth arbitrary functions of the coordinates – theframe gauge source functions.
Geometric interpretation
To gain intuition on the role played by the frame gauge source functions recall
that ΓAA B C = B B ∇AA C B ; see Equation (3.32). Equation (13.25) can be
rewritten asA B ∇P P ∇P P  B B + ∇P P A B ∇P P  B B = F A B (x).

(13.26)This is to be read as a quasilinear wave equation for the spin frame {B B }. Using
the symmetry of FAB and the wave Equation (13.26) one obtains
∇P P ∇P P  B B A B = 0,
so that by choosing
 B B  A B = δB A ,∇P P  B B A B = 0,

on some ﬁduciary hypersurface S one obtains a spin frame which is normalised
at later times.
Space spinor decomposition of the equation for the spin connection coeﬃcients
To obtain a suitable space spinor decomposition of Equations (13.24a), (13.24b)and (13.25), one deﬁnes
ΓABCD ≡ τB A ΓAA CD
and considers the split
ΓABCD =

1AB ΓCD + Γ(AB)CD ,2

ΓCD ≡ ΓQ Q CD .

Now, from∇QQ ΓQQ AB = −∇QQ τ P Q ΓQP AB= τ SQ ∇Q Q ΓQSAB − (∇QQ τ S Q )ΓQSAB√= ∇P Q ΓP QAB + 2χP R QR ΓP QAB ,346Hyperbolic reductions

it follows, using the split of ∇AB , that√PΓAB + 2DP Q Γ(P Q)AB + 2 2χP R QR ΓP QAB = 2FAB (x).

(13.27)In view of its symmetries, the zero quantity ΞABCC  EE  is decomposed as
ΞABCC  EE  = ΞABCE C  E  + ΞABC  E  CE ,with
ΞABCE ≡

1
ΞABCQ E Q ,21
ΞABQC  Q E  .
2ΞABC  E  ≡In terms of space spinors the latter decomposition can be rewritten as
ΞABCDEF = ΞABCE DF + Ξ∗ABDF CE ,
where




ΞABCDEF ≡ τD C τF E ΞABCC  EE  ,ΞABDF ≡ τD C τF E ΞABC  E  ,Ξ∗ABDF ≡ τD C τF E ΞABC  E  .

To expand ΞABCE and Ξ∗ABDF it is observed that∇(C Q ΓD)Q AB = −∇(C Q ΓD)SAB τ S Q= −τ S Q ∇(C Q ΓD)SAB − ∇(C Q τ S |Q | ΓD)SAB√= ∇(C S ΓD)SAB + 2χ(C|Q| SQ ΓD)SAB√
1= PΓ(CD)AB + D(C S ΓD)SAB + 2χ(C|Q| SQ ΓD)SAB
2
and that

τC C τD D ∇P (C  Γ|P |D )AB = ∇P (C Γ|P |D)AB1= − PΓ(CD)AB + DP (C Γ|P |D)AB .
2
From the above expressions it follows that1
11PΓ(CD)AB − DCD ΓAB + DC S Γ(DS)AB + DD S Γ(CS)AB2
2
2+ Γ(C P Q |A| ΓD)P QB − ΣC P Q D ΓP QAB − RABCD ,1
11Ξ∗ABCD = − PΓ(CD)AB + DCD ΓAB + DP C Γ(P D)AB + DP D Γ(P C)AB
2
2
2
PQ
∗
Γ
+ ΓP (C Q |A ΓP |D)QB + Σ+
D P QAB − RABCD .
CΞABCD =

Constraint equations are obtained from the combination
ΞABCD + Ξ∗ABCD = 0,

13.2 Hyperbolic reductions using gauge source functions

347

where
ΞABCD + Ξ∗ABCD = DP C Γ(P D)AB + DP D Γ(P C)AB+ Γ(C P Q |A| ΓD)P QB + ΓP (C Q |A ΓP |D)QB
PQ
PQ
+ Σ+
D ΓP QAB − ΣC
D ΓP QABC∗− RABCD − RABCD,

while the required evolution equations arise from
ΞABCD − Ξ∗ABCD = 0,with
ΞABCD − Ξ∗ABCD = PΓ(CD)AB − DCD ΓAB+ Γ(C P Q |A| ΓD)P QB − ΓP (C Q |A ΓP |D)QB
PQ
PQ
− Σ+
D ΓP QAB − ΣC
D ΓP QABC∗− RABCD + RABCD.

(13.28)It can be veriﬁed that the system composed by (13.27) and (13.28) leads
to a symmetric hyperbolic system for the independent components of ΓABand Γ(CD)AB – up to a suitable normalisation factor. A simple counting
argument shows that the system consists of 12 equations, three coming from
Equation (13.27) and nine from Equation (13.28). The characteristic polynomialof the system is given by−64(τ μ ξμ )6 (g νλ ξν ξλ )3 .
13.2.3 The conformal gauge source function
The third type of gauge source function to be considered arises from the analysisof the Cotton equation; see Equations (8.31e) and (8.35f). The starting point of
the analysis is the spinorial counterpart, Equation (8.37a), associated with thezero quantity


ΔCDBB  ≡ ∇(C Q LD)Q BB  + ∇Q B  ΞφCDBQ + ΞTCDBB  .
To deduce a symmetric hyperbolic system from this equation one needs tocomplete the symmetrised derivative ∇(C Q LD)Q BB  with the divergence∇QQ LQQ BB  . Information about this derivative is provided by the contracted
Bianchi identity for the Schouten tensor; compare Equation (8.17). In spinorial
notation one has1(13.29)∇QQ LQQ BB  = ∇BB  R.
6
Thus, using1∇(C Q LD)Q BB  = ∇C Q LDQ BB  + CD ∇QQ LQQ BB  ,2348

Hyperbolic reductions

one can rewrite the zero quantity ΔCDBB  as1
CD ∇BB  R
12+ ΣQ B  φCDBQ + ΞTCDBB  ,

ΔCDBB  = ∇C Q LDQ BB  +(13.30)where ΣAA ≡ ∇AA Ξ.
As discussed in Chapter 8, the conformal ﬁeld equations impose no diﬀerentialcondition on the unphysical Ricci scalar R. Accordingly, R can be speciﬁed freely
as a function of the coordinates. Thus, if the reduced rescaled Cotton spinor
TCDBB  can be rewritten so that it does not explicitly contain derivatives of thematter ﬁelds, one can deduce a symmetric hyperbolic system for the componentsof LAA BB  from Equation (13.30).
Geometric interpretation
The particular choice of the Ricci scalar ﬁxes the conformal gauge freedom.
Thus, it is natural to call R(x) the conformal gauge source function. Givena particular choice of R(x), the transformation law for the Ricci scalar implies awave equation for the conformal factor realising the prescribed Ricci scalar; see
Equation (8.30). This equation can always be solved locally if initial data on aﬁduciary hypersurface S is provided – namely, the values of the conformal factor
and its normal derivative on the hypersurface. Conversely, given an unphysicalspacetime (M, g) and a conformal factor Ξ linking it to a physical spacetime(M̃, g̃) via the standard relation g = Ξ2 g̃, one can compute the corresponding
conformal gauge source function R(x).
Space spinor decomposition of the equation for the componentsof the Schouten tensor
The space spinor decomposition of the equations for the Schouten tensor is basedon the expression
LAA CC  = ΦAA CC  +1
AC A C  R(x),24(13.31)

where ΦAA CC  denotes the spinorial counterpart of the trace-free part of the
Ricci tensor; see Section 3.2.4. The space spinor counterpart of LAA CC  isdeﬁned as

LABCD ≡ τB A τD C LAA CC  ,1= ΦABCD + AC BD R(x),24where ΦABCD ≡ τB A τD C ΦAA CC  so that
ΦABCD = ΦCBAD = ΦADCB ,13.2 Hyperbolic reductions using gauge source functions349

as a consequence of the symmetries of ΦAA CC  ; see Equation (3.44). A spinor
with these symmetries can be decomposed as
ΦABCD = Φ(ABCD) + 1
1A(B ΦD)C + C(B ΦD)A + ΦhACBD ,2
3(13.32)whereΦAB ≡ Φ(AB)Q Q ,Φ ≡ ΦABCD hACBD .

Now, using that
1
∇A Q LBQ CC  = ∇(A Q LB)Q CC  − AB ∇QQ LQQ CC  ,2together with the contracted Bianchi identity (13.29) one can rewrite the zero
quantity ΔABCC  as


ΔABCC  = ∇A Q LBQ CC  +1
AB ∇CC  R(x) + ΣQ C  φABCQ + ΞTABCC  .
12(13.33)Deﬁning

ΔABCD ≡ τD C ΔABCC  ,a calculation using (13.33) together with the deﬁnitions of the spinors LABCD
and χABCD , yields√√
ΔABCD = ∇A Q LBQCD + 2χAP QP LBQCD − 2χA QP D LBQCP
1+ AB ∇CD R(x) + ΣQ D φABCQ + ΞTABCD ,2
where ΣAB ≡ τB Q ΣAQ . Thus, using the decomposition of the operator ∇AB
one obtains
ΔABCD =√
1
PLBACD + DA Q LBQCD + 2χAP QP LBQCD
2√1− 2χA P Q D LBP CQ + AB ∇CD R(x) + ΣQ D φABCQ + ΞTABCD .
2

To extract the full information of ΔABCC  one also needs to consider








P
τB Q τC R τD S Δ̄P  Q R S  .
Δ+
ABCD ≡ τA

Proceeding as with ΔABCD one ﬁnds thatΔ+
ABCD =√
1
PLABCD − DQ A LQBDC + 2χQ A R B LQRDC
2√1++ 2χQ A P C LQBDP − AB ∇CD R(x) + Σ+R D φ+
ABCR + ΞTABCD .
2

350

Hyperbolic reductions

Given the above expressions for ΔABCD and Δ+
ABCD , suitable symmetric
hyperbolic evolution equations for the independent components of the ﬁelds
Φ(ABCD) , ΦAB and Φ can be found from the combinations
Δ(ABCD) + Δ+
(ABCD) = 0,

(13.34a)Q
Q
Δ+
Q (CD) − ΔQ (CD) = 0,
Q P
ΔQ Q P P + Δ+
= 0,
Q P

(13.34b)(13.34c)while constraint equations arise fromΔABCD − Δ+
ABCD = 0,
Q
Q
Δ+Q (CD) + ΔQ (CD) = 0,
Q P
ΔQ Q P P − Δ+
= 0.
Q P

The principal parts of Equations (13.34a)–(13.34c) are given, respectively, byPΦ(ABCD) − D(AB ΦCD) ,1
PΦAB + 2DP Q ΦP QAB − DAB Φ,
3
PΦ + DP Q ΦP Q .
The above expressions imply a symmetric hyperbolic system for the independent
components of the ﬁelds Φ(ABCD) , ΦAB and Φ. The explicit form of this system
will not be required in the subsequent discussion but can be readily computed.
13.2.4 The hyperbolic reduction of the Bianchi equation
This section discusses the hyperbolic reduction of the spinorial Bianchi identity.
This procedure leads to evolution equations for the components of the rescaled
Weyl spinor and is completely analogous to that for the Maxwell equations;
see Section 13.1.1. In particular, no gauge source functions are required for thissubsystem.
The spinorial Bianchi equation is encoded in the zero quantity
ΛA BCD ≡ ∇Q A φBCDQ + TCDBA .
In the following it will be convenient to work with a space spinor version of thiszero quantity, namely,
ΛABCD ≡ ∇Q A φBCDQ + TCDBA ,

TCDBA ≡ τA A TCDBA .

Using the decomposition (13.5) one can compute that1
ΛABCD = − PφABCD + DQ A φBCDQ + TCDBA .
2(13.35)

13.2 Hyperbolic reductions using gauge source functions

351

Suitable evolution equations are obtained from the above expression byconsidering−2Λ(ABCD) = PφABCD − 2DQ (A φBCD)Q + T(ABCD) = 0.

(13.36)In what follows, this system of evolution equations will be known as thestandard system. It gives rise to ﬁve independent equations for the ﬁve
independent components of φABCD . Contracting the indices A and B in
Equation (13.35) one obtains
ΛCD ≡ ΛQ QCD = DP Q φP QCD + TCDQ Q = 0,
the so-called Bianchi constraints. As in the case of the other constraint
equations discussed in the previous sections, the Bianchi constraints may containderivatives in the time direction.
The hyperbolicity of the standard system
The overall structure of Equation (13.36) suggests that it should imply asymmetric hyperbolic system. In analogy to the Maxwell equations, one considersa slightly modiﬁed version of Equation (13.36) given by
4−2Λ(ABCD) = 0.
A+B+C +D
The principal part of this equation can be written in matricial form as⎛τ μ + 2e01 μ⎜ 2e11 μ⎜μ
A ∂μ φ ≡ ⎜
0
⎜
⎝
0
0
⎛ ⎞
φ0
⎜ φ1 ⎟⎜ ⎟⎟× ∂μ ⎜⎜ φ2 ⎟ ,⎝ φ3 ⎠
φ4−2e00 μ4τ μ + 4e01 μ
6e11 μ000−6e00 μ6τ μ6e11 μ000−6e00 μ4τ μ − 4e01 μ
2e11 μ

⎞
0
⎟
0
⎟
⎟
0
⎟
μ ⎠
−2e00
τ μ − 2e01 μwith
φ0 ≡ φ0000 ,φ1 ≡ φ0001 ,φ2 ≡ φ0011 ,φ3 ≡ φ0111 ,φ4 ≡ φ1111 .

Using the reality conditions satisﬁed by the vectors eAB , it follows that thematrices of the system are Hermitian. Moreover, one has that Aμ τμ is positivedeﬁnite. Thus, the standard evolution system implies a symmetric hyperbolic
system for the independent components of φABCD . The characteristic matrix ofthe system is given by352

Hyperbolic reductions





2
det(Aμ ξμ ) = 36 τ μ ξμ g νλ ξν ξλ τ ρ τ σ + g ρσ ξρ ξσ .
3Thus, g-null hypersurfaces are characteristics of the standard system.
13.2.5 The hyperbolic reduction of the equations for the conformal
factor and its concomitants
Finally, one requires evolution equations for the conformal factor Ξ and its
concomitants ΣAA and s. The relevant zero quantities are given by
QAA ≡ ΣAA − ∇AA Ξ,(13.37a)
1 3
ZAA BB  ≡ ∇AA ΣBB  + ΞLAA BB  − sAB A B  − Ξ TAA BB  , (13.37b)2
1 2 CC CCZAA ≡ ∇AA s + LAA CC  ∇Ξ− Ξ ∇ΞTAA CC  .
(13.37c)
2
Their space spinor counterparts are deﬁned by

QAB ≡ τB A QAA ,







ZABCD ≡ τB A τD C ZAA CC  ,ZAB ≡ τB A ZAA .

It is also convenient to make use of the splitΣAB ≡ τB A ΣAA =1
AB Σ + Σ(AB) ,2Σ ≡ ΣQ Q .

From the condition QAB = 0 one obtains the equationsPΞ = Σ,DAB Σ = Σ(AB) ,which are, respectively, an evolution equation for Ξ and a constraint equation.
Next, using the identity

 √



τB A τD C ∇AA ΣCC  = ∇AB τD C ΣCC  − 2ΣCP χAB P D
and the split of ∇AB it follows thatZABCD =1
1
1
AB CD PΣ + AB PΣ(CD) + CD DAB Σ + DAB Σ(CD)
4
2
2
√
1
P
+ √ χABCD − 2ΣCP χAB D
2
1+ ΞLABCD − sAB CD − TABCD .
2

Evolution equations for Σ and Σ(AB) are obtained from
2ZAB AB = 0,

ZA A (CD) = 0,

13.2 Hyperbolic reductions using gauge source functions

353

where
√
√
2χAB AB Σ − 2 2χABP B Σ(AP ) + ΞLAB AB − 4s,√
1
ZA A (CD) = PΣ(CD) + √ χA A (CD) − 2Σ(C|P χA AP |D)2
1+ ΞLA A (CD) − Ξ3 TA A (CD) .
2
2ZAB AB = PΣ +The corresponding constraints arise from
Z(ABCD) = 0,

Z(AB)C C = 0,

with
√
1
Z(ABCD) = D(AB ΣCD) + √ χ(ABCD) Σ − 2Σ(C|P | χ|AB P D)2
1 3
+ ΞL(ABCD) − Ξ T(ABCD) ,2√
1
Z(AB)C C = DAB Σ + √ χ(AB)C C Σ − 2ΣP Q χ(AB) P Q
2
1C+ ΞL(AB)C − Ξ3 T(AB)C C .
2Finally, similar calculations lead to the expression
ZAB =

1
1
AB Ps + DAB s − LABC C Σ + LABCD ΣCD2
2
1
1+ Ξ2 TABC C − Ξ2 ΣCD TABCD .
4
2

The evolution and constraint equations for s are then given, respectively, by
ZA A = 0,

Z(AB) = 0,

with
1
1
ZA A = Ps − LA A C C + LA A CD ΣCD + Ξ2 ΣTA A C C2
4
1− Ξ2 ΣCD TA A CD ,2
1
1
Z(AB) = DAB s − L(AB)C C Σ + L(AB)CD ΣCD + Ξ2 ΣT(AB)C C
2
4
1− Ξ2 ΣCD T(AB)CD .
2
Remark. It should be observed that all the evolution equations obtained inthis section are transport equations – that is, they involve only the directional
derivative P. Accordingly the characteristic polynomial of each of them is justτ μ ξμ .

354Hyperbolic reductions
13.3 The subsidiary equations for the standard
conformal ﬁeld equations

After having discussed a set of evolution equations implied by the conformal
ﬁeld Equations (8.38a) and (8.38b), one is now in the position of analysing
the construction of the associated subsidiary system. The subsidiary equations
constitute a system of evolution equations for the zero quantities encoding theconformal ﬁeld equations. To prove the propagation of the constraints it
is necessary that these subsidiary evolution equations are homogeneous in thevarious zero quantities. If this is the case, then Corollary 12.1 implies a unique
vanishing solution to the subsidiary equations if the zero quantities are zero initially. The construction of the subsidiary system involves lengthy computations,parts of which are best carried out with spinorial expressions, while others are
more conveniently described in tensorial terms. The basic strategy behind theanalysis can be understood by ﬁrst discussing some model equations.
General setup
The general setup for the construction of the subsidiary equations for theconformal ﬁeld equations is similar to the one for the construction of the evolution
equations: one works in an open subset U ⊂ M of the unphysical spacetime
manifold; vector and spinor bases are introduced in a similar manner. The
key diﬀerence lies in the fact that the covariant derivative ∇ is, a priori, notassumed to be the Levi-Civita connection of the metric g. Thus, when considering
the commutator of covariant derivatives, one has to make use of the general
expression involving a non-vanishing torsion tensor. This is because the torsion
tensor is, in itself, a zero quantity of the conformal ﬁeld equations. On similar
grounds, one cannot regard the algebraic and geometric curvatures as being equal
to each other.
13.3.1 Hyperbolic reduction of model equations
The construction of a system of subsidiary equations for the conformal Einsteinequations leads to spinorial equations whose tensorial counterparts are of one of
the following forms∇[a Mb]K = NabK ,(13.38a)∇[a Pbc]L = QabcL ,(13.38b)where MaK and PabL are some zero quantities with
NabK = N[ab]K ,PabL = P[ab]L ,QabcL = Q[abc]L ,and K and L denote an arbitrary string of indices.
Equations (13.38a) and (13.38b) arise from the following observations concerning diﬀerential forms; see the Appendix to this chapter for a brief discussion

13.3 The subsidiary equations for the standard conformal ﬁeld equations

355

on this and related notions. The ﬁelds MaK and PabK can be regarded as thecomponents, respectively, of the 1-form and 2-form
M K ≡ MaK ω a ,P L ≡ PabL ω a ∧ ω b .

Accordingly, Equations (13.38a) and (13.38b) can be written asdM K = NabK ω a ∧ ω b ,dP L = QabcL ω a ∧ ω b ∧ ω c .

If τ denotes a timelike vector ﬁeld, then the Lie derivatives of M K and P L along
the direction of τ are given by the so-called Cartan’s formula£τ M K = iτ dM K + d(iτ M K ),£τ P L = iτ dP L + d(iτ P L ),where iτ denotes the operation of contraction between the vector τ and adiﬀerential form; see Frankel (2003). In terms of this notation the evolution
equations are given, respectively, byiτ M K = 0,iτ P L = 0,

so that£τ M K = iτ dM K ,£τ P L = iτ dP L .

The latter can be read as suitable evolution equations for the zero quantities
MaK and PabL . Their frame component version is given by∇[0 Mb]K = N0bK ,∇[0 Pbc]L = Q0bcL .

Detailed analysis of the ﬁrst model equation
The spinorial analogue of Equation (13.38a) is given by∇AA MBB  K − ∇BB  MAA K = 2NAA BB  K .
Exploiting the antisymmetry one obtains the equivalent expression
∇(A Q MB)Q K = NA Q BQ K ,NA Q BQ K = NB Q AQ K .


Deﬁning the space spinor counterpart MABK ≡ τB A MAA K and using thedeﬁnition of the spinor χABCD together with the decomposition (13.5) of ∇AB
one obtains the expression√
PM(AB)K + 2D(A P MB)P K + 2 2χ(A|Q| P Q MB)P K = NA Q BQ K .
Finally, assuming that the evolution equations implied by the zero quantity
MAA K are given by MQ Q K = 0, it follows that MBP K = M(BP )K and,moreover, that√

PM(AB)K + DA P M(BP )K + DB P M(AP )K + 2 2χ(A|Q| P Q MB)P K = NA Q BQ K .

356Hyperbolic reductions
This last expression is a suitable evolution equation for M(AB)K if NA Q BQ K
can be expressed as a linear combination of zero quantities. This computation depends on the particular structure of the conformal equation underconsideration.
Detailed analysis of the second model equation
In what follows, let PAA BB  L and QAA BB  CC  L denote, respectively, thespinorial counterparts of the ﬁelds PabL and QabcL . The spinorial counterpartof Equation (13.38b) can be conveniently written using the spinorial counterpartof the volume form as












AA BB CC DD ∇AA PBB  CC  L = AA BB CC DD QAA BB  CC  L .

(13.39)A convenient way of obtaining the space spinor version of this last equation is
to consider, alternatively, the expression
EF CDGH AB ∇EF PCDGHL ,where, following standard conventions, one deﬁnes

PCDGHL ≡ τD C τH G PCC  GG L ,








EF CDGHAB ≡ τF F τD D τH H τB B EF  CD GH  AB  .
A short computation using the expression of the volume form in terms of -spinors
yieldsEF CDGHAB = i(EG CA F B DH − EA CG F H DB ).
Now, exploiting the symmetries of PCDGHL one can write∗CG ,PCDGHL = PCGL DH + PDHL

where
PCGL ≡

1
PCQG Q L ,
2

∗
PDHL
≡

1
PQD Q HL .
2

A calculation shows that the above expressions lead toEF CDGH AB ∇EF PCDGHL∗= 2i ∇A Q PBQL− ∇Q B PAQL∗∗
= iP (PABL+ PABL ) + 2iDQ A PBQL
− 2iDQ B PAQL .

If the evolution equations associated with the zero quantity PABCDL are givenby the condition∗= 0,
PABL − PABL

13.3 The subsidiary equations for the standard conformal ﬁeld equations357

it follows thati
PPABL = − EF CDGH (AB) ∇EF PCDGHL .
2
It can be veriﬁed that the expression one obtains by working directly with theleft-hand side of Equation (13.39) diﬀers from the above expression by homogeneous terms involving PCDGHL and χABCD . To complete the constructionof a suitable subsidiary equation for PABCDL it is necessary to show that theright-hand side of Equation (13.39) can be expressed as a linear combination ofzero quantities – this computation is speciﬁc to each zero quantity.
13.3.2 The subsidiary equations for the equations governing theconformal factor and its concomitants
The zero quantities QAA , ZAA BB  and ZAA – see Equations (13.37a)–(13.37c)– lead to subsidiary equations which fall into the class described by the model
Equation (13.38a). Accordingly, one will have suitable subsidiary evolution
equations for the zero quantities QAA , ZAA BB  and ZAA if the derivatives
∇(A Q QB)Q ,∇(A Q ZB)Q CC  ,∇(A Q ZB)Q ,can be expressed as linear combinations of other zero quantities.
The subsidiary equation for QAAA direct computation using the deﬁnition of QAA shows that∇(A Q QB)Q = ∇(A Q ΣB)Q − ΣA QQ B ΣQQ ,where the deﬁnition of the torsion spinor – see Equation (8.35a) – has been used
to write∇(A Q ∇B)Q Ξ = ΣA QQ B ΣQQ ,ΣA QQ B ≡1
ΣA P QQ BP  .
2Finally, using the deﬁnition of the zero quantity ZAA BB  one can eliminate theterm ∇(A Q ΣB)Q . Observing that L(A Q B)Q = T(A Q B)Q = 0 – as these are
the spinorial counterparts of symmetric rank-2 tensors – one ﬁnds∇(A Q ΣB)Q = Z(A Q B)Q ,so that one concludes that∇(A Q QB)Q = Z(A Q B)Q − ΣA QQ B ΣQQ ,which is a linear combination of zero quantities as required.

358

Hyperbolic reductions
The subsidiary equation for ZAA BB 

A direct computation starting from the deﬁnition of ZAA BB  yields theexpression








∇(A Q ZB)Q CC  = ∇(A Q ∇B)Q ΣCC  + Σ(A Q LB)Q CC  + Ξ∇(A Q LB)Q CC 3
+ C(A ∇B)C  s − Ξ2 Σ(A Q TB)Q CC  − Ξ3 ∇(A Q TB)Q CC  .
2
Using the commutator∇AA ∇BB  ΣCC  − ∇BB  ∇AA ΣCC = −RP CAA BB  ΣP C  − R̄P C  A ABB  ΣCP − ΣAA QQ BB  ∇QQ ΣCC  ,one ﬁnds that∇(A Q ∇B)Q ΣCC  = −RP C(AB) ΣP C  − R̄P C  (AB) ΣCP − ΣA QQ B ∇QQ ΣCC  ,where
RABCD ≡
1
RABCQ D Q ,2

R̄A B  CD ≡


1
R̄A B  Q CD Q .
2Using the deﬁnitions of the zero quantities ΔCDBB  and ZAA to eliminate,
respectively, ∇(A Q LB)Q CC  and ∇BC  s, one obtains∇(A Q ZB)Q CC  = −RP C(AB) ΣP C  − R̄P C  (AB) ΣCP  − ΣA QQ B ∇QQ ΣCC + Σ(A Q LB)Q CC  + ΞΔABCC  − ΞΣQ C  φABCQ− Ξ2 TABCC  + C(A ZB)C  − C(A LB)C  QQ ΣQQ1
3− Ξ2 ΣQQ C(A TB)C  QQ − Ξ2 Σ(A Q TB)Q CC 22
1
− Ξ3 ∇(A Q TB)Q CC  .
2Next, one uses the zero quantity ΞABCC  DD to eliminate the geometriccurvature terms RP C(AB) and R̄P C  (AB) . Taking into account the expressionof the algebraic curvature in terms of the Schouten tensor and the rescaled Weyltensor one obtains∇(A Q ZB)Q CC  = −ΞP C(AB) ΣP C  − Ξ̄P C  (AB) ΣCP  − ΣA QQ B ∇QQ ΣCC + ΞΔABCC  − Ξ2 TABCC  + C(A ZB)C 
1
3− Ξ2 ΣQQ C(A TB)C  QQ − Ξ2 Σ(A Q TB)Q CC 22
1
− Ξ3 ∇(A Q TB)Q CC  ,2

13.3 The subsidiary equations for the standard conformal ﬁeld equations

359

where


1
1
ΞABCQ D Q ,Ξ̄A B  CD ≡ Ξ̄A B  Q CD Q .
2
2
Finally, observing that the deﬁnition of the rescaled Cotton tensor implies that
TABCC  = − 21 Ξ∇(A Q TB)Q CC  and exploiting the trace-freeness of the energymomentum tensor one ends up with the expression

ΞABCD ≡∇(A Q ZB)Q CC  = −ΞP C(AB) ΣP C  − Ξ̄P C  (AB) ΣCP  − ΣA QQ B ∇QQ ΣCC + ΞΔABCC  + C(A ZB)C  ,which, as required, is a linear combination of zero quantities.
The subsidiary equation for ZAAIn this case one needs to evaluate ∇(A Q ZB)Q . Making use of the deﬁnition of
ZAA one ﬁnds that












∇(A Q ZB)Q = ∇(A Q ∇B)Q s + ∇(A Q LB)Q P P  ΣP P + ∇(A Q ΣP P LB)Q P P 




1
− ΞΣ(A Q ΣP P TB)Q P P  − Ξ2 ∇(A Q ΣP P TB)Q P P 
2
1 2 PPQ− Ξ Σ∇(A TB)Q P P  .
2
Using the deﬁnition of the torsion tensor in the form∇(A Q ∇B)Q s = ΣA QQ B ∇QQ s,
and the deﬁnitions of ΔABCC  and ZAA BB  to eliminate, respectively,∇(A Q LB)Q P P  and ∇A Q ΣP P one obtains – after some simpliﬁcations
involving the symmetries of LAA BB  and TAA BB  –










∇(A Q ZB)Q = Σ(A QQ B) ∇QQ s + ΔABP P  ΣP P + Z(A Q P P LB)Q P P 1− Ξ2 Z(A Q P P TB)Q P P  .
2This expresion is a linear combination of zero quantities.
13.3.3 Subsidiary equation for the no-torsion condition
Following the general discussion of Section 13.3.1, one deﬁnesΣABCD a ≡ τB A τD C ΣAA QQ CC  eQQ a .
One can write
ΣABCD a = −ΣAC a BD − Σ+ BD a AC ,so that, if the evolution equation ΣAB a − Σ+ AB a = 0 holds, theniPΣAB a = − ∇EF ΣCDGH a EF CDGH (AB) .
2(13.40)360

Hyperbolic reductions

To conclude the argument one needs to express the right-hand side of the above
equation as a linear combination of zero quantities. To this end, one makes useof the ﬁrst Bianchi identity (2.10) to write∇[a Σb d c] = −Ξd [cab] − ρd [cab] − Σ[a e b Σc] d e ,where the zero quantity Ξd cab ≡ Rd cab − ρd cab . By construction, the algebraic
curvature has the same algebraic symmetries as the Riemann tensor of a LeviCivita connection so that, in particular, ρd [cab] = 0 and one has that∇[a Σb d c] = −Ξd [cab] − Σ[a e b Σc] d e .
This expression shows that the right-hand side of Equation (13.40) can be written
as a linear combination of zero quantities.
13.3.4 Subsidiary equation for the Ricci identity
It follows from the general discussion of Section 13.3.1 that, if the evolution
equations ΞABCD − Ξ∗ABCD = 0 are satisﬁed, theniPΞABCD = − ∇EF ΞCDGH EF CDGH (AB) .
2(13.41)

To express the right-hand side of this last equation as a linear combination ofzero quantities one makes use of the second Bianchi identity (2.11) to obtainf abc ∇[a Ξd |e|bc] = f abc ∇[a Rd |e|bc] − f abc ∇[a ρd |e|bc]= −f abc Σ[a g b Rd |e|c]g − f abc ∇[a ρd |e|bc] .

(13.42)

The ﬁrst term in the right-hand side of the last equation already has the desired
form. The second term needs to be examined in more detail. One considers
f abc ∇[a ρd |e|bc] = f abc ∇a ρd ebc= Ξf abc ∇a dd ebc + f abc ∇a Ξdd ebc + 2f abc Seb dh ∇a Lch ,where in the last line the expression of the algebraic curvature in terms of the
Weyl tensor and the Schouten tensor has been used. Now, a computation using
the properties of the Hodge dual and the deﬁnition of the zero quantity Λabcshows that
f abc ∇a dd ebc = −f abc ∇a ∗ d∗d ebc = −2∇a ∗ d∗∗d ef a
= 2∇a ∗ dd ef a = 2∇a d∗ f ad e
= e dgh ∇a da f gh = e dgh (Λf gh + Tf gh ).

(13.43)Using the above expression together with the deﬁnition of the zero quantities
Δabc and Qa to eliminate ∇[a Lc]f and ∇a Ξ, respectively, one ﬁnds that
f abc ∇[a ρd |e|bc] = f abc Qa dd ebc + Ξe dgh Λf gh + f abc Seb dh Δach .

13.3 The subsidiary equations for the standard conformal ﬁeld equations

361

Substituting this last expression into Equation (13.42) one obtains the required
expression for ∇[a Ξd |e|bc] as a linear combination of zero quantities. The spinorial
counterpart of this expression diﬀers from the right-hand side of Equation (13.41)
by terms homogeneous in zero quantities involving the spinor χABCD .
13.3.5 The subsidiary equations for the Cotton equation
Applying the general discussion of Section 13.3.1 to the zero quantity
ΔAA BB  CC  associated with the unphysical Bianchi identity leads to theexpression

1  +
Q
P ΔABKL + ΔABKL + DQ A Δ+BQKL − D B ΔAQKL
2i= − EF CDGH AB ∇EF ΔCDGHKL .
2(13.44)

To make use of the evolution equations in the above expression it is observedthat
1
1
ΔABCD = Δ(ABCD) + hABCD ΔP Q P Q + CD ΔABQ Q .
32
Thus, using the evolution equations for the various components of the Schouten
tensor one obtainsi
PΔ(ABKL) = − ∇EF ΔCDGH(AB EF CDGH KL) ,2i
PΔP Q P Q = − EF CDGHKL ∇EF ΔCDGHKL ,2i
PΔABQ Q = − EF CDGH AB ∇EF ΔCDGHQ Q .
2To analyse the right-hand sides of the above equations it is more convenient
to analyse AA BB CC DD ∇BB  ΔCC  DD EE  . This expression diﬀers from theright-hand side of Equation (13.44) by terms involving the spinor χABCD . For
conciseness, the analysis is carried out using tensorial notation. One has thatf ecd ∇e Δcdb = f ecd 2∇e ∇[c Ld]b − ∇e Σa da bcd − Σa ∇e da bcd
(13.45)− ∇e ΞTcdb − Ξ∇e Tcdb .
The ﬁrst term on the right-hand side of the above equation is manipulated using
the commutator of covariant derivatives by observing that2f ecd ∇e ∇[c Ld]b = 2f ecd ∇[e ∇c] Ldb= −2f ecd 2Rs (d|ec| Lb)s − Σe s c ∇s Ldb= −2f ecd 2Ξs (d|ec| Lb)s + f ecd ρs bec Lds + Σe s c ∇s Ldb ,where in the third line the identity ρa [bcd] = 0 has been used. The second
term in the last equation does not contain zero quantities. The third term in362Hyperbolic reductions

Equation (13.45) is now cast in a suitable form using the properties of Hodge
dual; compare the analogous argument leading to (13.43). One ﬁnds thatf ecd ∇e dabcd = −ab gh (Λf gh + Tf gh ).
Substituting the above two identities into Equation (13.45) and using the zero
quantity Zab to eliminate ∇a Σb one obtains
2f ecd ∇e ∇[c Ld]b
= −4f ecd Ξs (d|ec| Lb)s − 2Σe s c ∇s Lf b
− Σa ab gh Λf gh − f ecd Zea da bcd




1 3
Ξ Tea da bcd + ∇e ΞTcdb − Ξ∇e Tcdb − ab gh Σa Tf gh ,− f ecd2(13.46)
where the explicit expression of the algebraic curvature has been used to showthat
f ecd Ξda bcd Lea − 2ρs bec Lds = 0.
Expression (13.46) is, up to the matter terms in curly brackets, a linear
combination of zero quantities. Whether the terms in curly brackets can be
expressed as a linear combination of (matter) zero quantities depends on theparticular features of the matter model under consideration.
13.3.6 The subsidiary equations for the Bianchi identity
The construction of the subsidiary equation for the Bianchi identity is similar
to that of the subsidiary equation for the Maxwell equations. In this case therelevant zero quantity is given by
ΛA BCD ≡ ∇Q A φBCDQ + TCDBA ,

for which one computes ∇BB ΛB  BCD in two diﬀerent manners.
First, making use of the space spinor zero quantity ΛABCD ≡ τA A ΛA BCD
one has that∇BB ΛB  BCD = −∇BB τ P B  ΛP BCD= ∇AB ΛABCD − ∇BA τ A A ΛABCD .
Using Equation (13.8) for the derivative of the spinor τAA and the split of ∇AB
one obtains√
PΛCD − 2DAB ΛABCD − 2 2χB P P A ΛABCD = −2∇BB ΛB  BCD ,13.3 The subsidiary equations for the standard conformal ﬁeld equations

363

where it is recalled that ΛCD ≡ ΛQ QCD . Now, a calculation shows that thesymmetry ΛABCD = ΛA(BCD) implies the decomposition
3
ΛABCD = Λ(ABCD) − A(B ΛCD) ,4
so that√
PΛCD − 2DP (C ΛD)P − 2DAB Λ(ABCD) − 2 2χB P P A Λ(ABCD)
3+ √ χB P P A A(B ΛCD) = −2∇BB ΛB  BCD .
(13.47)2
As a second way of evaluating ∇BB ΛB  BCD one makes use of the deﬁnitionof the zero quantity so that∇BB ΛB  BCD = ∇BB ∇Q B  φBCDQ + ∇BB TCDBB  .
The ﬁrst term of the right-hand side is manipulated using the commutator∇AA ∇BB  φCDEF − ∇BB  ∇AA φCDEF= −RS CAA BB  φSDEF − RS DAA BB  φSCEF − RS EAA BB  φSCDF
− RS F AA BB  φSCDE + ΣAA P P BB  ∇P P  φCDEF .


Observe that the torsion ΣAA P P BB  , being one of the unknowns in thesubsidiary system, needs to be included in the commutator. Also, the curvatureterms in the above expression are understood to be those of the geometriccurvature. Contracting the expression of the commutator leads to










2∇BB ∇Q B  φBCDQ = −RS C BA Q A φSDBQ − RS D BA Q A φSCBQ− RS B BA Q A φSCDQ − RS Q BA Q A φSCDB+ ΣAA SS Q A ∇SS  φCDAQ .
Using the zero quantity
ΞC DAA BB  = RC DAA BB  − ρC DAA BB  ,to eliminate the geometric curvature and the decomposition (13.9) one obtains
an expression which is homogenous in zero quantities:










2∇BB ∇Q B  φBCDQ = −ΞS C BA Q A φSDBQ − ΞS D BA Q A φSCBQ− ΞS B BA Q A φSCDQ − ΞS Q BA Q A φSCDB+ ΣAA SS Q A ∇SS  φCDAQ .

(13.48)

In particular, all the terms coming from the algebraic curvature cancel out.

364

Hyperbolic reductions

Combining Equations (13.47) and (13.48) one obtains the required subsidiaryequation. Namely, one has that if Λ(ABCD) = 0, then3
PΛQ QCD − 2DP (C ΛQ D)P Q + √ χB P P A A(B ΛQ CD)Q
2= ΞS C BA Q A φSDBQ + ΞS D BA Q A φSCBQ+ ΞS B BA Q A φSCDQ + ΞS Q BA Q A φSCDB− 2ΣAA SS Q A ∇SS  φCDAQ − 2∇BB TCDBB  ,which is homogeneous in zero quantities if the matter term ∇BB TCDBB  canbe expressed, in turn, as a homogeneous expression of matter zero quantities.
Alternatively, one can perform the computation with tensorial objects. In thiscase one looks at∇b Λbcd = ∇b ∇a da bcd − ∇b Tcdb .
Again, using the properties of the Hodge dual one can write∇b ∇a da bcd = −∇a ∇b dabcd= ∇a ∇b∗ d∗abcd =1 abef gh cd ∇b ∇a def gh4
1 abef gh  s1 cd R eab df sgh + Rs gab dhsef − Σa s b ∇s def gh
42

1 abef gh  s
1 s
s
= 
 cd Ξ eab df sgh + Ξ gab dhsef − Σa b ∇s def gh .
42=

Hence, one concludes that ∇b Λbcd , except for the matter term ∇b Tcdb can be
written as a linear combination of zero quantities.
13.3.7 Summary
In most applications, the detailed form of the evolution and subsidiary equations
is not required; general structural properties suﬃce. These properties are
summarised in the following propositions.
It is convenient to group the independent components of the unknowns
appearing in the spinorial formulation of the conformal ﬁeld equations in thefollowing manner:σ independent components of Ξ, ΣAA , s;
υ independent components of eμAA , ΓAA BC , ΦAA BB  ;φ independent components of φABCD ;ϕ independent components of matter ﬁelds.
Moreover, let e and Γ denote, respectively, the independent components of theframe components and the connection coeﬃcients. In terms of these objects one
has the following:13.3 The subsidiary equations for the standard conformal ﬁeld equations

365

Proposition 13.1 (properties of the conformal evolution equations)
Given arbitrary smooth gauge source functions
F a (x),FAB (x),R(x),such that∇QQ eQQ a = F a (x),∇QQ ΓQQ AB = FAB (x),
1
∇QQ LQQ BB  = ∇BB  R(x),6and assuming that the components of the matter tensors Tab and Tabc can be
written in such a way that they do not contain derivatives of the matter ﬁelds,
then the conformal Einstein ﬁeld equations
Qa = 0,Zab = 0,Σa c b = 0,

Za = 0,Δabc = 0,Ξc dab = 0,

Λabc = 0,

imply a symmetric hyperbolic system of equations for the independent componentsof the geometric ﬁelds (σ, υ, φ) of the form

I + A0 (e) ∂τ φ + Aα (e)∂α φ = B(Γ)φ + C(σ, υ, ϕ),
I + D0 (e) ∂τ υ + Dμ (e)∂μ υ = E(Γ)υ + F(σ, υ, φ, ϕ),∂τ σ = G(Γ)σ + H(σ, υ, ϕ),where I denotes the identity matrix of the required dimensions,
Aμ (e),Dμ (e)

are smooth matrix-valued functions of the components of the frame components,
B(Γ),

E(Γ),

G(Γ)

are smooth matrix-valued functions of the connection coeﬃcients andC(σ, υ, ϕ),F(σ, υ, φ, ϕ),H(σ, υ, ϕ)

are smooth vector-valued functions with polynomial dependence on their arguments. The characteristics of this system satisfy a characteristic polynomial
involving factors of the form


2 ρσ
μ
νλ
ρ σ
g ξν ξλ ,τ τ + gξρ ξσ .
τ ξμ ,
3
Remarks(i) In the presence of matter, the symmetric hyperbolic system given in theabove proposition needs to the supplemented by a symmetric hyperbolic
system for the matter ﬁelds. As the rescaled Cotton tensor Tabc (and hence
also the spinor TABCC  ) is made up of derivatives of the energy-momentum
tensor, the matter evolution equations will need to include equations for thematter ﬁeld derivatives appearing in the geometric evolution equations.

366

Hyperbolic reductions(ii) The choice of the gauge source functions is dictated by the particular analysis
under consideration.
With regards to the subsidiary system one has the following:
Proposition 13.2 (properties of the subsidiary evolution system)
Assume that the evolution equations implied by the conformal Einstein ﬁeld
equations are satisﬁed and that the energy-momentum tensor Tab is such thatthe quantities
Nbf ≡ fecd

Mcd ≡ ∇b Tcdb ,1 3
Ξ Tea da bcd + ∇e ΞTcdb − Ξ∇e Tcdb
2− ab gh Σa Tf gh ,can be written as homogeneous expressions of the geometric and matter zero
quantities. Then the zero quantities encoding the constraint equations implied by
the conformal Einstein equations under the hyperbolic reduction procedure leading
to Proposition 13.1 satisfy a symmetric hyperbolic system which is a homogeneous
expression of zero quantities.
13.4 Hyperbolic reductions using conformal Gaussian systems
This section discusses a hyperbolic reduction procedure based on the propertiesof congruences of conformal geodesics. The approach discussed in this section
makes use of the formulation of the conformal ﬁeld equations in terms of Weyl
connections – the so-called extended conformal ﬁeld equations. As will be seen,
this procedure leads to simpler evolution equations than the ones obtained by
the reduction procedure discussed in Section 13.2.
For conciseness of the presentation, the discussion in the rest of this section isrestricted to the vacuum case.
13.4.1 Basic set up
In what follows, it is assumed one has a region U of a spacetime (M̃, g̃) whichis covered by a congruence of conformal geodesics (ẋ(τ ), β(τ )). For convenience,the vector ﬁeld tangent to the congruence will be denoted by τ . As discussed in
Section 5.5, a canonical representative g of the conformal class [g̃] is singled out
by the requirementg(τ , τ ) = 1,
so that g = Θ2 g̃ where the conformal factor Θ satisﬁes a third-order ordinary
diﬀerential equation along the congruence of conformal geodesics; see Equation (5.53b). In the case of a vacuum spacetime this equation can be explicitly
solved yielding a formula for Θ as a quadratic polynomial in the parameter τ .
The conformal factor is completely determined by the three coeﬃcients Θ , Θ̇ ,Θ̈ speciﬁed, say, on an initial hypersurface S̃ .

13.4 Hyperbolic reductions using conformal Gaussian systems

367

In the following, {ea } will denote a g-orthonormal frame which is Weylpropagated along the conformal geodesics and such that e0 = τ . As discussed
in Section 5.5, to every congruence of conformal geodesics one can associate aˆ This Weyl connection satisﬁes the relations
Weyl connection ∇.
ˆ τ ea = 0,∇L̂(τ , ·) = 0,

ˆ see Equation (5.41). In terms of
where L̂ denotes the Schouten tensor of ∇;frame components, the above conditions can be rewritten asΓ̂0 a b = 0,L̂0a = 0.

(13.49)In particular, it follows that the covector f which deﬁnes the Weyl connection
ˆ satisﬁes∇
f0 = 0.
The gauge choice can be reﬁned further by choosing the parameter of theconformal geodesics τ as the time coordinate. Thus, one has the additional gauge
conditione0 = ∂ τ ,so thate 0 μ = δ0 μ .

(13.50)In most applications, initial data for the congruence of conformal geodesics will
be prescribed on the initial hypersurface S . On S choose some local coordinates(xα ). Assuming that each curve of the congruence of conformal geodesics
intersects S only once, one can extend coordinates on S oﬀ the hypersurface byrequiring them to be constant along the conformal geodesic which intersects S
at the point with coordinates (xα ); see Figure 13.1. The spacetime coordinates(τ, xα ) one obtains by this procedure are known as conformal Gaussian
coordinates. More generally, the collection of the conformal factor Θ, Weylpropagated frame vectors {ea } and coordinates (τ, xα ) extended oﬀ some initial
hypersurface S using a congruence of conformal geodesics will be known as aconformal Gaussian gauge system.
(τ,xα)
conformal geodesic

S(0,xα)

Figure 13.1 Schematic depiction of the construction of conformal Gaussian
coordinates. The coordinates (xα ) of a point p ∈ S are propagated oﬀ thehypersurface along the unique conformal geodesic passing through p; see themain text for further details.

368Hyperbolic reductions

Remarks(i) The speciﬁc choice of the data for the conformal Gaussian gauge system
on an initial hypersurface S is dictated by the particular geometric setting
under consideration.
(ii) The discussion in this section can be adapted, with minor changes, to thecase of a congruence of so-called conformal curves in non-vacuum spacetimes;
see, for example, Lübbe and Valiente Kroon (2012) for further details.
13.4.2 Hyperbolic reduction of a model equation
The general ideas behind the procedure of hyperbolic reduction using conformal
Gaussian systems are best illustrated with a model equation. All the extended
conformal equations, except for the unphysical Bianchi identity, are of theformˆ b MaK = NabK ,ˆ a MbK − ∇∇(13.51)

where MaK and NabK = N[ab]K denote the components of some tensorial
quantities with respect to the frame {ea } and K denotes an arbitrary set of
tensor indices. To derive an evolution equation along the direction given by the
congruence of curves, one sets a = 0 so that
ˆ b M0K = N0bK ,ˆ 0 MbK − ∇∇or, more explicitly,e0 (MbK ) − eb (M0K ) = N0bK + Γ̂0 c b McK + Γ̂0 L K MbL − Γ̂b c 0 McK − Γ̂b L K M0L .
If the gauge conditions (13.49) are taken into account and coordinates are chosensuch that e0 = ∂ τ , then the above equation reduces to∂τ MbK − eb (M0K ) = N0bK − Γ̂b c 0 McK − Γ̂b L K M0L .

(13.52)This last equation is not a completely satisfactory evolution equation for thecomponents MaK as it does not yield information about ∂τ M0K – notice that by
setting a = b = 0 in (13.51) both sides of the equation vanish as a result of the
skew symmetry of the equation. To read Equation (13.52) as a suitable evolution
equation one needs to know the value of the time component M0K either asa result of symmetries of the tensor MaK or through some gauge condition.
In any of these cases, Equation (13.52) is just a transport equation along
the congruence of conformal curves, and, accordingly, it trivially gives rise to asymmetric hyperbolic subsystem of equations.

13.4 Hyperbolic reductions using conformal Gaussian systems

369

Analysis in terms of spinors
In view of subsequent applications, the properties of the spinorial counterpart ofEquation (13.51) are now analysed. In this case one hasˆ AA MBB  K − ∇ˆ BB  MAA K = NAA BB  K ,∇(13.53)where K denotes an arbitrary string of spinorial indices. In view of its antisymmetry, Equation (13.53) is completely equivalent to the pair of contracted
equationsˆ (A|P  | MB) P  K = 1 N(A|P  |B) P  K ,∇2
1Pˆ P (A MB  ) K = NP (A P B  )K .
∇2

(13.54a)(13.54b)Thus, not unsurprisingly, one has arrived at a situation similar to the one
analysed in Section 13.2. Namely, one has equations containing a symmetrised
spinorial curl. A symmetric hyperbolic system can then be obtained if suitableˆ P P MP P  K is available.
information about the divergence ∇The next step in the procedure consists of introducing the space spinor versionof MAA K , namely, MBB  K = −τ P B  MBP K so thatˆ AA MBB  K = −τP A τQ B  τ R B  ∇ˆ AA MBRK + MBRK ∇ˆ AA τ R B τP A τ Q B ∇√ˆ AP MBQK − 2MBRK χ̂AP R Q ,=∇√ˆ AA τCC  consistent with
where it has been used that 2χ̂ABCD ≡ τB A τD C ∇formula (4.17). From the above identity together with Equations (13.54a) and(13.54b) one obtains√
ˆ (A|P | MB) P K = 1 N(A|P |B) P K − 2M(A R |K| χ̂B)P R P ,∇2√1
ˆ A(P M A Q)K = NA(P A Q)K + 2M A RK χ̂A(P R Q) .
∇2
Using the decomposition
ˆ AB = 1 AB P̂ + D̂AB ,∇2
ˆ AA and D̂AB ≡ τ(A A ∇ˆ B)A – compare Equation (4.16) – andwith P̂ ≡ τ AA ∇writing MABK as

MABK =

1
AB mK + mABK
2

where mK ≡ MQ Q K and mABK ≡ M(AB)K , one obtains
1
1
P̂mABK − D̂AB mK − D̂P (A mB) P K
2
2√
1= − N(A|P |B) P K + 2M(A R |K| χ̂B)P R P
2

370

Hyperbolic reductions

1
1
P̂mP QK − D̂P Q mK + D̂A(P mA Q)K2
2√
1= NA(P A Q)K + 2M A RK χ̂A(P R Q) .
2
Taking linear combinations of the latter equations one ﬁnally arrives at
P̂mABK − D̂AB mK = EABK ,(13.55a)D̂P (A m(13.55b)PB)K = CABK ,where EABK and CABK are some expressions not involving derivatives of
MABK whose precise form is not relevant for the subsequent discussion.
Equation (13.55a) can be read as an evolution equation for the spatial components
mABK if the time component mK is known. Observe that the reduction proceduredoes not produce an equivalent equation for mK consistent with the discussionof Equation (13.51).
13.4.3 The evolution equations in the frame formalism
To obtain some intuition into the structural properties of the evolution equations,
it is convenient to look ﬁrst at the form of the equations in a tensor frame
formalism. Accordingly, one considers the vacuum extended conformal ﬁeld
equations as given in Section 8.4.1; see Equations (8.46).
The required evolution equations for the frame components, connection
coeﬃcients and components of the Schouten tensor are obtained from theconditions
Σ̂0b = 0,Ξ̂c d0b = 0,Δ̂0bc = 0.

In particular, the evolution equation for the covector f deﬁning the Weyl
connection is given byΞ̂c c0b = 0.
Using the deﬁnitions of the zero quantities given in Equations (8.44a)–(8.44c),recalling that in the vacuum case Tcdb = 0, and making use of the gauge
conditions (13.49) and (13.50), one obtains the evolution equations∂τ eb μ = −Γ̂b f 0 ef μ ,∂τ Γ̂b c d = −Γ̂f c d Γ̂b f 0 + δ0 c L̂bd + δd c L̂b0 − η0d η f c L̂bf + Θdc d0b ,∂τ L̂bc = −Γ̂b f 0 L̂f c + df df c0b .
These equations contain derivatives only in the τ direction – that is, they aretransport equations along the conformal geodesics.
The evolution equations for the components of the rescaled Weyl tensor are
obtained by resorting to an electric-magnetic decomposition; see Section 11.1.2.
Using Equations (11.9) and (11.10) for the decomposition of a Weyl candidatein the equations

13.4 Hyperbolic reductions using conformal Gaussian systems

371∇a d∗abcd = 0,

∇a dabcd = 0,
one obtains the expressions

Λ∗(b|0|d) = e0 (Ebd ) + Da Bc(b d) ac + 2aa ac (b Bd)c − 3χ(b c Ed)c
− b ac d ef Eac χcf + χEbd = 0,
Λ(b|0|d) = e0 (Bbd ) − Da Ec(b d) ac − 2aa ac (b Ed)c − 3χa (b Bd)a− b ac d ef Bac χcf + χBbd = 0,
where
hab ≡ gab − τa τb ,

χab = ha c ∇c τb ,

χ = hab χab ,aa ≡ τ b ∇b τa .

The above form of the equations is completely general. In the particular caseof a conformal Gaussian gauge system one has e0 = ∂ τ .
13.4.4 The evolution equations in the spinorial formalism
To discuss the spinorial version of the evolution equations one makes use of theextended conformal ﬁeld equationsΣ̂AA BB  = 0,

Ξ̂C DAA BB  = 0,

Δ̂CC  DD BB  = 0,Λ̂BB  CD = 0,

with the zero quantities as given in (8.53a)–(8.53e). These equations are regarded
as diﬀerential conditions on the ﬁelds
eAA μ ,Γ̂AA BC ,L̂AA BB  ,φABCD .

Moreover, one considers the spinor τ AA – the counterpart of the vector τ , with
normalisation τAA τ AA = 2. In terms of a spinor dyad {A A } adapted to τ AA
one hasτ AA = 0 A 0 A + 1 A 1 A .
In what follows, all spinorial objects will be expressed with respect to this basis.

In particular, the components of τ AA with respect to {A A } will be denoted
by τ AA .
The gauge conditions (13.49) and (13.50) in the spinorial formalism take the
form
√



τ AA Γ̂AA B C = 0,
τ AA L̂AA BB  = 0. (13.56)
τ AA eAA = 2∂ τ ,For future use, it is recalled that the reduced spin Weyl connection coeﬃcientsΓ̂CC  AB can be written in terms of the unphysical Levi-Civita connection
coeﬃcients ΓCC  AB and the covector fAA asΓ̂CC  AB = ΓCC  AB − AC fBC  .

372Hyperbolic reductions

Combining the above with the gauge conditions one obtainsτ CC ΓCC  AB = −τA C fBC (13.57)

and, furthermore, thatΓ̂CC  AB = ΓCC  AB − AC τ QQ ΓQQ P B τ P C  .
In the gauge given by conditions (13.56) the connection coeﬃcients Γ̂CC  AB
can be fully expressed in terms of the coeﬃcients ΓCC  AB , and vice versa.
Comparing Equation (13.57) with the deﬁnition in Equation (4.17), one seesthat the spinor fAA encodes the acceleration of the congruence of conformal
geodesics. In particular, if fAA = 0, then the congruence consists of standard
geodesics and one obtains a Gaussian gauge system.
The reduced symmetric hyperbolic system of evolution equations can be
deduced from the following contractions of the conformal ﬁeld equationsτ AA Σ̂AA P P BB  eP P  μ = 0,

τ CC Ξ̂ABCC  DD = 0,τ(A A Λ̂|A |BCD) = 0.

τ AA Δ̂AA BB  CC  = 0,

Explicitly, for the ﬁrst three equations one has
√




¯
2∂τ eAA μ = − Γ̂AA Q B τ BQ + Γ̂A A Q B  τ QB eQQ μ ,
√




¯
2∂τ Γ̂AA B C = − Γ̂AA P Q Γ̂P Q B C + Γ̂AA P Q Γ̂QP  B C τ QQ


+ L̂AA CQ τ BQ + ΘφB CQA τ Q A ,
√




¯
2∂τ L̂AA BB  = − Γ̂AA P Q L̂P Q BB  + Γ̂A A P Q L̂QP  BB  τ QQ

− dP P (φP AQB P  B  τ Q A + φ̄P  A Q B  P B τA Q ).
Following the same procedure discussed in Section 13.2.4 one ﬁnds, for the
Bianchi identity, thatPφABCD − 2D(A Q φBCD)Q = 0.

(13.58)Observe that this last expression is, for convenience, expressed in terms of theLevi-Civita connection ∇.
The space spinor split of the evolution equations
A more detailed version of the evolution equations is obtained by resorting to
the space spinor formalism, and, in particular, to the split of the connection
coeﬃcients as given in Section 4.3.1.
Following the general strategy behind the space spinor formalism, it isconvenient to deﬁneΓ̂ABCD ≡ τB A Γ̂AA CD ,

ΓABCD ≡ τB A ΓAA CD ,

ΘABCD ≡ τB A τD C L̂AA CC fAB ≡ τB A fAA ,13.4 Hyperbolic reductions using conformal Gaussian systems

373

In particular, one hasΓ̂ABCD = ΓABCD − CA fDB .
As a consequence of the gauge conditions (13.56) it follows thatfAB = f(AB) ,ΓQ Q AB = −fAB ,L̂Q Q AB = 0.

Deﬁning, as in Section 4.3.1, the spinors χABCD and ξABCD via
1 χABCD ≡ − √ ΓABCD + Γ+
ABCD ,
2

1
ξABCD ≡ √ (ΓABCD − Γ+
ABCD ),
2

one obtains from the metricity of the connection ∇ that1
ΓABCD = √ (ξABCD − χABCD )211= √ (ξABCD − χ(AB)CD ) − AB fCD .
2
2
Exploiting the gauge conditions, the spinor ΘABCD can be decomposed into1
ΘABCD = ΘAB(CD) + CD ΘABQ Q .
2
In addition, it is convenient to introduce the electric and magnetic partsof the rescaled Weyl spinor φABCD via
ηABCD ≡

1
(φABCD + φ+
ABCD ),
2i
μABCD ≡ − (φABCD − φ+
ABCD ).
2

A calculation using the above deﬁnitions yields the detailed system:∂τ eAB 0 = −χ(AB) P Q eP Q 0 − fAB ,(13.59a)∂τ eAB = −χ(AB)(13.59b)αPQ

αeP Q ,1∂τ ξABCD = −χ(AB) P Q ξP QCD + √ (AC χ(BD)P Q + BD χ(AC)P Q )f P Q
2√1E− 2χ(AB)(C fD)E − (AC ΘBDQ Q + BD ΘACQ Q )2(13.59c)− iΘμABCD ,1(13.59d)∂τ fAB = −χ(AB) P Q fP Q + √ ΘABQ Q ,2(13.59e)∂τ χ(AB)CD = −χ(AB) P Q χP QCD − ΘAB(CD) + ΘηABCD ,∂τ ΘCD(AB) = −χ(CD) P Q ΘP Q(AB) − ∂τ ΘηABCD√+ i 2dP (A μB)CDP ,√∂τ ΘABQ Q = −χ(AB) EF ΘEF Q Q + 2dP Q ηABP Q .

(13.59f)(13.59g)

Remark. The term ∂τ Θ in the second term of the left-hand side of Equation (13.59f) arises from the fact that, in a conformal Gaussian system, thetime component of the covector d is given by Θ̇; see Proposition 5.1.

374

Hyperbolic reductions

Setting
φ0 ≡ φ0000 ,φ1 ≡ φ0001 ,φ2 ≡ φ0011 ,φ3 ≡ φ0111 ,φ4 ≡ φ1111 ,the standard Bianchi system, Equation (13.58), explicitly reads√( 2 + 2e01 0 )∂τ φ0 − 2e11 0 ∂τ φ1 + 2e01 α ∂α φ0 − 2e11 α ∂α φ1√= −6Γ1111 φ2 + (4Γ1110 + 8Γ0111 )φ1 + (2Γ1100 − 8Γ0101 )φ0 ,( 2 + 2e01 0 )∂τ φ1 − 2e11 0 ∂τ φ2 − 2e11 α ∂α φ2 + 2e01 α ∂α φ1= −4Γ1111 φ3 + (6Γ(01)11 − 3f11 )φ2√+ (4Γ1100 − 4Γ(01)01 + 2f01 )φ1 − (2Γ(01)00 + f00 )φ0 ,2∂τ φ2 − e11 0 ∂τ φ3 + e00 0 ∂τ φ1 − e11 α ∂α φ3 + e00 α ∂α φ1= −Γ1111 φ4 − 2(Γ1101 + f11 )φ3 + 3(Γ0011 + Γ1100 )φ2√− 2(Γ0001 − f00 )φ1 − Γ0000 φ0 ,( 2 − 2e01 0 )∂τ φ3 + 2e00 0 ∂τ φ2 − 2e01 α ∂α φ3 + 2e00 α ∂α φ2= −(2Γ(01)11 + f11 )φ4 + (2Γ0011 − 4Γ(01)01 − 2f01 )φ3√+ (6Γ(01)00 + 3f00 )φ2 − 4Γ0000 φ1 ,( 2 − 2e01 0 )∂τ φ4 + 2e00 0 ∂τ φ3 − 2e01 α ∂α φ4 + 2e00 α ∂α φ3= (2Γ0011 − 8Γ1010 )φ4 + (4Γ0001 + 8Γ1000 )φ3 − 6Γ0000 φ2 .
For completeness, the constraintsΛAB ≡ DP Q φP QAB = 0
are also given in explicit form:
e11 0 ∂τ φ4 − 2e01 0 ∂τ φ3 + e00 0 ∂τ φ2 + e11 α ∂α φ4 − 2e01 α ∂α φ3 + e00 α ∂α φ2= −(2Γ(01)11 − 4Γ1110 )φ4 + (2Γ0011 − 4Γ(01)01 − 4Γ1100 )φ3+ 6Γ(01)00 φ2 − 2Γ0000 φ1 ,e11 ∂τ φ3 − 2e01 0 ∂τ φ2 + e00 0 ∂τ φ1 + e11 α ∂α φ3 − 2e01 α ∂α φ2 + e00 α ∂α φ1
0= Γ1111 φ4 − (4Γ(01)11 − 2Γ1101 )φ3 + 3(Γ0011 − Γ1100 )φ2− (2Γ0001 − 4Γ(01)00 )φ1 − Γ0000 φ0 ,e11 ∂τ φ2 − 2e01 0 ∂τ φ1 + e00 0 ∂τ φ0 + e11 α ∂α φ2 − 2e01 α ∂α φ1 + e00 α ∂α φ00= 2Γ1111 φ3 − 6Γ(01)11 φ2 + (4Γ0011 + 4Γ(01)01 − 2Γ1100 )φ1− (4Γ0001 − 2Γ(01)00 )φ0 .
These constraint equations contain time derivatives of the components of the
Weyl spinor. Furthermore, as the congruence of conformal curves is, in general,
not hypersurface orthogonal, the constraint equations are not intrinsic to theleaves of a foliation.

13.4 Hyperbolic reductions using conformal Gaussian systems

375

The boundary adapted system
The standard system (13.36) is not the only symmetric hyperbolic evolution
system that can be extracted from the Bianchi equation. In certain applications,such as the ones involving evolution domains with a timelike boundary, anotherform of the evolution equations is more convenient. In what follows, the system
extracted from1
(13.60a)−2Λ(0000) = 0, −2Λ(0001) − C00 = 0, −2Λ(0011) = 0,2
1−2Λ(0111) + C11 = 0, −2Λ1111 = 0,
(13.60b)2
will be known as the boundary adapted system. In the following, it will
be shown that it is, indeed, symmetric hyperbolic. The principal part of theboundary adapted system can be written as
⎞ ⎛ ⎞
⎛ μ
φ0
τ + 2e01 μ −2e00 μ
0
0
0
μ
μ
⎟ ⎜ φ1 ⎟⎜ 2e11 μ
2τ
−2e
0
0
00
⎟ ⎜ ⎟
⎜
⎟ ∂μ ⎜ φ2 ⎟ ,Aμ ∂ μ φ = ⎜2τ μ−2e00 μ002e11 μ
⎟ ⎜ ⎟
⎜
μ
μ
μ ⎠
⎝ φ3 ⎠
⎝
2τ
−2e00
0
0
2e11
0
0
0
2e11 μ τ μ − 2e01 μ
φ4(13.61)so that the matrices Aμ are Hermitian, and, in particular, Aμ τμ is positivedeﬁnite. The characteristic polynomial is given by




det(Aμ ξμ ) = 4 τ μ ξμ g νλ ξν ξλ lρσ ξρ ξσ ,where lρσ ≡ τ ρ τ σ + e00 (ρ e11 σ) . In Chapter 14, it will be seen that whenτ μ is tangent to a timelike hypersurface, then the pull-back of lμν gives thecomponents of the intrinsic three-dimensional Lorentzian metric implied by g on
the hypersurface.
Explicitly, the boundary adapted system takes the form√
( 2 + 2e01 0 )∂τ φ0 − 2e11 0 ∂τ φ1 + 2e01 α ∂α φ0 − 2e11 α ∂α φ1= −6Γ1111 φ2 + (4Γ1110 + 8Γ0111 )φ1 + (2Γ1100 − 8Γ0101 )φ0 ,√2∂τ φ1 − e11 0 ∂τ φ2 + e00 0 ∂τ φ0 − e11 α ∂α φ2 + e00 α ∂α φ4= −2Γ1111 φ3 − 3f11 φ2 + (2Γ1100 + 4Γ0011 + 2f01 )φ1 − (4Γ0001 − f00 )φ0 ,√2∂τ φ2 − e11 0 ∂τ φ3 + e00 0 ∂τ φ1 − e11 α ∂α φ3 + e00 α ∂α φ1= −Γ1111 φ4 − 2(Γ1101 + f11 )φ3 + 3(Γ0011 + Γ1100 )φ2− 2(Γ0001 − f00 )φ1 − Γ0000 φ0 ,√2∂τ φ3 − e11 0 ∂τ φ4 + e00 0 ∂τ φ2 − e11 α ∂α φ4 + e00 α ∂α φ2√= −(4Γ1110 + f11 )φ2 + (2Γ0011 + 4Γ1100 − 2f01 )φ3 + 3f00 φ2 − 2Γ0000 φ1 ,( 2 − 2e01 0 )∂τ φ4 + 2e00 0 ∂τ φ3 − 2e01 α ∂α φ4 + 2e00 α ∂α φ3= (2Γ0011 − 8Γ1010 )φ4 + (4Γ0001 + 8Γ1000 )φ3 − 6Γ0000 φ2 .

376Hyperbolic reductions
13.4.5 The construction of a subsidiary system

This section addresses the construction of a system of subsidiary equations forthe evolution equations discussed in the previous section. The particular problem
at hand consists of constructing evolution equations for the zero quantitiesΞ̂c dab ,Σ̂a c b ,Δ̂abc ,

Λabc ,

encoding the extended conformal ﬁeld equations. In addition, in the present
hyperbolic reduction procedure, one also needs to construct evolution equations
for the additional zero quantities
δa ,

γab ,

ςab ,

which play the role of constraints of the conformal equations; see Equations(8.47a)–(8.47c) for their deﬁnitions. The necessity of these extra zero quantities
can be traced back to Proposition 8.3.
As in the case of the analysis of the subsidiary equations for the hyperbolic
reduction procedure using gauge source functions, the subsidiary equations need
to be homogeneous in zero quantities so that the vanishing of the latter on
an initial hypersurface readily implies a unique vanishing solution. The basic
assumption in the construction of the subsidiary system is that the evolution
equations associated to the extended conformal ﬁeld equations are satisﬁed. Thatis, one assumes thatΣ̂0 c b = 0,Ξ̂c d0b = 0,Δ̂0bc = 0,

hold, together with either the standard or the boundary adapted system for thecomponents of the Weyl spinor. The aforementioned evolution equations havebeen constructed using the gauge conditions
f0 = 0,Γ̂0 b c = 0,L̂0b = 0,

which, therefore, can also be used in the construction of the subsidiary system.
Note also, that in the present gauge d0 = Θβ0 = ∇0 Θ so that one hasδ0 = 0.
Similarly,ˆ 0 βb − 1 S0b ef βe βf + λΘ−2 η0b = 0
γ0b = L̂0b − ∇2by virtue of the gauge conditions and the evolution equation
˜ 0 βa + β0 βa − 1 η0a (βe β e − 2λΘ−2 ) = 0,∇2
for the covector βa . Finally, one has
ˆ 0 fb + Γ̂b e 0 fe = 0,ς0b = −L̂b0 − ∇as a result of the evolution equation for the covector f .

(13.62)

13.4 Hyperbolic reductions using conformal Gaussian systems

377

The construction of subsidiary equations is similar in spirit to the one discussed
in Section 13.3. There are, however, certain diﬀerences. The most conspicuous
one is the fact that one is now working with a connection which is non-metric.
The subsidiary equation for the no-torsion condition
To construct the subsidiary equation for the no-torsion condition one considers
ˆ [a Σ̂b d c] and observes thatthe totally antisymmetric covariant derivative ∇ˆ [0 Σ̂b d c] = ∇ˆ 0 Σ̂b d c + ∇ˆ b Σ̂c d 0 + ∇ˆ c Σ̂0 d b
3∇ˆ 0 Σ̂b d c − Γ̂b e 0 Σ̂c d e − Γ̂c e 0 Σ̂e d b .
=∇(13.63)On the other hand, from the ﬁrst Bianchi identity, Equation (2.10), and thedeﬁnition of Ξ̂d cab one obtains
ˆ [a Σ̂b d c] = −Ξ̂d [cab] − Σ̂[a e b Σ̂c] d e ,∇(13.64)

where it has been used that ρ̂d [cab] = 0 by construction. The desired evolution
equation is obtained from combining Equations (13.63) and (13.64). Moreprecisely, one hasˆ 0 Σ̂b d c = − 1 Γ̂c e 0 Σ̂e d b − 1 Γ̂c e 0 Σ̂e d b − Ξ̂d 0bc .
∇3
3This evolution equation has the required homogeneous form.
The subsidiary equation for the Ricci identity
In this case, one considers the totally symmetrised covariant derivative
ˆ [a Ξ̂d |e|bc] . A direct computation shows that∇
ˆ [0 Ξ̂d |e|bc] = ∇ˆ 0 Ξ̂d ebc + ∇ˆ b Ξ̂d ec0 + ∇ˆ c Ξ̂d e0b3∇
ˆ 0 Ξ̂d ebc − Γ̂b f 0 Ξ̂d ecf − Γ̂c f 0 Ξ̂d ef b .
=∇Using the second Bianchi identity, Equation (2.11), and the deﬁnition of Ξ̂d ebc
one arrives at the expression
ˆ [a Ξ̂d |e|bc] = −Σ̂[a f b R̂d |e|c]f − ∇ˆ [a ρ̂d |e|bc] .
∇The ﬁrst term on the right-hand side is already of the required form. The second
one needs to be analysed in more detail. It is recalled thatρ̂d ebc ≡ C d ebc + 2Se[b df L̂c]f .
Thus,ˆ [a ρ̂d |e|bc] = ∇ˆ a L̂c]f .
ˆ [a C d |e|bc] + 2Se[b df ∇∇

378Hyperbolic reductions

ˆ a ρ̂d ebc . A direct
In order to further expand this expression one considers f abc ∇calculation shows that
ˆ [a C d |e|bc] = ∇[a C d |e|bc] + δ[a d f|f C f e|bc] + ηe[a f f C d |f |bc] .
∇(13.65)Moreover, one has
f abc ∇a C d ebc = −f abc ∇a ∗ C ∗d ebc= −2∇a ∗ C d ef a = 2∇a C ∗a f d e= −e dgh ∇a C a f gh .
Thus, using that C c dab = Θdc dab and the deﬁnition of the zero quantity Λabc
one concludes that
ˆ a C d ebc = Θe dgh Λf gh + 2∇g Θd∗d ef g + 2Θf g d∗ d + 2Θf g d∗d gf e .
f abc ∇gef
A similar computation using the deﬁnition of Δ̂abc yields
2f abc Seb dg Δ̂acg = 2Θβg d∗g ef d − 2Θβg d∗gd f e .
Thus, using the symmetries of d∗c dab and the deﬁnition of δa one concludes that
ˆ a ρ̂d ebc = Θe dgh Λf gh − 2Θδ g d∗d ef g + f abc Seb dg Δ̂acg .
f abc ∇Alternatively, using the properties of the generalised Hodge duals † and ‡ deﬁned
in Equation (2.24), one can writeˆ [a ρ̂d |e|bc] = 1 Θf abc e dgh Λf gh − 1 Θf abc δ g d∗d ef g − Se[b dg Δ̂ac]g .
∇6
3Combining the expressions, one obtains the required evolution equation.
Namely, one hasˆ 0 Ξ̂d ebc = Γ̂b f 0 Ξ̂d ecf + Γ̂c f 0 Ξ̂d ef b − Σ̂b f c R̂d e0f − 1 Θf 0bc e dgh Λf gh∇
2+ f 0bc δ g d∗d ef g + 3Se0 dg Δ̂cbg ,which is homogeneous in the zero quantities.
The subsidiary equation for the Cotton equation
ˆ [a Δ̂bc]d . A direct computation
In this case one considers the skew derivative ∇yields
ˆ 0 Δ̂bcd + ∇ˆ b Δ̂c0d + ∇ˆ c Δ̂0bd
ˆ [0 Δ̂bc]d = ∇3∇
ˆ 0 Δ̂bcd − Γ̂b e 0 Δ̂ced − Γ̂c e 0 Δ̂ebd .
=∇13.4 Hyperbolic reductions using conformal Gaussian systems

379

On the other hand, using the deﬁnition of Ξ̂e cab and the symmetries of ρ̂e cab
one obtains
ˆ b L̂c]d − ∇ˆ [a de |d|bc]ˆ [a Δ̂bc]d = 2∇
ˆ [a ∇
ˆ [a d|e de d|bc] − de ∇∇
ˆ |e| L̂c]d= −Ξ̂e [cab] L̂ed − Ξ̂e d[ab L̂c]e − ρ̂e d[ab L̂c]e + Σ̂[a e b ∇ˆ [a de |d|bc] .
ˆ [a d|e de d|bc] − de ∇−∇Using the deﬁnition of δa and γab one ﬁnds that
ˆ [a d|e de d|bc] = −Θδ[a β|e de d|bc] − Θγ[a|e de d|bc] − Θf[a β|e de d|bc] + ΘL̂[a|e de d|bc] .
∇Finally, a calculation similar to the one carried out in the previous subsection
shows that
f abc ∇a de dbc = d egh ∇a da f gh ,so that using Equation (13.65) and the properties of the generalised duals † and– see Equation (2.24) – one ﬁnds that‡

ˆ [a de |d|bc] = 1 abc f d egh Λf gh + δ[a e f|f df d|bc] + ηd[a f f de |f |bc] .
∇6
Combining the above expressions and using the properties of the decomposition of ρ̂d dab one obtains the expression
ˆ |e| L̂c]dˆ [a Δ̂bc]d = −Ξ̂e [cab] L̂ed − Ξ̂e d[ab L̂c]e + Σ̂[a e b ∇∇1+ Θδ[a β|e de d|bc] + Θγ[a|e de d|bc] − abc f d egh Λf gh βe ,6and, ﬁnally, the evolution equation
ˆ 0 Δ̂bcd = Γ̂b e 0 Δ̂ced + Γ̂c e 0 Δ̂ebd − Ξ̂e 0bc L̂ed + δb de de dc0 + δc de de d0b∇
1+ Θγbe de dc0 + Θγce de d0b − 0bc f d egh Λf gh βe ,2which is homogeneous in zero quantities as required.
The subsidiary equations for the physical Bianchi identity
The argument to show the propagation of the Bianchi identity in the present
context is similar to the one discussed in Section 13.3.6. In particular, the zero
quantity ΛABCD satisﬁes Equation (13.47). It remains to compute ∇b Λbcd andexpress it in terms of zero quantities associated with the extended conformal
ﬁeld equations. A calculation using the commutator of the covariant derivative∇ yields
2∇b Λbcd = 2∇b ∇a dabcd = 2∇[b ∇a] dabcd= 2Re [c ba dd]eab − 2Re a ba debcd + Σb e a ∇e dab cd .

380Hyperbolic reductions

ˆ − ∇ = S(f ), then Σ̂a c b = Σa c b . Moreover,Now, it is recalled that if ∇
ˆ and ∇,
using the formula relating the curvature tensors of the connections ∇cEquation (5.25b), the deﬁnitions of the zero quantities Ξ̂ dab and ςab and thesymmetries of dabcd , one concludes that1∇b Λbcd = Ξ̂e [c ba dd]eab − Ξ̂e a ba debcd + Σ̂b e a ∇e dab cd + ς ab dabcd .
2This expression is homogeneous in zero quantities and, thus, also its spinorialcounterpart ∇AA ΛA ACD . Consequently, if the standard evolution equations
hold, it follows from Equation (13.47) and the calculations in the previous
paragraph that

3
PΛAB − D(A P ΛB)P − √ χP Q SQ S(A ΛBP ) = 2∇QQ ΛQQ AB
2is homogeneous in zero quantities.
Finally, in the case of the boundary adapted system, one obtains a symmetric
hyperbolic system of evolution equations of the formPΛ00 + D00 Λ01 = U00 ,(13.66a)PΛ01 + D00 Λ11 − D11 Λ00 = U01 ,(13.66b)PΛ11 − D11 Λ01 = U11 ,(13.66c)where U00 , U01 and U11 are expressions homogeneous in zero quantities.
The subsidiary equations for the auxiliary zero quantities
Direct computations show that
ˆ [0 δb] = ∇ˆ 0 δb + Γ̂b e 0 δe ,2∇
ˆ 0 γbc + Γ̂b e 0 γec ,ˆ [0 γb]c = ∇2∇
ˆ 0 ςbc − Γ̂b e 0 ςce − Γ̂c e 0 ςeb .
ˆ [0 ςbc] = ∇3∇For δa one ﬁnds, using the deﬁnitions of the various zero quantities, that
ˆ [a ∇
ˆ b] Θ
ˆ [a δb] = ∇ˆ a βb − ∇ˆ a fb − Θ−1 ∇∇1ˆ e Θ.
= −γ[ab] + ςab − Θ−1 Σa e b ∇2
A lengthier computation yields
ˆ [a γb]c = 2∇
ˆ b] βc + 2Sc[a ef β|e| ∇ˆ b] βfˆ [a L̂b]c − 2∇
ˆ [a ∇2∇ˆ [a Θηb]c − 2λΘ−2 f[a ηb]c
− 2λΘ−3 ∇ˆ e βc + 2βc γ[ab] − 2β[a γb]c + ηc[a β e γb]e
= Δ̂abc + βe Ξ̂e cab − Σ̂a e b ∇+ 2λΘ−2 δ[a ηb]c + β[a ηb]c βe β e − 2λΘ−2 ηc[a βb] .

13.4 Hyperbolic reductions using conformal Gaussian systems

381

Similarly, using de [cab] = 0, one obtains
ˆ [a ςbc] = ∇ˆ b] fc]
ˆ [[a L̂b]c] − ∇ˆ [[a ∇∇
11
11
ˆ |e| fc]
= Δ̂[abc] − de de [cab] + R̂e [cab] fe − Σ̂[a e b ∇
2
2
2
2
1
1
1
ˆ |e| fc] .
= Δ̂[abc] + Ξ̂e [cab] fe − Σ̂[a e b ∇222Hence, one obtains the evolution equations
ˆ 0 δi = γi0 − Γ̂i e 0 δe ,∇
ˆ 0 γic = −γjc Γ̂i j 0 − β0 γic − βc γi0 + η0c (β e γie − 2λΘ−2 δi ),∇ˆ 0 ςjk = Γ̂j e 0 ςke + Γ̂k e 0 ςej + 1 Δ̂jk0 + 1 Ξ̂e 0jk fe + 1 Σ̂j e k Γ̂e f 0 ff ,∇2
2
2
where, in the last equation relation, (13.62) has been used to get further
cancellation of terms.
13.4.6 Summary of the analysis
It is convenient to group the independent components of the spinorial ﬁelds inthe extended conformal ﬁeld equations as:υ̂ independent components ofeAA μ , Γ̂AA BC , L̂AA BB  ,φ independent components ofφABCD .
Also, let e and Γ̂ denote, respectively, the independent components of the frameand connection coeﬃcients. In terms of the above deﬁnitions one has:
Proposition 13.3 (properties of the conformal evolution equations) The
extended conformal vacuum Einstein ﬁeld equations
Σ̂a c b = 0,Ξ̂c dab = 0,Δ̂abc = 0,

Λabc = 0,

expressed in terms of a conformal Gaussian gauge imply a symmetric hyperbolic
system for the components of (υ̂, φ) of the form∂τ υ̂ = Kυ̂ + Q(Γ̂)υ̂ + L(x)φ,
I + A0 (e) ∂τ φ + Aα (e)∂α φ = B(Γ̂)φ,where I is the 5×5 unit matrix, K is a constant matrix, Q(Γ̂) is a smooth matrixvalued function, L(x) is a smooth matrix-valued function of the coordinates,Aμ (e) are Hermitian matrices depending smoothly on the frame coeﬃcients e
and B(Γ̂) is a smooth matrix-valued function of the connection coeﬃcients. In
the case of the standard Bianchi system, the characteristic polynomial consistsof the factors


2 μν
μ
μν
μ ν
g ξμ ξν ,τ τ + gξμ ξν ,τ ξμ ,3

382

Hyperbolic reductions

while for the boundary-adapted Bianchi system one has the factors μ ν
τ μ ξμ ,g μν ξμ ξν ,
τ τ + e00 (μ e11 ν) ξμ ξν .
Remark. It is important to emphasise the relative simplicity of the evolution
system provided by Proposition 13.3 compared with the one given in Proposition
13.1. This structure reinforces the intuition that the Weyl tensor encodes thedegrees of freedom of the gravitational ﬁeld.
With regards to the subsidiary system one obtains a result analogous to
Proposition 13.2:
Proposition 13.4 (properties of the subsidiary evolution) Assume thatthe conditions
Σ̂0 c b = 0,Ξ̂c d0b = 0,Δ̂0bc = 0,

hold and that the associated evolution equations are expressed in terms of aconformal Gaussian gauge system. Moreover, let the independent componentsof the rescaled Weyl spinor satisfy either the standard or the boundary-adapted
evolution system. Then, the independent components of the zero quantitiesΣ̂a c b ,Ξ̂c dab ,Δ̂abc ,

Λabc ,

δa ,

γab ,

ςab ,

which are not determined by either the evolution equations or gauge conditions
satisfy a symmetric hyperbolic system which is homogeneous in zero quantities.
Controlling the conformal Gaussian gauge
The conformal Gaussian hyperbolic reduction procedure is based on theassumption of the existence of a non-singular (i.e. non-intersecting) congruenceof conformal geodesics. While this assumption may be valid close to an initial
hypersurface, it may fail at later times. To analyse the potential breakdown of
the gauge, one appends to the evolution system given in Proposition 13.4 anevolution equation for the components of the deviation vector of the congruence;see Section 5.5.7.
In what follows, let z denote a separation vector for the congruence ofconformal geodesics. Accordingly, one has[ẋ, z] = 0.
Thus, writing z = z a ea where {ea } is a Weyl propagated frame such thate0 = ẋ, it follows thate0 (z a )ea = z a [ea , e0 ].
Using the conformal ﬁeld equation Σ̂a c b = 0 and using that, in the present
gauge, e0 = ∂ τ and Γ̂0 c b = 0, the above expression can be rewritten as∂τ z a = Γ̂b a 0 z b .

13.5 Other hyperbolic reduction strategies

383

Now, let zAA denote the spinorial counterpart of z a . Deﬁning the space spinorcounterpart zAB ≡ τB A zAA and using the splitzAB =1zAB + z(AB) ,2

a computation similar to the one used to derive the evolution equations yields
the following evolution equations for the irreducible components of zAB :∂τ z = fAB z (AB) ,(13.67a)∂τ z(AB) = χCD(AB) z (CD) .

(13.67b)

The congruence of conformal geodesics will be non-intersecting as long asz(AB) = 0.
13.5 Other hyperbolic reduction strategies
The hyperbolic reduction procedures discussed in Sections 13.2 and 13.4 do not
exhaust the possible strategies to extract evolution equations from the conformal
Einstein ﬁeld equations. Indeed, other approaches have been put forward in theliterature.
13.5.1 Hyperbolic reductions for the metric conformal ﬁeld equations
Numerical evaluations of solutions to the vacuum conformal Einstein ﬁeld
equations have been carried out in Hübner (1999a,b, 2001a) using the metricformulation of the equations; see Equations (8.28a)–(8.28e). As the metricconformal ﬁeld equations contain no equation which can be read as a diﬀerential
equation for the components of the unphysical metric g, one needs to supplement
the equations in some manner. Assuming that suitable evolution equations can befound for the components of the conformal ﬁelds Ξ, Σa , s, Lab and da bcd in somelocal coordinates x = (xμ ), the conformal metric g can be computed from thecomponents of the Schouten tensor, Lμν , using generalised wave coordinates;
see the Appendix to this chapter for the vacuum Einstein ﬁeld equations and theremark at the end of Section 8.2.5. More precisely, the components gμν of g are
given as the solutions to the equations
gμν − 2∇(μ Fν) − 2gλρ g στ Γσ λ μ Γτ ρ ν − 4Γλ σ ρ g λτ gσ(μ Γν) ρ τ1= −4Lμν − R(x)gμν ,3that is,Γμ = F μ (x),xμ = −F μ (x),where F μ (x) are some suitable coordinate gauge source functions and it
has been used that Rab = 2Lab + 16 R(x)gab . Observe that in the right-hand sideof the ﬁrst of the above equations one has the Ricci scalar R, which, following
the discussion from previous sections, is to be treated as a further gauge source
function.

384

Hyperbolic reductions

From an analytic point of view, the approach described in the previous
paragraphs leads to an evolution system with equations of mixed order. This
type of system requires a more general notion of hyperbolicity than the one
discussed in Chapter 12: the so-called Leray hyperbolicity ; see, for example,
Choquet-Bruhat (2008) and Rendall (2008).
13.5.2 Wave equations for the conformal ﬁelds
One way of avoiding mixed-order evolution systems is to construct wave
equations for the components of the conformal ﬁelds Ξ, Σa , s, Lab and da bcd . This
strategy has been pursued in Paetz (2015) for the metric (vacuum) conformal
Einstein ﬁeld equations. More precisely, it has been shown that by introducing
suitable gauge source functions, the conformal ﬁeld equations can be rewritten
as a system of quasilinear wave equations for the conformal ﬁelds. An alternativereformulation can be obtained using spinors; see Gasperı́n and Valiente Kroon(2015). This approach is brieﬂy discussed in the remainder of this section.
Wave equations for the concomitants of the conformal factor
Wave equations for the ﬁelds Ξ, ΣAA and s can be obtained from the following
derivatives of the relevant zero quantities:∇AA ZAA BB  = 0,

∇AA QAA = 0,

∇AA ZAA = 0.

A direct computation renders the equationsΞ − ∇AA ΣAA = 0,ΣBB  + ΣAA LAA BB  + Ξ∇AA LAA BB  − ∇BB  s = 0,








s + ΣCC ∇AA LAA CC  + ∇AA ΣCC LAA CC  = 0.
The wave equation satisﬁed by the rescaled Weyl spinor
Recalling the deﬁnition of the zero quantity ΛB  BCD , one has∇A B ΛB  BCD = ∇A B ∇Q B  φBCDQ
1= −∇(A B ∇Q)B  φBCD Q + AQ ∇P P ∇P P  φBCD Q
2
1= AQ φBCD Q − φABCD ,2

where AB denotes the box operator discussed in Section 3.2.5. A further
calculation shows that1
AQ φBCD Q = 6ΞφP Q (AB φCD)P Q − R(x)φABCD .
4

13.6 Further reading385

Thus, the condition −2∇A Q ΛBQ CD = 0 implies the wave equation
1
φABCD − 12ΞφP Q (AB φCD)P Q + R(x)φABCD = 0
2
for the components of the rescaled Weyl spinor as long as the conformal gaugesource function R(x) is explicitly provided.
The wave equation satisﬁed by the components of the Schouten spinor
To construct an equation for the Schouten spinor, one considers the expression−2∇C C  ΔCDBB  = 0,
as given by Equation (13.30) together with the decomposition (13.31) for
the Schouten tensor in terms of the spinor ΦAA BB  and the Ricci scalar.
Accordingly, one has1
2∇C C  ΔCDBB  = ∇C C  ∇C Q ΦDQ BB  + DB ∇C C  ∇CB  R(x)2+ ∇C C  ΣQ B  φCDBQ + ΣQ B  ∇C C  φCDBQ ,
where




∇C C  ∇C Q ΦDQ BB  = −∇C (C  ∇|C|Q ) ΦD Q BB 1
− C  Q ∇C P  ∇C P ΦD Q BB 2¯ C  Q ΦD Q BB  − 1 ΦDC  BB  ,
= −
2
1
C 
∇ C ∇CB  R(x) = C  B  R(x).
2
Thus, using that¯ C  Q ΦD Q BB  = ΦP Q BB  ΦDC  P Q + ΦD Q P B  ΦBC  P Q+ Ξφ̄C  Q B  S  ΦD Q B S
11− R(x)ΦDC  BB  − R(x)ΦDB  BC  ,824
one obtains the desired wave equation for the components of ΦAA BB  . Finally, a
suitable subsidiary equation to ensure that the conformal gauge source function
R(x) is, indeed, the Ricci scalar of the connection ∇ can be obtained from thecontracted Bianchi identity (13.29).
13.6 Further reading
The original references for the hyperbolic reduction procedure based on the use ofspinors and gauge source functions are Friedrich (1983, 1991) – in particular, thelatter reference contains a discussion of the hyperbolic reduction of the EinsteinYang-Mills equations. The hyperbolic reduction procedure using a conformal
Gaussian system was ﬁrst discussed in Friedrich (1995, 1998c); extensions of386

Hyperbolic reductions

these ideas to the non-vacuum case using conformal curves have been given
in Lübbe and Valiente Kroon (2012). An alternative discussion of hyperbolic
reductions of the conformal ﬁeld equations using space spinors can be found in
Frauendiener (1998a,b). A gauge source function-based hyperbolic reduction of
the conformal Einstein-Euler system for a perfect ﬂuid with a radiation equationof state has been described in Lübbe and Valiente Kroon (2013b). A discussionof the hyperbolic reduction of the conformal Einstein-scalar ﬁeld system using
gauge source functions is given in Hübner (1995).
A general discussion of the procedure of hyperbolic reduction of the standard
Einstein ﬁeld equations in the vacuum and matter case can be found inFriedrich and Rendall (2000), where, for example, the case of the EinsteinDirac system is discussed. A related reference is Reula (1998). More speciﬁc
discussions of hyperbolic reductions for the vacuum Einstein ﬁeld equations andtheir associated subsidiary evolution systems can be found in Friedrich (1996,2005). A Lagrangian hyperbolic reduction for the Einstein-Euler system has been
discussed in Friedrich (1998b). Extensions of this Lagrangian approach havebeen obtained for the equations of relativistic magnetohydrodynamics coupled togravity – the so-called Einstein-Euler-Maxwell system – in Pugliese and Valiente
Kroon (2012) and for the Einstein-charged scalar ﬁeld system in Pugliese andValiente Kroon (2013).
Readers interested in the hyperbolic reductions of the Einstein ﬁeld equations
used in numerical relativity are referred to the monographs by Alcubierre (2008)
and Baumgarte and Shapiro (2010) as an entry point to the extensive literature.
Appendix A.1: the reduced Einstein ﬁeld equations
This chapter has been primarily focused on hyperbolic reductions for theconformal Einstein ﬁeld equations in their spinorial formulation. In order to
put the discussion into a more general context, it is useful to brieﬂy consider
the hyperbolic reduction procedure of the (standard) Einstein ﬁeld equations
using generalised wave coordinates. This procedure is essentially the oneused in the seminal work by Fourès-Bruhat (1952) where the well-posedness of
the Cauchy problem in general relativity was ﬁrst established.
For simplicity, in the following, the discussion is restricted to the vacuum caseso that the Einstein ﬁeld equations are equivalent toR̃ab = 0.

(13.68)Given general coordinates x = (xμ ), the Ricci tensor can be explicitly written interms of the components of the metric tensor g̃ and its ﬁrst and second partial
derivatives as1
˜ (μ Γ̃ν)R̃μν = − g̃ λρ ∂λ ∂ρ g̃μν + ∇2+ g̃λρ g̃ στ Γ̃σ λ μ Γ̃τ ρ ν + 2Γ̃λ σ ρ g̃ λτ g̃σ(μ Γ̃ν) ρ τ ,(13.69)13.6 Further reading387

where it is recalled that the Christoﬀel symbols Γ̃μ ν λ can be written in terms ofpartial derivatives of the metric tensor as1Γ̃μ ν λ = g̃ νρ (∂μ g̃ρλ + ∂λ g̃μρ − ∂ρ g̃μλ ),2and one has deﬁnedΓ̃ν ≡ g̃ μλ Γ̃μ ν λ ,the so-called contracted Christoﬀel symbols. The principal part of thevacuum Einstein ﬁeld equation (13.68) is given by the terms1
˜ (μ Γ̃ν) .
− g̃ λρ ∂λ ∂ρ g̃μν + ∇2
The ﬁrst term in the above expression is hyperbolic as it coincides with˜ μ acting on the
˜ μ∇
˜ ≡ ∇
the principal part of the D’Alambertian operator components g̃μν . If the second term in the principal part can be removed one
would obtain a system of non-linear wave equations for g̃μν .
Generalised wave coordinates
A systematic approach to the construction of coordinates x = (xμ ) is to require
the coordinates to satisfy the equation
˜ μ = −F μ (x),x(13.70)where the coordinate gauge source functions F μ (x) are arbitrary smoothfunctions of the coordinates x. In the particular case where F μ (x) = 0 one talks ofwave coordinates, called harmonic coordinates in older accounts. In orderto unravel the consequences of Equation (13.70), one treats the coordinates xμ
as scalar ﬁelds over M̃. Accordingly, a direct computation gives˜ ν x μ = ∂ ν x μ = δν μ ,∇˜ λ∇˜ ν xμ = ∂λ δν μ − Γ̃λ ρ ν δρ μ = −Γ̃ν μ λ ,∇so that˜ μ = g̃ νλ Γ̃ν μ λ = −Γ̃μ .
x(13.71)

A natural way of prescribing initial conditions for Equation (13.70) on a
hypersurface S̃ with normal ν a is to set x0 = 0 with ν μ ∂μ x0 = 1 while setting
the spatial coordinates (xα ) to be equal to some given coordinates on S̃ andrequiring that ν μ ∂μ xα = 0. Given this data, the general theory of hyperbolic
diﬀerential equations ensures the existence of a solution to Equation (13.70),
and as a result of Equation (13.71), one concludes thatΓ̃μ = F μ (x).

(13.72)Moreover, if the coordinate diﬀerentials dxμ are chosen initially to be pointwise
independent on the initial hypersurface S̃ , then they will also remain pointwise388

Hyperbolic reductions

independent close to S̃ . Thus, by a suitable choice of coordinates, the contracted
Christoﬀel symbols can be made to agree, locally, with any prescribed set offunctions F μ (x). These coordinate gauge source functions and the data for
Equation (13.71) uniquely determine the coordinates. Conversely, given any
metric g̃, any coordinate system is characterised by some suitable gauge source
function and initial data. The domain on which the coordinates x = (xμ ) form agood coordinate system depends on the initial data, the coordinate gauge source
functions and the metric g̃ itself. Consequently, there is little that can be said,a priori, about the domain of existence of the coordinates.
The reduced Einstein equation and the subsidiary evolution equation
Substituting Equation (13.72) into the Einstein ﬁeld equations in the form givenby (13.69) one ﬁnds that1˜ (μ Fν) (x) + g̃λρ g̃ στ Γ̃σ λ μ Γ̃τ ρ ν− g̃ λρ ∂λ ∂ρ g̃μν + ∇2+ 2Γ̃λ σ ρ g̃ λτ g̃σ(μ Γ̃ν) ρ τ = 0,(13.73)where Fμ (x) ≡ gμν F ν (x). This equation is a system of quasilinear wave equations
for the components of the metric tensor g̃. For this system, the local Cauchy
problem with data on a spacelike hypersurface S̃ is well posed – one can show
the existence and uniqueness of solutions and their continuous dependence on
the data; see, for example, Friedrich and Rendall (2000). Equation (13.73) is
known as the reduced Einstein ﬁeld equation.
The introduction of a speciﬁc system of coordinates via the gauge source
functions F μ (x) breaks the tensoriality of the Einstein ﬁeld equation (13.68).
Given a solution to the reduced Einstein ﬁeld equation (13.73) the latter will
imply a solution to the actual Einstein ﬁeld equations as long as the coordinates
x = (xμ ) satisfy Equation (13.71) for the chosen coordinate source function
F μ (x) appearing in the reduced equation. To prove that this is the case oneneeds to construct a suitable subsidiary evolution equation.
A suitable subsidiary equation for the hyperbolic reduction procedure underconsideration can be obtained by observing that the reduced Einstein ﬁeld
equation, Equation (13.73), can be written as
˜ (μ Qν) ,R̃μν = ∇Qμ ≡ Γ̃μ − Fμ (x),(13.74)

where Γ̃μ = g̃μν Γ̃ν . Now, from the contracted Bianchi identity in the form
˜ μ R̃μν − 1 R̃g̃μν = 0,∇
2
it follows, by substituting Equation (13.74), that˜ ν + R̃μ ν Qμ = 0.
Q

13.6 Further reading

389

From the homogeneity on Qμ of this wave equation, it follows that if Qν = 0 and˜ μ Qν = 0 on some initial hypersurface and if g̃μν satisﬁes the reduced Einstein∇ﬁeld equations, then Γ̃μ = F μ (x) at later times.
Appendix A.2: diﬀerential forms
Let M be a four-dimensional manifold. A p-form α on M is a totally
antisymmetric covariant tensor of rank p. Thus, if αi1 ···ip is the abstract index
version of α, one has that
αi1 ···ip = α[i1 ···ip ] .
Given q ∈ M, the space of p-forms at q is denoted by Λp |q (M), while the bundleof p-forms over M is denoted by Λp (M). In particular, 0-forms are scalar ﬁelds sothat Λ0 (M) = X (M) and 1-forms are covectors – accordingly, Λ1 (M) = T ∗ (M).
A counting argument readily shows that dim Λp |q (M) = 4!/p!(4 − p)! – thus, infour dimensions any 4-form is proportional to the volume form. Given a p-form
α and a q-form β, their wedge product α ∧ β is deﬁned, using abstract index
notation, as(α ∧ β)a1 ···ap b1 ···bq ≡(p + q)!
α[a1 ···ap βb1 ···bq ] .
p!q!

Given local coordinates x = (xμ ) in M, a 1-form α can be written as α = αμ dxμ .
More generally, for a p-form one has the expansion
α = αμ1 ···μp dxμ1 ∧ · · · ∧ dxμp .
It can be veriﬁed thatdxμ ∧ dxν = dxμ ⊗ dxν − dxν ⊗ dxμ .
Given a p-form α and a vector v = v μ ∂ μ , one deﬁnes the contraction iv αas the (p − 1)-formiv α ≡ v ν ανμ1 ···μp−1 dxμ1 ∧ · · · ∧ dxμp−1 .
The exterior derivative dα is the (p + 1)-form deﬁned via the relationdα ≡ ∂[μ1 αμ2 ···μp+1 ] dxμ1 ∧ · · · ∧ dxμp+1 .
It follows from the commutativity of partial derivatives that d2 α = 0.
Finally, it observed that the Lie derivative of a p-form can be computed using
Cartan’s formula:£v α = iv dα + div α.
Further details on the above expressions can be found in, for example, Frankel(2003).

14
Causality and the Cauchy problem
in general relativity

The study of the Cauchy problem in general relativity was initiated by theseminal work by Fourès-Bruhat (1952). The extensions and reﬁnements ofthis work and, in particular, the analysis of the existence of maximal Cauchy
developments by Choquet-Bruhat and Geroch (1969) bring to the forefront
the delicate interplay between geometry and the theory of partial diﬀerential
equations arising in Einstein’s theory of general relativity.
This chapter provides a discussion of two aspects of the Cauchy problem ingeneral relativity: (i) the connection between the notions of causality originatingfrom the theory of symmetric hyperbolic equations and those derived from theexistence of a Lorentzian metric on the underlying spacetime manifold – theso-called Lorentzian causality, and (ii) the existence and uniqueness of a socalled maximal Cauchy development of an initial value problem for the Einsteinﬁeld equations. This chapter sets the context for the discussion in Part IV ofthis book where asymptotically simple spacetimes are constructed by means ofsuitably posed initial value problems.

14.1 Basic elements of Lorentzian causality
In Section 2.5 some basic notions of Lorentzian geometry have already been
introduced. These ideas are now further elaborated to present the notions ofLorentzian causal theory . The summary presented here is adapted from thediscussion in Ringström (2009).
In what follows, the discussion is restricted to four-dimensional Lorentzianmanifolds (M̃, g̃) which are orientable and time orientable. In particular, timeorientability is equivalent to the existence of a smooth timelike vector t; see
Section 2.1. The Lorentzian manifold (M̃, g̃) is not assumed to satisfy the
Einstein ﬁeld equations.

14.1 Basic elements of Lorentzian causality

391

Chronological future, causal future, and so on
A vector v ∈ T (M̃) is said to be causal if v = 0 and v is either timelike or null.
Consistent with the discussion of Section 2.5, v is said to be future pointing ifg̃(v, t) > 0 and past pointing if g̃(v, t) < 0. A future-pointing causal curveon (M̃, g̃) is one for which its tangent vector is everywhere future pointing causal.
The notion of past-pointing causal curve is deﬁned in an analogous manner.
Causal curves can be used to deﬁne order relations between points of the
manifold M̃. Given p, q ∈ M̃, one writes p ≺≺ q if there is a future-pointing
timelike curve in M̃ from p to q; p ≺ q if there is a future causal curve fromp to q; and p  q if either p = q or p ≺ q. Given a subset U ⊂ M̃ one deﬁnes thechronological future and chronological past of U , respectively, as
I + (U ) ≡ p ∈ M̃ | q ≺≺ p for some q ∈ U ,
I − (U ) ≡ p ∈ M̃ | p ≺≺ q for some q ∈ U .
Moreover, the causal future and causal past of U are deﬁned, respectively, as
J + (U ) ≡ p ∈ M̃ | q  p for some q ∈ U ,J − (U ) ≡ p ∈ M̃ | p  q for some q ∈ U .
A schematic depiction of the sets I ± (U ) and J ± (U ) is given in Figure 14.1. The
sets I + (U ) and I − (U ) can be shown to be open. No general statements of thistype can be made about J + (U ) and J − (U ). However, one has that I + (U ) ⊆J + (U ) and I − (U ) ⊆ J − (U ).
Global hyperbolicity
The natural class of spacetimes for which an initial value problem can beformulated is that of globally hyperbolic ones.
A spacetime (M̃, g̃) without closed timelike curves is called causal . A causal
spacetime (M̃, g̃) is said to be globally hyperbolic if for any pair of points p, q ∈M̃ with p ≺ q the causal diamond J + (p) ∩ J − (q) is compact; see Figure 14.2.
The classical deﬁnition of global hyperbolicity as given, for example, in Wald

J+(U)

I+(U)

U

U

J−(U)

I−(U)

Figure 14.1 Schematic representation of the sets I ± (U) and J ± (U) for a subset
U ⊂ M̃.

392

Causality and the Cauchy problem in general relativityq

J+(p) ∩ J−(q)pFigure 14.2 The causal diamond J + (p) ∩ J − (q): the points p, q ∈ M̃ satisfyp ≺ q. In a globally hyperbolic spacetime any such diamond is compact.
q+

O

p

γ

A

q–

Figure 14.3 The edge of a closed achronal set A: for the point p ∈ A there
exists an open neighbourhood O containing p and q+ ∈ I + (p), q− ∈ I − (p)such that q− and q+ can be joined by a timelike curve γ not intersecting A.

(1984), makes use of the stronger notion of strongly causal spacetimes, thatis, the non-existence of “almost closed” causal curves. The classical deﬁnition
and the one given here have been shown to be equivalent in Bernal and Sánchez(2007).
In physical terms, global hyperbolicity is closely connected to the idea ofclassical determinism, that is, the prediction or retrodiction of future or paststates, respectively, from a set of initial conditions. Pathologies like the existenceof closed timelike curves are not present in globally hyperbolic spacetimes.
Cauchy surfaces
A subset A of a Lorentzian manifold (M̃, g̃) is said to be achronal if there isno pair of points p, q ∈ A that can be connected by a timelike curve. Spacelike
hypersurfaces are examples of achronal sets. For A closed and achronal, onedeﬁnes its edge as the set of points p ∈ A such that every open neighbourhood
O of p contains points q+ ∈ I + (p), q− ∈ I − (p) and a timelike curve γ from q−
to q+ which does not interset A; see Figure 14.3.
Given A ⊂ M̃ achronal, the future domain of dependence of A is theset D+ (A) of all points p ∈ M̃ such that every past inextendible causal curvethrough p intersects A. The past domain of dependence of A is deﬁned in an
analogous manner by considering future inextendible causal curves. The (full)14.1 Basic elements of Lorentzian causality

393

J+(A)

D+(A)
A
D−(A)
J−(A)

Figure 14.4 Domain of dependence of an achronal set A and its relation to
the causal past and future J ± (A).

domain of dependence of A is then deﬁned as
D(A) ≡ D+ (A) ∪ D− (A).
In some accounts, the sets D+ (A) and D− (A) are called, respectively, the future
and past Cauchy development of A. The reason for these alternative names isclariﬁed by the discussion in Section 14.2. It can be veriﬁed that A ⊂ D+ (A) ⊂J + (A); see Figure 14.4. From the achronality of A it follows that D+ (A) ∩I − (A) = Ø.
Since information travels along causal curves, a point p ∈ D+ (S) receives
information only from S. Accordingly, if physical laws are causal – as in the caseof general relativity – initial data should determine the physics in D+ (S) – and,in fact, in all of D(S).
A Cauchy hypersurface in M̃ is a hypersurface S̃ such that
D(S̃) = M̃.
Cauchy hypersurfaces are characterised by the fact that they are intersected
exactly once by every inextendible timelike curve in M̃; see, for example,
Ringström (2009). Cauchy hypersurfaces are continuous three-dimensional submanifolds of the spacetime manifold M̃; see, for example, Wald (1984). Cauchy
hypersurfaces provide an alternative description of globally hyperbolic spacetimes: any globally hyperbolic spacetime possesses a Cauchy hypersurface. Global
hyperbolicity restricts the topology of a spacetime. More precisely, one has that:
Proposition 14.1 (topology of globally hyperbolic spacetimes) Let (M̃, g̃)denote a connected, time-oriented globally hyperbolic Lorentzian manifold and let
S̃ be a Cauchy hypersurface thereof. Then
M̃ ≈ R × S̃.
In other words, M̃ can be foliated by Cauchy hypersurfaces. Moreover, if S̃  isanother Cauchy hypersurface, then S̃ ≈ S̃  .

394

Causality and the Cauchy problem in general relativity

The above result is complemented by the following:
Proposition 14.2 (existence of a global time function) Let (M̃, g̃) be anoriented, time-oriented, connected and globally hyperbolic spacetime and let S̃ be
a Cauchy hypersurface thereof. Then there is a smooth function t on M̃ suchthat dt is timelike and future directed everywhere and satisﬁes the property thatt−1 (T• ) is a Cauchy hypersurface for every T• ∈ R. Furthermore, t−1 (0) = S̃and for every inextendible causal curve γ : (s− , s+ ) → M̃ one has t(γ(s)) → ±∞as s → s± .
For a proof of these results, see, for example, Ringström (2009), proposition
11.3 and theorem 11.27. Finally, one has the following:
Proposition 14.3 (asymptotic simplicity and global hyperbolicity ) An
asymptotically simple and empty spacetime (M̃, g̃) is globally hyperbolic.
The reader interested in a proof is referred to Hawking and Ellis (1973),
proposition 6.9.2.
Cauchy horizons
In what follows let D+ (A) denote the closure of the future domain of dependenceof an achronal set A ⊂ M̃. This set is characterised by the fact that forp ∈ D+ (A) every past inextendible timelike curve from p intersects A; see
proposition 8.3.2 in Wald (1984). The achronal set A is not necessarily a
Cauchy hypersurface. To characterise how much A deviates from being a Cauchy
hypersurface, it is convenient to introduce the set
H + (A) ≡ D+ (A) \ I − D+ (A) ,the so-called future Cauchy horizon of A. The past Cauchy
horizonis
deﬁned in an analogous manner as H − (A) ≡ D− (A) \ I + D− (A) . It canbe shown that H + (A) is achronal. Moreover, one has that A ⊂ D+ (A) and∂D+ (A) = H + (A) ∪ A; see Figure 14.5. Similar properties hold for D− (A).
+−The (full) Cauchy horizon
 is then deﬁned as H(A) ≡ H (A) ∪ H (A). It
can be proved that H(A) = ∂ D(A) and that the achronal set A is a Cauchy
surface for (M̃, g) if and only if H(A) = Ø; see proposition 8.3.6 and its corollaryin Wald (1984).
The following property of Cauchy horizons will be used at various points inthis book (cf. theorem 8.3.5 in Wald (1984)):
Proposition 14.4 (structure of Cauchy horizons) Every point p ∈ H + (A)lies on a null geodesic contained entirely in H + (A) which is either inextendibleor has a past endpoint on the edge of A.

14.2 PDE causality versus Lorentzian causality

H+(A)

395

H+(A)
D+(A)

A

Figure 14.5 The Cauchy horizon of an achronal set A. Observe that ∂D+ (A) =H + (A) ∪ A.

˜
D+(S)
D+(R)
R
S̃

Figure 14.6 Schematic representation of the causal domains of Theorem 14.1.
The hypersurface S̃ is a Cauchy hypersurface and R ⊆ S̃ is a region withinsuch that u = 0.

14.2 PDE causality versus Lorentzian causality
Two diﬀerent notions of causality have been discussed so far in this book:partial diﬀerential equation (PDE) causality based on the uniqueness of solutions
to symmetric hyperbolic systems – Theorem 12.1 – and Lorentzian causality,
discussed in the ﬁrst sections of this chapter. These notions of causality areconceptually diﬀerent from each other. However, they are linked by the following
result (see also Figure 14.6):
Theorem 14.1 (the relation between PDE and Lorentzian causalities)
Let (M̃, g̃) be a connected, oriented, time-oriented, globally hyperbolic spacetimeand let S̃ be a smooth spacelike Cauchy hypersurface. Let R ⊆ S̃and let U bean open set containing D+ (R). Assume that u : U → CN solves the symmetric
hyperbolic system
Aμ (x, u)∂μ u + B(x, u) = 0.
Moreover, assume that the above equation has a characteristic polynomial whichcontains the factor (g̃ μν ξμ ξν ) where g̃ μν denotes the contravariant components ofthe metric g̃. If u vanishes on R, then u vanishes on D+ (R). There is a similarstatement for D− (R).

396

Causality and the Cauchy problem in general relativity

p
D+(R)

J−(p) ∩ J+(R)R

Figure 14.7 Schematic representation of the causal domains in the proofof Theorem 14.1. On D+ (R) one considers for arbitrary p ∈ D+ (R) the
associated domain D(p) ≡ J − (p) ∩ J + (R).

The interested reader is referred to chapter 12 of Ringström (2009) fora detailed account of the proof of an analogous result for quasilinear wave
equations. It is, nevertheless, useful to discuss some of the ideas behind the proof.
Theorem 12.1 and its Corollary 12.1 – ensuring the uniqueness of solutions tosymmetric hyperbolic systems – can be applied only on lens-shaped domains. The
main idea behind Theorem 14.1 is then to construct a cover of D+ (R) consistingof lens-shaped domains. The metric g̃ provides a natural way of constructing therequired cover. Accordingly, the Lorentzian metric allows the introduction of thenotions of Lorentzian causality discussed in the ﬁrst sections of this chapter.
One begins by considering points p ∈ D+ (R) which are suitably close to R
and aims to conclude that u = 0 on D(p) ≡ J − (p) ∩ J + (R);—see Figure 14.7.
By means of the exponential map expp : T |p (M̃) ⊃ V̄ → V ⊂ M̃ – see Section
11.6.2 – the metric g̃ allows the introduction of normal coordinates in someneighbourhood of p – these coordinates can be seen as providing a diﬀeomorphism
between a neighbourhood V̄ of the origin in T |p (M̃) to a neighbourhood V of p.
By considering p suﬃciently close to R one can ensure that D(p) is compact andcompletely contained in V. On T |p (M̃) one can deﬁne a function f¯ : T |p (M̃) → R
via f¯(v) = g̃(v, v). Hence, for the present purposes, the neighbourhood V canbe regarded as a subset of the Minkowski spacetime coordinatised by standard
Cartesian coordinates. One also deﬁnes f : V → R such that f ≡ f¯◦ exp−1
p . Now,
given a constant c > 0, the condition g̃(v, v) = c deﬁnes (spacelike) hyperboloids
on V. More precisely, for given c, the locus of points in V corresponding to the−1
hyperboloid is given by f −1 (c) ≡ {q ∈ V | g̃(exp−1
p (q), expp (q)) = c} – observe−1
is a set consisting of morethat both f¯ and f are not injective so that f−1than a single point. Now, f (c) has two components: one associated to futuredirected vectors and the other associated to past-directed vectors. For c > 0, let
Qc (p) denote the component of f −1 (c) associated to past-directed vectors andlet Q0 (p) denote the past null cone through p. One can use the hyperboloids

14.2 PDE causality versus Lorentzian causality

397

p

Q(p)

D(p)

Qc(p)
Dc(p)

R

Figure 14.8 Schematic representation of the causal domains in the proofof Theorem 14.1. Given p ∈ D+ (R), the sets Q(p) and D(p) describe,
respectively, the past light cone through p and the region between the lightcone and R. For c > 0, the set Qc (p) describes a hyperboloid inside the past
light cone of p, while Dc (p) is the region between the hyperboloid and R. The
set Dc (p) is a lens-shaped domain. See the main text for further details.

Qc (p) to foliate the interior of the past light cone passing through p. One deﬁnes
Dc (p) ≡ J − (Qc (p))∩J + (R). The domain Dc (p) describes the region between thehyperboloid Qc (p) and R, while D(p) describes the region between the past lightcone and R; see Figure 14.8. Now, D(p) can be shown to be compact. Moreover,it can be seen that Qc (p) ⊂ I − (p) for c > 0 so that J − (Qc (p)) ⊂ J − (p) and, in
addition, that Dc (p) ⊂ D(p). A further argument allows one to verify that Qc (p)for c > 0 is a lens-shaped domain on which, modulo some technical details, Corollary 12.1 can be applied. Thus, if u = 0 on R, one concludes that u = 0 on Dc (p).
To show that u = 0 on D(p) one now considers a sequence {cl }, l ∈ N, ofpositive numbers converging to zero. It can then be shown that
int D(p) ⊂ ∪l Dcl (p) ⊂ D(p)– intuitively, by choosing smaller and smaller cl ’s one obtains hyperboloids whichare, successively, “closer” to the light cone Qp thus “ﬁlling” D(p). From thisobservation and given that u = 0 on each of the Dcl (p) one can conclude that,indeed, u = 0 on D(p).
Now, an adaptation of Proposition 14.2 ensures the existence of a time function
t on D+ (R). Given c > 0, t−1 ([0, c]) denotes a slab in D+ (S̃) ⊃ D+ (R).
Considering points suitably close to S̃, it is possible to construct a slab Kε ≡t−1 [0, ε] ∩ D+ (R) in D+ (R) for some ε > 0 – this slab can be thought of as
the union of domains of the type D(p) on each of which one already knows thatu = 0. The top of the slab, t−1 (ε) ∩ D+ (R) – on which u = 0 – can now be used
as a new initial surface from which one constructs a further slab. The rest of theproof consists of showing that D+ (R) can be fully covered by slabs of the typedescribed above so that u = 0 everywhere on D+ (R).

398

Causality and the Cauchy problem in general relativity

Remark. An important observation is that
D+ (R) ∩ I + (S̃ \ R) = Ø.
Accordingly, the value of u on D+ (R) is determined only by the data on R
– that is, whatever data is prescribed on S̃ \ R, it has no inﬂuence on the
behaviour of u on D+ (R). The proof of this statement follows by contradiction:
let q ∈ D+ (R) ∩ I + (S̃ \ R); on the one hand we have that q ∈ I + (S̃ \ R) so that
there exists a future timelike curve γ from p ∈ S̃ \ R to q. On the other handq ∈ D+ (R) so that every past inextendible causal curve through q intersects R.
As a consequence one has that p ∈ R. This is a contradiction since p ∈ S̃ \ R.
14.3 Cauchy developments and maximal Cauchy developments
For ease of presentation, the subsequent discussion is restricted to the case ofstandard Cauchy initial value problems where initial data is prescribed on a
Cauchy hypersurface S̃. For a detailed account of the Cauchy problem in generalrelativity the reader is referred to the monograph by Ringström (2009).
As discussed in Section 11.3, the (say, vacuum) Einstein ﬁeld equations imply
on S̃ a set of constraint equations: the so-called Hamiltonian and momentum
constraints for a Riemannian metric h̃ and a symmetric trace-free tensor K̃.
Assume one is given a solution (h̃, K̃) to the Hamiltonian and momentum
constraint Equations (11.13a) and (11.13b) on S̃. To discuss the relation between
a solution to the Einstein constraint equations and a solution to the Einstein ﬁeld
equations one needs to introduce the notion of a Cauchy development:
Deﬁnition 14.1 (Cauchy development) A Cauchy development of theinitial data set (S̃, h̃, K̃) consists of a solution (M̃, g̃) of the vacuum Einsteinﬁeld equations, an embedding ϕ of S̃ into M̃ and a choice of a unit normal vector
such that ϕ(S̃) is a Cauchy hypersurface and the pull-backs by ϕ of the induced
metric and the second fundamental form for the prescribed unit normal coincidewith h̃ and K̃.
The seminal work in Fourès-Bruhat (1952) has shown that, given a solution
to the constraint equations on S̃, it is always possible to obtain a Cauchy
development. More precisely, one has the following (see also Figure 14.9):

S˜

Figure 14.9 Schematic representation of a Cauchy development (in gray) of
some initial data set (S̃, h̃, K̃) for the Einstein ﬁeld equations.

14.3 Cauchy developments and maximal Cauchy developments

399

Theorem 14.2 (existence of a development of an initial data set) Given
an initial data set (S̃, h̃, K̃) for the Einstein ﬁeld equations it is always possible
to ﬁnd a corresponding development.
The above result is a cornerstone of the mathematical study of the Einsteinﬁeld equations as it shows that it is meaningful to formulate a Cauchy problem
for the Einstein ﬁeld equations. The original proof of the theorem used thehyperbolic reduction of the Einstein ﬁeld equations based on wave coordinates;
see the Appendix to Chapter 13. The hyperbolic reductions for the conformal
Einstein ﬁeld equations discussed in Chapter 13 readily lead to an alternative
proof which is brieﬂy sketched for completeness.
Proof The proof of the theorem amounts to a local existence result for the
Cauchy problem for the Einstein ﬁeld equations. For convenience, consider the
spinorial version of the standard conformal Einstein ﬁeld equations; see Section
8.3.2. Setting Ξ = 1 and ΦAA BB  = 0 one obtains a spinorial representationof the vacuum Einstein ﬁeld equations. For these equations, the hyperbolic
reduction procedure summarised in Proposition 13.1 shows that given a choiceof coordinate and frame gauge source functions F a (x) and FAB (x), the Einsteinﬁeld equations imply a symmetric hyperbolic system for the frame coeﬃcients,
connection coeﬃcients and the Weyl spinor. Smooth initial data u for theseevolution equations can be obtained from the pair (h̃, K̃) using the procedureleading to Lemma 11.1. The basic existence and uniqueness result for symmetric
hyperbolic systems given in Theorem 12.2 ensures the existence of a solution
u to the evolution equations in a slab of the form M̃T ≡ (−T, T ) × S̃ forsome T > 0. In what follows, for conceptual clarity, the Riemannian 3-manifold
S̃ regarded as a submanifold of M̃T will be denoted as S̃ ; that is, one has
S̃ = ϕ(S̃). On S̃ the solution u coincides with the initial data u . In view ofthe homogeneous structure of the subsidiary evolution equations as describedin Proposition 13.2, the solution u implies a solution to the conformal Einsteinﬁeld equations with Ξ = 1 and ΦAA BB  = 0. From the components of u one
can construct a Lorentzian metric g̃ which will be a solution to the Einsteinﬁeld equations on M̃T ; compare Proposition 8.1. To conclude, it is observedthat the hyperbolic procedure leading to the evolution equations is based on an
adapted frame tetrad {ea } such that e0 on S̃ gives the unit normal of the initial
hypersurface; see Section 11.4. From this observation it follows that the pull-backof g̃ to S̃ coincides with the Riemannian metric h̃. Moreover, by construction,
the extrinsic curvature of S̃ coincides with the tensor K̃. Accordingly (M̃T , g̃)provides the required Cauchy development.
An important aspect of the notion of a Cauchy development is its nonuniqueness. A diﬀerent choice of gauge source functions will, in general, leadto a diﬀerent Cauchy development for the same initial data. Observe, however,that as one is constructing solutions to tensorial equations in the regions where
two diﬀerent Cauchy developments (M̃, g̃) and (M̃ , g̃  ) overlap M̃ ∩ M̃ , these400

Causality and the Cauchy problem in general relativity

must be related to each other by a diﬀeomorphism, that is, a coordinatetransformation. This non-uniqueness of Cauchy developments of a given initial
data creates a tension with the notion of geometric uniqueness, that is, theexpectation that a given initial data set should give rise to a unique solution to
the Einstein ﬁeld equations. To deal with this issue one introduces the notion ofa maximal Cauchy development:
Deﬁnition 14.2 (maximal Cauchy development) Let (S̃, h̃, K̃) be an initial
data set for the vacuum Einstein equations. A Cauchy development (M̃, g̃) withembedding ϕ : S̃ → M̃ of this data is said to be maximal if for any other
Cauchy development (M̃ , g̃  ) with embedding ϕ : S̃ → M̃ , there is a smoothmap ψ : M̃ → M̃ which is a diﬀeomorphism onto its image such that ϕ = ψ ◦ ϕand ψ ∗ g̃ = g̃  .
The maximal Cauchy development describes the biggest spacetime that can berecovered from a given initial data set for the Einstein ﬁeld equations. Any other
Cauchy development must be contained in it. For this notion to be of utility
it should satisfy some existence and uniqueness properties. Indeed, one has thefollowing result, ﬁrst proven in Choquet-Bruhat and Geroch (1969):
Theorem 14.3 (existence of a maximal Cauchy development) Given
some initial data (S̃, h̃, K̃) for the Einstein ﬁeld equations, there exists a maximal
Cauchy development which is unique up to isometries.
The original proof of this theorem famously relies on Zorn’s lemma. Alternative
proofs not depending on this axiom of set theory have been given more recently
in Sbierski (2013) and Wong (2013).
Remark. The maximal Cauchy development of an initial data set is, in general,
diﬀerent from the so-called maximal analytic extension of the solution to
the initial value problem, that is, the biggest spacetime that can be associatedto a given metric allowing for analytic changes of coordinates. As an example,
compare the Penrose diagram of the maximal analytical extension of theReissner-Nordström spacetime given in Figure 6.14 and the Penrose diagramof its maximal Cauchy development in Figure 14.10.
The characterisation and construction of the maximal Cauchy developmentof an arbitrary initial data set (S̃, h̃, K̃) is a challenging endeavour. It requires
controlling the evolution dictated by the Einstein ﬁeld equations under very
general conditions. Generically, one expects the following to be true:
Conjecture 14.1 (strong cosmic censorship) The maximal Cauchy development of generic initial data for the vacuum Einstein ﬁeld equations is
inextendible.

14.4 Stability of solutions

401

H−

H+

I+

H+

I−

H−

Figure 14.10 Maximal Cauchy development of the Reissner-Nordström spacetime. In this case, the spacetime extends only up to the Cauchy horizon H − .
Notice that the timelike singularities of the spacetime do not appear in the
diagram.

A concise discussion of the above conjecture and its various caveats can befound in Chruściel (1991), Rendall (2008) and Ringström (2009).
14.4 Stability of solutions
A problem simpler than cosmic censorship is the construction of the developmentof initial data sets which are, in some sense, close to initial data for some exact
solution (the background solution) whose global structure is well known.
Such initial data are called a perturbation of the initial data for theexact solution. Under suitable circumstances one expects the maximal Cauchy
development of the perturbed initial data to have a global structure similar tothat of the maximal Cauchy development of the exact solution. The resulting
spacetime is called a perturbation of the exact solution. This notion of
perturbations is a non-linear one: the perturbed solutions are required to satisfy
the Einstein ﬁeld equations without any approximation – as opposed, say, tolinearised perturbations where one considers solutions to evolution equations
which are linearised with respect to some background exact solution. The
underlying strategy behind the analysis of non-linear perturbations is to use
the knowledge of the global properties of a solution to the equations of generalrelativity to infer the existence of other solutions with analogous properties. This
point of view leads to the notion of stability.
When discussing the stability of solutions to the Einstein ﬁeld equations one
typically distinguishes between the notions of orbital and asymptotical stability.
A solution is said to be orbitally stable if the global geometry of the perturbed
evolution exhibits the same features as the original (background) solution –for example, the existence of a complete null inﬁnity. The stronger notion

402

Causality and the Cauchy problem in general relativityof asymptotic stability requires, in addition, that the perturbed solution
converges to the background solution for late times. The stability results tobe discussed in the remainder of this book will be of the orbital type.
Remark. Although the notion of stability has a strong physical motivation –see, for example, the discussion in Section 12.3.2 – the precise formulation ofcloseness to a certain exact solution is dictated by the details of the PDE theory
used to analyse the evolution equations – for example, Sobolev norms – and it
may be diﬃcult to provide it with a direct physical interpretation. In particular,statements about closeness may not be gauge independent.
14.5 Causality and conformal geometry
Let (M, g) denote a conformal extension of a physical spacetime (M̃, g̃) withg = Ξ2 g̃. As a Lorentzian manifold in its own right, the unphysical spacetime(M, g) gives rise to its own causal notions. The causal notions in (M̃, g̃) and(M, g) are, however, related to each other – it is not hard to see that the causal
notions introduced in Section 14.1 are conformally invariant. More precisely, if
p, q ∈ M̃ are connected to each other via some particular causal relation withrespect to the metric g̃ (e.g. p ≺ q, p  q or p ≺≺ q), then they are also
causally related in the same way with respect to the metric g. Special care isneeded, however, when discussing points which lie on the conformal boundaryof the conformal extension (M, g) as these points do not exist in the physicalspacetime manifold M̃. Moreover, any compact set in the unphysical manifold(M, g) which intersects the conformal boundary will be, from the perspectiveof the physical manifold M̃, non-compact. This observation is of importance forthe notion of global hyperbolicity as it is formulated in terms of compactness of
domains in the physical spacetime (M̃, g̃).
A further cautionary note concerns Cauchy horizons in the unphysicalspacetime (M, g) which may not correspond to domains in M̃. The prototypicalcase of this situation arises in the discussion of Minkowski-like spacetimes.
From the point of view of (M, g), the conformal boundary of these spacetimes
corresponds to the Cauchy horizon of hyperboloidal hypersurfaces – which fromthe conformal point of view are compact domains. From the physical perspectiveof (M̃, g̃) the hyperboloids are non-compact and there are no Cauchy horizons.
The correspondence between the conformal boundary and Cauchy horizons for
Minkowski-like spacetimes is analysed in some detail in Section 16.3.
Penrose diagrams provide a convenient way of visualising the causal propertiesof spacetimes. For example, an inspection of the Penrose diagram of the antide Sitter spacetime, Figure 14.11, readily shows that the spacetime cannot be
globally hyperbolic: causal diamonds intersecting the conformal boundary of
the conformal representation correspond to non-compact causal diamonds inthe physical spacetime. Alternatively, by looking at the Penrose diagram it
is easy to draw a timelike curve which does not intersect a putative Cauchy

14.5 Causality and conformal geometry403qJ−(q) ∩ J+(p)
γ
I

I

p

S

Figure 14.11 Non-global hyperbolicity of the anti-de Sitter spacetime. To theleft: causal diamonds intersecting the conformal boundary are non-compact in
the physical picture. To the right: given a putative Cauchy hypersurface S, it
is always possible to ﬁnd a timelike curve γ not intersecting S.
I+
D+(R2)
R2

D+(R1)
R1

I−

Figure 14.12 Examples of some domains of dependence in the de Sitter
spacetime.

hypersurface S – it is only necessary that in the conformal picture the curvestarts at some point of the conformal boundary which lies in the future of S.
A second example of the insights provided by the inspection of the Penrose
diagrams involves the de Sitter spacetime; see Figure 14.12. A peculiarity ofthis spacetime is that there exist regions in the spacetime whose domain of
dependence is non-compact – to see this, it is only necessary to consider domains
which are, from the conformal point of view, suﬃciently close to the conformalboundary I .

404

Causality and the Cauchy problem in general relativity
14.6 Further reading

Detailed accounts of the theory of Lorentzian causality can be found in
Hawking and Ellis (1973), chapter 6; O’Neill (1983), chapter 5; or Wald (1984),
chapter 8.
An extensive discussion of the Cauchy problem in general relativity can befound in Ringström (2009). A concise presentation is given in Rendall (2008).
A related discussion is contained in Friedrich and Rendall (2000). A discussion ofvarious aspects of strong cosmic censorship as well as a number of ancillary results
concerning the Cauchy problem can be found in the monograph by Chruściel(1991).

Part IV
Applications

15
De Sitter-like spacetimes

This chapter discusses the global existence and stability of de Sitter-likespacetimes, that is, vacuum spacetimes with a de Sitter-like value of thecosmological constant. This class of spacetimes admits a conformal extension
with a spacelike conformal boundary; see Theorem 10.1. The construction ofde Sitter-like spacetimes provides, arguably, the simplest application of theconformal ﬁeld equations to the analysis of global properties of spacetimes. The
original discussion of the analysis presented in this chapter was given in Friedrich(1986b). The results of this seminal analysis were subsequently generalised to thecase of Einstein equations coupled to the Yang-Mills ﬁeld in Friedrich (1991).
The methods used in the proof of the stability of the de Sitter spacetime can beadapted to analyse the future non-linear stability of Friedman-Robertson-Walker
cosmologies with a perfect ﬂuid satisfying the equation of state of radiation; seeLübbe and Valiente Kroon (2013b).
The global existence and stability theorem proven in this chapter can beformulated as follows:
Theorem (global existence and stability of de Sitter-like spacetimes).
Small enough perturbations of initial data for the de Sitter spacetime give rise
to solutions of the vacuum Einstein ﬁeld equations which exist globally towards
the past and the future. The solutions have the same global structure as the deSitter spacetime. Thus, perturbations of the de Sitter spacetime are asymptoticallysimple.
Intuitively, the last statement in the theorem can be read as saying that theresulting spacetimes have a Penrose diagram similar to the one of the de Sitter
spacetime; see Figure 15.1. Accordingly, these spacetimes provide non-trivial (i.e.
dynamic) examples of asymptotically simple spacetimes. A detailed formulationof the above result is given in the main text of the chapter; see Theorem 15.1.
To illustrate the comparative advantages of the hyperbolic reduction procedures discussed in Chapter 13, two versions of the proof are provided. The
ﬁrst one makes use of gauge source functions and follows the original proofs

408

De Sitter-like spacetimes

Figure 15.1 Penrose diagram of a de Sitter-like spacetime.

in Friedrich (1986b, 1991). The second proof makes use of conformal Gaussian
systems and is based on the analysis given in Lübbe and Valiente Kroon (2009).
In both approaches, and as a consequence of the use of the conformal ﬁeld
equations, it is possible to formulate initial value problems for the perturbed deSitter-like spacetime not only on a standard initial hypersurface at a ﬁduciaryﬁnite time, but also on a hypersurface corresponding to the conformal boundaryof the spacetime.
The basic strategy used in this chapter to analyse the global existence ofsolutions to the Einstein ﬁeld equations had been previously used in ChoquetBruhat and Christodoulou (1981) to establish the global existence of solutions
to the Yang-Mills equations.
15.1 The de Sitter spacetime as a solution to the conformal
ﬁeld equations
The basic conformal properties of the de Sitter spacetime have already been
discussed in Section 6.3. In this section the de Sitter spacetime is recast as asolution to the conformal ﬁeld equations. This is a ﬁrst step in the constructionof an existence and global stability result.
15.1.1 Basic representation in the Einstein cosmos
For simplicity of the exposition, the cosmological constant will be assumed totake the value λ = −3 so that the conformal factor realising the embedding of
the de Sitter spacetime (given in standard coordinates) into the Einstein static
universe is given byΞ̊ ≡ ΞdS = cos τ,(15.1)

where τ , the aﬃne parameter of the geodesics introduced in Equation (6.9), isused as a time coordinate. Here, and in the rest of the chapter, the symbol ˚is used to indicate that the associated object is treated as a background ﬁeld.

15.1 The de Sitter spacetime as a solution to the conformal ﬁeld equations 409
As a consequence of the conformal embedding, the geometry of the conformalde Sitter spacetime is given by the corresponding expressions for the Einstein
static universe as discussed in Section 6.1.3. The various conformal ﬁelds on the
Einstein cylinder (R × S3 , g E ), withg E = dτ ⊗ dτ − h̄,
will be expressed in terms of an orthonormal frame {e̊a } such thate̊0 = ∂ τ ,e̊i = ci ,(15.2)

where {ci } denotes the globally deﬁned frame on S3 discussed in Section 6.1.2;
see Equations (6.2a)–(6.2c).
In what follows, the manifold ME ≡ R × S3 will be described locally in termsof Gaussian coordinates (τ, xα ) where (xα ) are some local coordinates on S3
which are extended to coordinates on a subset U ⊂ ME by requiring them toremain constant along the geodesics parametrised by τ . As a consequence of theg E -orthonormality of the vector ﬁelds {∂ τ , ci }, it follows thate̊a = δa b cb ≡ e̊a b cb .

(15.3)

Using the structure equations – see Section 2.7.3 – on S3 , it can be veriﬁed thatthe connection coeﬃcients γ̊i j k of the Levi-Civita connection D of the standard
metric of S3 , h̄, with respect to the spatial frame {ci } are given byγ̊i j k = −i j k ,(15.4)

where ijk denotes the components of the volume form on S3 ; see Section 6.1.2.
Now, observing that e̊0 = ∂ τ is a Killing vector of the Einstein cylinder, it follows
that the connection coeﬃcients associated to the frame {e̊a } are given byΓ̊a b c = 0a b c .

(15.5)The sign diﬀerence between Equations (15.4) and (15.5) arises from the fact that
the Riemannian metric implied by g E on S3 is negative deﬁnite.
Using the expressions (6.8a) and (6.8b) for the Schouten tensor of the Einstein
cylinder, it follows that, in terms of the frame {e̊a } described above, one has
1L̊ab = δa 0 δb 0 − ηab ,2d˚a bcd = 0.

(15.6a)(15.6b)For later use, it is also observed that the components of the trace-free Ricci
tensor are given by1Φ̊ab = δa 0 δb 0 − ηab .
4

410

De Sitter-like spacetimes

Finally, a further computation yields thatΣ̊ = − sin τ,Σ̊i = 0,1s̊ = − cos τ.
4(15.7a)(15.7b)Spinorial expressions
To compute the spinorial counterpart of the ﬁelds discussed√in the previous
section let τ AA denote the spinorial counterpart of the vector 2∂ τ so that one
has the normalisation τAA τ AA = 2.
The spinorial counterpart of the frame coeﬃcients e̊a b = δa b – compare
Equation (15.3) – is given bye̊AA b = σAA b ,where σAA b denotes the Infeld-van der Waerden symbols; see Section 3.1.9. In
general, the coeﬃcients e̊AA a can be decomposed ase̊AA a =1
τAA e̊a − τ Q A e̊(AQ) a ,
2

with


e̊a ≡ τ AA e̊AA a ,

e̊AB a ≡ τB A e̊AA a .

By construction it follows thate̊0 = 1,e̊(AB) 0 = 0,e̊i = 0,e̊(AB) i = σAB i ,with σAB i the spatial Infeld-van der Waerden symbols; see Section 4.2.2.
The spinorial counterpart of the trace-free Ricci tensor is given byΦ̊AA BB  =1
1
τAA τBB  − AB A B  ,
2
4




so that its space spinor version Φ̊ABCD = τB A τD C Φ̊AA CC  is given byΦ̊ABCD =11
AB CD − AC BD .
2
4

From this last expression is can be veriﬁed that the irreducible components ofΦ̊ABCD are given byΦ̊(ABCD) = 0,Φ̊AB = 0,3Φ̊ = − ;4compare the expressions in Equation (13.32). The rescaled Weyl spinor is triviallygiven byφ̊ABCD = 0.

15.1 The de Sitter spacetime as a solution to the conformal ﬁeld equations 411
Let Γ̊AA BB CC  denote the spinorial counterpart of the connection coefﬁcients Γ̊a b c . Using the spinorial expression of the spacetime volume form,
Equation (3.25), one ﬁnds that
1Γ̊AA BB CC  = √ τ DD DD AA BB CC  ,2i  B= √ τC C  A δA B − τ B C  CA δA B .
2

The reduced spin connection coeﬃcients can be obtained from the expression
Γ̊AA B C = 12 Γ̊AA BQ CQ . One obtainsiΓ̊AA BC = √ A(B τC)A .
2
The space spinor version of the above expression is given byiΓ̊ABCD = τB A Γ̊AA CD = − √ hABCD ,2

where it is recalled that hABCD ≡ −A(C D)B . It can be veriﬁed that Γ̊†ABCD =−Γ̊ABCD ; that is, the spin connection coeﬃcients are the components of animaginary spinor. From here it follows that˚
ξABCD = −ihABCD ,χ̊ABCD = 0.

Gauge source functions
The gauge source functions associated to the considered conformal representationof the de Sitter spacetime can be computed from the expressions given in theprevious section.
Treating the frame component e̊a b as the component of a covariant tensor one
ﬁnds that˚ b e̊b a = η cb e̊c (δb a ) − η cb Γ̊c e b δe a∇= −η cb Γ̊c a b = −η cb 0c a b = 0.
It follows that the coordinate gauge source function is given by˚ AA e̊AA a = 0.
F̊ a (x) = ∇
Similarly, treating the connection coeﬃcients as the components of a (1, 2)tensor one has
˚ d Γ̊a b c = η da ed (Γ̊a b c ) + η da Γ̊d b e Γ̊a e c − η da Γ̊d e a Γ̊e b c − η da Γ̊d e c Γ̊a b e
η da ∇= η da 0d b e 0a e c − η da 0d e c 0a b e − η da 0d e a 0e b c = 0.

412

De Sitter-like spacetimes

It follows that the frame gauge source functions for the present representationof the de Sitter spacetime are given by˚ AA Γ̊AA BC = 0.
F̊BC (x) = ∇Finally, the conformal gauge source function is given by the value of the Ricci
scalar. That is, one has
R̊(x) = −6.
Summary
The results from the previous analysis are summarised in the following:
Lemma 15.1 (de Sitter spacetime as a solution to the conformal
Einstein ﬁeld equations) The ﬁelds(Ξ̊, Σ̊, Σ̊i , e̊a b , Γ̊a b c , L̊ab , d˚a bcd )as given by Equations (15.1)–(15.5), (15.6a), (15.6b), (15.7a) and (15.7b) or,
respectively, their spinorial counterparts(Ξ̊, Σ̊, Σ̊AA , e̊AA b , Γ̊AA B C , Φ̊AA BB  , φ̊ABCD )deﬁned over the Einstein cylinder R × S3 constitute a solution to the standard
frame vacuum conformal Einstein ﬁeld Equations (8.32a) and (8.32b) and,
respectively, the spinorial vacuum conformal Einstein ﬁeld Equations (8.38a)and (8.38b). The gauge source functions associated to this solution are given byF̊ a (x) = 0,F̊AB (x) = 0,R̊(x) = −6.

15.1.2 Representation using conformal Gaussian systems
In Section 6.1.3 it has been shown that an alternative conformal representationof the de Sitter spacetime is given by the conformal metric

2
τ̄ 2
ḡ E = dτ̄ ⊗ dτ̄ − 1 +h̄,4where τ̄ is an aﬃne parameter of the g̃ dS -conformal geodesics as in Equation(6.32); that is, ẋ = ∂ τ̄ . The associated covector is given by
β dS (τ̄ ) = −

2τ̄
dτ̄ .
4 − τ̄ 2

ˆ = ∇+S(β˜This covector is exact, thus indicating that the Weyl connection ∇dS )is, in fact, a Levi-Civita connection. Now, the metric ḡ E is related to the physicalde Sitter metric g̃ dS via
ḡ E = Θ2dS g̃ dS ,

ΘdS = 1 −

τ̄ 2
,
4

15.1 The de Sitter spacetime as a solution to the conformal ﬁeld equations 413
so that a calculation shows that
β dS = Θ−1dS dΘdS .
That is, the Weyl connection associated to the congruence of conformal geodesics¯ of the metric ḡ E . Recalling(6.32) coincides with the Levi-Civita connection ∇that g E and ḡ E are related to each other by ḡ E = Θ̄2 g E withΘ̄ ≡ 1 +τ̄ 2,
4

one ﬁnds that an adapted ḡ E -orthonormal frame {ēa } is given byē0 = ∂ τ̄ = Θ̄−1 ∂ τ ,ēi = Θ̄−1 ci .

This frame can be veriﬁed to be Weyl propagated. It follows that the framecoeﬃcients ēi b , with ēi = ēi b cb , are given byēi b =

4
δi b .
4 + τ̄ 2(15.8)In terms of the above, the components of the covector d̄ = ΘdS β dS with respect
to the frame {ēa } are given byτ̄d¯0 = Θ̇dS = − ,2d¯i = 0.

The computation of the connection coeﬃcients Γ̄a b c requires a certain amountˆ =∇¯ and ∇ are related to each otherof care. Recalling that the connections ∇−1ˆ
via ∇ − ∇ = S(Ῡ) with Ῡ ≡ Θ̄ dΘ̄, it follows by deﬁnition that¯ a ēc cΓ̄a b c = ω̄ b c ēa a ∇= ω̄ b c ēa a ∇a ēc c + ω̄ b c ēa a ēc d Sad ec Υe ,where Ῡa ≡ Ῡ, ēa . Using that ēa a = Θ̄−1 ea a one computesω̄ b c ēa a ∇a ēc c = −Θ̄−1 ω b c ea a ec c ∇a Θ̄ + Θ̄−1 ω b c ea a ∇a ec c= −Ῡa δc b + Θ̄−1 Γa b candΘ̄−1 ω̄ b c ēa a ēc d Sad ec ∇e Θ̄ = Ῡa δc b + Ῡc δa b − ηac Ῡb .
Accordingly, one concludes that
Γ̄a b c = Θ̄−1 Γa b c + Υc δa b − ηac Υb .
Using
Υa =

2τ̄
δa 0 ,
4 + τ̄ 2

414

De Sitter-like spacetimes

it can be veriﬁed that Γ̄0 b c = 0, as one would expect from the Weyl connection
associated to a congruence of conformal geodesics. Moreover,1f¯a = Γ̂a b b = 0.
4

(15.9)A direct computation shows thatR[ḡ E ] = −36,
4 + τ̄ 2

Schouten[ḡ E ] =
1
1
1 + τ̄ 2 h̄,
24W eyl[ḡ E ] = 0,
where the last expression follows simply by the conformal invariance of the Weyltensor. The components of the Schouten tensor with respect to the frame {ēa }are given byL̄0a = 0,2
δij .
4 + τ̄ 2

L̄ij =(15.10)Furthermore, one has thatd¯a bcd = 0.

(15.11)

Spinorial expressions√
In what follows, let τ̄ AA denote the spinorial counterpart of the vector 2∂ τ̄ .
One has the normalisation τ̄AA τ̄ AA = 2. Denoting the spinorial counterpart of
the frame coeﬃcients by ēAA a and making use of the standard space spinor
decomposition
ēAA a =1
τ̄AA ēa − τ̄ Q A ē(AQ) a ,2

one obtainsē0 = 1,ē(AB) 0 = 0,ēi = 0,ē(AB) i =(15.12a)4
σAB i .
4 + τ̄ 2(15.12b)Now, let L̄AA BB  denote the spinorial counterpart of the components of theSchouten tensor L̄ab . Setting L̄ABCD ≡ τ̄B A τ̄D C L̄AA CC  one ﬁndsΘ̄CD = 0,Θ̄ABCD = −2hABCD ,
4 + τ̄ 2with Θ̄CD ≡ L̄Q Q CD and Θ̄ABCD ≡ L̄(AB)(CD) . For the rescaled Weyl spinor
one has thatφ̄ABCD = 0.

15.1 The de Sitter spacetime as a solution to the conformal ﬁeld equations 415
To obtain a spinorial expression for the connection coeﬃcients one observes
that the spinorial counterpart of ζa b c ≡ δc 0 δa b − ηac δ0 b is given by



1 
ζAA BB CC  = √ δA B δA B τ̄CC  − AC A C  τ̄ BB ,2

so that the associated reduced coeﬃcients are
1
ζAA BQ CQ
2
1 = √ δA B τ̄CA + AC τ̄ B A .
2 2

ζAA B C ≡The space spinor version ζABCD ≡ τ̄B A ζAA CD takes the form
1
ζABCD = − √ hABCD .
2
From the expressions computed in the previous paragraph it follows that2(τ̄ + 2i)
Γ̄ABCD = − √hABCD
2(1 + 4τ̄ 2 )and, consequently,
4i
hABCD ,
4 + τ̄ 22τ̄χ̄ABCD =hABCD ,
4 + τ̄ 2
fAB = 0.

ξ¯ABCD = −(15.13a)(15.13b)(15.13c)

To keep track of the behaviour of the conformal Gaussian gauge system,
one considers separation ﬁelds measuring the deviation of the congruence ofconformal geodesics. The separation ﬁelds are governed by Equations (13.67a)and (13.67b). Assume, without loss of generality, a separation vector ﬁeld z thatis spatial on the ﬁduciary hypersurface S described by the condition τ̄ = 0, so
that
zAA  = −τ Q A z(AQ) .
Using Equations (15.13b) and (15.13c) one can integrate Equations (13.67a) and(13.67b) to ﬁnd
1 2(15.14)z = 0,z(AB) = 1 + τ̄ z(AB) .
4
Observe that z(AB) = 0 for all τ̄ . Thus, the congruence of conformal geodesics
remains non-singular. This observation is key to ensure the non-singularbehaviour of the gauge in the perturbed spacetime.

416

De Sitter-like spacetimes
Summary

The results of the analysis of the last two sections are summarised in thefollowing:
Lemma 15.2 (de Sitter spacetime as a solution to the extended
conformal Einstein ﬁeld equations) The ﬁelds(Θ̄ , d¯a , ēa b , Γ̄a b c , L̄ab , d¯a bcd )as given by Equations (15.8)–(15.11) or, equivalently, their spinorial counterparts(Θ̄, d¯AA , ēAA b , Γ̄AA BC , L̄AA BB  , φ̄ABCD )deﬁned over R × S3 constitute a solution to the extended conformal Einstein ﬁeld
Equations (8.46) and the associated gauge constraints (8.48) and, respectively,
the spinorial vacuum conformal Einstein ﬁeld Equations (8.54a) and (8.54b) and(8.55).
15.2 Perturbations of initial data for the de Sitter spacetime
This section clariﬁes the notion of perturbations of initial data for the de Sitter
spacetime. In what follows, let S denote a three-dimensional manifold withthe topology of S3 . On S one considers a solution to the vacuum conformal
Hamiltonian and momentum constraint equations (S, h, K, Ω, Σ) with a deSitter-like value of the cosmological constant, that is, Equations (11.15a) and(11.15b) with  = 0 and jk = 0.
Remark. For conceptual clarity it is often convenient to distinguish between the
3-manifold S and its embedding, S , in the spacetime arising as the developmentof the initial data set (S, h, K, Ω, Σ).
15.2.1 Initial data on a standard initial hypersurface
Using the procedure described in Section 11.4.3, the tensor ﬁelds h and K canbe used to construct a solution to the vacuum conformal constraint Equations(11.35a)–(11.35j). As the 3-manifold S is assumed to be compact, one can,without loss of generality, assume that Ω = 1 and Σ = 0.
As S ≈ S3 , there exists a diﬀeomorphism ψ : S → S3 which can be used topull back coordinates x = (xα ) in S3 to S. In this way one obtains a system of
coordinates x ≡ x ◦ ψ on S and can write x = (xα ). The diﬀeomorphism ψ canbe used to push forward the vector ﬁelds {ci } on T (S3 ) to vector ﬁelds {ψ∗−1 ci }on T (S) and to pull back their dual covectors {αi } on T ∗ (S3 ) to covectors {ψ∗ αi }on T ∗ (S). For simplicity of the presentation, in a slight abuse of notation, thevectors and covectors {ψ∗−1 ci } and {ψ∗ αi } will be written (except for the next
subsection) as {ci } and {αi }, respectively.

15.2 Perturbations of initial data for the de Sitter spacetime

417

Gauge ﬁxing
The construction described in the previous paragraph depends strongly on theparticular choice of the diﬀeomorphism ψ. This gauge freedom can be ﬁxed
by considerations similar to those used in the discussion of the coordinate gaugesource functions of Section 13.2.1.
Given an h-orthonormal frame {ei } on S, one can write ei = ei j (ψ∗−1 cj ) and
use the frame coeﬃcients ei j to introduce a spatial coordinate gauge source
function F i (x ) via the relation
Dj ej i = F i (x ),where D denotes the Levi-Civita covariant derivative of h. Writingψ∗ αi = (ψ∗ αi )α dxα
and noticing that ei j = ψ∗ αj , ei  one ﬁnds that Dj ej i = Dβ (ψ∗ αi )β .
Expressing the coordinates x in S3 in terms of the coordinates x on S in theform xα = xα (x ) one ﬁnds, by a calculation similar to the one discussed in
Section 13.2.1, that the diﬀeomorphism ψ : S → S3 is a harmonic map. Thatis, one has thatDβ Dβ xα = 0,ifhαβ D γ αi δ∂xγ ∂xδ= F i (x ),∂xα ∂xβ

where D denotes the Levi-Civita connection of the metric h̄ on S3 and αi =iαα α dx . Finally, if one lets xα = xα (x) be the identity map so that x = x, one
concludes that
F i (x) = δ jk γ̊j i k = 0,
where the last equality follows from (15.4). This construction and the resulting
spatial gauge source function ﬁxes the gauge freedom in the diﬀeomorphism ψ;
see Figure 15.2.

Figure 15.2 Construction of coordinates on a compact three-dimensional
manifold describing perturbations of standard de Sitter initial data. The
identiﬁcation of the 3-manifolds S and S3 is realised through a harmonic map;see main text for further details.

418

De Sitter-like spacetimes
Parametrising the perturbation data

While the frame {ci } is orthonormal with respect to the standard metric h̄ ofS3 , in general, this will not be the case with respect to the 3-metric h on S.
Now, let {ei } denote an h-orthonormal frame over T (S) and let {ω i } denote its
corresponding cobasis. In what follows, it will be assumed that one can write
ei = ci + ĕi ,(15.15)for some vectors {ĕi }. This is essentially equivalent to saying that one has
introduced coordinates x = (xα ) on S such thath = h̄ + h̆.
It is important to emphasise that the above statement depends on the gauge.
From the split in Equation (15.15), it follows that the solution to the conformal
constraint equations implied by (Ω = 1, Σ = 0, h, K) on S can be written asea b = δa b + ĕa b ,γi j k = i j k + γ̆i j k ,Lij = δij + L̆ij ,dij = d˘ij ,χij = K̆ij ,Li = δi 0 + L̆i ,d∗ = d˘∗ ,ij

ij

where the components of the various ﬁelds are expressed as components withrespect to the frame {ei } as given in (15.15) and one has
ĕa b = 0,γ̆i j k = 0,L̆ij = 0,L̆i = 0,ĕi = 0,

K = 0.

d˘∗ij = 0,d˘ij = 0,

if and only if

Accordingly, the ﬁelds topped with a ˘ together with Kij describe the deviationof a solution to the conformal constraint equations from data for the exact deSitter spacetime. It is important to observe that asĕa b ,

γ̆i j k ,K̆ij ,

L̆ij ,L̆i ,d˘ij ,d˘∗ij

are scalars, by virtue of the diﬀeomorphism ψ : S → S3 , they can be considered
as ﬁelds over S3 . As such, for m ≥ 0, one deﬁnes the Sobolev norms
ĕa b

S,m ≡

ĕa b
a,b

S3 ,m ,

γ̆i j kS,m ≡γ̆i j kS3 ,m ,

i, j, kand, similarly, for the other ﬁelds – the sums in the previous expressions arecarried out over the independent components of the particular ﬁeld underconsideration. In terms of these norms, it will be said that the initial data for15.2 Perturbations of initial data for the de Sitter spacetime
the conformal ﬁeld equations are ε-close in the norm
for the de Sitter spacetime if
ĕa b

S,m +

γ̆i j k
+

S,m +

L̆i

K̆ij

S,m +

S,m +

d˘ij

L̆ij

S,m

S,m +

d˘∗ij

419

S,m to initial dataS,m < ε.

(15.16)This notion of closeness to initial data is gauge dependent. Nevertheless, it is theappropriate one to exploit the existence and stability theorems of Chapter 12.
15.2.2 Initial data on the conformal boundary
An important property of de Sitter-like spacetimes is that the individualcomponents of the conformal boundary can serve as Cauchy hypersurfacesof the unphysical spacetimes. Accordingly, it is possible to formulate for thesespacetimes an asymptotic initial value problem where initial data areprescribed on a 3-manifold corresponding to, say, I − .
The solutions to the conformal constraint equations at the conformal boundary
have been discussed in Section 11.4.4. In particular, it has been shown that one
needs to prescribe on I − a 3-metric h, a symmetric trace-free and divergencefree tensor corresponding to the initial value of the electric part of the rescaled
Weyl tensor and a function κ. From these free data it is possible to compute thevalues of the remaining conformal ﬁelds. In the particular case of the exact deSitter spacetime it can be veriﬁed that the asymptotic free data are given by
h

h̄,

dij

0,

κ

0,

where components are expressed with respect to the h̄-orthonormal frame {ci }.
From the above one ﬁnds
1
δij , d∗ij 0.
ei j δi j , γi j k i j k , Kij 0, Li 0, Lij
2
Perturbations of the above asymptotic initial data for the de Sitter spacetime
are discussed in a manner similar to that of perturbations of standard Cauchy
data. Accordingly, assuming that I − ≈ S3 , one can make use of diﬀeomorphisms
ψ : I − → S3 to introduce coordinates on the conformal boundary and topull back the components of the various conformal ﬁelds to S3 . Initial data
corresponding to perturbations of asymptotic de Sitter initial data will then be
described in terms of ﬁelds
h = h̄ + h̆,

dij = d˘ij ,κ̆,where d˘ij are the components of a symmetric, h-trace-free and h-divergencefree tensor expressed in terms of the components of the h-orthonormal frame{ei } = {ci + ĕi }. Mimicking the standard Cauchy case, the perturbation of
asymptotic data for the de Sitter spacetime will be said to be ε-close to exact
asymptotic de Sitter data in them -norm if the various conformal ﬁelds
on I − satisfy an inequality of the form of (15.16). In principle, it is possible to420

De Sitter-like spacetimes

express this smallness requirement in terms of a smallness condition on the basic
perturbation data ĕi j , d˘ij and κ̆; this idea will not be further pursued here.
15.3 Global existence and stability using gauge source functions
In this section a ﬁrst proof of the global existence and stability of de Sitterlike spacetimes is provided. This proof makes use of the hyperbolic reduction of
the spinorial conformal ﬁeld equations using gauge source functions as discussed
in Section 13.2 and of the conformal representation of the de Sitter spacetime
discussed in Section 15.1.1. This approach can be readily generalised to include
trace-free matter. The discussion presented here follows the seminal work byFriedrich (1986b, 1991).
15.3.1 Gauge considerations
The ﬁrst step in the construction of de Sitter-like spacetimes consists of the ﬁxingof the gauge in the evolution equations. This gauge ﬁxing allows one to relate,in an unambiguous manner, ﬁelds in the background de Sitter spacetime withﬁelds in the perturbed spacetime; see Figure 15.3.
As the (unphysical) spacetime (M, g) to be constructed will be of the form
M ≈ [a, b]×S3 ⊂ R×S3 with a, b ∈ R, it is natural to make use of the coordinatesand frames in the background spacetime (R × S3 , g E ) to coordinatise andconstruct a suitable gauge in the perturbed spacetime. Following the discussionof Section 13.2.1, coordinates x = (τ, xα ) on the Einstein cylinder ME = R × S3
can be regarded as coordinates on a perturbed spacetime (M, g) if one identiﬁes
the manifolds M and ME . This coordinatisation is equivalent to the coordinate
gauge source choice
F a (x) = −η bc Γ̊b a c = 0,

Figure 15.3 Schematic representation of the construction of coordinates on a
perturbation of the de Sitter spacetime (M, g) using coordinates on the exact
de Sitter spacetime (ME , g E ) and a diﬀeomorphism ϕ : M → M̊ as describedin the main text. The particular realisation of the diﬀeomorphism identiﬁes
the manifolds M and ME in such a way that ϕ is a wave map.

15.3 Global existence and stability using gauge source functions

421

where the last equality follows from the discussion leading to Lemma 15.1. Thisparticular choice of coordinate gauge source function makes the identiﬁcation
between M and ME a wave map; see Section 13.2.1.
By similar considerations, the vectors {c̊a } = {∂ τ ,c̊i } originally deﬁned on
ME can be regarded as vectors on the perturbed spacetime (M, g) in terms ofwhich the g-orthonormal frame {ea } can be expanded by writing ea = ea b c̊b .
In an analogous manner, the ﬁeldsů = (Ξ̊, Σ̊, Σ̊AA , e̊AA a , Γ̊AA B C , Φ̊AA BB  , φ̊ABCD ),as given by Lemma 15.1, can be regarded as ﬁelds over M. It is important toemphasise that all of the above ﬁelds (except for e̊AA a ) are, in fact, componentsof tensors with respect to the background frame {e̊a } = {δa b c̊b }.
The gauge ﬁxing is completed by setting the frame gauge source function
FAB (x) and the conformal gauge source function R(x) equal to their values inthe background spacetime (R × S3 , g E ). That is, one sets
R(x) = −6;

FAB (x) = 0,compare Lemma 15.1.

15.3.2 The evolution system
The hyperbolic reduction procedure discussed in Section 13.2 and summarised in
Proposition 13.1, leads to an evolution system which, in terms of local coordinates
x = (τ, xα ) of an open domain U ⊂ R × S3 , takes the form∂τ σ = G(Γ)σ + H(σ, υ),
I + D0 (e) ∂τ υ + Dα (e)∂α υ = E(Γ)υ + F(σ, υ, φ),
I + A0 (e) ∂τ φ + Aα (e)∂α φ = B(Γ)φ,(15.17a)(15.17b)(15.17c)where σ encodes the conformal factor Θ and the independent components of its
concomitants; υ collects the independent components of the frame components,
the connection coeﬃcients and the trace-free Ricci spinor and φ groups theindependent components of the rescaled Weyl spinor.
To apply the methods of the theory of hyperbolic partial diﬀerential equations(PDEs) discussed in Chapter 12 it is convenient to split the various ﬁeld
unknowns into a background part and a perturbation part. More precisely,
one setsΞ = Ξ̊ + Ξ̆,ΣAB = Σ̆AB ,Σ = Σ̊ + Σ̆,
0

0

0

0

e = e̊ + ĕ ,iie = ĕ ,ΓAB = Γ̆AB ,is = s̊ + s̆,0eAB = ĕAB ,(15.18b)i(15.18c)ieAB = e̊AB + ĕAB ,Γ(AB)CD = Γ̊(AB)CD + Γ̆(AB)CD ,Φ(ABCD) = Φ̆(ABCD) ,(15.18a)ΦAB = Φ̆(AB) ,φABCD = φ̆ABCD ,Φ = Φ̊ + Φ̆,(15.18d)(15.18e)(15.18f)

422

De Sitter-like spacetimes

where
Ξ̊,

Σ̊,

e̊0 ,

s̊,

e̊AB i ,Γ̊(AB)CD ,Φ̊are the non-vanishing components of the ﬁelds describing the background deSitter solution as discussed in Section 15.1.1, while
Ξ̆,

Σ̆,

Σ̆AB ,s̆,Φ̆(ABCD) ,ĕa ,ĕAB a ,Φ̆(AB) ,Φ̆,Γ̆AB ,

Γ̆(AB)CD ,φ̆ABCD(15.19a)(15.19b)describe the perturbations away from the de Sitter solution. The split between
background and perturbations given by Equations (15.17a)–(15.17c) depends
strongly on the choice of gauge.
By construction, the background ﬁelds are a solution to the conformal
evolution Equations (15.17a)–(15.17c). Consequently, one has∂τ σ̊ = G(Γ̊)σ̊ + H(σ̊, υ̊),

I + D0 (e̊) ∂τ υ̊ + Dα (e̊)∂α υ̊ = E(Γ̊)υ̊.
Accordingly, substituting now the ansatz (15.18a)–(15.18f) in the evolution
system (15.17a)–(15.17c), one obtains equations for the independent componentsof the perturbation ﬁelds (15.19a) and (15.19b):∂τ σ̆ = G(Γ̊)σ̆ + G(Γ̆)σ̊ + G(Γ̆)σ̆+ H(σ̊, ῠ) + H(σ̆, υ̊) + H(σ̆, ῠ),(15.20a)α
I + D (e̊ + ĕ) ∂τ ῠ + D (e̊ + ĕ)∂α ῠ = E(Γ̆)υ̊ + E(Γ̊)ῠ + E(Γ̆)ῠ + F(σ̊, υ̊, φ̆)0+ F(σ̊, ῠ, φ̆) + F(σ̆, υ̊, φ̆) + F(σ̆, ῠ, φ̆)− I + D0 (ĕ) ∂τ υ̊,(15.20b)
I + A0 (e̊ + ĕ) ∂τ φ̆ + Aα (e̊ + ĕ)∂α φ̆ = B(Γ̊ + Γ̆)φ̆.

(15.20c)

In view of the properties of the original conformal evolution Equation (15.17a)–(15.17c) the above equations constitute a symmetric hyperbolic evolution system
for the components of ŭ = (σ̆, ῠ, φ̆). Accordingly, the theory of hyperbolic PDEs,
as discussed in Chapter 12, can be applied in domains of the form [0, τ• ]×S3 with
τ• > 0, to guarantee the existence of solutions and to assert Cauchy stability.
In particular, as the background solution (σ̊, υ̊, φ̊) is well deﬁned on the wholeof R × S3 one obtains the following existence, uniqueness and Cauchy stability
result:Proposition 15.1 (existence of solutions to the standard conformal
evolution equations) Let u = ů + ŭ denote de Sitter-like initial data forthe conformal ﬁeld equations prescribed on a 3-manifold S ≈ S3 . Given m ≥ 4
and τ• > 34 π, then:15.3 Global existence and stability using gauge source functions

423(i) There exists ε > 0 such that ifŭm < ε,

then there exists a C m−2 unique solution to the conformal evolution
Equations (15.20a)–(15.20c) deﬁned on [0, τ• ] × S3 .
(n)
(n)
(n)
(ii) Given a sequence of initial data u = ů + ŭ such that
(n)

ŭ

m< ε

and

(n)

ŭ

m→ 0

as

n → ∞,

then for the corresponding solutions ŭ(n) ∈ C m−2 ([0, τ• ] × S3 ) one has thatŭ(n) m → 0 uniformly in τ ∈ [0, τ• ] as n → ∞.
Proof The above proposition is a direct consequence of Theorem 12.4. To apply
this theorem it is necessary to ensure that both
I + A0 (e̊ + ĕ )andI + D0 (e̊ + ĕ )(15.21)are both positive deﬁnite away from zero in a uniform manner over S3 . An explicit
calculation shows that
I + A0 (e̊ )andI + D0 (e̊ )are positive deﬁnite away from zero. Thus, by setting ε suﬃciently small,condition (15.21) can be guaranteed. By further reducing ε, if necessary, one
can ensure that all solutions with ŭ m < ε have a minimum existence timeτ• > 34 π. Taking into account the above, point (i) follows from points (i)–(iii) of
Theorem 12.4 while point (ii) follows from point (iv) in the same theorem.
Remark 1. The purpose of point (i) in Proposition 15.1 is to guarantee aminimum existence time of solutions to the evolution system (15.20a)–(15.20c)
containing the conformal boundary of the perturbed solution.
Remark 2. Point (ii) in Proposition 15.1 is a statement of Cauchy stability.
It ensures that data suﬃciently close to data for the de Sitter spacetime giverise to solutions with an existence time similar to that of the backgroundsolution. Moreover, within the established existence time, the solutions are
suitably close to the background solution. Observe, however, that this result
makes no statement about whether a particular solution converges in time to thebackground solution. Thus, one has obtained only an orbital stability result
for the conformal evolution Equations (15.20a)–(15.20c).
The solutions ŭ provided by Proposition 15.1 give rise, in turn, to a solution
to the conformal ﬁeld equations. More precisely, one has:
Proposition 15.2 (propagation of the constraints for the standard
conformal evolution system) Given a solution u = ů + ŭ to the conformal

424

De Sitter-like spacetimes

evolution Equations (15.20a)–(15.20c) on [0, τ• ] × S3 such that the conformal
constraint equations are satisﬁed on S , thenZa = 0,Zab = 0,Δcdb = 0,

Σa c b = 0,Ξc dab = 0,Λbcd = 0,

on [0, τ• ] × S3 .
Proof From the discussion in Chapters 11 and 13 it follows that if the conformal
constraint equations and the conformal evolution equations are satisﬁed on theinitial hypersurface S , then one obtains
Za |S = 0,

Zab |S = 0,
Δcdb |S = 0,

Σa c b |S = 0,Ξc dab |S = 0,Λbcd |S = 0.

Now, from Proposition 13.2 it follows that the above zero quantities satisfy asymmetric hyperbolic subsidiary evolution system. As the initial data for thisevolution system vanish and the evolution system is homogeneous in the zero
quantities, it follows from Corollary 12.1 that the zero quantities must vanish on[0, τ• ) × S3 so that the result follows.
Locating the conformal boundary
The existence of solutions to the evolution Equations (15.20a)–(15.20c) for aminimum existence interval [0, τ• ) ⊃ [0, 34 π) provides room enough for thedevelopment of the conformal boundary. That this does indeed happen is crucial
for the interpretation of the solution to the conformal evolution equations as a
global solution to the Einstein ﬁeld equations. This property is ensured by thefollowing:
Lemma 15.3 (structure of the conformal boundary ) Given a solution ŭ,
as given by Proposition 15.1, with ŭ m < ε suﬃciently small, there exists afunction τ+ = τ+ (x), x ∈ S3 such that 0 < τ+ (x) < τ• and
Ξ>0on M̃ ≡ (τ, x) ∈ R3 | 0 ≤ τ < τ+ (x) ,
1on I + ≡ (τ+ (x), x) ∈ R × S3 .
Ξ = 0 and Σa Σa = − λ < 03Remark. The above lemma ensures the existence, at least for suﬃciently small
perturbations, of a complete spacelike component of the conformal boundary.
Observe also that the function τ+ (x) provided by Lemma 15.3 deﬁnes a
diﬀeomorphism between S3 and I + . Consequently I + ≈ S3 .
Proof The key observation to prove this result is that Ξ̊|τ =3π/4 < 0. Using
Proposition 15.1 (ii), for suﬃciently small ε > 0 one has (Ξ̊ + Ξ̆)|τ =3π/4 < 0. As(Ξ̊ + Ξ̆)|τ =0 > 0 there must exist a τ+ for which Ξ = 0. By reducing ε further – if

15.3 Global existence and stability using gauge source functions

425

necessary – one has that τ is unique, and, hence, the function τ+ (x) is welldeﬁned. Now, from the conformal Equation (8.28e) it follows that1∇a Ξ∇a Ξ = − λ > 0,3

if Ξ = 0.

Accordingly, τ = τ+ (x) deﬁnes a regular spacelike hypersurface I + .
The last step in the present analysis is to show that the obtained solutions tothe conformal evolution Equations (15.20a)–(15.20c) give rise to a global solution
to the vacuum Einstein ﬁeld equations. One has the following:
Theorem 15.1 (global existence and stability of de Sitter-like spacetimes: gauge source functions version) Given m ≥ 4, a solution u =ů + ŭ to the conformal constraint equations with de Sitter-like cosmologicalconstant such that ŭ m < ε for ε > 0 suitably small gives rise to a unique
C m−2 solution to the conformal Einstein ﬁeld equations on
M ≡ M̃ ∪ I +with M̃ and I + as deﬁned in Lemma 15.3. This solution implies, in turn, asolution (M̃, g̃), to the Einstein ﬁeld equations with de Sitter-like cosmologicalconstant for which I + represents conformal inﬁnity.
Remark. The above theorem together with Propositions 15.1 and 15.2 andLemma 15.3 constitute a technical version of the main theorem of this chapter.
As the component of the conformal boundary obtained by this procedure is aspacelike hypersurface with the topology of S3 , one concludes that the solution(M̃, g̃) to the Einstein ﬁeld equations has the same global structure as the exact
de Sitter spacetime; see Figure 15.4.
Proof From Proposition 8.2; it follows that a solution to the conformal Einsteinﬁeld equations implies the existence of a metric g̃ satisfying the Einstein ﬁeld

Figure 15.4 Penrose diagram of a perturbation of the de Sitter spacetime
given by Theorem 15.1. The spacetime is obtained as a result of an initialvalue problem on the Cauchy hypersurface S .

426

De Sitter-like spacetimes

equations wherever Ξ = 0. The statement about the interpretation of theconformal boundary follows from Lemma 15.3.
15.4 Global existence and stability using conformal Gaussian systems
This section provides an alternative proof of the main theorem of this chapter
using the extended conformal Einstein ﬁeld equations expressed in terms of aconformal Gaussian system. This alternative proof allows one to contrast thestrengths and weaknesses of the two diﬀerent hyperbolic reduction methods
discussed in Chapter 13. As will be seen in the following, the use of propertiesof conformal geodesics greatly simpliﬁes the analysis of the conformal boundaryof the spacetime. Generalising this approach to include matter ﬁelds is, however,more complicated than if one were to use gauge source functions.
The details of the construction of a conformal Gaussian system for theextended conformal ﬁeld equations have already been discussed in Section 13.4.1.
To apply this general discussion to the analysis of perturbations of the de Sitter
spacetime, one needs to specify the particular form of the conformal factor Θand the covector d associated to the congruence of conformal geodesics. This isdone in the following subsection.
15.4.1 A priori analysis of the structure of the conformal boundaryof perturbations of the de Sitter spacetime
One of the advantages of the formulation of the conformal evolution equations interms of conformal Gaussian systems is that it provides an a priori knowledge ofthe location and structure of the conformal boundary; that is, one has an explicit
description of the locus of its points, even before knowing that a solution actuallyexists. This a priori knowledge provides valuable insight into the nature of theunderlying initial value problem.
In what follows, let (M̃, g̃) denote a vacuum spacetime with de Sitter-likecosmological constant. It will be assumed that (M̃, g̃) can be covered with a nonintersecting congruence of conformal geodesics (x(τ̄ ), β̃(τ̄ )) with aﬃne parameterτ̄ and that the data for the congruence is prescribed on a ﬁduciary spacelike
hypersurface S described by the condition τ̄ = 0. From Proposition 5.1 it follows
that, associated to this congruence of conformal geodesics, one has a canonicalconformal factor Θ of the form1
Θ = Θ + Θ̇ τ̄ + Θ̈ τ̄ 2 ,2(15.22)

with the constraintsΘ̇ = d , ẋ ,Θ Θ̈ =1 
1g (d , d ) + λ,2
6(15.23)where the coeﬃcients Θ , Θ̇ and Θ̈ are constant along a given conformal
geodesic. The conformal factor Θ allows one to obtain a conformal extension

15.4 Global existence and stability using conformal Gaussian systems

427(M̄, ḡ) of the physical spacetime (M̃, g̃) with ḡ ≡ Θ2 g̃. The speciﬁc detailsof the conformal factor Θ depend on the location of the hypersurface S withrespect to the conformal boundary and give rise to two diﬀerent initial value
problems.
Standard Cauchy problem
First, consider a situation where the initial hypersurface S does not coincidewith the conformal boundary. As one is interested in the construction of
spacetimes whose spatial sections have the topology of S3 it is natural to set,
without loss of generality, thatΘ = 1,Θ̇ = 0,so that no further distortion is introduced in the 3-manifold and the congruenceof conformal geodesics is symmetric with respect to the initial hypersurface.
Moreover, one can setβ̃  = 0,
so that d = Θ β̃  = 0 and the general expression for the conformal factor
reduces to
Θ=1+1
λτ̄ 2 .
12

Now, using Proposition 5.1 one ﬁnds that the components da of the covectord respect to a Weyl propagated frame {ea } along the congruence of conformal
geodesics and such that e0 = ẋ are given by
d0 = Θ̇,di = 0.

A direct computation shows that the conformal factor Θ vanishes for12τ̄± = ±.
|λ|The above expression gives the location of the conformal boundary. Accordingly,
it is natural to deﬁne
I ± ≡ {τ̄± } × S,
and one has I ± ≈ S3 . Finally, recalling the constraint d = Θf + dΘ andassuming that f is regular at I ± one ﬁndsg(dΘ, dΘ) = η ab da db = Θ̇2 > 0

at I ± .

Thus, if the conformal boundary exists and is regular, it must be a spacelike
hypersurface.

428

De Sitter-like spacetimes
Asymptotic Cauchy problem at I −The conformal ﬁeld equations allow the formulation of an alternative initial value
problem in which initial data are prescribed on a spacelike hypersurface S ≈ S3
representing one of the components of the conformal boundary, say, I − – anasymptotic initial value problem. In this spirit, it is natural to prescribe theinitial data for the congruence of conformal geodesics directly at the conformalboundary. This is made possible by the conformal invariance of the conformal
geodesic equations.
By assumption, on an asymptotic initial value problem one has that Θ = 0 on
I − . Thus, one necessarily has that Θ = 0 and the conformal factor takes theform1Θ = Θ̇ τ̄ + Θ̈ τ̄ 2 .
2
The second expression in Equation (15.23) implies that g  (d , d ) = −λ/3 > 0
so that d must be timelike. Now, taking into account the further constraintd = Θf + dΘ and requiring ẋ to be normal to I − , it follows that|λ|,di = 0,d0 = Θ̇ =3
where the positive root has been chosen so that Θ is positive in the future of
I − . Accordingly, oﬀ I − one has#$|λ|d0 = Θ̇ =+ Θ̈ τ̄ ,di = 0.
3
So far, the coeﬃcient Θ̈ has remained unspeciﬁed. Accordingly, it will beconsidered as free data. These data are, in fact, related to value of the Friedrichscalars≡

1
1∇a ∇a Θ + RΘ424

on I − . On the one hand, a calculation givess
1ea Σa + Γa a b Σb4
1Θ̈ + Θ̇ Γa a 0 ,4

where the last equality follows from the fact that Σi vanishes at I − . On theother hand, the solution to the conformal constraints at the conformal boundary,as discussed in Section 11.4.4, shows that s Θ̇ κ where κ is a scalar ﬁeld overI − ; compare Equation (11.40). A further calculation using a frame adapted toI − readily yields
Γa a 0Γi i 0

χi i

κδi i

3κ.

15.4 Global existence and stability using conformal Gaussian systems

429

One thus concludes thatΘ̈ = κ Θ̇ .
In practice, it is convenient to set κ to be constant on I − . The choice κ = 0
gives a representation in which I − is a hypersurface with vanishing extrinsic
curvature; see Equation (11.41). This representation does not involve a second
component of the conformal boundary.
To have a second component of the conformal boundary one hence needs κ = 0.
Adopting the simple choice Θ̈ = −1/2, the conformal factor vanishes atτ̄− = 0,

τ̄+ = 4|λ|.
3In this conformal representation, the two components of the conformal boundaryof the de Sitter-like spacetime (M̃, g̃) are given by the sets
%  &
|λ|
−
+
I = 4
I = {0} × S,
× S.
3More generally, keeping Θ̈ unspeciﬁed, one ﬁnds that the location of I + isdetermined by the free data Θ̈ . Finally, on I + one hasg(dΘ, dΘ) = η ab da db = −

λ
> 0,
3

so that both I ± , if they exist, are spacelike hypersurfaces.
Remark. In what follows, the analysis of both the standard and the asymptotic
initial value problems will be simpliﬁed by making use of the choice λ = −3 forthe cosmological constant.
15.4.2 The extended conformal evolution system
Once the conformal factor Θ and the covector d associated to the conformal
Gaussian system have been speciﬁed, one can proceed to the formulation of an
initial value problem. In Proposition 13.3 it has been shown that the extended
conformal Einstein ﬁeld equations expressed in terms of a conformal Gaussian
system imply a symmetric hyperbolic evolution system of the form∂τ̄ υ̂ = Kυ̂ + Q(Γ̂)υ̂ + L(x)φ,
I + A0 (e) ∂τ̄ φ + Aα (e)∂α φ = B(Γ̂)φ,(15.24a)(15.24b)for û = (υ̂, φ) where υ̂ encodes the independent components of the frame, theconnection coeﬃcients and the Schouten tensor, while φ contains the componentsof the rescaled Weyl spinor.

430

De Sitter-like spacetimes

Mimicking the analysis of Section 15.3, one considers solutions of the formeAB 0 = ĕAB 0 , eAB α = ēAB α + ĕAB α ,ξABCD = ξ¯ABCD + ξ˘ABCD , χABCD = χ̄ABCD + χ̆ABCD ,ΘABCD = Θ̄ABCD + Θ̆ABCD ,fAB = f˘AB ,φABCD = φ̆ABCD ,whereēAB μ ,ξ¯ABCD ,χ̄ABCD ,Θ̄ABCDare the values of the exact de Sitter solution expressed in a conformal Gaussian
system as discussed in Section 15.1.2; see, in particular, Proposition 15.2.
Accordingly, one can writeυ̂ = ῡ + ῠ,φ = φ̆,(15.25a)e = ē + ĕ,Γ̂ = Γ̄ + Γ̆.

(15.25b)On the initial hypersurface S one hasυ̂  = ῡ  + ῠ  ,φ = φ̆ ,where ů = (ῡ  , 0) is the exact de Sitter data discussed in Section 15.2 andŭ = (ῠ  , φ̆ ).
As the conformal factor Θ and the covector d are universal – that is, they
possess the same form for either the exact de Sitter data or the perturbations
thereof – one has∂τ̄ ῡ = Kῡ + Q(Γ̄)ῡ.
Substituting the ansatz (15.25a) and (15.25b) into Equations (15.24a) and(15.24b) yields the following evolution equations for ŭ = (ῠ, φ̆):∂τ̄ ῠ = Kῠ + Q(Γ̄ + Γ̆)ῠ + Q(Γ̆)ῡ + L(x)φ̆,
I + A0 (ē + ĕ) ∂τ̄ φ̆ + Aα (ē + ĕ)∂α φ̆ = B(Γ̄ + Γ̆)φ̆.

(15.26a)(15.26b)

The above equations are already in a form where the theory of hyperbolic
PDEs, as discussed in Chapter 12, can be applied. In particular, existence andCauchy stability of Equations (15.26a) and (15.26b) are given by Theorem 12.4.
The natural domains for solutions to these equations are of the formMτ̄• ≡ [0, τ̄• ] × S,

S ≈ S3 ,for some τ̄• > 0. The analogue of Propositions 15.1 and 15.2 for the conformal
evolution system (15.26a) and (15.26b) is given by:
Proposition 15.3 (existence of de Sitter-like solutions to the extended
conformal evolution equations and propagation of the constraints) Letû = ū + ŭ denote de Sitter-like (standard or asymptotic) initial data for theconformal ﬁeld equations prescribed on a 3-manifold S ≈ S3 . Given m ≥ 4:

15.4 Global existence and stability using conformal Gaussian systems

431(i) There exists ε > 0 such that ifŭm < ε,

then there exists a C m−2 unique solution ŭ to the conformal evolution
equations (15.26a) and (15.26b) deﬁned on (− 52 , 52 ) × S3 for the standard
Cauchy problem and on [0, 92 ) × S3 for the asymptotic Cauchy problem.
(ii) IfΞ̂c dab S = 0,Σ̂a c b S = 0,Δ̂abc S = 0,Λabc S = 0,and
δa S = 0,

γab S = 0,

ςab S = 0,then the solution ŭ to the conformal evolution equations given by (i) implies,
by reducing ε if necessary, a C m−2 solution û = ū + ŭ to the extended
conformal ﬁeld equations on (− 52 , 52 ) × S3 and, respectively, on [0, 92 ) × S3 .
(n)
(n)
(n)
(iii) Given a sequence of initial data û = ū + ŭ such that
(n)

ŭ

m < ε,

and

(n)

ŭ

m→ 0

as n → ∞,then for the corresponding solutions ŭ(n) ∈ C m−2 (− 52 , 52 ) × S3 and,ŭ(n) m → 0 uniformly inrespectively, C m−2 ([0, 92 ) × S3 ), one hasτ̄ ∈ (− 52 , 52 ) and, respectively τ̄ ∈ [0, 92 ), as n → ∞.
Proof The proof of points (i) and (iii) of the above proposition is, again, a direct
application of Theorem 12.4 along the lines of Proposition 15.1. The proof ofpoint (ii) concerning the existence of an actual solution of the extended conformal
ﬁeld equations follows from the homogeneity of the subsidiary evolution system asgiven in Proposition 13.4 together with Corollary 12.1 by an argument identicalto that used in Proposition 15.2.
Remark. By an argument similar to the one leading to Proposition 15.3, using
the expression (15.14) for a separation vector in the background de Sitter
spacetime, it can be shown that if ε is suﬃciently small, then the separationﬁelds for the perturbed spacetime remain non-zero in (− 52 , 52 ) and, respectively,[0, 92 ). Thus, the conformal Gaussian system used in the hyperbolic reduction
remains well behaved throughout.
Constructing solutions to the Einstein ﬁeld equations
The discussion of this section can be summarised in the following two technicalversions of the main theorem of this chapter:

432

De Sitter-like spacetimes

Theorem 15.2 (global existence and stability of de Sitter-like spacetimes: conformal Gaussian systems version) Given m ≥ 4, a solutionu = ū + ŭ to the conformal constraint equations with λ = −3 on a standard
Cauchy hypersurface S ≈ S3 such that ŭ m < ε, for ε > 0 suitably small,gives rise to a solution u to the conformal ﬁeld equations on
M ≡ [−2, 2] × S3 .
This solution implies, in turn, a solution (M̃, g̃) to the Einstein ﬁeld equations
with cosmological constant λ = −3 whereM̃ ≡ (−2, 2) × S3 ,for whichI ± ≡ {±2} × S3 ,represent future and past conformal inﬁnity, respectively.
In the case of asymptotic Cauchy data one obtains a similar statement:
Theorem 15.3 (global existence and stability for the asymptotic initialvalue problem) Given m ≥ 4, a solution u = ū + ŭ to the conformal
constraint equations with λ = −3 on a 3-manifold S ≈ S3 representing the pastcomponent of the conformal boundary such that ŭ m < ε, for ε > 0 suitably
small, gives rise to a solution u to the conformal ﬁeld equations on
M ≡ [0, 4] × S3 .
This solution implies, in turn, a solution (M̃, g̃) to the Einstein ﬁeld equations
with cosmological constant λ = −3 whereM̃ ≡ (0, 4) × S3 ,for whichI − ≡ {0} × S3 ,I + ≡ {4} × S3 ,represent future and past conformal inﬁnity, respectively.
The proofs of Theorems 15.2 and 15.3 are identical to that of Theorem 15.1.
Penrose diagrams of the spacetimes thus obtained can be seen in Figure 15.5.
Observe that in the gauge being considered, the Penrose diagrams for the exact
de Sitter spacetime and the perturbations are identical!

15.4 Global existence and stability using conformal Gaussian systems

433

Figure 15.5 Penrose diagrams of de Sitter-like spacetimes obtained from
Theorems 15.2 and 15.3: on the left is the spacetime obtained from a standard
Cauchy initial value problem; on the right is the spacetime obtained from theasymptotic initial value problem.

15.4.3 Geodesic completeness and asymptotic analysis
The analysis of the existence and stability of de Sitter-like spacetimes developedin Sections 15.3 and 15.4 can be reﬁned to include geodesic completeness. As
the exact de Sitter spacetime is geodesically complete, it is to be expected thatsuitably small perturbations thereof will also share this property. More precisely:
Proposition 15.4 (geodesic completeness of de Sitter-like spacetimes)
Suitably small perturbations (M̃, g̃) of the de Sitter spacetime are null andtimelike g̃-geodesically complete.
In particular, the above proposition together with the existence and stability
results obtained in the previous sections show that suitably small perturbationsof the de Sitter spacetime are asymptotically simple spacetimes.
It is convenient to divide the analysis of Proposition 15.4 into two cases.
Null geodesics
The key observation required to prove null geodesic completeness is the following:
given the conformal representation (R × S3 , ḡ E ) any null ḡ E -geodesic starting
within the unphysical spacetime reaches the conformal boundary for a ﬁnite valueof its aﬃne parameter.
In what follows, let (M̄, ḡ) be one of the de Sitter-like spacetimes obtained
from, say, a standard Cauchy initial value problem with data prescribed on a
hypersurface S . Making use of a perturbative argument similar to the ones
employed in Propositions 15.1 and 15.3 and by reducing ε, if necessary, it canbe shown that given a point p ∈ S and a ﬁxed δ > 0, for all points q ∈ S lying

434

De Sitter-like spacetimes

in an h-metric ball of radius δ centred at p, the future directed null ḡ-geodesics
starting at q will reach I + in a ﬁnite value of their aﬃne parameter. As S is acompact hypersurface, it can be covered with a ﬁnite number of such h-metricballs of radius δ, and, accordingly, there exist non-trivial perturbations of thede Sitter spacetime for which all null ḡ-geodesics starting on S reach I + ina ﬁnite value of their aﬃne parameter. Now, every ḡ-null geodesic is (up to a
reparametrisation) also a g̃-null geodesic. Moreover, making use of the discussionin Chapters 7 and 10, one can ﬁnd an aﬃne parameter s of the g̃-null geodesic
such that s → ∞ as Θ → 0. Hence, one concludes null g̃-completeness.
Timelike geodesics
In the case of timelike geodesics, following Lemma 5.2 every timelike g̃-geodesicis, up to a reparametrisation, a timelike ḡ E -conformal geodesic. It can be
explicitly checked – starting, for example, from the general solution to theconformal geodesic equations in the Minkowski spacetime as discussed in Section
6.2.3 – that every ḡ E -conformal geodesic starting inside the region of the Einstein
cylinder associated to the conformal de Sitter spacetime reaches the conformalboundary of the spacetime in a ﬁnite amount of its unphysical proper timeτ̄ . Using, as in the case of the null geodesics, a perturbative argument, thisproperty is seen to be preserved for suitably small perturbations of the de Sitter
spacetime. Of course, not every ḡ-conformal geodesic can be reparametrised toa g̃-geodesic. This is the case only for those curves reaching the conformallyboundary orthogonally – as can be checked using Lemma 5.3. Finally, using theproperties of conformal geodesics in Einstein spaces as discussed in Section 5.5.6,
the physical proper time of g̃ satisﬁes τ̃ → ∞ as Θ → 0. This implies geodesic
completeness.
15.5 Extensions
The results of this chapter can be extended to the case of the Einstein equations
coupled with suitable trace-free matter; see Chapter 9.
For simplicity, the subsequent discussion will be restricted to the standard
conformal ﬁeld equations. One of the main diﬃculties when attempting a direct
extension to include matter is the presence of the rescaled Cotton tensor Tcdb in
the Cotton and Bianchi equations. As discussed in Chapter 9 this tensor involves
derivatives of the matter ﬁelds. As the Cotton and the Bianchi equations areinterpreted as diﬀerential conditions on the components of the Schouten tensor
and the rescaled Weyl tensor, the inclusion of matter in the analysis brings
further terms into the principal part of the conformal evolution equations which,in principle, destroy the symmetric hyperbolicity. In general, these derivatives
cannot be eliminated using the ﬁeld equations for the matter model. Thus, one
introduces the derivatives of the matter variables as new unknowns into the
problem. Equations for these auxiliary variables can be obtained by applying

15.5 Extensions

435

a covariant derivative to the matter equation and commuting derivatives. This
procedure has been described, for the Maxwell ﬁeld, in Section 9.2. For suitable
matter models – such as the Maxwell ﬁeld, the conformally invariant scalar ﬁeld
and radiation ﬂuids – the resulting ﬁeld equations for the auxiliary ﬁeld admit
a symmetric hyperbolic reduction without the need of introducing further gaugesource functions.
The construction of suitable symmetric hyperbolic evolution equations forthe auxiliary ﬁelds needs to be supplemented with their associated subsidiaryevolution equations and a further subsidiary equation for the deﬁnition of theauxiliary variable. This procedure is similar in spirit to the construction ofsubsidiary equations for the geometric ﬁelds described in Sections 13.3 and 13.4.5.
The procedure brieﬂy described in the previous paragraph has been implemented by Friedrich (1991) for the Einstein-Yang-Mills system, using thestandard conformal ﬁeld equations and a hyperbolic reduction involving gaugesource functions to obtain a generalisation of the existence and stability result
given in the main theorem of this chapter. The same basic ideas can be used toobtain a future global existence and stability result for perturbations of radiation
perfect ﬂuid Friedman-Robertson-Walker cosmological models; see Lübbe andValiente Kroon (2013b).
Matter and the extended conformal ﬁeld equations
The implementation of the ideas discussed in the previous paragraphs to theextended conformal ﬁeld equations requires further consideration. The matterﬁeld equations are usually expressed in terms of the Levi-Civita connection ∇ ofthe unphysical metric g. However, the conformal ﬁeld equations provide direct
ˆ Equation (5.30a),
access only to the Riemann tensor of the Weyl connection ∇.
ˆ
relating the Riemann tensors of the connections ∇ and ∇, involves the covariant
derivatives of the covector fa . Thus, further derivatives of the conformal ﬁelds
enter the principal part of the evolution system in a way which destroys theˆ [a fb] can besymmetric hyperbolicity. The antisymmetric part of the derivative ∇replaced by terms not containing derivatives using the equation
ˆ b fa = L̂ab − L̂ba ;ˆ a fb − ∇∇compare Equation (8.45). However, a similar substitution is not possible forˆ (a fb) . In order to obtain a suitable symmetric hyperbolic
the symmetric part ∇ˆ (a fb) as an unknown of the system – or,system one needs to introduce ∇alternatively, the components Lab of the Schouten tensor of the unphysical LeviCivita connection ∇. In the case of the Einstein-Maxwell equations it is possible
to ﬁnd suitable evolution equations for the auxiliary ﬁeld ψAA BC ≡ ∇AA φBC
ˆ (a fb) ; see Lübbe and Valiente
which do not contain the symmetrised derivative ∇Kroon (2012). This, however, is an exceptional situation.

436

De Sitter-like spacetimes
15.6 Further reading

The results discussed in this chapter were ﬁrst obtained in Friedrich (1986b).
Similar results starting from asymptotic Cauchy data were ﬁrst discussed inFriedrich (1986a). These results have been extended to the case of the Einsteinequations coupled to a Yang-Mills ﬁeld in Friedrich (1991). Alternative proofs,
which make use of the extended conformal ﬁeld equations and conformal gaugesystems, in the vacuum and Einstein-Maxwell case, have been given in Lübbe
and Valiente Kroon (2009, 2012).
A diﬀerent way of generalising the global existence and stability results
discussed in this chapter is to consider higher dimensions. In this case one
cannot make use of the conformal Einstein ﬁeld equations of Chapter 8,
which are valid only for four-dimensional spacetimes. Alternative ﬁeld equations
are required. Global existence and stability results for de Sitter-like vacuum
spacetimes of arbitrary dimension have been given in Anderson (2005a) andAnderson and Chruściel (2005) using the conformal equations implied by theFeﬀerman-Graham obstruction tensor.
The methods of this chapter can be adapted to analyse perturbations ofcosmological models with radiation perfect ﬂuids and an asymptotic structure
similar to that of de Sitter spacetime. An example of this type of analysis canbe found in Lübbe and Valiente Kroon (2013b).

16
Minkowski-like spacetimes

This chapter studies the existence and stability of Minkowski-like spacetimes,that is, solutions to the vacuum Einstein ﬁeld equations with vanishing cosmological constant. The main result of this chapter is very similar in spirit to
the main result concerning the global existence and stability of de Sitter-likespacetimes of Chapter 15. There is, however, a key diﬀerence: while the resultsin Chapter 15 are global in nature, the ones in the present chapter are semiglobal . More precisely, the spacetimes to be discussed arise as the developmentof suitable initial data on hyperboloidal hypersurfaces – an examination of thePenrose diagram of the Minkowski spacetime in Figure 16.1 reveals that these
hypersurfaces are not Cauchy hypersurfaces of the spacetime. Accordingly, only
a portion of the whole spacetime can be recovered from this type of initial value
problem. The main result of this chapter can be formulated as follows:
Theorem (semiglobal existence and stability of Minkowski-like spacetimes). Small enough perturbations of hyperboloidal initial data for the
Minkowski spacetime give rise to solutions to the vacuum Einstein ﬁeld equations
which exist globally towards the future and have an asymptotic structure similar
to that of the Minkowski spacetime.
This result was ﬁrst proved in Friedrich (1986b) and subsequently extended
to the Einstein-Yang-Mills equations in Friedrich (1991). The original proof ofthe result made use of the standard conformal Einstein ﬁeld equations and issimilar to the argument given for the de Sitter spacetime in Section 15.3. In
this chapter a proof of the theorem is given which makes use of the extended
conformal ﬁeld equations and conformal Gaussian systems following the ideas inLübbe and Valiente Kroon (2009). This approach allows for a more detailed andexplicit discussion of the structure of the conformal boundary.
The restriction of the analysis of the present chapter to the hyperboloidal
initial value problem may seem mysterious at ﬁrst sight. As will be discussed
in some detail in Chapter 20, the initial data for the conformal Einstein ﬁeld

438

Minkowski-like spacetimes

Figure 16.1 Penrose diagram of the Minkowski spacetime and the regionsthat can be recovered from data on the standard hyperboloid H . The future
and past domains of dependence D+ (H ) and D− (H ) are depicted in grey
shading. Observe that H is not a Cauchy hypersurface as there are portionsof the conformal diagram that cannot be recovered from the data on H .

equations on an asymptotically Euclidean (Cauchy) hypersurface is generically
singular at spatial inﬁnity – the various issues associated to this singularbehaviour are usually known as the problem of spatial inﬁnity.
Despite the above limitation, hyperboloidal initial value problems arise
naturally in evolution problems in which the behaviour of gravitational radiationis the main concern; see, for example, Rinne and Moncrief (2013) or Zenginoglu(2008). While the ADM mass – which is computed on asymptotically ﬂat
hypersurfaces (see Section 11.6.1) – is a conserved quantity, the notion of mass
associated to hyperboloidal hypersurfaces, the so-called Bondi mass, showsa monotonic behaviour, and so it describes the process of mass loss due togravitational radiation; see Section 10.4.
16.1 The Minkowski spacetime and the conformal ﬁeld equations
The ﬁrst step of the stability analysis is a study of the Minkowski spacetime in
the gauge used to deduce the conformal evolution equations of Proposition 13.3.
16.1.1 The basic representation
As discussed in Section 6.2, the Minkowski spacetime (R4 , η̃) can be conformallyembedded into the expanding Einstein cylinder (R × S3 , ḡ E ) where

2
τ̄ 2
h̄,
ḡ E ≡ dτ̄ ⊗ dτ̄ − 1 +4by means of the conformal rescalingḡ E = Θ2M η̃,ΘM ≡ 2 cos2



ψ
ψ
1
1 − tan2 τ̄ 2 .
2
4
2(16.1)

16.1 The Minkowski spacetime and the conformal ﬁeld equations

439

The coordinate τ̄ is an aﬃne parameter of the conformal geodesics (xE (τ̄ ), β̄ E (τ̄ ))withxE (τ̄ ) = ∂ τ̄ ,xE (τ̄ ) = (τ̄ , xα ),β̄ E (τ̄ ) =

2τ̄
dτ̄ .
4 + τ̄ 2

The underlying geometry of the conformal representation of the Minkowski
spacetime described in the previous paragraph is that of the expanding cylinder. Accordingly, the geometric ﬁelds for this conformal representation of the
Minkowski spacetime coincide with those of the conformal representation of thede Sitter spacetime discussed in Section 15.1.2. That is, one has
ē0 = 1,ē(AB) 0 = 0,ēi = 0,ē(AB) i =(16.2a)4
σAB i ,
4 + τ̄ 2

fAB = 0,(16.2c)4ihABCD ,
ξ¯ABCD = −4 + τ̄ 22i
hABCD ,
χ̄(AB)CD =4 + τ̄ 2
ΘAB = 0,φABCD = 0.

(16.2b)

ΘABCD = −(16.2d)(16.2e)2hABCD ,
4 + τ̄ 2(16.2f)(16.2g)

It is important to emphasise, however, that the conformal gauge ﬁelds ΘM anddM , relating the conformal representation to the physical Minkowski spacetime,
are diﬀerent from those of the de Sitter spacetime.
A schematic representation of the conformal boundary associated to the above
conformal representation of the Minkowski spacetime is given in Figure 16.2.
It is observed that, as a consequence of the explicit time symmetry of the

Figure 16.2 Plot of the conformal boundary of the Minkowski spacetimein the conformal Gaussian gauge given by Equation (16.1). This conformal
representation is explicitly time symmetric and does not contain the points i±
representing future and past null inﬁnity. The image has been cropped. Thisﬁgure is a coordinate plot, not a conformal diagram; thus, null geodesics do
not have a slope of 45 degrees.

440

Minkowski-like spacetimes

representation, the points i± representing future and past timelike inﬁnity are notincluded. This representation is not the most convenient one to use in analysinga hyperboloidal initial value problem. A related, more convenient representation
is given in the next subsection.
16.1.2 A conformal representation adapted to thestandard hyperboloid
As discussed in Section 6.2, the Minkowski spacetime can be embedded into the
Einstein cylinder using the conformal factorΞM ≡ cos τ + cos ψ.
In the following it will be convenient to shift the above standard embeddingby π/2 to the past with the replacement τ → τ̌ + π/2, so that the standard
Minkowski hyperboloid H which is given by the condition τ = π/2 is now
located at τ̌ = 0; see Equation (6.26). Accordingly, one obtains the shifted
conformal factorπΞ̌M ≡ cos τ̌ ++ cos ψ = cos ψ − sin τ̌ .
(16.3)
2
In particular, the conformal factor embedding the hyperboloidal 3-manifold into
S3 is given byΩ̄ ≡ cos ψ.
One has that Ω̄ = 0 at ψ = π/2. Hence, it is natural to deﬁneπ∂H ≡ p ∈ S3 | ψ(p) =.
2
Observe that dΩ̄ = 0 at ∂H .
To relate the conformal representation of the Minkowski spacetime given by the
conformal factor in Equation (16.3) to the so-called expanding Einstein cylinder
discussed in Section 16.1.1, it is recalled that g E = Ξ̌2M η̃ and ḡ E = Θ̄2E g E so thatḡ E = Θ̌2M η̃,Θ̌ ≡ Θ̄E Ξ̌M .

The relation betweenthe shifted coordinateτ̌ and the aﬃne parameter τ̄ of theconformal geodesics xE (τ̄ ), β E (τ̄ ) in the Einstein cylinder is, formally, thesame as the one between the original coordinate τ and τ̄ ; in particular, one
has that τ̄ = 0 if τ̌ = 0. Thus, using the conformal transformation properties
ofas described in Section 5.5.2, one ﬁnds that the pair conformal geodesicsx̌M (τ̌ ), β̌ M (τ̌ ) with


2 τ̌
α
x̌M ≡ (τ̄ , xα
,
x
)
=
cos
,

2 


1
τ̌
dτ̌ +
cos τ̌ dτ̌ + sin ψdψβ̌ M ≡ β E + Ξ̌−1
M dΞ̌M = tan
2sin τ̌ − cos ψ16.1 The Minkowski spacetime and the conformal ﬁeld equations

441

and
τ̌ = 2 arctanτ̄2(16.4)

gives rise to a congruence of conformal geodesics in the Minkowski spacetime
adapted to the conformal factor Ξ̌M in Equation (16.3). This congruence canbe used to construct a conformal Gaussian gauge system for the Minkowski
spacetime.
A calculation using standard trigonometric identities and the relation (16.4)
between the parameters τ̌ and τ̄ yields the expressionτ̄ 2Θ̌ = cos ψ 1 − sec ψτ̄ +4


(16.5)

for the conformal factor associated to the new conformal Gaussian gauge system.
This conformal factor vanishes wheneverτ̄ =2 ± sin ψ.
cos ψ

A plot of this conformal factor can be seen in Figure 16.3. Moreover, thecomponents of the covector ďM ≡ Θ̌β̌ M with respect to a Weyl propagated˙ M are given byframe {ēa } such that ē0 = x̌β̌0 = ∂τ̄ Θ̌M ,β̌i = ēi (Ω̄).

Finally, it follows from the discussion from the previous paragraphs that thegeometry of the conformal representation of the Minkowski spacetime given by
the conformal factor (16.5) is described by the ﬁelds (16.2a)–(16.2g); that is,
the geometry of this representation coincides with that of the expanding Einstein

Figure 16.3 Plot of the conformal boundary for the Minkowski spacetime ina conformal Gaussian gauge adapted to the standard hyperboloid. In thisparticular representation timelike inﬁnity i+ is at a ﬁnite location. The set∂H denotes the intersection of the conformal boundary I + with the initial
hyperboloid H . As in the case of Figure 16.2, this plot is not a conformal
diagram.

442

Minkowski-like spacetimes

cylinder. In particular, a suitable Jacobi ﬁeld zAB measuring the deviation of
the curves of the congruence of conformal geodesics is given byτ̄ 2z(AB) = 1 +z(AB) ,4

z = 0,

where z(AB) is some ﬁduciary initial value at the standard hyperboloid H ;compare Equation (15.14).
16.1.3 Initial data for the Minkowski spacetimeon the standard hyperboloid
The congruence of conformal geodesics giving rise to the conformal Gaussian
system is speciﬁed on the standard hyperboloid H by the dataΘ̌ = cos ψ,dˇ0 = −1,dˇi = ci (Ω̄).

On H , the conformal ﬁelds satisfy the conditionsē0 = 1,ifAB = 0,ē(AB) 0 = 0,i

(16.6a)iē(AB) = σAB ,ē = 0,ξ¯ABCD = −ihABCD ,χ̄(AB)CD = 0,Θ̄AB = 0,

1
Θ̄ABCD = − hABCD ,2
φABCD = 0.

(16.6b)(16.6c)(16.6d)(16.6e)16.2 Perturbations of hyperboloidal data for the
Minkowski spacetime
In what follows, it is assumed that one has a solution (H, h, K, Ω, Σ) to theconformal Hamiltonian and momentum constraints, Equations (11.15a) and(11.15b), with hyperboloidal boundary conditions as discussed in Section 11.7. It
is convenient to regard the hyperboloidal manifold H as a region of a 3-manifold
S ≈ S3 . Following the conventions of the previous chapters, when regarding the
3-manifolds H and S as hypersurfaces of a four-dimensional spacetime one writes
H and S , respectively. One can use coordinates (xα ) on S3 as coordinates on
H and introduce reference frame and coframe ﬁelds {ci } and {αi } by requiring
the identiﬁcation between S and S3 to be a harmonic map; see the discussion in
Section 15.2.1.
Initial data for the conformal evolution equations can be obtained from thebasic initial data (H, h, K, Ω, Σ) using the procedure described in Section 11.4.3.
It will be assumed that the data can be written on the initial hyperboloid H in
the form

16.2 Perturbations of hyperboloidal data for the Minkowski spacetime 443
e(AB) 0 = 0,e0 = 1,iie = 0,(16.7a)iie(AB) = σAB + ĕ(AB) ,(16.7b)fAB = 0,(16.7c)ξABCD = ξ¯ABCD + ξ˘ABCD ,(16.7d)χ(AB)CD = χ̄(AB)CD ,(16.7e)ΘAB = Θ̆AB ,(16.7f)

ΘABCD = Θ̄ABCD + Θ̆ABCD ,φABCD = φ̆ABCD ,(16.7g)withξ¯ABCD ,χ̄(AB)CD ,Θ̄ABCD

as given by Equations (16.6a)–(16.6e), while the ﬁeldsĕ(AB) i ,ξ˘ABCD ,χ̆(AB)CD ,Θ̆AB ,

Θ̆ABCD ,φ̆ABCDdescribe the perturbation from standard hyperboloidal Minkowski data andσAB i are the spatial Infeld-van der Waerden symbols.
While the background ﬁelds σAB i , ξ¯ABCD , χ̄(AB)CD , Θ̄ABCD are
deﬁned on the whole of S ≈ S3 , the perturbation ﬁelds ĕ(AB) i , ξ˘ABCD ,χ̆(AB)CD , Θ̆AB , Θ̆ABCD , φ̆ABCD are deﬁned only on H. To apply thebasic existence and stability result, Theorem 12.4, to the present situation
one extends the hyperboloidal initial data set on H to data on S. Using theextension theorem, Proposition 12.2, and given m ≥ 4 there exists a linear
operator E : H m (H, CN ) → H m (S, CN ) such that for ŭ ∈ H m (H, CN ) then(E ŭ )(x) = ŭ (x) almost everywhere in H andE ŭm,S ≤ K

ŭ

m,H ,

with K a universal constant for ﬁxed m. As in the case of the de Sitter spacetime,
the background initial data ů is deﬁned on the whole of S so that the extendeddatau = ů + E ŭ(16.8)

is a well-deﬁned function in H m (S, CN ). The extension of the hyperboloidal
data given by (16.8) is non-unique and, in general, will not satisfy the conformal
constraint equations on S \ H. As the norm E ŭ m,S is dominated by theE ŭ m,S can be made as small as necessary bynormŭ m,H , thenmaking u m,H suitably small. In complete analogy to the case of the deSitter spacetime, one says that a hyperboloidal initial data set of the form
(16.7a)–(16.7g) is ε-small (in the normS,m ) ifĕ(AB) i

S,m +ξ˘ABCD
+

S,m +

Θ̆AB

S,m +

χ̆ABCD

S,mΘ̆ABCDS,m +φ̆ABCDS,m < ε,444

Minkowski-like spacetimes

Figure 16.4 Domain of dependence D+ (H ) of data for the conformal evolution equations on a hyperboloid H : on the left, a schematic representation of
the setup; on the right, a three-dimensional depiction. To make use of Kato’s
existence theorem, the data have to be extended to S \H where S ≈ S3 . The
domain of dependence of the extended data D+ (S ) corresponds, in principle,
to the cylinder [0, ∞)×S3 . The chronological future of the extension on S \H ,denoted by I + (S \ H ), does not intersect the domain of dependence of thehyperboloidal data D+ (H ), and, thus, it is independent of the particularextension being used.

where it is understood that each of the terms in the above expression comprises
a sum over all the independent components of the spinorial ﬁeld underconsideration.
The extended data (16.8) is non-unique. This non-uniqueness does not pose
any problem for the considerations of this chapter. While it is true that thedevelopment D+ (S ) is clearly dependent on the particular extension of theinitial data, one has
D+ (H ) ∩ I + (S \ H ) = Ø;compare the Remark at the end of Section 14.2. Thus, the particular choice ofextension of the data on H has no eﬀect on D+ (H ); see Figure 16.4 for further
details.
16.3 A priori structure of the conformal boundary
This section discusses the available a priori knowledge of the structure of theconformal boundary of the development of hyperboloidal initial data.
In what follows, assume that H can be regarded as an open subset of acompact manifold S ≈ S3 . Moreover, assume that ∂H ≈ S2 . On S one
considers a conformal factor Ω such that Ω > 0 in the interior of H and Ω = 0
on ∂H ; that is, the conformal factor Ω can be thought of as a boundary-deﬁning
function. Consistent with the hyperbolic reduction procedure for the extended
conformal ﬁeld equations as described in Section 13.4, it is assumed that thedomain of dependence D+ (S ) can be covered by a non-singular congruence of16.3 A priori structure of the conformal boundary445

conformal geodesics with data prescribed on S . In particular, it is required thatthe conformal geodesics are initially orthogonal to S .
Determining the conformal factor
From Proposition 5.1, it follows that the general form of the conformal factor
associated to the congruence of conformal geodesics is given by1
Θ = Θ + Θ̇ τ̄ + Θ̈ τ̄ 2 ,2(16.9)

where the coeﬃcients Θ , Θ̇ and Θ̈ are functions of the spatial coordinates andare subject to the constraintsΘ̇ = d , ẋ ,Θ Θ̈ =1 
g (d , d ).
2(16.10)

It is convenient to setΘ = Ω
and to make the spatial part of d equal to dΩ. Accordingly, one ﬁnds thatd0 = Θ̇ ,di = Di Ω.

Now, lettingα ≡ Ω−1 Θ̇ ,one ﬁnds from the constraints in (16.10) that2ΩΘ̈ = h (dΩ, dΩ) + α2 ,where it is recalled that h (dΩ, dΩ) < 0 as a consequence of the signature
convention. Whenever Ω = 0, it follows from the constraints (16.10) that d
must be a null covector as dΘ = 0.
Making use of the above observations, one ﬁnds that Equation (16.9) takes
the particular form
 


1 2
1
α − 2 τ̄ 2 ,Θ = Ω 1 + ατ̄ +
4
ω
where
ω≡'

2Ω

.

|h (dΩ, dΩ)|A calculation shows that Θ = 0 forτ̄± ≡2αω 2 ± 4ω.
4 − α2 ω 2(16.11)

446

Minkowski-like spacetimes

Accordingly, it is natural to deﬁne the future (and, respectively, past) nullinﬁnity of the development associated to the hyperboloid H as
I

±



3
≡ (τ̄ , x) ∈ R × S | τ̄ = τ̄± (x) .

This expression shows how the location of the conformal boundary is predetermined by the initial data Ω and d as long as the underlying congruenceof conformal geodesics remains non-singular. As Ω → 0, one has that eitherτ̄± → 0 or τ̄± → −2Θ̇ /Θ̈ . It follows that I + and I − are smooth hypersurfaces
whenever dΘ = 0. Moreover, ∂H is the intersection of I ± with H = {0} × H
as is to be expected for hyperboloidal data. In analogy to the model caseof the hyperboloids in the Minkowski spacetime, the development of generic
hyperboloidal data has a conformal boundary which corresponds to either I +or I − , but not both; see Figure 16.5. This information is contained in the signof the free datum Θ̇ . By convention, the conformal factor is positive in theregion corresponding to the physical spacetime (M̃, g̃). Accordingly, if Θ̇ > 0
on ∂H , then M̃ lies to the future of the conformal boundary and one speaks of a
hyperboloid which intersects past null inﬁnity, and, thus, the conformal boundary
is identiﬁed with I − . If, by contrast, Θ̇ < 0 on ∂H , then M̃ lies to the pastof the conformal boundary. In this case, the hyperboloid intersects future nullinﬁnity and I + gives the conformal boundary. Without loss of generality, inthe following, attention will be restricted to hyperboloids intersecting future nullinﬁnity so that Θ̇ < 0 on ∂H .

Figure 16.5 The two possible conﬁgurations of the conformal boundary for
hyperboloidal data as discussed in the main text: on the left, one has thecase Θ̇ < 0 at ∂H where the conformal boundary given by the conformal
Gaussian gauge system corresponds to I + ; on the right, one has the situation
corresponding to Θ̇ > 0 at ∂H so the realised component of the conformalboundary is given by I − .

16.3 A priori structure of the conformal boundary

447

Timelike inﬁnity
To identify the points which can be regarded as representing timelike inﬁnity,
one needs to investigate the critical points of Θ on the conformal boundary, thatis, the points where dΘ = 0 and τ̄ = τ̄± (x). A calculation shows that


 



1 2
1 2
1
1
2
dΘ = 1 + ατ̄ +α − 2 τ̄ dΩ + Ω α + 2τ̄
α − 2
dτ̄
4
ω
4
ω


1
2
αdα + 3 dω .
+ Ωτ̄ dα + Ωτ̄ 22ω
Thus, a necessary condition for having a critical point of Θ on I ± is


1 2
1
α + 2τ̄±α − 2 = 0.
4
ωA short computation shows that the above is equivalent to h(dΩ, dΩ) = 0.
That is, the critical points of Θ can occur only along conformal geodesics forwhich dΩ = 0 on the initial hypersurface S . The standard hyperboloid in the
Minkowski spacetime contains precisely one such point. By continuity, suitably
small perturbations of this data will have only one point for which dΩ = 0.
Now, for points lying along a conformal geodesic for which dΩ = 0, Equation(16.11) yields τ̄± = −2/α. Note that τ̄± > 0 if α < 0, that is, if Θ̇ < 0.
To obtain a conformal representation which includes timelike inﬁnity one needs
to set α = 0. This condition will be assumed in the remainder of this chapter.
Moreover, one deﬁnes
τ̄i+ ≡ −2/α.
In particular, for the conformal representation of the Minkowski spacetime givenby the conformal factor of Equation (16.5) one ﬁnds that τ̄i+ = 2.
To conclude the discussion of timelike inﬁnity, it is necessary to analyse theHessian of the conformal factor Θ. In what follows it is assumed that one has
obtained a solution to the conformal ﬁeld equations and that the associatedunphysical metric g has been determined.
From the general discussion of the conformal ﬁeld equations in Chapter 8 it
follows that Θ satisﬁes the equations∇a Θ = Σa ,(16.12a)∇a Σb = sηab − ΘLab ,(16.12b)∇a s = −Lac Σ ,(16.12c)cwhere s denotes the Friedrich scalar, ∇ is the Levi-Civita connection of theunphysical metric g ≡ Θ2 g̃ and Lab are the components of the Schouten tensorof ∇ with respect to the Weyl propagated frame {ea }. If s and Lab are regularat the points for which τ̄ = τ̄i+ , one ﬁnds that
Hess Θ|i+ = s|i+ g|i+ .

448

Minkowski-like spacetimes

If, in addition, s|i+ = 0 – which, as will be seen, is the case for perturbationsof the Minkowski spacetime – one concludes that the Hessian of the conformal
factor Θ is non-degenerate at i+ , and, consequently, the point i+ can be rightfully
regarded as the timelike inﬁnity of the development of the hyperboloidal initial
data prescribed on H .
The Cauchy horizon of the hyperboloidal data and the conformal boundary
The discussion in the previous two subsections can be further reﬁned to showthat the conformal boundary I + coincides with the Cauchy horizon H + (H )of the initial data prescribed on H . General results of Lorentzian causal
theory as described in Chapter 14 imply that the Cauchy horizon H + (H ) is
generated by null geodesic segments with endpoints on ∂H ; see Proposition
14.4. Since ∂H is assumed to be a smooth two-dimensional manifold, it follows
that H + (H ) is, in a neighbourhood of ∂H , a g-null hypersurface.
Setting Σa ≡ ∇a Θ it follows from the initial data on H thatΩ = 0,andΣa Σa = η ab da db = 0,on ∂H ,where the various ﬁelds are expressed in terms of their components with respect
to the Weyl propagated frame {ea }. Accordingly, the null directions tangent to
H + (H ) – the so-called null generators of null inﬁnity – are given on ∂H
by Σa . On ∂H one can deﬁne g-null vectors l and n by requiring
la = Σa ,n ⊥ ∂H ,g(l, n) = 1,

on ∂H .

Moreover, on suitable open sets O ⊂ ∂H one can supplement l and nwith complex vectors m and m̄ tangent to ∂H with g(m, m̄) = −1. The
resulting Newman-Penrose frame {l, n, m, m̄} can be propagated along thenull generators of H + (H ) which terminate on O ⊂ ∂H by parallel transport
in the direction of l; that is, one hasla ∇a lb = 0,la ∇a nb = 0,la ∇a mb = 0.

Assuming now that the conformal ﬁeld equations are satisﬁed on H + (H ), it
follows from transvecting Equations (16.12a) and (16.12b) with la and ma thatla ∇a Θ = la Σa ,la ∇a lb Σb = −Θ Lab ,
la ∇a mb Σb = −Θ Lab la mb .
These equations can be regarded as ordinary diﬀerential equations for the scalars
Θ, lb Σb and mb Σb along the generators of I + . By construction, these ﬁelds
vanish on O ⊂ ∂H . Therefore, following a generator on H + (H ) oﬀ ∂H
one ﬁnds that Θ = 0, lb Σb = 0, mb Σb = 0 until, possibly, one arrives at acaustic point. Consequently, there is at least a portion of H + (H ) where the16.3 A priori structure of the conformal boundary449

conformal factor vanishes. It follows from the above that on O the ﬁeld Σa must
be proportional to la – more precisely, one can writeΣa = nb Σb la

on O ⊂ ∂H .

(16.13)The portion of the Cauchy horizon where Θ vanishes can be identiﬁed witha portion of I + as given by Equation (16.11). On this part of H + (H ), theconformal ﬁeld Equations (16.12b) and (16.12c) imply thatla ∇a nb Σb = s,la ∇a s = − nc Σc Lab la lb .
Since na Σa = 1 on ∂H it follows from the homogeneity of the above equations
that s and nb Σb cannot vanish simultaneously. Moreover, contracting ∇a Θ = Σawith ma , m̄a one obtains
(16.14)s = − na Σa ρ,
where ρ ≡ ma m̄b ∇b la is the Newman-Penrose spin coeﬃcient associated to the
expansion of the congruence of null generators. Thus, ρ is a measure of its convergence; see, for example, Stewart (1991), section 2.7. It follows from Equation(16.14) that ρ → ∞ if dΘ = 0 at some point p ∈ H + (H ); see Figure 16.6.
The discussion of the previous subsection shows that the development ofhyperboloidal data suitably close to Minkowski data will contain an isolated
point i+ on the conformal boundary for which dΘ = 0. As I + and H + (H )coincide wherever there are no caustics, it follows that the null geodesics on
H + (H ) must converge to i+ . Accordingly, H + (H ) is the past light cone ofi+ , and the causal past J − (i+ ) and the future domain of dependence D+ (H )coincide.

Figure 16.6 Null generators of I + meeting at i+ , as discussed in the maintext. The causal past of the caustic point i+ corresponds to the future domainof dependence of hyperboloidal data; that is, J − (i+ ) = D+ (H+ ).

450

Minkowski-like spacetimes
16.4 The proof of the main existence and stability result

Once the background Minkowski spacetime has been analysed in terms of aconformal Gaussian system adapted to the standard hyperboloid, a proof ofsemiglobal existence and stability is obtained by a procedure almost identical tothe one used for the de Sitter spacetime in Section 15.4.
As in the case of the analysis of the de Sitter spacetime, it is convenient toconsider an ansatz for the solutions to the conformal evolution equations of theformeAB 0 = ĕAB 0 , eAB α = ēAB α + ĕAB α ,ξABCD = ξ¯ABCD + ξ˘ABCD , χABCD = χ̄ABCD + χ̆ABCD ,ΘABCD = Θ̄ABCD + Θ̆ABCD ,fAB = f˘AB ,φABCD = φ̆ABCD ,whereēAB μ ,ξ¯ABCD ,χ̄ABCD ,Θ̄ABCDare the values of the exact conformal Minkowski spacetime as discussed in Section
16.1. For conciseness, the above ansatz will be written schematically as u =ū + ŭ. Taking into account that the background ﬁelds are also a solution to
the conformal evolution equations and writing the (explicitly known) conformal
gauge ﬁelds Θ and da in the form
Θ = Θ̌ + Θ̆,da = dˇa + d˘a ,one ﬁnds evolution equations for the perturbation ﬁelds of the form∂τ̄ ῠ = Kῠ + Q(Γ̄ + Γ̆)ῠ + Q(Γ̆)ῡ + L(x)φ̆,
I + A0 (ē + ĕ) ∂τ̄ φ̆ + Aα (ē + ĕ)∂α φ̆ = B(Γ̄ + Γ̆)φ̆,(16.15a)(16.15b)in the conventions of Proposition 13.3. The natural domains for solutions to theabove equations are sets of the formMτ̄• ≡ [0, τ̄• ] × S3
for some τ̄• > 0.
Using the evolution Equations (16.15a) and (16.15b) one obtains the following
technical version of the main theorem of this chapter:
Theorem 16.1 (semiglobal existence and stability for perturbations ofhyperboloidal data) Let u = ū + ŭ be hyperboloidal initial data for theconformal Einstein ﬁeld equations given on a hyperboloidal manifold H. Givenm ≥ 4 and τ̄• > 2 there exists ε > 0 such that:16.4 The proof of the main existence and stability result451(i) Forŭ m < ε there exists a solution u = ū + ŭ to the conformal
propagation equations with a minimal existence interval [0, τ̄• ] such thatu ∈ C m−2 ([0, τ̄• ] × S3 ),and the associated congruence of conformal geodesics contains no conjugate
points in [0, τ̄• ].
(ii) For every ŭ with ŭ m < ε there is a unique point p+ in the interior of
H such that dΩ = 0 with τi+ ≡ τ̄+ (p+ ) ∈ [0, τ̄• ].
The solution u = ū + ŭ is unique on D+ (H ) and implies, wherever Θ =
0, a C m−2 solution to the vacuum Einstein ﬁeld equations with a vanishingcosmological constant for which the set I + , as deﬁned by
I + ≡ (τ, p) ∈ R × S3 | τ = τ± (p+ ) ,represents null inﬁnity, while the point i+ ≡ (τ̄i+ , xα (p+ )) represents timelike
inﬁnity. Moreover, one has
D+ (H ) = J − (i+ ).
Proof The assertion in (i) follows from the general existence result fromsymmetric hyperbolic systems in Theorem 12.4 along lines similar to the ones
used in the proofs of Propositions 15.1 and 15.3. The key observation in thisrespect is thatas (I + A0(ē))| is positive deﬁnite and bounded away fromzero, then I + A0 (ē + ĕ) | can also be made positive deﬁnite and bounded
away from zero by choosing ε > 0 small enough. This observation and thegeneral structure of the evolution Equations (16.15a) and (16.15b) ensure theexistence of C m−2 solutions ŭ with ŭ m < ε on [0, τ̄• ] × S3 with τ̄• > 2.
The regularity of the congruence of conformal geodesics deﬁning the gauge is
obtained by supplementing the conformal evolution equations with evolution
equations for the conformal deviation ﬁelds, Equations (13.67a) and (13.67b),
and recalling that the deviation ﬁelds for the expanding Einstein cylinder are
given by Equation (15.14).
The proof of point (ii) follows from the discussion in Section 16.3 andby observing that the spatial conformal factor Ω̄ for the exact (background)
hyperboloidal data has an isolated critical point (in fact, a maximum) at ψ = 0.
Accordingly, by continuity, any suitably small perturbations of this data will
also have a unique isolated critical point of its spatial conformal factor. Again,
choosing ε > 0 suﬃciently small, one can ensure that τ̄+ < τ̄• .
The ﬁnal remarks in Theorem 16.1 follow from a propagation of the constraints
argument using the properties of the subsidiary evolution system as given by
Proposition 13.4 and the assumption that the initial data satisfy the conformal
constraint equations on H . The solution to the conformal ﬁeld equations
obtained by the above argument implies a solution to the vacuum Einstein ﬁeld
equations whenever Θ = 0 as a consequence of Proposition 8.3. Finally, the452

Minkowski-like spacetimes

statements about the interpretation of I + as the conformal boundary and thestructure of i+ follow from the analysis in Section 16.3.

Remarks(i) For conciseness, Theorem 16.1 is restricted to perturbations of the data
implied by the Minkowski spacetime on the standard hyperboloid. An
inspection of the argument, however, shows that this simplifying assumption
is non-essential and that an analogous result can be obtained, at theexpense of some further technical details, for perturbations of Minkowski
data on arbitrary hyperboloids. In other words, the location of the initial
hyperboloid within null inﬁnity is irrelevant. A more subtle consequence of
this observation is that it is, in principle, hard to quantify how far away agiven hyperboloidal initial data set lies from spatial inﬁnity or even whether
there is any (asymptotically Euclidean) Cauchy initial data for the Einsteinﬁeld equations whose development contains the hyperboloidal data.
(ii) Theorem 16.1 can be combined with the method of exterior gluing discussed
in Section 11.8.2 to show the existence of a large class of asymptoticallysimple spacetimes with a complete conformal boundary, that is, whose nullgenerators are inextendible geodesics starting at i0 and ending at i+ and,
respectively, i− . These ideas are discussed in more detail in Section 20.5.
(iii) The future domain of dependence D+ (H ) as given by Theorem 16.1
provides an inﬁnite portion of spacetime where the framework of asymptopia, as discussed in Chapter 10, can be applied; see also, for example,
chapter 3 of Stewart (1991). In particular, if the hyperboloidal initial data
are constructed using the methods of Theorem 11.2, one can obtain adevelopment which has any desired degree of smoothness and, accordingly,
satisﬁes the peeling behaviour ; see the discussion in Section 10.2.

16.5 Extensions and further reading
The ﬁrst semiglobal existence and stability result for hyperboloidal vacuum dataof Minkowski-like spacetimes was obtained in the seminal work by Friedrich(1986b). This analysis used the standard vacuum conformal ﬁeld equations
and gauge source functions. The approach adopted in this chapter, employing
the extended conformal ﬁeld equations and a gauge based on the propertiesof conformal geodesics, is adapted from the discussion given in Lübbe andValiente Kroon (2009). Similar semiglobal existence and stability results havebeen obtained in Anderson and Chruściel (2005) for arbitrary even-dimensional
spacetimes using the conformal equations given by the Graham-Feﬀerman
obstruction tensor.
The main result of this chapter can be extended to the case of the EinsteinMaxwell and Einstein-Yang-Mills equations. This was done in Friedrich (1991)

16.5 Extensions and further reading453

where the standard conformal ﬁeld equations and a hyperbolic reduction procedure based on gauge source functions were used. An alternative proof of the semiglobal existence and stability result for the Einstein-Maxwell equations has beenobtained in Lübbe and Valiente Kroon (2012) using an approach similar in spirit
to the one used in this chapter, that is, employing the extended conformal ﬁeld
equations and a conformal gauge based on the properties of conformal curves.
Conformal curves were preferred in this analysis as they provide an explicit
expression for the conformal factor. In the presence of matter, a standard conformal Gaussian system does not provide an explicit expression for the conformal
factor. There is, however, no reason why a semi-global result of the type discussed
in this chapter cannot be obtained using a gauge based on conformal geodesics.
Another way of generalising the main result of this chapter is to consider the
Einstein-conformally invariant scalar ﬁeld system; see Hübner (1995).
The methods in this chapter can be adapted to analyse semiglobal existence
and stability of asymptotically simple spacetimes with vanishing cosmologicalconstant which are neither the Minkowski spacetime nor perturbations thereof– so-called purely radiative spacetimes. These vacuum spacetimes consist ofgravitational radiation (hence the name) which is not necessarily weak, but stilltame enough to not form a black hole; see, for example, Friedrich (1986c) andthe discussion in Chapter 19. Stability of these types of spacetimes from theperspective of a hyperboloidal initial value problem has been analysed, for thevacuum case, in Lübbe and Valiente Kroon (2010) and, for the Einstein-Maxwell
case, in Lübbe and Valiente Kroon (2012).
The main theorem of this chapter has been beautifully veriﬁed in numericalsimulations in Hübner (2001a). In particular, the numerical results show how
the null generators of the conformal boundary converge, to machine precision,
at timelike inﬁnity. These numerical simulations are further discussed in
Section 21.3.

17Anti-de Sitter-like spacetimes

This chapter discusses the construction of anti-de Sitter-like spacetimes, thatis, solutions to the vacuum Einstein ﬁeld equations with an anti-de Sitter-likevalue of the cosmological constant λ. Following the general discussion in Chapter
10, an anti-de Sitter-like value of the cosmological constant implies a timelike
conformal boundary. This feature of anti-de Sitter-like spacetimes marks theessential diﬀerence between the analysis contained in this chapter and the ones
given in Chapters 15 and 16 for de Sitter-like and Minkowski-like spacetimes,
respectively.
While the de Sitter and Minkowski spacetimes are both globally hyperbolic,and, accordingly, perturbations thereof can be constructed by means of suitable
initial value problems, the anti-de Sitter spacetime is not-globally hyperbolic; see
the discussion in Section 14.5. Consequently, anti-de Sitter-like spacetimes cannot
be solely reconstructed from initial data. One needs to prescribe some boundary
data on the conformal boundary. Thus, the proper setting for the construction ofanti-de Sitter-like spacetimes is that of an initial boundary value problem. In thisspirit, one of the key objectives of this chapter is to identify suitable boundary
data for the conformal Einstein ﬁeld equations.
For both the de Sitter and Minkowski spacetimes it is possible to obtain
conformal representations which are compact in time so that global existenceof perturbations can be analysed in terms of problems which are local in time.
However, the conformal representations of the anti-de Sitter spacetime discussed
in Chapter 6 involve an inﬁnite range of time. As a consequence, the main resultof this chapter is local in time and makes no assertions about the stability of theanti-de Sitter spacetime. The main result of this chapter can be formulated asfollows:
Theorem (local existence of anti-de Sitter-like spacetimes). Givensmooth anti-de Sitter-like initial data for the Einstein ﬁeld equations on athree-dimensional manifold S with boundary and a smooth three-dimensional
Lorentzian metric on a cylinder [0, τ• )×∂S for some τ• > 0, and assuming that17.1 General properties of anti-de Sitter-like spacetimes

455

these data satisfy certain corner conditions, there exists a local-in-time solution
to the Einstein ﬁeld equations with an anti-de Sitter-like cosmological constantsuch that on {0}×S it implies the given anti-de Sitter-like initial data. Moreover,this solution to the Einstein ﬁeld equations admits a conformal completion suchthat the intrinsic metric of the resulting (timelike) conformal boundary belongs
to the conformal class [ ].
Thus, the conformal class of the intrinsic metric of the conformal boundary constitutes suitable boundary data for the construction of anti-de Sitter
spacetimes. This insight was ﬁrst obtained in Friedrich (1995).
17.1 General properties of anti-de Sitter-like spacetimes
In what follows, by an anti-de Sitter-like spacetime it will be understood
an asymptotically simple spacetime (M̃, g̃) with positive (i.e. anti-de Sitter-like)cosmological constant. The basic intuition on this type of spacetimes is obtained
from the paradigmatic example discussed in Section 6.4. In particular, it has
been shown that making use of the conformal factor
ΞadS = a cos ψ,a a constant,the anti-de Sitter spacetime (R4 , g̃ adS ) is conformal to the regionπ3M̃adS ≡ p ∈ R × S 0 ≤ ψ(p) <2of the Einstein cylinder R × S3 described in standard coordinates (T, ψ, θ, ϕ).
Moreover, the conformal boundary of the spacetime is given byπ3
I ≡ p ∈ R × S ψ(p) =,2which can be veriﬁed to be timelike.
17.1.1 General setting for the construction of anti-deSitter-like spacetimes
Let (M, g, Ξ) denote a conformal extension of an anti-de Sitter-like spacetime(M̃, g̃) with g = Ξ2 g̃. It will be assumed that the spacetime is causal (i.e.
it contains no closed timelike curves) and that it contains a smooth, oriented
and compact spacelike hypersurface S with boundary ∂S which intersects theconformal boundary I in such a way that S ∩ I = ∂S . It is convenient todeﬁne S̃ ≡ S \∂S . The portion of I in the future of S will be denoted by I + .
Furthermore, it will be assumed that the causal future J + (S ) coincides with the
future domain of dependence1 D+ (S ∪ I + ) and that S ∪ I + ≈ [0, 1) × S
1 In Chapter 14 the domain of dependence has been deﬁned for achronal sets. However, thatS ∪ I + is not achronal. This feature will not play a role in the subsequent discussion.

456

Anti-de Sitter-like spacetimes

Figure 17.1 Penrose diagram of the set up for the construction of anti-deSitter-like spacetimes as described in the main text. Initial data prescribed on
S \ ∂S allow one to recover the dark shaded region D+ (S \ ∂S ). In orderto recover D+ (S ∪ I + ) it is necessary to prescribe boundary data on I + .
Notice that D+ (S ∪ I + ) = J + (S ).

so that, in particular, I + ≈ [0, 1) × ∂S . A schematic depiction of the above
setting is given in Figure 17.1. One of the key objectives of the present chapter
is to address the question: what data on S ∪ I + are needed to reconstruct theanti-de Sitter-like spacetime (M̃, g̃) in a neighbourhood U ⊂ J + (S ) of S ?
As a consequence of the properties of the standard Cauchy problem andthe localisation property of hyperbolic equations, the solutions to the conformal
Einstein ﬁeld equations on D+ (S̃ ) are determined, up to diﬀeomorphisms,
in a unique manner by solutions to the constraint equations on S . To
recover J + (S ) \ D+ (S̃ ) one needs to prescribe suitable data on the conformalboundary I . The analysis of the suitable boundary data requires the prescriptionof some appropriate gauge near I . As will be seen, conformal geodesics are
ideally suited to provide such a gauge.
The conformal constraints at the conformal boundary
Because for anti-de Sitter-like spacetimes the conformal boundary is a g-timelike
hypersurface, it follows that the metric g induces on I a three-dimensional
Lorentzian metric . As discussed in Section 11.4.4, the conformal Einsteinﬁeld equations satisﬁed by the (unphysical) spacetime (M, g) imply on I a
simpliﬁed set of interior (constraint) equations. It is recalled that a solution
to these conformal constraints at the conformal boundary can be computedfrom the metric , a smooth scalar function κ and a symmetric -tracefree
three-dimensional tensor on I ; see Proposition 11.1. The scalar function is,in particular, a conformal gauge-dependent quantity which can be set to zero byconsidering a diﬀerent metric in [ ].

17.1 General properties of anti-de Sitter-like spacetimes

457

17.1.2 Conformal geodesics at the conformal boundary
In Section 6.4.2 it has been shown that the anti-de Sitter spacetime can becovered by a congruence of (non-intersecting) conformal geodesics. In thiscongruence, curves which for some value of their aﬃne parameter τ̄ are tangent
to I remain on I for all values of τ̄ . It will be shown that this observation is,in fact, a generic property of anti-de Sitter-like spacetimes.
On the conformal boundary of an anti-de Sitter-like spacetime consider
an adapted g-orthonormal frame {ea } such that e3 is inward pointing andorthogonal to I . This frame can then be extended to a neighbourhood U of
I by requiring the frame to be parallely propagated in the direction of e3 . It
follows that the connection coeﬃcients of ∇ associated to this frame satisfy
Γ3 a b = 0

on U .

If one uses Gaussian coordinates x = (xμ ) based on I such that
I = p ∈ U x3 (p) = 0 ,it follows from writing ea = ea μ ∂ μ thate3 μ = δ 3 μ ,ea 3 = 0.

To analyse the behaviour of conformal geodesics at the conformal boundary
it is convenient to consider the equations for these curves expressed in terms ofthe connection ∇. These equations can be decomposed in components using theadapted frame discussed in the previous paragraph. One writesẋ = z a ea ,β = βa ω a .

The conformal curve equations split into two groups. Firstly, one has the normal
equations:ẋ3 = z a ea 3 = z 3 ,ż 3 = −Γa 3 b z a z b − 2(βc z c )z 3 + (zc z c )β 3 ,1
β̇3 = Γa c 3 z a βc + (βc z c )β3 − (βc β c )z3 + L33 z 3 + Li3 z i .
2Secondly, for i, α = 0, 1, 2 one has the intrinsic equations:ẋα = ea α z a ,ż i = −Γc i b z c z b − 2(βc z c )z i + (zc z c )β i ,1
β̇i = Γb c i βc z b + (βc z c )βi − (βc β c )zi + L3i z 3 + Lci z c .
2
To simplify the analysis of the above equations one can exploit the conformal
freedom and choose an element of the conformal class of the intrinsic 3-metricof I for which11
s = ∇c ∇c Ξ + RΞ = 0.
4
24

458

Anti-de Sitter-like spacetimes

Following the discussion of Section 11.4.4, this can always be done locally. Under
this choice of conformal gauge, the solution of the conformal constraint equations
on I implies thatΓa 3 b = 0,

Γa c 3 = 0,

L3a = 0.

Moreover, one has
L3a = 0,

Lij = lij .

That is, the spacetime (unphysical) Schouten tensor on I is determined by the
Schouten tensor of the intrinsic metric .
From the previous discussion it follows that the normal subset of the conformal
geodesic equations reduces to:ẋ3 = z 3 ,ż 3 = −2(βc β c )z 3 + (zc z c )β3 ,1
β̇3 = (βc z c )β3 − (βc β c )z 3 + l33 z 3 .
2These equations are homogeneous in the unknowns (x3 , z 3 , β3 ). Thus, bychoosing initial data such thatx3 = 0,ẋ3 = 0,β3 = 0,(17.1)

one obtains thatx3 (τ ) = 0,z 3 (τ ) = 0,β3 (τ ) = 0

for later times. Accordingly, conformal curves with initial data given by (17.1)
will remain on I . Looking now at the intrinsic part of the conformal geodesic
equations one observes that the equations reduce toẋα = z i ei α ,ż i = −Γk i j z k z j − 2(βk z k )z i + (zk z k )β i ,1
β̇i = Γj k i z j βk + (βk z k )βi − (βk β k )zi + lki z k .
2
These are the conformal geodesic equations for the 3-metric on I .
To verify the consistency between the construction described in the previous
paragraphs and the adapted g-orthonormal frame {ea }, consider a vector v
satisfying the Weyl propagation equation∇ẋ v = −β, vẋ − β, ẋv + g(v, ẋ)β  ,along I . Making the ansatz v = αe3 , where α denotes a scalar function on I ,one ﬁnds the equation α̇ = −β, ẋα. Thus, if initially one has α = 0, thenα = 0 at later times. Accordingly, if one prescribes at some point of the conformal
geodesic in I an orthonormal frame {ea } containing a vector which is normal to
I , one ﬁnds that the solution to the Weyl propagation equations will be a frame

17.1 General properties of anti-de Sitter-like spacetimes

459

Figure 17.2 Representation of conformal geodesics on the conformal boundaryof an anti-de Sitter-like spacetime: those curves that at some point are tangent
to I remain in the conformal boundary and are conformal geodesics for theconformal structure implied by the intrinsic metric ; see Lemma 17.1. The
conformal geodesics are depicted by black lines.

along the conformal geodesic which contains a vector normal to I . Moreover, as
the Weyl propagation preserves the orthogonality of vectors, it follows that theelements of the frame which are at some point intrinsic to I will remain so atlater times; see Figure 17.2.
A more general result
The results obtained in the previous paragraphs make use of a particular metricin the conformal class [ ]. Thus, it is of interest to reformulate them in anarbitrary conformal gauge. As in Chapter 10, the symbol
denotes equality
on I . Now, consider on M a conformal factor ϑ > 0 such that ϑ 1 to performa rescaling of the form g  ≡ ϑ2 g. This rescaling leaves the metric unchanged. Furthermore, one ﬁnds thatin the sense that  ϑ2s(∇a Ξ∇a ϑ)

e3 (ϑ),

with e3 = (dΞ) as Ξ = x3 in local coordinates. The comparison of the above
expression with the solution to the conformal constraints at the conformalboundary as given in Section 11.4.4 suggests deﬁningκ ≡ 3/λ e3 (ϑ) I .
Deﬁning the covector k ≡ ϑ−1 dϑ, and taking into account the transformation
properties of conformal geodesics as given in Section 5.5.2, it follows that(x(τ ), β  (τ )),with β  ≡ β − k,is a solution to the conformal geodesic equations associated to the connection∇ ≡ ∇ + S(k). From the deﬁnition of k it follows that ∇ is the Levi-Civita
connection of the metric g  = ϑ2 g. Observe, in particular, thatβ3 (τ )−k3 (τ )−e3 (ϑ)−s .

The discussion of this section can be summarised as follows:460

Anti-de Sitter-like spacetimes

Lemma 17.1 A conformal geodesic in an anti-de Sitter-like spacetime which
passes through a point p ∈ I , is tangent to I at p and which satisﬁes
β, ν|p = −s,with ν the unit normal to I , remains in I and deﬁnes a conformal geodesic
for the conformal structure of I . Furthermore, the Weyl propagation equations
admit a solution containing a vector ﬁeld normal to I .
17.2 The formulation of an initial boundary value problem
The properties of conformal geodesics in anti-de Sitter-like spacetimes will now be
exploited to construct a conformal Gaussian system for the extended conformal
Einstein ﬁeld equations. As will be seen, the hyperbolic reduction associated tothis gauge leads to an initial boundary value problem for the conformal evolution
equations.
17.2.1 Construction of a boundary adapted gauge
Following the discussion of Chapter 14, the solution to the Einstein ﬁeld
equations on the domain of dependence D+ (S̃ ) = D+ (S \∂S ) is determined ina unique manner, up to diﬀeomorphisms, by a pair of tensors (h̃, K̃) satisfying
the Einstein constraint equations on S̃ . On S , letΩ ≡ Θ S̃ ,Σ̃ ≡ ν̃(Θ)|S̃ ,

with ν̃ the future-directed unit normal ﬁeld of S̃ with respect to g̃. In addition
to the usual smoothness and positivity assumptions, the ﬁelds Ω and Σ̃ arerestricted by their behaviour near ∂S where one requires that Σ ≡ ν(Θ) S =Ω−1 Σ̃ , with ν the future-directed g-unit normal, to be smooth. Using the aboveﬁelds one can use Equations (11.1a) and (11.1b) to compute the unphysicalﬁelds (h, K).
To simplify the subsequent discussion, it is assumed that the initial hypersurface S is such that the unit normal ν is tangent to I on ∂S . Accordingly, onehasΣ ≡ ν(Θ) S = 0
on ∂S .

Moreover, recalling that at the conformal boundary s can be made to vanish bya convenient choice of conformal gauge, it is assumed thats = 0,

on ∂S .

In what follows, each p ∈ S will be considered as the starting point of a futuredirected conformal geodesic (x(τ ), β(τ )) and an associated Weyl propagatedframe {ea }. The parametrisation of the curves is naturally chosen so that τ = 0
on S . For points p ∈ S̃ , the data for these curves are set in terms of g̃ and its
˜ by the conditions:Levi-Civita connection ∇17.2 The formulation of an initial boundary value problem

461(i) ẋ is future directed, orthogonal to S̃ and satisﬁes the normalisation
conditiong̃(ẋ, ẋ) = Θ−2 .
(ii) β  = Ω−1 dΩ so that β  , ẋ  = 0 —as Σ = 0 by assumption.
(iii) e0 = ẋ and g̃(ea , eb ) = Θ−2 ηab .
On suitable neighbourhoods W ⊂ J + (S ) of S , the conformal geodesics x(τ )deﬁne a smooth timelike congruence in W, {ea } a smooth frame ﬁeld and β,a smooth covector. The conformal geodesics can be used to ﬁx a conformal
Gaussian coordinate system on W by setting x0 = τ and then extending
local coordinates x = (xα ) on S by requiring them to remain constant along
conformal geodesics. The coeﬃcients ea μ = dxμ , ea  of the frame {ea } withrespect to the Gaussian coordinates satisfy on W the condition e0 μ = δ0 μ .
Observe, however, that in general ea 0 = 0 only on S . The conformal factor
Θ is then ﬁxed on W by requiringg(ea , eb ) = ηab .
The discussion of the conformal geodesics in the conformal boundary needs tobe done in terms of the metric g and its Levi-Civita connection ∇. In termsof these, the conformal geodesics are represented by a pair (x(τ ), f (τ )) withf ≡ β − Θ−1 dΘ. Accordingly, one has
f = 0,

on S .

As a result of Lemma 17.1, conformal geodesics which start on ∂S remain on I .
As s = 0 on ∂S one can writes = Ως ,with ς a smooth function on ∂S . It follows from Proposition 5.1 that
1 2
Θ = Ω 1 − ς τ ,2while for da ≡ d, ea  one obtains the explicit expressionei (Ω) ≡ (ei α ∂α Ω) ,da = Θ̇, ei (Ω) ,(17.2)(17.3)(17.4)

where the functions Ω, ς and ei (Ω) deﬁned initially on S are extended to W
so that they are constant along conformal geodesics.
Remark. Insight into the behaviour of the conformal factor (17.3) canbe obtained from the constraint Equation (11.35c). Using Equation (17.2),
exploiting that in an adapted gauge (dΩ) = −e3 and evaluating at ∂S one
concludes
ς−L03 Σ − L33 .

462

Anti-de Sitter-like spacetimes

Finally, from Equations (11.40) and (11.41) it follows that in a conformal gauge
for which s 0 one also necessarily has L03 0. Thus, one obtains the simple
expressionς−L33 .

In particular, if L33 > 0, then from Equation (17.3) the conformal factor Θvanishes only if Ω vanishes. This observation is consistent with the discussion ofSection 17.1.2 – conformal geodesics which start normal to S and away from∂S cannot enter the conformal boundary. Ideally, one would like to deduce theproperty L33 > 0 from an analysis of the conformal constraint equations. For
data for the exact de Sitter spacetime, Equation (6.8b) implies L33 = 12 on ∂S .
Suitable perturbations of data for the anti-de Sitter spacetime should preserve
this property.
17.2.2 The conformal evolution system
Combining the gauge construction with the hyperbolic reduction for the extended
conformal ﬁeld equations discussed in Section 13.4 one obtains an evolution
system of the form∂τ υ̂ = Kυ̂ + Q(Γ̂)υ̂ + L(x)φ,
I + A0 (e) ∂τ φ + Aα (e)∂α φ = B(Γ̂)φ,(17.5a)(17.5b)where the notation of Proposition 13.3 is retained and the matrix-valued function
L(x) is given explicitly in terms of the conformal gauge ﬁelds Θ and da as
given by Equations (17.3) and (17.4). In the above system, Equation (17.5b)is understood to correspond to the boundary-adapted Bianchi evolution system(13.60a) and (13.60b) in Chapter 13. The evolution system (17.5a) and (17.5b) is
ideally suited to the formulation of a boundary value problem, as the equations
described by the subsystem (17.5a) are mere transport equations along
the conformal boundary which do not need to be supplemented by boundary
conditions. Hence, all the boundary conditions arise from the subsystem (17.5b)
associated to the evolution of the Weyl tensor.
Following the discussion of initial boundary value problems for symmetric
hyperbolic equations as described in Section 12.4, the identiﬁcation of suitableboundary conditions for Equation (17.5b) stems from an analysis of thenormal matrix A3 at the conformal boundary. Making use of the explicit
expression for the principal part of the boundary-adapted Bianchi system given
in Equation (13.61) and taking into account that, in the boundary adapted
conformal Gaussian gauge, one has
e00 3

0,

e11 3

0,

17.2 The formulation of an initial boundary value problem

463

it follows that
⎛

A3

−1
⎜ 0
⎜
2e01 3 I ⎜
⎜ 0
⎝ 0
0

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

⎞
0
0 ⎟
⎟
0 ⎟
⎟.
0 ⎠
1

This normal matrix is almost in the form required by the theory ofChapter 12. It needs only to be veriﬁed that the evolution of the framecoeﬃcient e01 3 on I can be decoupled from that of the components of the Weyltensor. An inspection of the conformal evolution Equations (13.59a)–(13.59g)– of which Equation (17.5a) above is a schematic representation – shows thatwhenever Θ = 0, the evolution equations for certain components of the ﬁelds
eAB α , χ(AB)CD , ΘCD(AB) decouple from the evolution of φABCD . Thus, it ispossible to determine the frame coeﬃcient e01 3 directly from the initial data at∂S – hence, it is independent of any boundary value prescriptions on I . Thisobservation will be discussed in some detail in the following subsection.
Remark. The normal matrix for the standard Bianchi system is given by⎛⎞−1 0 0 0 0⎜ 0 −2 0 0 0 ⎟⎜⎟A3 2e01 3 I ⎜0 0 0 0 ⎟
⎜ 0
⎟,
⎝ 0
0 0 2 0 ⎠
0

0

0

0 1

so that this normal matrix leads to a much more complicated analysis ofboundary conditions.
17.2.3 Behaviour of the frame at the conformal boundary
In this section, the discussion is restricted to a suitable open neighbourhood W ofa point on ∂S such that the intersection with conformal geodesics is connected.
Consistent with the discussion in Section 17.2.1, one introduces on S ∩ W an
adapted three-dimensional spatial frame {ei } such that e3 is orthogonal andinward directed at ∂S and such that ∇3 ea = 0 on S ∩ W. One introduces
coordinates x = (xα ) on S ∩ W so that x3 vanishes on ∂S and dxα , e3  = δ3 α
on S ∩W. A conformal Gaussian gauge system satisfying the above assumptions
near ∂S will be called a boundary adapted gauge.
For future reference it is observed that the conformal evolution Equations(13.59b), (13.59e) and (13.59f) reduce, on the conformal boundary, to−χ(AB) P Q eP Q α ,(17.6a)∂τ χ(AB)CD−χ(AB)(17.6b)∂τ ΘCD(AB)−χ(CD)∂τ eAB αPQ

χP QCD − ΘAB(CD) ,√ΘP Q(AB) + i 2dP (A μB)CDP .

PQ(17.6c)464

Anti-de Sitter-like spacetimes

The above evolution equations at the conformal boundary are conveniently
analysed in terms of a 1+1+2 spinorial formalism. Given a spinorial basis{A A } such that

τ AA = δ0 A δ0 A + δ1 A δ1 A ,

it is convenient to introduce a spatial spinor ρAA with components with respect
to the basis {A A } given byρAA ≡ δ0 A δ0 A − δ1 A δ1 A .


The space spinor counterpart of ρAA is given byρAB ≡ τB A ρAA = −2δ(A 0 δB) 1 .
It can be veriﬁed that, in addition to the condition√
2e0 = τ AA eAA , one has√
2e3 = ρAA eAA = ρAB eAB = 2e01 ,on S ∩ W.

(17.7)

In particular, one has
eAB (Θ) = dAB = −

λ/6ρABon ∂S .

The spinor ρAB will be used to split space spinor ﬁelds into parts orthogonal
and tangent to I . Accordingly, one deﬁnes
e3⊥ ≡ ρAB eAB 3 ,eAB 3 ≡ ρ(A C eB)C 3

χ⊥⊥ ≡ ρAB ρCD χ̂ABCD ,χ ⊥ AB ≡ ρ(A E χ̂B)ECD ρCD ,χ⊥ CD ≡ ρAB χ̂ABE(C ρE D) ,Θ⊥⊥ ≡ ρAB ρCD Θ̂ABCD ,χ

ABCD ≡ ρ(AEχ̂B)EF (C ρF D) ,Θ ⊥ AB ≡ ρ(A E Θ̂B)ECD ρCD ,whereχ̂ABCD ≡ χ(AB)CD ,Θ̂ABCD ≡ ΘAB(CD) .

Observing that ∂τ ρAB = 0, it follows from Equations (17.6a)–(17.6c) that∂τ eAB 3∂τ χ̂ABCD ρCD∂τ Θ̂CDAB ρAB
−χ̂AB P Q eP Q 3 ,−χ̂AB P Q χ̂P QCD ρCD − Θ̂ABCD ρCD ,−χ̂AB P Q Θ̂P QCD ρAB ,17.2 The formulation of an initial boundary value problem

465

where it has been used that dP (A μB)CDP ρAB = 0 as dAB and ρAB are
proportional to each other. By further contractions with ρAB one ﬁnds thatthe above equations split into the subsystems
1 ⊥χ AB e3⊥ + χ eAB 3 ,2
1 ⊥χ AB χ⊥⊥ + χ ABP Q χ ⊥P Q − Θ ⊥ AB ,2
1 ⊥
χ AB Θ⊥⊥ + χ ABP Q Θ ⊥P Q ,2∂τ eAB 3∂τ χ ⊥ AB∂τ Θ ⊥ AB(17.8a)(17.8b)(17.8c)and∂τ e3⊥∂τ χ⊥⊥∂τ Θ⊥⊥

1 ⊥⊥ 3⊥
χ e + χ ⊥ P Q e3 P Q ,2
1  ⊥⊥ 2χ+ χ⊥ P Q χ ⊥P Q − Θ⊥⊥ ,2
1 ⊥⊥ ⊥⊥Θ χ+ χ⊥ P Q Θ ⊥P Q .
2(17.9a)(17.9b)(17.9c)Initial data for e3⊥ and eAB 3 at ∂S follow directly from (17.7). Namely, one
has√eAB 3 ∂S = 0.
(17.10)e3⊥ ∂S = 2,


⊥



⊥⊥

For χ AB and χ , initial data can be extracted from the conformal constraint
Equation (11.35b) which, taking into account that by assumption Σ = 0 and
La = 0 on S , takes the form χa c Dc Ω = 0 on ∂S . It follows then thatχ⊥⊥ = 0,χ ⊥ AB = 0,

on ∂S .

(17.11)

Finally, to compute the data for Θ ⊥ AB and Θ⊥⊥ one considers the conformal
constraint (11.35c) which, in the present context, takes the form
D3 s = −Db ΩLb3 .
Recalling that s = Ως and that, in local Gaussian coordinates, Ω = x3 one
concludes thatΘ⊥⊥ = 2ς ,Θ ⊥ AB = 0,

on ∂S .

(17.12)Using the initial conditions (17.10), (17.11) and (17.12) together with thehomogeneity of the subsystem (17.8a)–(17.8c), it follows directly that
eAB 3

0,

χ ⊥ AB0,Θ ⊥ AB0.

The solution to the subsystem (17.9a)–(17.9c) is given by
√
2 2
4τ ς
4ς
,
χ⊥⊥ = −,Θ⊥⊥ =.
e3⊥ = −
2 + τ 2 ς
2 + τ 2 ς
2 + τ 2 ς

466

Anti-de Sitter-like spacetimes

The discussion in this section is summarised in the following:
Lemma 17.2 For any solution to the conformal evolution Equations (17.5a)and (17.5b) satisfying on ∂S the conditions (17.10), (17.11) and (17.12), one
has that the normal matrix A3 I of the boundary adapted Bianchi system isgiven by
⎛

A3

−1
√
⎜ 0
2 2 ⎜
⎜ 0
2 + τ 2 ς ⎜
⎝ 0
0

0
0
0
0
0

⎞
0 0 0
0 0 0 ⎟⎟
0 0 0 ⎟⎟,
0 0 0 ⎠
0 0 1

irrespectively of the value of φABCD on W ∩ I .
17.2.4 Identiﬁcation of boundary conditions
The results of the previous paragraphs allow the identiﬁcation of maximallydissipative boundary conditions for the conformal evolution equations. Following
the discussion in Section 12.4, the basic condition to be satisﬁed by the normal
matrix is the inequality
φ, A3 I φ ≤ 0,
which, assuming that 2 + τ 2 ς > 0, implies that
|φ4 |2 − |φ0 |2 ≤ 0.

(17.13)To characterise the subspaces of C5 satisfying the above condition consider twosmooth complex-valued functions c1 and c2 on I and let
φ4 = c1 φ0 + c2 φ̄0 .
Exploiting that (c1 φ0 − c2 φ¯0 )(c̄1 φ̄0 − c̄2 φ0 ) ≥ 0 one ﬁnds that
|φ4 |2 − |φ0 |2 ≤ (|c1 |2 + |c2 |2 − 1)|φ0 |2 .
Thus, condition (17.13) is satisﬁed if one requires|c1 |2 + |c2 |2 ≤ 1.
The above discussion shows that suitable inhomogeneous maximallydissipative boundary conditions for the conformal evolution equations are
given byφ4 − c1 φ0 − c2 φ̄0 = q,|c1 |2 + |c2 |2 ≤ 1,with c1 , c2 , q smooth complex-valued functions on I .

(17.14)

17.2 The formulation of an initial boundary value problem

467

Corner conditions
As seen in Section 12.4, the smoothness of a solution to an initial boundary
value problem requires certain compatibility conditions between the initial data
and the boundary conditions at the edge ∂S – so-called corner conditions.
Following the general discussion given in Section 12.4, one can use the boundaryadapted Bianchi system (17.5b) to determine a formal expansion in terms of τof the vector φ on I near ∂S . This expansion implies, in turn, an expansion
for φ4 − c1 φ0 − c2 φ̄0 and must be consistent with the prescription of the freely
speciﬁable function q. The explicit form of these corner conditions is rather
cumbersome. In what follows, it will be assumed that these corner conditions
are satisﬁed to any order.
17.2.5 The local existence result
The analysis of the boundary conditions leads to a local existence result for an
initial boundary value problem for the conformal evolution system (17.5a) and(17.5b) with boundary conditions of the form (17.14). This result is a direct
application of Theorem 12.6. More precisely, one has the following:
Proposition 17.1 (local existence for the initial boundary value problem) Given an initial boundary value problem for Equations (17.5a) and (17.5b)with smooth initial data
on S ,υ̂  (x), φ (x) ,and inhomogeneous maximally dissipative boundary data
φ4 − c1 φ0 − c2 φ̄0 = q,|c1 |2 + |c2 |2 ≤ 1,

on I ,

with c1 , c2 , q smooth complex-valued functions on I and assuming that therequired corner conditions at ∂S between initial and boundary data are satisﬁed
to any order, there exists τ• > 0 such that the initial boundary value problem hasa unique smooth solution (υ̂(τ, x), φ(τ, x)) deﬁned onMτ• ≡ [0, τ• ) × S.
Remark. Although the above result is local in time, it is nevertheless global inspace. As already mentioned, existence on D+ (S \∂S ) follows from the standard
Cauchy problem. The solutions away from the boundary and those close to theboundary are then patched together to render the full solution.
17.2.6 Propagation of the constraints
In order to transform the existence result given by Proposition 17.1 into an
assertion about the Einstein ﬁeld equations it is necessary to provide an analysisof the propagation of the constraints.

468

Anti-de Sitter-like spacetimes

The subsidiary evolution system associated to the conformal evolution
Equations (17.5a) and (17.5b) has been discussed in Proposition 13.4. The key
structural feature of these subsidiary equations is that they are homogeneous in
the zero quantities. A further crucial feature is that the equations for the zero
quantitiesΣ̂a c b ,Ξ̂c dab ,Δ̂abc ,

δa ,

γab ,

ςab

are all transport equations, and, accordingly, they do not give rise to boundary
conditions on I . For the zero quantity Λabc associated to the Bianchi identity,
the subsidiary system implied by the boundary-adapted system contains noderivatives with respect to the coordinate x3 and, thus, has a vanishing normal
matrix; compare Equations (13.66a)–(13.66c). It follows that the subsidiaryevolution equations require no boundary condition on I . From the uniquenessresult for initial boundary value problems, Theorem 12.5, if the conformal
Einstein equations are satisﬁed on S – that is, the zero quantities vanish – thenthey are also satisﬁed on Mτ• . Combining this discussion with Proposition 8.3
one obtains the following existence result for the Einstein ﬁeld equations:
Theorem 17.1 (propagation of the constraints for the initial boundary
value problem) Consider smooth anti-de Sitter-like initial data for the extended
conformal Einstein ﬁeld equations on a three-dimensional manifold S andboundary initial data of the form (17.14) on I . Assume that the above datasatisfy the required corner conditions to all orders on ∂S = S ∩ I . Then thesolution of the initial boundary value problem given by Proposition 17.1 implies
a solution to the extended conformal Einstein ﬁeld equations on Mτ• . Thissolution, in turn, implies an anti-de Sitter-like solution to the vacuum Einsteinﬁeld equations onM̃τ• ≡ Mτ• \ I ,for which I represents the conformal boundary.
Remark. For an anti-de Sitter-like initial data set it is understood acollection of conformal ﬁelds satisfying the conformal constraint equations withthe required anti-de Sitter asymptotic behaviour; see Section 11.7.
17.3 Covariant formulation of the boundary conditions
From a geometric point of view, the formulation of the boundary conditionsin Proposition 17.1 is not satisfactory. The ﬁelds appearing in the maximallydissipative boundary conditions (17.14) are expressed with respect to a certain
boundary adapted gauge. This gauge speciﬁcation is an integral part of theboundary conditions: changes on the adapted boundary imply changes in thedata. It is therefore important to recast the conditions (17.14), or at least17.3 Covariant formulation of the boundary conditions469

a subclass thereof, in a covariant manner. In what follows, attention will be
restricted to the subclass
φ4 − cφ̄0 = q,c constant, |c| ≤ 1.

(17.15)17.3.1 Space spinor split of the boundary data
To recast the boundary condition (17.15) in a covariant manner, it is ﬁrst
necessary to express the ﬁelds in terms of objects intrinsic to the conformalboundary I . It is convenient to make use of a timelike spinor formalism
based on the spacelike spinorρAA = δ0 A δ0 A − δ1 A δ1 A ,as deﬁned in Section 17.2.3, to project spinorial ﬁelds into I in analogy to
the space spinor splits with respect to τ AA . The spinor ρAA is the spinorial
counterpart of the inward-pointing normal ν = e3 to I . Notice, however, the
normalisation ρAA ρAA = −2. Deﬁne the space spinor version τAB of τAA asτAB = ρB B τAB  = 2δ(A 0 δB) 1 .
Now, taking into account the decomposition of the spinorial counterpart of the
Weyl spinor one can compute its electric and magnetic parts with respect toρAA asEABCD ≡


1 A EE  C  F F 
1
ρB ρ
ρD ρ
dAA EE  CC  F F  = φABCD + φ‡ABCD ,22

BABCD ≡


1 A EE  C  F F  ∗
i
ρB ρ
ρD ρ
dAA EE  CC  F F  = − φABCD − φ‡ABCD ,
2
2

with








φ‡ABCD ≡ ρA A ρB B ρC C ρD D φ̄A B  C  D .
By construction EABCD = E(ABCD) and BABCD = B(ABCD) .
The spinors EABCD and BABCD can be decomposed in a 1 + 2 mannerwith respect to the spinor τAB . The subsequent discussion will be restricted
to BABCD , but an identical analysis can be carried out for EABCD . This
decomposition is best carried out using tensor frame components and thentranslating the result into spinors. One obtains

1 
BABCD = μABCD + μAB τCD + τAB μCD + μ 3τAB τCD − 2A(C D)B ,4(17.16)
with the ﬁelds
μABCD = μ(ABCD) ,μAB = μ(AB) ,μ = μ̄,470

Anti-de Sitter-like spacetimes

satisfying
τ AB μABCD = 0,

τ AB μAB = 0.

The geometric interpretation of the various spinors follows from the above properties. By inspection, it can be shown that the only non-vanishing components ofthe spinor μABCD are given by μ1111 = μ0000 . Similarly, for the rank-2 spinor
μAB one has the non-vanishing components and μ00 = μ11 . From the deﬁnitionsof the magnetic parts of φABCD it follows thatμ1111 = −



i
i
φ1111 − φ̄0 0 0 0
μ11 = − φ0111 − φ̄1 0 0 0 ,22 μ = −i φ0011 − φ̄1 1 0 0 .

It follows from the above expressions and their analogues for EABCD that theboundary condition (17.15) can be rewritten in terms of the components of the
spinors EABCD and BABCD . Of particular interest are the cases
c=1:

B1111 = q,(17.17a)c = −1 :E1111 = q.

(17.17b)

The Bianchi constraints at the conformal boundaryNow, assume that one is provided with boundary data in the form (17.17a)or (17.17b). A natural question is whether it is possible to recover the fullspinor EABCD and, respectively, BABCD . It is recalled that the conformal ﬁeld
equation∇A A φABCD = 0
implies on I the constraint equations
DP Q ηP QAB = 0,

DP Q μP QAB = 0,

(17.18)with DAB ≡ ρ(A A ∇B)A ; see Section 11.4. The above equations are thespinorial versions of the conformal constraints (11.39f) and (11.39g). They canbe decomposed by introducing the directional derivativesP ≡ τ AA ∇AA ,δAB ≡ τ(A Q DB)Q ,along, respectively, the direction dictated by the conformal geodesics threading
the conformal boundary and the direction orthogonal to them. A directcomputation gives
DAB =1
τAB P + δAB .
2

17.3 Covariant formulation of the boundary conditions

471

Combining this split with the decomposition (17.16) of the spinor BABCD one
ﬁnds that the constraint Equations (17.18) imply the system
2Pμ + 4δ AB μAB = 2μAB Pτ AB − 3μDAB τAB+ 2τ EF DAB μABEF ,4PμCD + 2δCD μ = 4(μCD D

EF

E

F

+ 4(δC δD(17.19a)τEF + μEF D+ τCD τEFτCD ) − 3μPτCD
AB)D μABEF .
(17.19b)EF

A similar system is satisﬁed by the components of EABCD . Direct inspection
reveals that the above equations constitute a linear symmetric hyperbolic system(intrinsic to I ) for the ﬁelds μ and μAB if the ﬁeld μABCD is provided; thatis, μABCD plays the role of source terms. The terms involving derivatives withrespect to the spinor ﬁeld τAB appearing in the right-hand sides of the above
equations can be simpliﬁed if one assumes a boundary-adapted gauge on I .
The discussion of the previous paragraphs can be summarised in the following
manner: suppose one is given boundary data on I of the form (17.17a) andsuppose one knows the values of the ﬁelds μ and μAB on ∂S ; then, at leastin a neighbourhood of the edge ∂S , it is possible to determine the components
μ and μAB by solving the hyperbolic system (17.19a) and (17.19b). A similar
discussion holds for the electric part.
17.3.2 Prescribing the Cotton tensor of the conformal boundary
Despite the formal symmetry between the boundary conditions (17.17a) and(17.17b), the former condition possesses a much stronger geometric content.
As a consequence of Equation (11.42), the magnetic part of the rescaled Weyltensor corresponds, essentially, to the components of the Cotton tensor yijk of
the intrinsic Lorentzian metric of I . Thus, one can ask whether, given thecomponents yijk of a tensor on I with the symmetries of the Cotton tensor, it
is possible to ﬁnd a Lorentzian metric on I such that yijk are the components,
with respect to a boundary-adapted frame, of the Cotton tensor of . If this ispossible, then, as a consequence of its conformal transformation properties, one
has obtained a way of reexpressing a subset of the general maximally dissipative
boundary conditions for the conformal ﬁeld equations in terms of the conformal
structure on I . One has the following result, adapted from lemma 7.1 inFriedrich (1995):
Proposition 17.2 (geometric formulation of boundary conditions)Suppose one has a solution to the extended conformal ﬁeld equations with antide Sitter-like cosmological constant on Mτ• = [0, τ• ) × S for τ• > 0 for which
I = [0, τ• ) × ∂S represents the conformal boundary. Let g denote the metric onMτ• obtained from the solution to the conformal ﬁeld equations and let denotethe 3-metric induced on I by g. Assume that the boundary-adapted conformal
Gaussian gauge system can be extended to all of Mτ• . One then has:

472

Anti-de Sitter-like spacetimes(i) Given the restriction to ∂S of the data for the conformal Einstein ﬁeld
equations in the boundary-adapted gauge and given the conformal class [ ],it is possible to compute the function q appearing in the boundary condition(17.17a).
(ii) Conversely, given on ∂S the restriction of the data for the conformal
Einstein ﬁeld equations in the boundary-adapted gauge and the boundary
condition (17.17a), it is possible to determine, in a unique manner, theconformal class [ ].
Proof To prove (i) it is observed that as a consequence of Lemma 17.1, theboundary-adapted conformal Gaussian gauge at the conformal boundary can beconstructed by solving the conformal geodesic equations for the metric . Oncethe associated Weyl-propagated frame {ei } has been obtained, one can directlycompute the components yijk of the Cotton tensor. Using the discussion of theprevious subsection one can, in turn, compute the function q appearing in theboundary condition (17.17a).
The proof of (ii) is much more involved and only a sketch of the main ideas will
be provided. Here, one has to verify whether a given three-dimensional tensor
is the Cotton tensor of a three-dimensional Lorentzian metric. In view of theLorentzian nature of this problem, one can address this question by formulatinga suitable initial value problem on I with data on ∂S for the evolution equations
implied by the structural equations on I . Formulated in this manner one has asituation which is very similar to the Cauchy problem for the extended conformal
Einstein ﬁeld equations.
In what follows, let D denote the Levi-Civita covariant derivative of themetric , and let D̂ denote a Weyl connection in the conformal class of . As
in the four-dimensional case, the connections are related to each other via a
relation of the form D̂ − D = S(f ), with f representing a three-dimensional
covector and S the three-dimensional version of the transition tensor discussed
in Section 5.2.1. Let {ei } denote an -orthogonal frame on I , and let γ̂i j k
be the associated connection coeﬃcients of the connection D̂. Moreover, let ˆlij
denote the components of the Schouten tensor of the connection D̂. In analogy
to the discussion of the conformal ﬁeld equations, it is convenient to introduce
a number of zero quantities encoding the structure equations to be satisﬁedby the various geometric ﬁelds:Σ̂i k j ek ≡ [ei , ej ] − (γ̂i k j − γ̂j k i )ek ,Ξ̂k lij ≡ ei (γ̂j k l ) − ej (γ̂i k l ) + γ̂m k l (γ̂j m i − γ̂i m j )+ γ̂j m l γ̂i k m − γ̂i m l γ̂j k m − 2Sl[i km ˆlj]m ,Δ̂ijk ≡ D̂i ˆljk − D̂j ˆlik − yijk ,Λj ≡ Di yij ,17.3 Covariant formulation of the boundary conditions

473

where
1
yij ≡ − j kl yikl ,2

yi i = 0,

yij = yji ,is the so-called Bach tensor. The zero quantity Σ̂i k j encodes the vanishingof the torsion of the connection D̂, Ξ̂k lij contains the relation between thegeometric and algebraic curvatures (the Ricci identities), Δ̂ijk describes thesecond Bianchi identity for D̂ while Λj corresponds to the so-called third
Bianchi identity – the diﬀerential identity satisﬁed by the Bach tensor.
To obtain a hyperbolic reduction of the above equations one considers theconformal Gaussian system implied by the conformal geodesics on I . Usingarguments similar to the ones in the four-dimensional case one has
ei α = δiα ,ˆl0j = 0,γ̂0 k j = 0,(17.20)and one considers the evolution equations
Σ̂0 k j ek = 0,Ξ̂k l0j = 0,Δ̂0jk = 0,Λ̂j = 0.

(17.21)Taking into consideration the gauge conditions (17.20), it can be veriﬁed thatthe ﬁrst three equations in (17.21) are transport equations on I . The fourth
equation requires a more careful discussion: using the solution to the conformal
constraint equations as given by Equation (11.42) some components of yij can be
expressed in terms of the boundary conditions; for the remaining components one
has that Equations (17.19a) and (17.19b) imply a symmetric hyperbolic system.
Thus, one has obtained a symmetric hyperbolic system for the ﬁelds ei α , γ̂i k j ,ˆlij and for the components of yij not determined by the boundary conditions.
Initial data on ∂S for these ﬁelds can be computed from the restriction to ∂S
of the initial data for the conformal evolution equations. Hence, using the general
theory of symmetric hyperbolic systems as discussed in Chapter 12, one obtains
a solution to Equations (17.21) in a neighbourhood U in I of ∂S . To show that
this solution implies, in turn, a solution to the equations
Σ̂i k j ek = 0,Ξ̂k lij = 0,Δ̂ijk = 0,

Λj = 0,on U

provided that they are satisﬁed at ∂S, one needs to discuss the propagation of
the constraints along the lines of Section 13.4.5. The resulting frame {ei } canbe used to construct on U ⊂ I a Lorentzian metric . This metric characterises
the conformal class of the intrinsic metric of the conformal boundary.
Reﬂective boundary conditions
An important class of boundary conditions covered by the prescription (17.17a)is that of the so-called reﬂective boundary conditions. These correspond to
the particular choice of q = 0 so that one has
φ11111 = φ̄0 0 0 0 ,on I .

474

Anti-de Sitter-like spacetimes

In what follows, this boundary condition will be supplemented by the conditions
φ0111 = φ̄0 0 0 1 ,φ0011 = φ̄0 0 1 1 ,on ∂S .

Accordingly, from the discussion in Section 17.3.1 it follows that BABCD = 0
on ∂S . Furthermore, using the interior evolution system (17.19a) and (17.19b)
one hasBABCD = 0,

on I .

As BABCD corresponds to the Cotton tensor of I , it follows that reﬂective
boundary conditions together with some supplementary conditions at the edge
imply that the intrinsic metric on I is conformally ﬂat.
As pointed out in Friedrich (2014a), despite the above neat geometriccharacterisation of reﬂective boundary conditions, if one wants to construct
a smooth solution to the initial boundary value problem, one still needs tosatisfy an inﬁnite hierarchy of corner conditions. Whether this requirement iscompatible with the known procedures for constructing anti-de Sitter-like initial
data remains an open question.
Comparison with other initial boundary value problems for the
Einstein ﬁeld equations
Initial boundary value problems in general relativity arise in a natural mannerin numerical applications. There exists a number of treatments of the wellposedness of this type of partial diﬀerential equation problem for the Einsteinﬁeld equations; see, for example, Friedrich and Nagy (1999) and Kreiss et al.
(2009). The approach and formulation of the Einstein equations considered in
the former reference are similar to the ones discussed in this book.
The analysis in Friedrich and Nagy (1999) makes use of a frame formulationof the Einstein ﬁeld equations. The equations employed in this reference canbe obtained from the standard conformal Einstein ﬁeld equations discussed in
Section 8.3.1 by setting Ξ = 1. Given these equations, the question is what typeof boundary data need to be prescribed on a, in principle, arbitrary, timelike
hypersurface to obtain a well-posed initial boundary value problem and to ensure
the propagation of the constraints. It turns out that the allowed boundary data
are essentially expressed as a combination of components of the Weyl tensor (withrespect to a boundary adapted frame) of the form given in Equation (17.14).
Despite these parallels, the situation of the initial boundary value problem
analysed in Friedrich and Nagy (1999) and the one discussed in this chapter
diﬀer in a key aspect: the boundary hypersurface in anti-de Sitter spacetimes
has a canonic character. As a consequence, it is possible to formulate covariant boundary conditions, and one ends up with a setting where geometricuniqueness of the solutions can be ensured. In Friedrich and Nagy (1999) it
was not possible to obtain a geometric formulation of the boundary conditions

17.5 Further reading475

on the timelike hypersurface. Thus, they remain tied to the prescription of theboundary-adapted gauge. As geometric uniqueness cannot be asserted, it is, inprinciple, not possible to determine whether two seemingly diﬀerent boundary
conditions will lead to the same spacetime, modulo diﬀeomorphisms. A further
discussion can be found in Friedrich (2009).
17.4 Other approaches to the construction of anti-deSitter-like spacetimes
The analysis of this section has been concerned with the construction of fourdimensional anti-de Sitter-like spacetimes by means of an initial boundary value
problem for the conformal Einstein ﬁeld equations. There are, however, otherapproaches to this problem if, for example, one assumes the existence of a static
Killing vector on the spacetime. The assumption of staticity is a strong one andrenders results of a global nature. As an example of this type of statement one
has the following theorem from Anderson et al. (2002):
Theorem 17.2 (existence of static anti-de Sitter-like spacetimes) Let
denote a smooth strictly globally static Lorentzian metric of non-negative scalar
curvature on R × S2 . Then (R × S2 , ) is the conformal boundary of a complete
strictly globally static vacuum Lorentzian metric on R4 with anti-de Sitter-likecosmological constant.
A strictly globally static spacetime is a spacetime containing an everywhere timelike vector which is orthogonal to the level sets of a globally deﬁned
time function. The proof of this result relies on the use of the FeﬀermanGraham obstruction tensor; see Feﬀerman and Graham (1985, 2012). Related
to the above theorem is the rigidity result given in Anderson (2006), in which
it is shown that complete non-singular anti-de Sitter-like spacetimes with a
globally stationary conformal inﬁnity and an asymptotically stationary bulk must
be globally stationary. This result seems to suggest the instability of anti-deSitter-like spacetimes, at least for certain types of boundary conditions. This
expectation has been reinforced by the evidence of turbulent instability observedin numerical simulations of spherically symmetric solutions of the Einstein-scalarﬁeld system with anti-de Sitter-like boundary conditions reported by Bizon and
Rostworowski (2011).
17.5 Further reading
The approach to the construction of anti-de Sitter-like spacetimes discussed inthis chapter has been adapted from the seminal analysis in Friedrich (1995).
Boundary conditions for a range of test ﬁelds in the anti-de Sitter spacetime
have been studied in Ishibashi and Wald (2004). General properties of the exact
anti-de Sitter spacetime are examined in detail in Griﬃths and Podolský (2009),

476

Anti-de Sitter-like spacetimes

while properties of anti-de Sitter-like spacetimes are discussed in Henneaux andTeitelboim (1985) and Frances (2005). An issue which has not been touched on inthis chapter is that of the deﬁnition of the mass for anti-de Sitter-like spacetimes.
Conformal approaches to this question have been discussed, for example, in
Ashtekar and Magnon (1984) and Ashtekar and Das (2000). Readers interested ina discussion of the issue of the stability/instability of the anti-de Sitter spacetime
are referred to the reviews by Bizon (2013) and Maliborski and Rostworowski(2013) and references within.
A considerable part of the interest on anti-de Sitter-like spacetime stems fromthe so-called AdS/CFT correspondence; see, for example, Maldacena (1998),
Witten (1998) and Witten and Yau (1999). A good discussion of the issues
involved from a mathematician’s point of view are presented in Anderson(2005b).

18
Characteristic problems for the conformal
ﬁeld equations

This chapter discusses the basic theory of characteristic problems for theconformal ﬁeld equations. Characteristic problems have been of great conceptual
value in the development of the modern theory of gravitational radiation. Indeed,
the seminal works by Bondi et al. (1962) and Sachs (1962b), in which themodern understanding of gravitational waves was established, were carried out
in a setting based on a characteristic initial value problem; see also Sachs(1962c) and Newman and Penrose (1962). The connection between characteristic
problems and the notion of asymptotic ﬂatness, already present in the seminal
work by Penrose (1963), was further elaborated in Penrose (1965, 1980). From amathematical point of view, the realisation that the characteristic initial value
problem for the Einstein ﬁeld equations leads to a symmetric hyperbolic evolution
system for which the machinery of the theory of partial diﬀerential equations(PDEs) is available was ﬁrst established in Friedrich (1981b). In Friedrich (1981a,
1982) these ideas were subsequently extended to a situation in which part ofthe data is prescribed at null inﬁnity –a so-called asymptotic characteristic
initial value problem, the subject of this chapter. These results establishedthe local existence of analytic solutions and were later extended to the smoothcase by Kánnár (1996b) using the method of reduction to a standard Cauchy
problem by Rendall (1990); see Section 12.5.3.
There are two basic types of asymptotic characteristic problem for theconformal Einstein ﬁeld equations. The ﬁrst type is the so-called standard
asymptotic characteristic problem – introduced in Friedrich (1981b) – where
initial data are prescribed on null inﬁnity and a null hypersurface intersecting
null inﬁnity in a two-dimensional surface with the topology of a 2-sphere; see
Figure 18.1, left. In the second type – the so-called characteristic problem
on a cone, ﬁrst discussed in Friedrich (1986c) – one prescribes information ona null cone down to its vertex; see Figure 18.1, right. For reasons discussed in
Section 12.5, characteristic problems on a cone are more technically involved.
Existence results have been obtained in Chruściel and Paetz (2013).

478

Characteristic problems for the conformal ﬁeld equations

Figure 18.1 Two possible asymptotic characteristic problems for the conformal
ﬁeld equations: on the left, initial data are prescribed on an outgoing null
hypersurface N and null inﬁnity I − ; on the right, data are prescribedon a null cone representing past null inﬁnity I − . The vertex of the cone
corresponds to past timelike inﬁnity, i− .

The standard and characteristic initial value problems have several structuralproperties in common. Moreover, the characteristic problem on a cone can be
regarded as a limiting case of the standard characteristic problem. In bothcases, the Einstein ﬁeld equations on the initial hypersurfaces split into a setof interior (or intrinsic) equations and a set of transverse equations.
The interior equations split, in turn, into constraint equations which need tobe satisﬁed only on some subsets of the initial hypersurface (the intersection of
the null hypersurfaces or the vertex of the cone) and transport equations
which propagate information along the generators of the null hypersurfaces.
The transverse equations dictate the evolution oﬀ the initial hypersurfaces.
One of the key aspects of the analysis of asymptotic characteristic problems
is the identiﬁcation of freely speciﬁable data from which the full data for theevolution equations can be derived. An appealing feature of this type of settingis the natural interpretation of the free data in terms of radiation ﬁelds so that a
clear-cut connection with the theory of asymptotics as discussed in Chapter 10
can be established.
The discussion in the present chapter is mostly concerned with standard
characteristic problems. Certain aspects of the characteristic problem on a coneare brieﬂy considered. The existence results discussed are local in nature. Thatis, one obtains existence of solutions in a neighbourhood of the intersection of thenull hypersurfaces or the vertex of the initial cone. From the perspective of thephysical spacetime these local neighbourhoods represent unbounded domains in
the asymptotic region.
18.1 Geometric and gauge aspects of the standard characteristic
initial value problem
This section provides a discussion of the geometric setting and the gauge
ﬁxing procedure for the standard asymptotic characteristic problem. Taking into
account the general theory of characteristic problems described in Section 12.5.1
one can consider two possible conﬁgurations (see Figure 18.2): (i) that of a

18.1 Geometric and gauge aspects of the initial value problem

479

Figure 18.2 The two possible standard asymptotic characteristic problems forthe conformal Einstein ﬁeld equations. Case (i) where data are prescribed on afuture-oriented (outgoing) null hypersurface N  and future null inﬁnity I + ,and case (ii) where data are prescribed on a past-oriented (incoming) null
hypersurface N and past null inﬁnity I − .

future-oriented (i.e. outgoing) null hypersurface intersecting future null inﬁnityor (ii) a past-oriented (i.e. incoming) null hypersurface intersecting past nullinﬁnity. In order to compare with the characteristic problem on a cone, thepresent discussion focuses in the latter case. A careful inspection of the setting
discussed here leads to the formulation of case (i).
18.1.1 Geometric setting
In what follows, let (M, g, Ξ) denote a conformal extension of an asymptoticallysimple spacetime (M̃, g̃) satisfying Ric[g̃] = 0 which contains past null inﬁnity
I − . Let W denote a region of M with W ≈ R+ × R+ × S2 bounded by anincoming null hypersurface N and past null inﬁnity I − . It will be assumed
that both N and I − have the topology of R+ × S2 . Let Z ≡ N ∩ I − withZ ≈ S2 . One has that W ⊂ J + (Z ). A schematic representation of the geometricsetting can be seen in Figure 18.3.
An adapted coordinate system (xμ ) and an associated null tetrad {eAA }will
be used to describe the geometry of the region W. Let {ω AA } denote theassociated coframe and require thatg(eAA , eBB  ) = AB A B  .

(18.1)On Z one considers some coordinate system (xA ) where A = 2, 3. The complex
vectors e01 and e10 = e01 of the null tetrad {eAA } will be chosen so thatthey span the tangent bundle T (Z ) – recall that in standard Newman-Penrose
notation the vectors e01 and e10 correspond to m and m.
Now, choose e00 so that, on I − , it is tangent to the null generators of
the conformal boundary – in standard Newman-Penrose notation this vector
corresponds to l. Let v denote an aﬃne parameter of these generators with the∂ v where, following theproperty that v|Z = 0. Thus, one has that e00conventions of Chapter 10, the symbol
denotes equality at I − . The vectors
e01 and e10 can be extended to the rest of I − by parallel propagation along
the null generators. Accordingly, one has∇00 e000,∇00 e010,∇00 e100,

on I − ,(18.2)

480

Characteristic problems for the conformal ﬁeld equations

Figure 18.3 Schematic representation of the set up for the standard asymptotic
characteristic problem. The existence results are restricted to a neighbourhood
U of Z in J + (Z ).

where ∇00 ≡ e00 a ∇a is the directional derivative in the direction of e00 . Givenv• ∈ [0, ∞), let Zv• ⊂ I − denote the two-dimensional surfaces given byZv• ≡ {p ∈ I − | v(p) = v• }.
As a result of their parallel propagation, the vectors e01 and e10 span T (Zv• ).
Having ﬁxed the vectors e00 , e01 and e10 on I − , regarding the conformalboundary as a submanifold of M, and given that the spacetime metric g isassumed to be known, it follows that at every point p ∈ I − , there exists a uniquefuture-pointing null vector linearly independent to {e00 , e01 , e10 }. This vector
is used to complete the null frame {eAA } on I − – accordingly, it will be denotedby e11 , or n in Newman-Penrose notation. The vector e11 is ﬁxed by the four
conditions
g(e11 , eBB  ) = 1B 1 B  .
Now, for ﬁxed v• , there exists (at least locally) a unique null hypersurface Nv•
in M satisfying Nv• ∩ I − = Zv• such that at Zv• the vector e11 is tangent toNv• – this involves solving the eikonal equation g(dΦ, dΦ) = 0 for some scalarΦ ∈ X (W) near I − with the appropriate initial conditions; for further details
see, for example, Stewart (1991), section 4.3. By varying v• one thus obtains (atleast locally) a foliation of null hypersurfaces intersecting I − . Thence, the aﬃneparameter v along the null generators of I − can be used as a coordinate on W.
Accordingly, one sets x0 = v, and has
Nv• ≡ {p ∈ W | v(p) = v• },18.1 Geometric and gauge aspects of the initial value problem

481

so that the normal to Nv• is given by dv. The vector e11 can now be extended
into W by requiring it to be tangent to the generators of these hypersurfaces;that is, one has
e11 = g  (dx0 , ·).

(18.3)

Let r denote an aﬃne parameter of the integral curves of e11 so that one canwrite e11 = ∂ r . Without loss of generality one can choose r 0. The coordinate
system (xμ ) on W is then completed by setting x1 = r and by extending thecoordinates (xA ) on Zv so that they are constant along the integral curves of
e00 and e11 . As a consequence of this construction one has
N ≡ {p ∈ W | x0 (p) = 0},I − ≡ {p ∈ W | x1 (p) = 0}.

The vectors e00 , e01 and e10 can be extended oﬀ I − by parallel propagation
along the direction of e11 . Accordingly, one has∇11 e11 = 0,∇11 e01 = 0,∇11 e10 = 0,

on W.

(18.4)

To obtain an explicit expression for the frame {eAA } in the coordinates(x ) = (v, r, xA ), it is observed that from Equation (18.3) – rewritten in theform g(∂ r , ·) = dv, · – one obtains the pairings
μg(∂ r , ∂ v ) = 1,g(∂ r , ∂ r ) = 0,g(∂ r , ∂ A ) = 0.

(18.5)Taking into account the above, the most general form for the frame {eAA }consistent with Equations (18.1) and (18.3) is given bye00 = ∂ v + U ∂ r + X A ∂ A ,e11 = ∂ r ,e01 = ω∂ r + ξ A ∂ A ,e10 = ω̄∂ r + ξ¯A ∂ A ,where U and X A are real functions and ω and ξ A are complex functions. Observe,in particular, that because of the conditions in (18.5), e01 and e10 cannot
have a v-component. Using, again, relation (18.1) one ﬁnds that the componentsg μν = g  (dxμ , dxν ) are of the form⎛

01
(g μν ) = ⎝ 1 g 11
0 g A10⎞

g 1A ⎠ ,
g AB

where
g 11 = 2(U − ω ω̄),

g 1A = X A − (ξ A ω̄ + ξ¯A ω),

g AB = − ξ A ξ¯B + ξ¯A ξ B ).

482

Characteristic problems for the conformal ﬁeld equations

In particular, one has that U 0, X A 0, ω 0, consistent with the fact thate11 is tangent to the generators of null inﬁnity and that v is an aﬃne parameter;
hence, e11 ∂ v . Observe also that e01 ξ A ∂ A . Thus, the pull-back to Zv ofg AB ∂ A ⊗ ∂ B = − ξ A ξ¯B + ξ¯A ξ B )∂ A ⊗ ∂ B ,to be denoted by ς  , corresponds to the two-dimensional (contravariant) metricof the sections of null inﬁnity. Now, by assumption Zv ≈ S2 so that ς is conformal
to the standard metric of S2 .
Finally, combining the propagation conditions (18.2) and (18.4) with thedeﬁnition of the spin connection coeﬃcients – see Equations (3.31) and (3.33) –in the form
ΓAA BC =
1BP ω P Q , ∇AA eCQ ,
2

one ﬁnds
Γ00 100,

Γ00 000,andΓ00 11 = Γ̄10 1 0 + Γ10 10 ,Γ01 11 = Γ̄10 1 1 ,

Γ11 AB = 0,

on W ⊂ M.

The discussion of this section is summarised in the following
Lemma 18.1 (frame gauge conditions for the standard characteristic
problem) Let (M̃, g̃) denote an asymptotically simple spacetime satisfying
Ric[g̃] = 0 and let (M, g, Ξ) with g = Ξ2 g̃ be a conformal extension thereof forwhich the condition Ξ = 0 describes past null inﬁnity I − . The frame {eAA }can be chosen so that, given a null hypersurface N intersecting I − on Z ≈ S2 ,one hasΓ00 11 = Γ̄10 1 0 + Γ10 10 ,Γ01 11 = Γ̄10 1 1 ,

Γ11 AB = 0,

on W ⊂ M.

In addition, one has thatΓ00 01 = Γ00 10 = Γ00 00 = U = X A = ω = 0,

on I − .

Remark. The conventions used here for the vectors e00 and e11 are theopposite of those used in Kánnár (1996b). They have been chosen to agree withthe standard conventions in the treatment of asymptotics as given in Penrose
and Rindler (1986) and Stewart (1991) and to ease the comparison with thecharacteristic problem on a cone.

18.1 Geometric and gauge aspects of the initial value problem

483

18.1.2 The choice of conformal gauge
The geometric setting discussed in the previous section has an inherent conformal
gauge freedom which can be exploited to simplify the analysis.
As discussed in Section 8.2.5, the Ricci scalar R[g] plays the role of aconformal gauge source function for the conformal ﬁeld equations. A
possible choice in the present setting is to ﬁx the conformal factor Ξ linking
the metrics g̃ and g in such a manner that R[g] = 0. To see that this can always
be done, consider ﬁrst a situation involving a generic conformal factor Ξ forwhich R[g] = 0, and let
g  ≡ ϑ2 g,(18.6)

with ϑ a positive function on W. Deﬁning Ξ ≡ ϑΞ one ﬁnds that g  = Ξ2 g̃.
Consistent with the above conformal rescaling one considers the following
transformation behaviour for the g-orthonormal frame {eAA }:e00 = e00 ,e11 = ϑ−2 e11 ,e01 = ϑ−1 e01 ,e10 = ϑ−1 e10 .

Using the transformation law under conformal rescalings for the Ricci scalar,
Equation (5.6c), one ﬁnds that the requirement R[g  ] = 0 is equivalent to thewave equation∇a ∇a ϑ =

1
R[g];
6

(18.7)

see also Equation (8.30). The general theory of the characteristic problem forwave equations ensures the existence of a unique solution to this equation ina neighbourhood U of Z in J + (Z ) if some suitable data are prescribed on
N ∪ I − ; see, for example, Rendall (1990). A natural requirement on the initialdata for Equation (18.7) is to have ω 11 = dΞ on Z where {ω AA } denotes
the coframe dual to {eAA }. This is equivalent to settingω 11 = ϑdΞ

on Z .

By choosing ϑ−1 |Z = e11 (Ξ)|Z = dΞ, e11 |Z one can, in fact, ensure thate11 (Ξ ) = dΞ , e11  = 1

on Z .

The principal part of the wave Equation (18.7), expressed in terms of frame
derivatives, is given by








e00 e11 (ϑ) + e11 e00 (ϑ) − e01 e10 (ϑ) − e10 e01 (ϑ) .
Thus, Equation (18.7) implies an intrinsic propagation equation on N for e00 (ϑ)
if e11 (ϑ) is known on N . Analogously, one has an intrinsic propagation equation
on I − for e11 (ϑ) if e00 (ϑ) is known on I − . The freedom in the speciﬁcation

484

Characteristic problems for the conformal ﬁeld equationsof characteristic data can be exploited by observing that under the conformal
rescaling (18.6) one obtains the transformation rulesΓ01 11 = Γ01 11 − ϑ−1 e11 (ϑ),Γ10 00 = ϑ−2 Γ10 00 − ϑ−3 e00 (ϑ).

Accordingly, by setting
e11 (ϑ) = ϑΓ01 11 ,e00 (ϑ) = ϑΓ10 00

on Z ,

one obtains
Γ01 11 = 0,Γ10 00 = 0,

on Z .

To propagate the freely speciﬁable components of ∇AA ϑ along N and I − itis convenient to consider the transformation law under conformal rescalings ofthe trace-free part of the Ricci tensor
−1∇a ∇b ϑ − 2ϑ−1 ∇a ϑ∇b ϑ
Φab − Φab = −2ϑ

1
c
−1
c
(18.8)− gab (∇ ∇c ϑ − 2ϑ ∇c ϑ∇ ϑ) .
4
Now, recalling that
ΦAA BB  = eAA a eBB  b Φab ,ΦAA BB  = eAA a eBB  b Φab ,one can consider the propagation equations

2e11 e11 (ϑ) − 2ϑ−1 e11 (ϑ) = ϑΦ22
2e00 e00 (ϑ) − 2ϑ−1 e00 (ϑ) = ϑΦ00

on N ,(18.9a)on I − .

(18.9b)

These two equations can be read as ordinary diﬀerential equations along thegenerators of N and I − for e11 (ϑ) and e00 (ϑ), respectively. Accordingly,
a solution exists in a neighbourhood of Z on N and, respectively, on I − .
Comparing with Equation (18.8), one sees that these solutions ensureΦ22 = 0

on N ,

Φ00 = 0−

on I .

(18.10a)(18.10b)

Once the solutions e11 (ϑ) and e00 (ϑ) to the propagation conditions (18.9a)and (18.9b) have been obtained, one can use the intrinsic equations implied by(18.7) on N ∪ I − to obtain e00 (ϑ) on N and e11 (ϑ) on I − .
The analysis of this section can be summarised in the following:
Lemma 18.2 (conformal gauge conditions for the standard characteristic problem) Let (M̃, g̃) denote an asymptotically simple spacetime satisfying
Ric[g̃] = 0 and let (M, g, Ξ) with g = Ξ2 g̃ be a conformal extension thereof
for which the condition Ξ = 0 describes past null inﬁnity I − . Given the frame

18.2 The conformal evolution equations in the initial value problem

485{eAA } of Lemma 18.1, the conformal factor Ξ can be chosen so that given anull hypersurface N intersecting I − on Z ≈ S2 one hasin a neighbourhood W of Z on J + (Z ).

R[g] = 0,

Moreover, one has the additional gauge conditions
e11 (Ξ) = 1,
Φ22 = 0

Γ01 11 = Γ10 00 = 0,

on Z ,

on N ,


ΣAA = e11 (Ξ)δA 1 δA 1 ,Φ00 = 0,on I − .

Remark. In the gauge given by Lemma 18.2 one has that LAA BB  = ΦAA BB  .
This fact will be used repeatedly in the following without any further mention.
18.2 The conformal evolution equations in the standard
characteristic initial value problem
This section analyses general aspects of the standard characteristic initial value
problem for the conformal Einstein ﬁeld equations with data prescribed on
the null hypersurfaces N and I − . The spinorial conformal ﬁeld equations, as
discussed in Section 8.3.2, will be used to formulate this problem. Accordingly,
on W it will be required that
ΣAA BB  = 0,
ΞAA = 0,
ΔCDBB  = 0,

ΞC DAA BB  = 0,
ZAA BB  = 0,

ZAA = 0,

ΛBB  CD = 0,

(18.11a)Z = 0,(18.11b)(18.11c)where, for convenience, one deﬁnes
ΞAA ≡ ΣAA − ∇AA Ξ.
Following the conventions of Chapter 13 let u denote the collection of independent components of the unknowns appearing in the conformal ﬁeld Equations(18.11a)–(18.11c) and let u be its value on N ∪ I + .
Strictly speaking, as no hyperbolic reduction procedure has yet been applied
to equations (18.11a)–(18.11c) – that is, the equations do not constitute a
symmetric hyperbolic system – one does not directly obtain a characteristic
problem in the sense described in Section 12.1.2. Nevertheless, the structure of
the conformal evolution equations can be used to obtain a symmetric hyperbolic
system for which the theory of Section 12.5 can be applied. Thus, it is necessary
to analyse the properties of the conformal ﬁeld equations on the hypersurfaces N
and I + . When evaluated on N ∪ I + the system (18.11a)–(18.11c) splits into a
set of interior and a set of transverse equations. As the name suggests, interior
equations contain only derivatives which are intrinsic to the null hypersurfaces.
The interior equations divide, in turn, into transport equations containing thedirectional derivative along the generators of the hypersurface and constraint486

Characteristic problems for the conformal ﬁeld equations

equations which do not contain this derivative. In the transverse equations one
deals with the directional derivative transverse to the surface.
To see how this split comes about, it is convenient to recall some aspects of
the hyperbolic reduction procedure for the Equations (18.11a)–(18.11c). Givena timelike vector τ μ and a suitable set of gauge source functions F a (x) andFAB (x) on W, one obtains a symmetric hyperbolic system for the independent
components of the various conformal ﬁelds. As discussed in Proposition 13.1,
the characteristic polynomial of this system contains factors of the form g μν ξμ ξν .
Accordingly, the combined null hypersurface N ∩I + is a null hypersurface of the
reduced evolution system. Following the discussion of Section 12.1.2, it follows
that the reduced system contains equations which are intrinsic to N ∩ I + andequations which are transverse to the initial hypersurface. In the following, it isshown how this observation can be extended to the full conformal ﬁeld equations.
The interior equations on N
The interior equations on the null hypersurface N should contain only the
directional derivatives along the directions given by e11 , e01 and e10 .
Inspection shows that the subset of (18.11a)–(18.11c) with this property is givenby the equations
Ξ11 = 0,Z11 AA = 0,C

Z11 = 0,Σ11 BB  = 0,ΞD11 BB  = 0,

Δ1DBB  = 0,

ΛB1 CD = 0.

(18.12a)(18.12b)(18.12c)More explicitly, taking into account the gauge conditions given by Lemmas
18.1 and 18.2 one has the equations
e11 (Ξ) = Σ11 ,(18.13a)e11 (Σ00 ) = −ΞΦ11 − s,e11 (Σ01 ) = −ΞΦ12 ,e11 (Σ11 ) = 0,e11 (s) = −Φ11 Σ11 + 2Φ12 Σ01 ,(18.13c)e11 (eBB  μ ) = −ΓBB  C 1 eC1 μ − Γ̄B  B C 1 e1C  μ ,e11 (ΓBB  CD ) = −ΓF 1 CD ΓBB

F

(18.13b)

1 − Γ1F  CD Γ̄B  B(18.13d)F1

− ΞφBCD1 1 B  − ΦC1 DB  1B ,(18.13e)e11 (ΦD0 BB  ) = ∇10 ΦD1 BB  − ∇D1 Φ10 BB  + ∇D0 Φ11 BB 
− 2Σ1B  φBD01 + 2Σ0B  φBD11 ,e11 (φABC0 ) = ∇01 φABC1 ,(18.13f)(18.13g)

with the understanding that equations for quantities already determined bygauge conditions are dropped from the list. Despite their apparent complexity,
the above equations possess a delicate hierarchical structure which allows one tosolve them sequentially from some basic data on Z and N . This structure is
brieﬂy described in the following paragraphs.

18.2 The conformal evolution equations in the initial value problem

487

One starts by combining Equation (18.13a) with the third equation in (18.13b)
and then using that e11 = ∂ r to ﬁnd that ∂r2 Ξ = 0. Hence, taking into accountLemma 18.2 one concludes that Ξ = r along N . Next, one can consider Equation(18.13e) for Γ01 11 and Γ10 11 (in standard Newman-Penrose (NP) notation γ
and λ) which, in view of the gauge conditions, gives the subsystem∂r Γ01 11 = −(Γ01 11 )2 − Γ10 11 Γ̄1 01 1 ,∂r Γ10 11 = −2Γ01 11 Γ10 11 + Ξφ4 .
The above Riccati system can be solved if φ4 is known along N . With Γ01 11
and Γ10 11 known, one can then make use of Equation (18.13d) for e01 A = ξ A
which takes the form∂r ξ A = −Γ01 11 ξ A − Γ̄1 01 1 ξ¯A .
This equation together with its complex conjugate constitute a system ofordinary diﬀerential equations for ξ A and ξ¯A which can be solved with theinformation already available. To determine the frame coeﬃcient ω one considers
Equation (18.13d) for e01 1 = ω so that∂r ω = −Γ01 11 ω − Γ̄1 01 1 ω̄ + Γ01 01 + Γ̄1 00 1 .
Accordingly, one also needs to consider the equations for Γ01 01 and Γ10 01(β and α in NP notation), namely,∂r Γ01 01 = −Γ01 01 Γ10 11 − Γ10 01 Γ̄1 01 1 + Φ12 ,∂r Γ10 01 = −Γ01 01 Γ10 11 − Γ10 01 Γ̄0 11 1 + Ξφ3 ,so that, in addition, one requires equations for φ3 and Φ12 . These can be foundto be given by∂r φ3 = ω∂r φ4 + ξ A ∂A φ4 − 4Γ01 11 φ3 + 4Γ01 01 φ4 ,∂r Φ12 = Σ01 φ4 − Σ11 φ3 .
Thus, to close the system one considers the third equation in (18.13b). The key
observation is that for a given choice of φ4 on N and with the knowledge ofΓ01 11 and Γ10 11 from a previous integration one obtains a system of ordinary
diﬀerential equations along the generators of N for the unknowns ω, ξ A , Γ01 01 ,
Γ10 01 , φ3 , Φ12 and Σ01 .
At this point, one considers Equation (18.13d) for e11 A . One has∂r X A = −Γ00 11 ξ A − Γ̄0 01 1 ξ¯A .
Recalling the gauge condition Γ00 11 = Γ̄10 1 0 + Γ10 10 one has enoughinformation to integrate along the generators of N . Next, one considers theequations for Γ01 00 and Γ10 00 (σ and ρ in NP notation):∂r Γ01 00 = −Γ01 00 Γ01 11 − Γ10 00 Γ̄01 1 1 + Φ02 ,∂r Γ10 00 = −Γ01 00 Γ10 11 − Γ10 00 Γ̄10 1 1 + Ξφ2 .

488

Characteristic problems for the conformal ﬁeld equations

Hence, one has to couple the above to the equations for φ2 and Φ02 :∂r φ2 = ω∂r φ3 + ξ A ∂A φ3 + Γ01 00 φ4 + 2Γ01 01 φ3 − 3Γ01 11 φ2 ,∂r Φ02 = ∇10 Φ12 + Σ01 φ3 − Σ00 φ4 .
Thus, it is then necessary to consider simultaneously the ﬁrst equation in (18.13b)
and Equation (18.13c) to determine Σ00 and s – notice that at this stage one
already knows all the frame and connection coeﬃcients appearing in ∇10 . In
turn, this forces the coupling with the equation for Φ11 obtained from (18.13f):∂r Φ11 = ∇10 Φ12 − Σ11 φ2 + Σ01 φ3 .
Recapitulating, one has obtained a further closed subsystem of ordinary diﬀerential equations along the generators of N for the ﬁelds Γ01 00 , Γ10 00 , φ2 , Φ02 ,Φ11 , s and Σ00 . With the information obtained from the solution to this system,
one can also solve for the frame coeﬃcient U and the connection coeﬃcient Γ00 01( in NP notation) via the equations∂r U = −Γ00 11 ω − Γ̄0 01 1 ω̄ + Γ00 01 + Γ̄0 00 1 ,∂r Γ00 01 = −Γ01 01 Γ00 11 − Γ10 01 Γ̄0 01 1 + Ξφ2 + Φ11 .
The integration of the connection coeﬃcients can now be completed with theequation for Γ00 00 (κ in NP notation) dictated by (18.13e), that is,∂r Γ00 00 = −Γ01 00 Γ00 11 − Γ10 00 Γ̄0 01 1 + Ξφ1 + Φ01 ,which needs to be supplemented by the equations for φ1 and Φ01 :∂r φ1 = ∇01 φ2 ,∂r Φ01 = ∇10 Φ11 − Σ01 φ2 + Σ00 φ3 .
Again, one has a subsystem of ordinary diﬀerential equations along the generatorsof N . The integration of the interior equations on N is completed by considering
the equation for the rescaled Weyl spinor component φ0∂r φ0 = ∇01 φ1 ,which, too, is an ordinary diﬀerential equation, and by that for Φ00 :∂r Φ00 = ∇10 Φ01 − ∇01 Φ01 + 2Σ00 φ2 − 2Σ01 φ1 + ∇00 Φ11 .
This last equation is diﬀerent from the other ones in the hierarchy as its last termin the right-hand side (i.e. ∇00 Φ11 ) contains transverse derivatives. However,using the evolution equations in Section 18.2.2, this term can be formallycomputed on N from the available data.

18.2 The conformal evolution equations in the initial value problem

489

The interior equations on I −On I − the intrinsic equations should contain only the derivatives along thedirections given by e00 , e10 and e01 . The relevant subset of (18.11a)–(18.11c)
is, in this case, given by
0,

ΞAA

ZAA

0,
C

for AA = 11 ,(18.14a)for AA , BB  = 11 ,(18.14b)

ZAA BB ΣAA BB 0,Ξ

Δ0DBB 0,

ΛB0 CD

0,

DAA BB 

0,0.

(18.14c)More explicitly, taking into account the gauge conditions given by Lemmas
18.1 and 18.2 the above equations encode the following transport equations:e00 (Ξ)

0,(18.15a)−s,e00 (Σ11 )(18.15b)−Φ11 Σ11 ,e00 (s)μCC (18.15c)01 eCCμ− Γ01CC e00 (e01 )Γ00

00 eCC

PCD00 BB 

−ΞφBCD0 0 B  − ΦC0 DB  0B∇00 φABC1∇10 φABC0 ,μ,(18.15d)BB  = 11 ,(18.15e)(18.15f)∇00 ΦD1 BB  + ∇D0 Φ01 BB  − ∇01 ΦD0 BB 
− ∇D1 Φ00 BB 

2Σ1B  φ0DB1 ,(18.15g)

where, following the notation of Chapter 8, the ﬁeld PCDAA BB  denotes thegeometric curvature. In addition to the above, Equations (18.14a)–(18.14c) also
contain the constraint equations
e01 (Ξ)

0,e01 (Σ11 )e01 (e10 ) − e10 (e01 )
μ

μ

Γ01

PCD01 10

0,
CC 

e01 (s)10 eCCμ−Φ01 Σ11 ,− Γ10CC 

Ξφ1CD0 1 0 − ΦC1 D0 .

01 eCC(18.16a)μ,(18.16b)(18.16c)

If Equations (18.16a)–(18.16c) hold in a certain section of I − , then using anargument similar to that of the propagation of the constraints in the standard
Cauchy problem, it can be shown that they will hold everywhere else on nullinﬁnity by virtue of the transport Equations (18.15a)–(18.15g). Thus, they need
to be solved only on Z .
In analogy to the transport equations on N , the transport Equations (18.15a)–(18.15g) can be solved along the generators of I − exploiting a hierarchicalstructure if some basic data are provided. Some inspection reveals that thebasic data are given by either the connection coeﬃcient Γ10 11 or the rescaled
Weyl spinor component φ0 . The details of this construction will not be further
elaborated.

490

Characteristic problems for the conformal ﬁeld equations
18.2.1 The freely speciﬁable data

The discussion of the hierarchical structure of the interior equations on N ∪ I −allows the identiﬁcation of the basic reduced initial data set r from which
the full initial data u on N ∪ I − for the conformal Einstein ﬁeld equations
can be computed. As already observed, the choice of reduced initial data sets is
not unique. Two possible ways of specifying the reduced data are given in thefollowing:
Lemma 18.3 (freely speciﬁable data for the standard characteristic
problem) Assume that the gauge conditions given by Lemmas 18.1 and 18.2
are satisﬁed in a neighbourhood U of Z on N ∪ I − . Initial data u for theconformal Einstein ﬁeld equations on N ∪ I − can be computed from either of
the two following reduced initial data sets:(i) r1 consisting ofΓ10 11

on I − ,φ4on N ,

φ3 ,φ2 + φ̄2 ,ξA,on Z ;

φ3 ,φ2 + φ̄2 ,ξA,(ii) r2 consisting of
φ0

on I − ,φ4on N ,

Γ10 11 ,Φ20 ,on Z .

In both cases the ﬁeld ξ A is chosen so that −(ξ A ξ¯B +ξ¯A ξ B )∂ A ⊗∂ B is conformal
to the standard (contravariant) metric on S2 .
Remark. The reduced set r2 in (ii) has the advantage of being symmetric withrespect to N and I − .
Proof The proof of this lemma follows from the discussion in the previous
subsection. Further discussion can be found in Friedrich (1981a).
18.2.2 The reduced conformal ﬁeld equations
To apply the theory on the characteristic initial value problem discussed in
Section 12.5 one has to extract a suitable symmetric hyperbolic system out of theconformal ﬁeld Equations (18.11a)–(18.11c). Given the split between intrinsic
and transverse equations, a hyperbolic reduction procedure such as the one
discussed in Chapter 13 is not required. Instead, a suitable choice of reduced
conformal ﬁeld equations is given by the combinations
Ξ11 = 0,Σ11 BB  = 0,Z11 = 0,ΞCZ11 AB  = 0,

D11 BB  = 0,

(18.17a)(18.17b)18.3 A local existence result for characteristic problems− Δ1BC0 = 0,

Δ0BC0 − Δ1BC1 = 0,

− Λ01 00 = 0,Λ00 BC − Λ11 BC = 0,

Δ0BC  1 = 0,
Λ10 11 = 0.

491
(18.17c)(18.17d)A more explicit form of the equations is discussed in Section 18.3. Fromthese expressions, adopting the matricial notation of Chapter 12 and considering
suitable multiples of the equations, the reduced conformal ﬁeld equations can be
written schematically in the form
Aμ (x, u)∂μ u + B(x, u) = 0,

(18.18)with Aμ Hermitian matrices andAμ (ω 00 μ + ω 11 μ )positive deﬁnite.

(18.19)Thus, one obtains a symmetric hyperbolic system for the components of u.
Using the expressions for the principal part of the system (18.17a)–(18.17d),
a computation shows that the characteristic polynomial of the reduced system
contains factors of the form g μν ξμ ξν so that the null hypersurfaces N and I −are indeed characteristics of the system. It follows from (18.19) that the surfaceswith normal ω 00 + ω 11 are spacelike for the symmetric hyperbolic system.
Although the coordinates x0 = v and x1 = r have been constructed so that they
have non-negative values, the reduced Equations (18.17a)–(18.17d) also hold fornegative values of the coordinates. It follows that the hypersurfaceS ≡ p ∈ R × R × S2 | x0 (p) + x1 (p) = 0(18.20)is spacelike for Equation (18.18) in a neighbourhood of Z .
18.3 A local existence result for characteristic problems
As discussed in Section 12.5, the existence and uniqueness of solutions to acharacteristic initial value problem can be obtained via an auxiliary Cauchy
initial value problem on a spacelike hypersurface – in the present case thehypersurface S deﬁned by (18.20). The formulation of this auxiliary Cauchy
problem crucially depends on Whitney’s extension theorem so that initial dataon N ∪ I − can be extended to a spacetime neighbourhood U of Z . In turn, theapplication of Whitney’s theorem depends on being able to evaluate all (interior
and transverse) derivatives of the initial data on N ∪ I − .
18.3.1 Computation of the formal derivatives on N ∪ I −To verify that one can compute all derivatives of the initial data on N ∪ I − oneneeds to inspect the principal part of the reduced Equations (18.17a)–(18.17d).

492

Characteristic problems for the conformal ﬁeld equations

Borrowing the notation of Proposition 13.1, the reduced Equations (18.17a)–(18.17b) take the form∂r σ = G(σ, Γ, Φ, φ),(18.21a)∂r e = H(e, Γ),(18.21b)∂r Γ = K(Γ, Φ, φ);(18.21c)that is, they are transport equations along the direction given by e00 . For theequations in (18.17c) one has∂r Φ20 − ω̄∂r Φ21 − ξ¯A ∂A Φ21 = L20 (Γ, Φ, φ),∂r Φ10 − ω̄∂r Φ11 − ξ¯A ∂A Φ11 = L10 (Γ, Φ, φ),(18.22b)∂r Φ00 − ω̄∂r Φ01 − ξ¯A ∂A Φ01 = L00 (Γ, Φ, φ),(18.22c)(18.22a)A∂r Φ21 + ∂v Φ21 + U ∂r Φ21 + X ∂A Φ21− ω∂r Φ20 − ξ A ∂A Φ20 − ω̄∂r Φ22 − ξ¯A ∂A Φ22 = M21 (Γ, Φ, φ),(18.22d)A∂r Φ11 + ∂v Φ11 + U ∂r Φ11 + X ∂A Φ11− ω∂r Φ10 − ξ A ∂A Φ10 − ω̄∂r Φ12 − ξ¯A ∂A Φ12 = M11 (Γ, Φ, φ),(18.22e)A∂r Φ01 + ∂v Φ01 + U ∂r Φ01 + X ∂A Φ01− ω∂r Φ00 − ξ A ∂A Φ00 − ω̄∂r Φ02 − ξ¯A ∂A Φ02 = M01 (Γ, Φ, φ),A(18.22f)

A∂v Φ22 + U ∂r Φ22 + X ∂A Φ22 − ω∂r Φ21 − ξ ∂A Φ21 = N22 (Γ, Φ, φ), (18.22g)∂v Φ12 + U ∂r Φ12 + X A ∂A Φ12 − ω∂r Φ11 − ξ¯A ∂A Φ11 = N12 (Γ, Φ, φ), (18.22h)∂v Φ02 + U ∂r Φ02 + X A ∂A Φ02 − ω̄∂r Φ01 − ξ¯A ∂A Φ01 = N02 (Γ, Φ, φ), (18.22i)
where L20 , L10 , L00 , M21 , M11 , M01 , N22 , N12 and N02 are smooth functionsof their arguments – their explicit form will not be required. Finally, for theEquations (18.17d) involving the components of the rescaled Weyl tensor one has∂r φ0 − ω∂r φ1 − ξ A ∂A φ1 = W0 (Γ, φ),A∂r φ1 + ∂v φ1 + U ∂r φ1 + X ∂A φ1− ω̄∂r φ0 − ξ¯A ∂A φ0 − ω∂r φ2 − ξ A ∂A φ2 = W1 (Γ, φ),A∂r φ2 + ∂v φ2 + U ∂r φ2 + X ∂A φ2− ω̄∂r φ1 − ξ¯A ∂A φ1 − ω∂r φ3 − ξ A ∂A φ3 = W2 (Γ, φ),A∂r φ3 + ∂v φ3 + U ∂r φ3 + X ∂A φ3− ω̄∂r φ2 − ξ¯A ∂A φ2 − ω∂r φ4 − ξ A ∂A φ4 = W3 (Γ, φ),A∂v φ4 + U ∂r φ4 + X ∂A φ4 − ω̄∂r φ3 − ξ¯A ∂A φ3 = W4 (Γ, φ),(18.23a)(18.23b)(18.23c)(18.23d)(18.23e)(18.23f)(18.23g)(18.23h)with W0 , W1 , W2 , W3 and W4 smooth functions of their arguments – again, their
explicit form will not be required.
In what follows, it is shown that all formal partial derivatives on N ∪ I − canindeed be computed from the above equations.

18.3 A local existence result for characteristic problems

493

Computation of formal derivatives on I −To compute the formal derivatives on I − one ﬁrst observes that the partial
derivatives ∂v , ∂2 , ∂3 are interior, while ∂r is transverse. In this case, direct
inspection shows that except for∂r φ4 ,

∂r Φ22 ,∂r Φ12 ,∂r Φ02 ,all ∂r -derivatives of the unknown u can be computed using Equations (18.21a)–(18.21c), (18.22a)–(18.22f) and (18.23a)–(18.23g). The exceptional cases shown
above arise due to the fact that ω = U = 0 on I − so that Equations (18.22g)–(18.22i) and (18.23h) evaluated at I − do not, in fact, contain ∂r -derivatives.
To get around this problem one computes the ∂r -derivative of (18.22g)–(18.22i)and (18.23h) and then evaluates on I − to obtain the system∂v ∂r Φ22 + ∂r U ∂r Φ22 + ∂r X A ∂A Φ22 − ∂r ω∂r Φ12








− ∂r ξ A ∂A Φ12 − ξ A ∂A ∂r Φ12∂r N22 ,A∂v ∂r Φ12 + ∂r U ∂r Φ12 + ∂r X ∂A Φ12 − ∂r ω∂r Φ11− ∂r ξ A ∂A Φ11 − ξ A ∂A ∂r Φ11∂r N12 ,A∂v ∂r Φ02 + ∂r U ∂r Φ02 + ∂r X ∂A Φ02 − ∂r ω∂r Φ01− ∂r ξ A ∂A Φ01 − ξ A ∂A ∂r Φ01

∂r N02 ,A∂v ∂r φ4 + ∂r U ∂r φ4 + ∂r X ∂A φ4 − ∂r ω̄∂r φ3− ∂r ξ¯A ∂A φ3 − ξ¯A ∂A ∂r φ3∂ r W4 .

The latter can be interpreted as a system of ﬁrst-order linear ordinary diﬀerential
equations for ∂r φ4 , ∂r Φ22 , ∂r Φ12 , ∂r Φ02 . The initial data on Z for theseequations can be computed from the data on N ∪ I − . General results of thetheory of ordinary diﬀerential equations ensures that this system of equations canbe solved in a neighbourhood of Z on I − . Accordingly, all the ﬁrst transverse
derivatives on I − can be explicitly computed. The argument described in thisparagraph can be generalised, by repeatedly diﬀerentiating the reduced equations
with respect to ∂r , to iteratively compute higher order ∂r -derivatives as thesolution to a system of algebraic equations and linear PDEs.
Computation of formal derivatives on N
The analysis of the formal derivatives on N is almost the mirror image of thaton I − . In this case ∂r , ∂2 , ∂3 are interior derivatives, while ∂v is transverse.
After an inspection of the list of Equations (18.21a)–(18.21c), (18.22a)–(18.22i)and (18.23a)–(18.23h) one ﬁnds that only∂v φ4 ,∂v Φ22 ,∂v φ3 ,∂v Φ12 ,∂v φ2 ,∂v Φ11 ,∂v φ1 ,∂v Φ02 ,∂v Φ01

494

Characteristic problems for the conformal ﬁeld equations

are algebraically determined by the initial data on N . To obtain the remaining
transverse derivatives, one computes the ∂v -derivatives of Equations (18.21a)–(18.21c), (18.22a)–(18.22c) and (18.23a) and evaluates them on N to obtain aﬁrst-order system of ordinary diﬀerential equations along the generators of N for
∂v σ,

∂v e,

∂v Γ,

∂v Φ02 ,∂v Φ01 ,∂v Φ00 ,∂v φ0 .

Supplementing this system with the information on Z implied by the initial
data for the reduced equations, one ﬁnds that the general theory of ordinary
diﬀerential equations ensures the existence of solutions in a neighbourhood ofZ on N . In this manner one obtains a complete set of ﬁrst-order transverse
derivatives on N . Higher order transverse derivatives can be obtained iteratively
by computing higher order ∂v -derivatives of the reduced conformal ﬁeld equations
as required.
The analysis described in the previous paragraphs can be summarised in thefollowing:
Lemma 18.4 (computation of formal derivatives) Any arbitrary formal
derivatives (∂ α u) of the vector unknown u on N ∪I − can be computed from theprescribed initial data u for the reduced conformal ﬁeld equations on N ∪ I − .
18.3.2 The subsidiary system
To show that the solutions of the reduced equations imply a solution to thefull conformal ﬁeld equations if initial data satisfying the constraints on N and
I − are prescribed, it is necessary to obtain a suitable subsidiary system for thezero quantities encoding the conformal ﬁeld equations. The propagation of theconstraints is ensured by the following:
Proposition 18.1 (propagation of the constraints) A solution u of the
reduced conformal ﬁeld Equations (18.17a)–(18.17d) on a neighbourhood U of Z
on J + (Z ) that coincides with initial data on N ∪ I − satisfying the conformal
equations is a solution to the conformal ﬁeld Equations (18.11a)–(18.11c) on U .
A subsidiary system adapted to the geometry of the characteristic problem
described in the previous sections is obtained from the following derivatives ofthe zero quantities associated to the conformal ﬁeld equations:∇11 ΞAA ,∇11 ZAA ,∇11 ΣAA BB  ,(∇00 + ∇11 )ΔCDBB  ,∇11 ZAA BB  ,∇11 ΞCDAA BB (∇00 + ∇11 )ΛBB  CD .

Using arguments similar to those employed in Sections 13.3 and 13.4.5 one
rewrites the above derivatives as homogeneous expressions in the zero quantities.
Further details of these lengthy calculations can be found in Friedrich (1981a).

18.3 A local existence result for characteristic problems

495

Once a subsidiary system of the required form has been obtained, the propagationof the constraints follows from the uniqueness of solutions to the characteristic
problem.
In addition to Proposition 18.1 one has the following:
Corollary 18.1 (preservation of the conformal gauge) Let u denote asolution to the characteristic problem for the conformal ﬁeld equations on aneighbourhood U of Z on J + (Z ) which satisﬁes the gauge conditions given in
Lemmas 18.1 and 18.2. Then the metric g constructed from the components ofthe solution u satisﬁes the vacuum Einstein ﬁeld equations R[g] = 0.
This result follows from an argument similar to the one used to prove thepropagation of the algebraic conformal ﬁeld equation encoding the transformation rule for the Ricci scalar in Lemma 8.1. Here one considers thederivative
∇11 Ξ∇AA ∇AA Ξ − 2∇AA Ξ∇AA Ξ
and makes use of the conformal ﬁeld equations to rewrite it as a homogeneous
expression in zero quantities. In view of the transformation law of the Ricci
scalar under conformal rescalings, the term in brackets coincides with R[g]. Now,
from the discussion leading to Lemma 18.2 one concludes that R[g] = 0 on
N ∪ I − . The corollary then follows from the uniqueness of the solutions to thecharacteristic problem.
18.3.3 The existence result
Combining the analysis developed in the previous subsections with the theoryof characteristic initial value problems for symmetric hyperbolic systems of
Section 12.5, one obtains the following existence result:
Theorem 18.1 (existence and uniqueness to the standard asymptotic
characteristic problem) Given a smooth reduced initial data set r for theconformal Einstein ﬁeld equations on N ∪ I − , there exists a unique smooth
solution of the conformal ﬁeld equations in a neighbourhood U of Z in J + (Z )which implies the prescribed initial data on N ∪ I − .
Proof It follows from Lemma 18.4 that the formal derivatives of u can becomputed to any arbitrary order from the reduced data r on N ∪ I − .
Hence, it is possible to formulate an auxiliary Cauchy problem for the reduced
conformal ﬁeld Equations (18.17a)–(18.17d) with data implied by the extension
to a neighbourhood of Z given by Whitney’s theorem. Thus, using Theorem
12.7 and the discussion in Section 12.5.3 there is a neighbourhood W of Z in
J + (Z ) in which there exists a unique solution u to the reduced conformal ﬁeld
equations which on N ∪I − coincides with the data u implied by the prescribed

496

Characteristic problems for the conformal ﬁeld equations

reduced initial data – as Z ≈ S2 , it is necessary to combine solutions in two
diﬀerent patches. Finally Proposition 18.1 and Corollary 18.1 imply that thesolution to the reduced equations is, in fact, a solution to the full conformal ﬁeld
equations.
The characteristic problem on N  ∪ I +The analysis leading to Theorem 18.1 can be adapted to analyse the dual
asymptotic characteristic problem with data on N  ∪ I + where N  is a futureoriented null hypersurface. In this case one endeavours to ﬁnd a solution in aneighbourhood U  of Z  = N  ∩ I + in J − (Z  ). All the relevant expressions
can be obtained from those for the characteristic problem on N ∪ I − throughthe replacements 0 → 1 , 1 → 0 in the spinorial frame indices so thate00 → e11 ,e11 → e00 ,e01 → e10 ,e10 → e01 .

In particular, one has
φ0 → φ4 ,φ1 → φ3 ,φ2 → φ2 ,φ3 → φ1 ,φ4 → φ0and
ω → ω̄,ξ A → ξ¯A .

Similarly, for the connection coeﬃcients and the components of the trace-free
Ricci spinor one has
Γ01 00 → Γ10 11 ,

Φ12 → Φ10 = Φ01 ,and so on.

For consistency, one should replace the coordinate v along the generators of I −with a coordinate u along the generators of I + .
18.4 The asymptotic characteristic problem on a cone
As discussed in the introduction, an alternative characteristic problem for theconformal Einstein ﬁeld equations consists of a conﬁguration where initial data
is prescribed in a neighbourhood of the vertex of a cone representing thetimelike inﬁnity of a Minkowski-like spacetime; see Figure 18.1, right. This
type of geometric setup for a characteristic initial value problem was originallyintroduced in Friedrich (1986c) and is intended to model purely radiative
spacetimes, that is, a system describing gravitational radiation from pastnull inﬁnity which interacts non-linearly with itself and eventually escapes tofuture null inﬁnity. Intuitively, one would expect this type of solution to the
Einstein ﬁeld equations to have a smooth structure at null inﬁnity. To ensurethat the gravitational ﬁeld consists only of gravitational radiation one requiresthat the generators of null inﬁnity are complete and that past timelike inﬁnity isrepresented by a point i− which is regular from the point of view of the conformal
completion.

18.4 The asymptotic characteristic problem on a cone

497

To discuss the geometric setting in a more precise manner it is convenient tointroduce some deﬁnitions.
Deﬁnition 18.1 (spacetimes with a cone past boundary ) A spacetime(M, g) is said to have a cone past boundary if:(i) There exists a causal, oriented and time-oriented spacetime (M , g  ) (theambient manifold).
(ii) There exists a point o ∈ M such that the set consisting of o and all pointsof M which can be joined to o by a causal curve in M – to be denoted by
J + (o, M ) – is closed in M .
(iii) Given No ≡ ∂J + (o, M ), then No \ {o} is a smooth null hypersurface ofM .
(iv) The set M corresponds to J + (o, M ) together with the structures it inherits
from (M , g  ) – in particular, g is the pull-back of g  to M.
Given p ∈ M, the set Np ⊂ M is called the future null cone of p.
In terms of the above notions one introduces the further notion:
Deﬁnition 18.2 (spacetimes with a complete past null inﬁnity cone)
A vacuum spacetime (M̃, g̃) is said to be a solution to the Einstein ﬁeld
equations with complete null cone at past timelike inﬁnity i− if there
exists a conformal extension (M, g, Ξ) with cone-like past boundary Ni− suchthat the conformal factor satisﬁes
Ξ>0

on M \ Ni− ,Ξ=0

on Ni− ,dΞ = 0dΞ = 0,(18.24a)(18.24b)−on Ni− \ {i },(18.24c)−

Hess Ξ non-degenerate at i ,(18.24d)and there is a diﬀeomorphism by means of which the manifolds M̃ and M \ Ni−
can be identiﬁed so that g = Ξ2 g̃ on M \ Ni− . The set Ni− \ {i− } is swept by
the future-directed null geodesics through i− and represents the past null inﬁnity
I − of the spacetime.
Equipped with the above deﬁnitions, one can formulate a pure radiation
problem in which one asks: given data on a cone No , is there a unique solution
to the Einstein ﬁeld equations with complete past null inﬁnity implying ﬁelds on
I − which can be identiﬁed with the data prescribed on No and such that the
point o can be identiﬁed with i− ?
18.4.1 Gauge conditions
This section gives a brief discussion of the gauge speciﬁcation process for thecharacteristic initial value problem on a cone. As is the case in all initial value

498

Characteristic problems for the conformal ﬁeld equations

problems concerning the conformal ﬁeld equations, one has to consider three
diﬀerent types of gauges: conformal, coordinate and frame gauges. These are
analysed in turn.
The conformal gauge
Given a null cone No with vertex o, let l denote the vector tangent to the nullgenerators of No . Consistent with conditions (18.24a)–(18.24d), it is assumed
that one has a conformal factor Ξ such thatΞ = 0,s = 0dΞ = 0,at o.

Mimicking the discussion of Section 16.3, one can transvect the conformal ﬁeld
equations∇a ∇b Ξ = −ΞLab + sgab ,∇a s = −∇b ΞLba ,(18.25)

with l to ﬁnd that Ξ = 0 and s = 0 on No and, moreover, that dΞ = 0 on No \{o}.
It is also observed that if s|o = 0, then dΞ = 0 on No . The behaviour of theconformal gauge at o can be reﬁned by considering a rescaling as in Equation(18.6) with ϑ > 0. Making use of the transformation formula for the Friedrichscalar s, Equation (8.29b), one ﬁnds that s |o = (sϑ−1 )|o . Let γ(ς) with ς ∈ Rdenote a future-directed null geodesic on No with γ(0) = o such that l = γ̇and, consequently, ∇l l = 0. Setting l ≡ ϑ−1 l, one ﬁnds that g  (l , l ) = 0 and∇l l = 0 as well. Using the transformation formula for the trace-free Ricci tensor
Φab , Equation (18.8), one ﬁnds that along γ it holds that
ϑ3 la lb Φab = ϑ−1 la lb Φab + 2lb ∇b la ∇a (ϑ−1 ) .
Thus, if the value of the component la lb Φab is prescribed, the above equation
can be read as an ordinary diﬀerential equation for ϑ along the null geodesic γ.
The initial value of ϑ can be ﬁxed through the speciﬁcation of s|o . Using the ﬁrst
of the equations in (18.25) one ﬁnds thats|o g  (l , l )|o = ∇l ∇l Ξ |o .
In order to have a local minimum of Ξ at o, one needs that ∇l ∇l Ξ |o > 0 forcing
s|o > 0 – in the signature (+ − −−). Without loss of generality, one can then set
s=2

at o,

(18.26)andla lb Φab = 0

on No near o.

(18.27)In this construction there is still the freedom of specifying the value of dϑ at o.
Adapting the arguments of Section 18.1.2 one can set a characteristic initial value
problem on No for the wave Equation (18.7) in such a way thatR[g] = 0

on J + (No ) near o.

(18.28)

18.4 The asymptotic characteristic problem on a cone499

The coordinates and the frame near o
A convenient four-dimensional description of the null cone No is obtained using
g-normal coordinates y = (y μ ) centred at o; see Sections 2.4.5 and 11.6.2.
Accordingly, one has that y μ (o) = 0, gμν (o) = ημν , ∂λ gμν (o) = 0 and Γμ ν λ (o) = 0.
These properties can be more concisely summarised in the expressiony μ gμν = y μ ημν

in a neighbourhood of o.

(18.29)In these coordinates, for ﬁxed (y μ ) = 0 one has that the curve γ : ς → ςy μ is a
geodesic through o and thatNo = {y μ ∈ R4 | ημν y μ y ν = 0, y 0 ≥ 0}.
Thus, in these coordinates the null cone No can be thought of as being the nullcone through the origin in Minkowski spacetime.
Associated to the g-normal coordinates, it is natural to consider a normal
frame centred at o, that is, a frame {ea } which, in a neighbourhood U of o,satisﬁes g(ea , eb ) = ηab and ∇γ̇ ea = 0 for any geodesic passing through o.
Without loss of generality, one can assume that the frame coeﬃcients in ea =ea μ ∂ μ satisfy ea μ (o) = δa μ . Using the properties of the exponential function, it
can be shown that the frame coeﬃcients ea μ depend smoothly on the coordinates(y μ ). It can then be veriﬁed that g(γ̇, ea ) is constant along γ. Moreover, usingthat gμν = ηab ω a μ ω b ν , it can be shown thaty μ δμ a ea ν (y) = y ν ,y μ ημν ea ν (y) = y μ ημν δa ν .

(18.30)

The above conditions can be regarded as an alternative deﬁnition of normal
coordinates. More precisely, if a set of coordinates y = (y μ ) and frame coeﬃcients{ea μ } satisfy the conditions in (18.30) the metric components gμν will satisfy
condition (18.29).
To complete the discussion, it is convenient to introduce the vector ﬁeld
y(y) = y μ ∂ μ tangent to the geodesics through o. One then has
y(o) = 0,(∇μ y ν )|o = δμ ν ,∇y y = y.

Writing y in terms of a g-normal frame one has that y = y a ea where y a (y) =δν a y ν . Furthermore, using ∇y ea = 0 one concludes thaty a (y)Γa b c (y) = δν a y ν Γa b c (y) = 0,close to o.

The coordinates y = (y μ ) and the frame {ea } satisfying the conditions
discussed in the previous paragraphs will be collectively known as a normal
gauge. This gauge system is supplemented by a normalised spin frame {A A }satisfying y AA ∇AA A B = 0 such that {eAA } = {A ¯A } with eAA =σAA a ea – here y AA is the spinorial counterpart of the vector y. In what follows,
all spinors will be expressed in components with respect to this type of frame.

500

Characteristic problems for the conformal ﬁeld equations
Adapted coordinates on No

The coordinates y = (y μ ) introduced in the previous subsections provide
a convenient spacetime description of No . However, to analyse the intrinsic
geometry of the cone, one needs adapted coordinates. The construction of thesecoordinates is similar to that of the coordinates (v, r, xA ) used in the analysisof the characteristic problem on N ∪ I − in Section 18.1.1. The fundamental
diﬀerence is that, in the case of a cone, these adapted coordinates degenerate
at the vertex o. More precisely, one can consider adapted coordinates x = (xμ )such that No is given as a level surface by the condition r ≡ x1 = 0 and v ≡ x0
is a parameter along the generators with tangent l – thus, l = ∂ v . The twodimensional spacelike surfaces Zv• ≡ {p ∈ N | v(p) = v• } satisfy Zv• ≈ S2 ,except for the limit case Z0 = {o} which is a point. As in Section 18.1.1, (xA )denote local coordinates on Zv• . On No the covector n ≡ dr is null and normal
to No . The coordinate r can be chosen so that one has the usual normalisationg(l, n) = 1. Finally, the vectors l and n can be completed to a frame by choosing
a pair of complex conjugate vectors m, m̄ ∈ T (Zv• ), for v• = 0, such thatg(m, m̄) = −1. As in Section 18.1.1 the vectors m and m̄ can be parallelly
propagated along the generators of No oﬀ some ﬁduciary section Zv• .
18.4.2 Null data on the cone
As in the case of the characteristic problem on N ∪I − , there are several ways ofprescribing the free data. The most physically meaningful speciﬁcation consistsof the so-called radiation ﬁeld encoding information on the two componentsof the Weyl tensor with the slowest fall-oﬀ at null inﬁnity, and can be thoughtof as describing the two polarisation states of incoming radiation.
To describe the null data, let, as in previous sections, l denote the vector
tangent to the generators of the null cone No . As l is a null vector, there exists a
spinor κA such that lAA = κA κ̄A with lAA the spinorial counterpart of l. The
spinor κA is deﬁned up to a phase κA → eiϑ κA with ϑ ∈ R constant along thenull generators. The radiation ﬁeld is then deﬁned as the component
φ0 ≡ κA κB κC κD φABCDof the rescaled Weyl spinor. Due to the phase ambiguity in κA , the radiationﬁeld is a spin-weighted quantity. The information encoded in the radiation ﬁeld isequivalent to information on the pull-back of dabcd la lc to No . More precisely, if m
and m̄ are complex vectors tangent to the sections of No such that g(l, m) = 0,then it follows from the symmetries of the Weyl tensor that φ0 = dabcd la mb lc md .

Solving the constraints on No
In analogy to the characteristic problem on N ∪ I − , and making use of theadapted coordinates x = (v, r, xA ) and of the frame {l, n, m, m̄}, the conformal

18.4 The asymptotic characteristic problem on a cone

501

Einstein ﬁeld equations split into equations transverse and intrinsic to No . The
intrinsic equations divide, in turn, into propagation equations (i.e. ordinary
diﬀerential equations) along the generators of the cone and constraints which
need to be solved only at a particular cut. Assuming the conformal gauge
discussed in Section 18.4.1, the knowledge of the radiation ﬁeld φ0 on No allows
one to compute the value of the remaining conformal ﬁelds in a neighbourhoodof o on No . More precisely, one has the following:
Proposition 18.2 (reduced initial data for the asymptotic characteristic
problem on a cone) In the conformal gauge given by conditions (18.26),(18.27) and (18.28), the transport equations induced by the conformal Einsteinﬁeld equations and the structure equations on No uniquely determine the ﬁelds Ξ,s, ΦAA BB  and φABCD on No once the radiation ﬁeld φ0 has been prescribed.
The resulting ﬁelds satisfy the constraint equations on No .
Details on this result can be found in Friedrich (2014b).
Evaluating formal derivatives on No
In addition to solving the constraint equations on No , and in order to apply thetheory of characteristic problems on a cone, given a choice of radiation ﬁeld, it isnecessary to show that the (formal) derivatives of any order of the conformalﬁelds can be determined on the null cone along the generators of No . This
analysis is analogous to the one discussed in Section 18.3.1 for the characteristic
problem on N ∪ I − . In the present case, however, the analysis is more delicate
as the set Z = N ∩ I − shrinks to a point, so that the information for the
integration along the generators has to be extracted solely from the null data.
The key result is the following (see Friedrich (2014b)):
Proposition 18.3 (computation of formal derivatives at the vertex ) Ina neighbourhood of the point o let the ﬁelds Ξ, s, ΦAA BB  φABCD , eAA μ ,ΓAA BC be smooth and be expressed in an o-centred normal gauge and aconformal gauge satisfying Equations (18.26), (18.27) and (18.28). If the aboveﬁelds satisfy the conformal ﬁeld equations, then the Taylor expansions of the ﬁelds
Ξ, s, ΦAA BB  and φABCD in a suitable neighbourhood of o are determined by
the null datum φ0 .
Remark. In the above proposition, the neighbourhoods of o are spacetime
neighbourhoods in the ambient manifold M containing the cone No .
18.4.3 The existence result
The setting described in the previous paragraphs leads to the following existence
result, adapted from Chruściel and Paetz (2013):

502

Characteristic problems for the conformal ﬁeld equations

Figure 18.4 Schematic representation of the set up for the asymptotic
characteristic problem on a cone. The existence results are restricted to aneighbourhood U of o in J + (o).

Theorem 18.2 (local existence for the asymptotic characteristic problem on a cone) For any smooth prescription of the radiation ﬁeld φ0 on the nullcone at the origin of the Minkowski spacetime, No , there exists a neighbourhood
U ⊂ J + (o) of o, a smooth metric g and a smooth function Ξ such that:(i) No is the light cone of o for g.
(ii) Ξ = 0 on No .
(iii) dΞ = 0, Hess Ξ = 0 on o.
(iv) dΞ = 0 on ∂J + (o) ∩ U \ {o}.
(v) The function Ξ has no zeros on U ∩I + (o) and the metric g̃ = Ξ−2 g satisﬁes
the vacuum Einstein ﬁeld equations on U ∩ I + (o).
Moreover, the rescaled Weyl spinor φABCD of the pair (g, Ξ) extends smoothly
across No and the restriction of φABCD 0 A 0 B 0 C 0 D to No \{o} coincides withthe prescribed radiation ﬁeld φ0 . The solution is unique up to isometries.
Remark. It follows from points (ii), (iii) and (iv) that the set No \ {o}corresponds to the past null inﬁnity I − of the resulting spacetime, while thevertex o is its past timelike inﬁnity i− . A schematic representation of the set upof the above theorem is given in Figure 18.4.
The proof of the above theorem, as given in Chruściel and Paetz (2013), makes
use of the metric version of the conformal ﬁeld equations and the associated wave
equations discussed in Paetz (2015); see also Section 13.5.2. The reason behindthe use of a hyperbolic reduction based on wave equations – as opposed, say, tothe ﬁrst-order symmetric hyperbolic systems used throughout this book – liesin the fact that the available theory of characteristic problems on a cone is wellunderstood for this type of equations; see Dossa (1986, 2002).
18.5 Further reading
Characteristic problems in general relativity have a long history. The ﬁrst systematic discussion has been given in Sachs (1962c). Further classical discussions
can be found in Penrose (1965, 1980) and Müller zu Hagen and Seifert (1977).

18.5 Further reading503

A review on the various approaches to the problem, including an analysis of thepossible choices of free data, can be found in Chruściel and Paetz (2012); thisreference provides a convenient point of entry to the literature on the subject.
The basic theory of asymptotic characteristic initial value problems forthe conformal ﬁeld equations has been developed in the articles by Friedrich(1981a,b). A version of Theorem 18.1 in the analytic setting was given in Friedrich(1982). This result has been extended to the smooth setting in Kánnár (1996b)
using the reduction to an auxiliary Cauchy problem given in Rendall (1990).
The geometric set up for the asymptotic characteristic problem on a cone has
ﬁrst been given in Friedrich (1986c). The relation between Taylor expansions atthe vertex of the null cone and the interior equations implied by the conformal
Einstein ﬁeld equations has been examined in Friedrich (2014b). The existence
result for the characteristic problem in the cone has been given in Chruściel
and Paetz (2013). Characteristic problems on a cone are less studied than thoseon intersecting null hypersurfaces. A good point of entry to the literature is
Choquet-Bruhat et al. (2011).
Characteristic problems provide a natural approach to the constructionof solutions to the Einstein equations by means of numerical methods. An
advantage of this formulation is its clear-cut connection with the notion ofgravitational radiation; see, for example, Damour and Schmidt (1990). A review
on the subject can be found in Winicour (2012).
The characteristic initial value problem has been used in the seminal work by
Christodoulou on the collapse of a spherically symmetric self-gravitating scalar
ﬁeld; see Christodoulou (1986).

19
Static solutions

In this chapter a study of static solutions to the vacuum Einstein ﬁeld equations
from the point of view of conformal methods is undertaken. Static and, moregenerally, stationary solutions provide valuable physical and mathematicalintuition concerning the behaviour of solutions to the Einstein ﬁeld equations.
Static solutions describe the exterior region of time-independent, non-rotating,isolated bodies. Accordingly, they provide an interesting class of solutions toanalyse the structure of spatial inﬁnity; see Chapter 20. In addition, someparticular static solutions (the Schwarzschild spacetime) are expected to describe
the asymptotic state of the evolution dictated by the Einstein ﬁeld equations.
From a mathematical point of view, the results discussed in this chapter areof particular interest as they lie at the interface of classical potential theory,
conformal geometry and general relativity. Throughout this chapter, the focus isrestricted to the asymptotic region of an asymptotically ﬂat static spacetime.
Several of the key results for static spacetimes admit a suitable stationary
counterpart; the interested reader is referred to the literature for further details.
These generalisations of the theory are much more technically involved than theoriginal static version and they will not be considered here.
19.1 The static ﬁeld equations
For a static spacetime it will be understood a solution to the Einstein ﬁeld
equations Ric[g̃] = 0 endowed with a hypersurface orthogonal Killing vector ξwhich, in a suitable asymptotic region, is timelike. Using coordinates (t, y) =(t, y α ) adapted to this Killing vector, one has that ξ = ∂ t . As ξ is hypersurface
orthogonal, then there exists a function v = v(y) such thatξ  = g̃(ξ, ·) = v 2 dt.
Thus, v 2 = g̃(∂ t , ∂ t ) is the square of the norm of ξ. It follows that thehypersurfaces of constant coordinate t deﬁne a foliation of the spacetime. In what

19.1 The static ﬁeld equations

505

follows, it will be convenient to consider a frame {ea } adapted to the static
Killing vector and set e0 to be parallel to ξ; that is, one has ξ = ve0 . The
spatial part of the frame, {ei }, spans the tangent bundle of the hypersurfacesof constant t. Without loss of generality, the spatial frame can be parallely
propagated along the direction of e0 so that using the deﬁnition of the connection
coeﬃcients one has that Γ̃0 a i = 0. Let {ω a } be the associated coframe. One
readily ﬁnds that ω 0 = vdt. It follows from this discussion that the metric g̃takes the formg̃ = v 2 dt ⊗ dt + h̃,v = v(y) > 0,h̃ = h̃αβ (y)dy α ⊗ dy β ,(19.1)

where h̃ denotes a (negative deﬁnite) Lorentzian metric on the hypersurfaces of
constant time coordinate.
Derivation of the static equations
The equations satisﬁed by the ﬁelds v and h̃ appearing in the metric (19.1) canbe deduced using the frame formalism introduced in the previous paragraphs.
Observing that ξ0 ≡ ξ  , e0  = v one concludes that ξa = vδa 0 . It follows that
the Killing equation
˜ b ξa = 0
˜ a ξb + ∇∇takes the form
˜ a vδb 0 + ∇˜ b vδa 0 + v Γ̃a 0 b + Γ̃b 0 a = 0.
∇As v is time independent, one concludes from setting a, b = 0 that Γ̃0 0 0 = 0.
Setting a = i and b = j one ﬁnds that Γ̃i 0 j + Γ̃j 0 i = 0 so that from the deﬁnitionof the extrinsic curvature, Equation (2.45), one concludes that Kij = 0; thatis, the surfaces of constant coordinate t are time symmetric. Accordingly, theEinstein constraint Equations (11.13a) and (11.13b) reduce to the conditionr[h̃] = 0.

(19.2)

A further condition can be obtained from the equation
˜ a ξb + ∇˜ b ξa = 0.
˜0 ∇∇Commuting covariant derivatives and using that the Killing equation implies
˜ 0 ξb = − ∇˜ b ξ0 = − ∇˜ bv∇
one ﬁnds that˜ b v + v R̃0a0b = 0.
˜ a∇∇From the last equation, using the Gauss-Codazzi identity, Equation (2.47), one
concludes that˜ j v = vr̃ij
˜ i∇∇(19.3)

506

Static solutions

where rij denotes the components with respect to {ei } of the Ricci tensor of the
3-metric h̃.
Equations (19.2) and (19.3) provide the required static Einstein ﬁeld
equations for the ﬁelds v and h̃. After some further slight manipulations they
can be rewritten in tensorial form as
Δh̃ v = 0,1
r̃ij = D̃i D̃j v,
v(19.4a)(19.4b)where D̃i and r̃ij denote, respectively, the Levi-Civita connection and the Ricci
tensor of the 3-metric h̃. In what follows, a pair (v, h̃) solving the static
equations (19.4a) and (19.4b) will be called a static solution. A static
solution, expressed in terms of h̃-harmonic coordinates is analytic; see Müller
zu Hagen (1970).
Observe that discarding the ﬁeld v, a solution to the static equations gives rise
to a solution to the time-symmetric Einstein constraints. This dual perspective of
static solutions as a spacetime and as time-symmetric initial data for a spacetime
will be used often. The context will dictate the appropriate point of view.
Equations (19.4a) and (19.4b) can be regarded as a three-dimensional analogueof the Einstein ﬁeld equations in which the curvature is coupled to a ﬁctitious
matter ﬁeld described by v. This interpretation also holds for other symmetry
reductions of the vacuum Einstein ﬁeld equations, say, axial symmetry; see, forexample, Geroch (1971a, 1972a).
Asymptotic conditions and the Licnerowicz theorem
Of special interest are static solutions describing the asymptotic region of isolated
systems. For simplicity, it will be assumed that S̃ has a single asymptotic regionin which coordinates y = (y α ) can be found such thatm
+ Ok (|y|−(1+ε) ),
|y|


2m
h̃αβ = − 1 +δαβ + Ok (|y|−(1+ε) ),
|y|

v =1−

(19.5a)(19.5b)as |y| → ∞ where m = 0 denotes the Arnowitt-Deser-Miser (ADM) mass andε > 0. The notation Ok has been described in the Appendix to Chapter 11.
The above decay conditions can be deduced from more primitive assumptions
which make no reference to asymptotic ﬂatness; see Reiris (2014a,b). In orderto describe an isolated system – say, the exterior of a star – Equations (19.4a)and (19.4b) need to be supplemented with suitable boundary conditions at aninterior boundary ∂ S̃ – say, the surface of a star. An analysis of this type has
been carried out by Reula (1989) and Miao (2003). The role played by boundary
conditions in the determination of static solutions is nicely exhibited in the case19.1 The static ﬁeld equations

507

where S̃ ≈ R3 . In this case, it follows from Equation (19.4a) by integration byparts thatvΔh̃ vdμ = − D̃i v D̃i vdμ,
0=
S̃

S̃

so that D̃i v = 0 on S̃. This implies that v has to be constant on S̃. Moreover,using (19.5a) one concludes that v = 1. Substituting into Equation (19.4a) one
ﬁnds that r̃ij = 0 so that h̃ must be ﬂat – recall that in three dimensions
the curvature is fully determined by the Ricci tensor. Consequently, in order tohave static solutions other than the Minkowski solution one needs hypersurfacesS̃ with a non-trivial topology or with some inner boundary ∂ S̃. This result is
usually known as Licnerowicz’s theorem.
19.1.1 The conformal static ﬁeld equations
In the remainder of this chapter, the discussion of static solutions will be
restricted to a suitable asymptotic region where the decay conditions (19.5a)and (19.5b) hold. Accordingly, it is convenient to make use of the deﬁnition of
asymptotically Euclidean and regular manifolds given in Section 11.6.2. Hence,
one considers a function Ω on S ≡ S̃ ∪ {i} with Ω ∈ C 2 (S̃) ∩ C ∞ (S̃), Ω > 0 on
S̃ which conformally extends h̃ to a smooth metrich ≡ Ω2 h̃on S,

in such a way thatΩ = 0,Di Ω = 0,

Di Dj Ω = −2hij ,at i.

(19.6)In order to exploit the above conformal setting, it is convenient to rewrite
the static Equations (19.4a) and (19.4b) in terms of ﬁelds satisfying regularequations in a neighbourhood of i. The procedure of constructing a system ofregular conformal static equations is similar in spirit to the one carried out
in Chapter 8 to obtain the conformal ﬁeld equations. The key idea is to identify
quantities which in the conformally rescaled picture are both suitably regular andwhich satisfy equations that are formally regular at i. In this spirit, the equation
obtained from combining the static Equation (19.4b) with the transformation
law of the three-dimensional Ricci tensor, Equation (5.16a), should be read not
as a diﬀerential condition for the components of a conformally rescaled metric
but rather as diﬀerential equations involving second derivatives of a quantity
associated to the conformal factor. Similar considerations need to be taken into
account when attempting to construct a conformal equation for the scalar ﬁeld v.
Using the transformation law for the Yamabe operator, Equation (11.23), one
obtains1Δh − r[h] Ω−1/2 v = 0.
8

508

Static solutions

This equation is formally singular at i unless it is possible to tie the behaviourof Ω with that of v. Alternatively, one could try to ﬁnd a regular equation for aquantity which indirectly allows one to gain knowledge about v. These ideas are
explored in the following subsections.
Fixing the conformal gauge
The standard approach to obtain a set of regular conformal static ﬁeld equations
relies on a speciﬁc choice of conformal gauge which explicitly prescribes theconformal factor Ω in terms of the norm of the static Killing vector v; see,for example, Beig and Simon (1980a) and Friedrich (1988, 2004, 2007). In thefollowing, the approach taken in the last two references will be followed. A generalversion of the conformal static equations which retains the whole conformal
freedom has been given in Friedrich (2013).
It can be veriﬁed that the conditions (19.6) expressed in terms of physicalcoordinates y = (y α ) require Ω to behave like 1/|y|2 as |y| → 0. This observationsuggests, in turn, considering a conformal factor of the form

2
1−v
Ω=
.
(19.7)m
As will be seen in Section 19.2, this is not the only possible way of ﬁxing theconformal freedom. The choice in Equation (19.7) ﬁxes the value of the Ricci
scalar of the conformal metric h. This can be seen from the transformation lawof the Yamabe operator, Equation (11.23), by setting u = Ω1/2 and making the
replacements φ → Ω1/2 , h → h, h → h̃ so that, on the one hand, one has


1−v
1
Lh [1] = Ω−5/2 Lh̃= − Ω−5/2 Δh̃ v = 0,mmwhile, on the other hand,Lh [1] =11
Δh − r[h] [1] = − r[h].
88Hence, one concludes that r[h] = 0.
A decomposition of the conformal factor
Following the general discussion of Section 11.6.3, one has that the conformal
factor Ω satisﬁes




1
on S̃,
Δh − r[h] Ω−1/2 = 0,8
and |x|Ω−1/2 → 1 as |x| → 0. Here, and in what follows, let x = (xα ) denote
some coordinates in a neighbourhood U ⊂ S with xα (i) = 0. Close to i one has
the representation
Ω−1/2 = ζ −1/2 + W,

(19.8)

19.1 The static ﬁeld equations

509

with ζ, W smooth – confront this decomposition with the discussion in Section
11.6.4. In particular, one hasm1
W (i) = ,(19.9)
Δh − r[h] W = 0,
8
2
and




1
Δh − r[h] (ζ −1/2 ) = 4πδ[i].
8(19.10)From this last equation it follows that
ζ(i) = 0,Di ζ(i) = 0,

Di Dj ζ(i) = −2hij (i).

(19.11)

One has that ζ is essentially the Green function of the Yamabe operator
and describes the local geometry in a neighbourhood of i, while W encodes
information of a global nature – in particular, its ADM mass m. Accordingly,
the functions ζ and W will be called, respectively, the massless part and masspart of the conformal factor Ω. Given a conformal metric h, the decomposition(19.8) is unique. Moreover, using the so-called Hadamard’s parametrix
construction, it can be shown that ζ and W are analytic if h is analytic;
see Friedrich (1998c, 2004) for further details about this last statement andGarabedian (1986) for the underlying PDE theory. In particular, for the choiceof the conformal factor (19.7) it follows that the parametrisation (19.8) takes the
form

2
m2
m
1 1−v
μ≡
.
(19.12)
,
W = ,
ζ=
μ 1+v
2
4
It can be veriﬁed that the function ζ satisﬁes the asymptotic conditions (19.11)
and that W is, indeed, a solution of Equation (19.9).
Using the chain rule to rewrite Equation (19.10) as an equation for Δh ζ andtaking into account the asymptotic conditions (19.11), one ﬁnds that1
Δh ζ,
(19.13)3
which is a regular equation in a suitable neighbourhood of i. In particular,
it can be veriﬁed that ς(i) = −2. Equation (19.13) is the analogue of theconformal Einstein ﬁeld Equation (8.24). It encodes a regularised version of thetransformation equation for the Ricci scalar. As will be seen in the following,
it can be interpreted as a constraint which is automatically satisﬁed if otherequations hold.
2ζς = Di ζDi ζ,

with ς ≡Equations for the curvature
To exploit the fact that one is working with a gauge for which r[h] = 0, it isconvenient to introduce an h-tracefree tensor sij such thatrij [h] = sij .

510

Static solutions

Recalling that in three dimensions the Riemann curvature tensor ri jkl is fully
determined by the Ricci tensor, it is then natural to interpret the tensors
rij and sij as describing, respectively, the geometric and algebraic threedimensional curvatures; see Section 8.3.1 for further discussion on thesenotions in the context of the conformal Einstein ﬁeld equations. If the zero
quantityΞij ≡ rij [h] − sijvanishes, then the three-dimensional (contracted) Bianchi identity takes the form
Di sij = 0.

(19.14)The ﬁelds ζ, ς and sij can be used to obtain a regular version of the formally
singular transformation law for the Ricci tensor; see Equation (5.16a). Rewriting
derivatives of the conformal factor Ω as derivatives of ζ one obtains
Sij ≡ Di Dj ζ − ςhij + ζ(1 − μζ)sij = 0.

(19.15)Equation (19.15) will be read as a diﬀerential equation for ζ. To close the system
one needs diﬀerential equations for ς and sij . Suitable equations can be obtained
from the integrability conditions
1
D[k Si]j + Dl Sl[k hi]j = 0,2

Di Sij = 0,

(19.16)

encoding the three-dimensional second Bianchi identity in contracted anduncontracted form, respectively. The identities (19.16) can be veriﬁed through a
direct computation and introducing the zero quantities
Si ≡ Di ς + (1 − μζ)sij Dj ζ = 0,
Hkij ≡ (1 − μζ)D[k si]j − μ 2D[k ζsi]j + Dl ζhl[k si]j .
It follows from a further computation that Si = 0 and Hkij = 0 are equivalent
to the integrability conditions (19.16). The condition Hkij = 0 can be read as
an expression for the Cotton tensor
1
bjki ≡ D[k ri]j − D[k rhi]j = 2D[k li]j ,4where lij denotes the three-dimensional Schouten tensor. In the remainder ofthis chapter it is often more convenient to work with the dualised version of bjki ,bij ≡ 12 bikl j kl . A computation shows thatbij =μ
1 − μζ1sli j kl Dk ζ − slm ji l Dm ζ .
2

19.1 The static ﬁeld equations

511

Summary: the conformal static equations
In what follows, the conditions
Ξij = 0,Si = 0,

Sij = 0,

Hkij = 0(19.17)will be known as the conformal static equations. They provide an overdetermined system of diﬀerential conditions for the ﬁelds hij , sij , ζ and ς. As will beseen, the equations in (19.17) imply an elliptic system for the components of thevarious conformal ﬁelds.
Remark. A direct computation yields the identity
Di 2ζς − Dk ζDk ς = 2ζSi − 2Sik Dk ζ.
Thus, if Si = 0 and Sij = 0, then 2ζς − Dk ζDk ς is a constant. Evaluating at
i and using the known values of the various ﬁelds at this point, one concludes
that the expression in brackets must vanish. This argument shows that Equation(19.13) plays the role of a constraint. Hence, it has not been included in the list
(19.17).
19.1.2 Spinorial version of the equations
To write the spinorial version of the conformal static equations, let sABCD =s(ABCD) denote the spinorial counterpart of the trace-free tensor sij . The Bianchi
identity (19.14) takes the form
DAB sABCD = 0.
In terms of the spinor sABCD the equation Hkij = 0 takes, after exploiting theantisymmetry in the pair ki , the simple form
DA Q sBCDQ =2μsQ(ABC DD) Q ζ.
1 − μζ

(19.18)The spinorial transcription of equations Si = 0 and Sij = 0 is completely
direct. When working with spinors, the equation rij = sij is replaced by the
Cartan structure equations, Equations (2.41) and (2.42), for the 3-geometry withan algebraic 3-curvature given by sij . These structure equations provide,
respectively, diﬀerential conditions for the coeﬃcients of a frame and for the
associated spin connection coeﬃcients; see, for example, Friedrich (2007). It willoften be convenient to express the various spinorial ﬁelds and the associatedequations in terms of their components (e.g. sABCD ) with respect to some spin
dyad {A A }.
For later use, let bABCDEF denote the spinorial counterpart of the Cotton
tensor bijk . Exploiting the antisymmetry in the indices jk , one obtains thedecomposition
bABCDEF = bABCE DF + bABDF CE ,bABCD ≡ DQ (A sBCD)Q .

(19.19)

512

Static solutions

Consequently, one has the symmetry bABCD = b(ABCD) . Moreover, it can be
veriﬁed that
DAB bABCD = 0.
In what follows, bABCD will be referred to as the Cotton spinor.
19.2 Analyticity at inﬁnity
The conformal static Equations (19.17) allow one to show that, under some basic
regularity assumptions, there exist coordinates in a neighbourhood of the point
at inﬁnity for which all the conformal ﬁelds are analytic. This result brings to theforefront the inherent ellipticity of the equations and constitutes the foundationof any further analysis of static solutions from a conformal perspective. The
result was originally proven by Beig and Simon (1980a). In the following, an
adaptation of this result will be given. One has:
Theorem 19.1 (analyticity of static solutions at inﬁnity ) Let (v, h̃)denote a solution to the static Equations (19.4a) and (19.4b) such that Ω asdeﬁned by Equation (19.7) satisﬁes the conditions (19.6) with hαβ = Ω2 h̃αβ thecomponents of a C 4,α metric for some coordinates x = (xα ) in a neighbourhoodof i. Then there exist coordinates x = (xα ) deﬁned in a neighbourhood of i suchthat hαβ , ζ  , ς  and sαβ are analytic.
Remark. The regularity assumptions in this result are expressed in terms ofHölder spaces; see the Appendix to this chapter.
Proof The proof exploits the fact that the Ricci operator of a Riemannian
metric expressed in harmonic coordinates is elliptic – the Lorentzian counterpartof this observation has been discussed in the Appendix to Chapter 13. The
general theory of elliptic equations – see, for example, Garabedian (1986) – showsthat it is always possible to ﬁnd a neighbourhood of i in which the equations
Δh xα = 0(19.20)have a solution xα = xα (x). The coeﬃcients of the diﬀerential operator in
Equation (19.20) consist of hαβ and its derivatives so that they are of class C 3,α .
The general theory of elliptic partial diﬀerential equations (PDEs) shows thatsolutions of second-order elliptic equations gain two derivatives with respect tothe coeﬃcients of the equation. Accordingly, one has that xα = xα (x) is C 5,α .
This regularity is suﬃcient to invert the coordinates. Taking into account thetransformation law of the metric tensor under change of coordinates,hαβ =∂xγ ∂xδhγδ ,∂xα ∂xβ

it follows that hαβ is C 4,α . Similarly, the ﬁeld ζ  can be veriﬁed to be C 4,α , whileς  and sαβ are C 2,α . To conclude the proof, one needs to construct a system

19.2 Analyticity at inﬁnity

513

of elliptic equations for the various ﬁelds. In the remainder of the proof it isassumed that all the ﬁelds are expressed in terms of the coordinates x , andthe primes will be dropped from the expressions. Let γα β γ denote the Christoﬀelsymbols of the metric h and denote by γ β ≡ hαγ γα β γ the associated contracted
Christoﬀel symbols. A discussion analogous to that of the hyperbolic reductionof the Einstein ﬁeld equations in generalised wave coordinates – see the discussion
in the Appendix of Chapter 13 – shows thatrαβ − D(α γβ) = sαβ
is an elliptic equation for the components hαβ of the metric h in the coordinates(xα ) if sαβ are known. To close the system one considers the equations
S α α = 0,Dα Sα = 0,Dγ Hγαβ = 0.

Using the Bianchi identity (19.14) and the conformal static ﬁeld equations toremove all the second derivatives of the conformal ﬁelds which are not Laplacians,
one obtains a system of the form
(19.21)Δh hαβ , ζ, ς, sαβ = F(hαβ , ζ, ς, sαβ , Dγ hαβ , Dα ζ, Dα ς, Dγ sαβ ),with F an analytic vector-valued function of its entries. Despite having a
Laplacian operator on the left-hand side, it is a priori not clear that the system(19.21) is elliptic as Δh applied to hαβ and sαβ gives rise to further secondorder derivatives of hαβ which come from derivatives of the Christoﬀel symbols.
To verify ellipticity one needs to compute the determinant of the symbol of(19.21). A calculation shows that this determinant is, in fact, proportional to(hαβ ξα ξβ )13 so that one, indeed, has an elliptic system as hαβ are the components
of a Riemannian metric; compare the deﬁnition of ellipticity in Section 11.2. The
general theory of the regularity of solutions of elliptic systems shows that if one
has a C 2,α solution to the above equation, then it must, in fact, be analytic in
U ; a discussion of this result is given in Morrey (1958).
Remarks(i) The original proof in Beig and Simon (1980a) was carried out in a conformal
gauge obtained from writing the static metric (19.1) in the formg̃ = e2U dt ⊗ dt − e−2U ĥαβ dy α ⊗ dy β ,where U is a scalar ﬁeld and ĥαβ denote the components of a Riemannian
3-metric. Their analysis shows that ω ≡ (U/m)2 and h ≡ ω 2 ĥ are analytic
in h -harmonic coordinates. This gauge and the one used to prove Theorem
19.1 can be related by letting Ω ≡ ωeU . One has by analogy to Equation(19.8) the split Ω−1/2 = ζ −1/2 + W  with
ζ =

ω
,
cosh2 U/2

W =m sinh U/2
.
U

514

Static solutions

It can be veriﬁed that the conformal metrics h and h are related to eachother via h = ϑ4 h with ϑ ≡ 2W  /m.
(ii) Kenneﬁck and O’Murchadha (1995) have shown that the smoothness
assumption on the conformal metric made in Theorem 19.1 can be
deduced from weaker diﬀerentiability and decay conditions on the physical
3-metric h̃.
(iii) Theorem 19.1 can be further strengthened by considering h-normal coordinates based on i. It can be veriﬁed that the coordinate transformation
relating the analytic coordinate system x and normal coordinates is also
analytic.
A remark concerning the notion of analyticity at i
As a consequence of the analytic behaviour ensured by Theorem 19.1 one has
that, for example, the ﬁeld ζ can be expressed in a suitably small neighbourhood
U of i as a convergent series of the form∞

ζα2 ···αp xα2 · · · xαp ,ζ=ζα2 ···αp ∈ R.

(19.22)p=2The other conformal ﬁelds have similar expansions. An alternative descriptionof the above expansion can be obtained by introducing polar coordinates.
Accordingly, one deﬁnes
ρ2 ≡ |x|2 = δαβ xα xβ ,

ρα ≡

xα
.
|x|

(19.23)The unit position vector ρα can be parametrised by means of some coordinatesθ = (θA ) on the 2-sphere S2 so that one can write ρα = ρα (θ). In what follows it
will be assumed, for convenience, that the coordinates (θA ) are analytic functionsof the original coordinates x – clearly, the coordinates (θA ) cannot cover thewhole of S2 . This fact will not play a role in the sequel. In terms of the coordinates(ρ, θA ) the expansion (19.22) takes the form∞

ζα2 ···αp ρα2 · · · ραp ρp .

ζ=
p=2

Accordingly, ζ is also an analytic function of the coordinates (ρ, θA ). Decomposing the product ρα2 · · · ραp (which depends only on the angular coordinates)
into symmetric, trace-free terms one obtains the usual expansion in terms ofspherical harmonics Ylm . This computation can be conveniently performed inspace spinors; see, for example, Torres del Castillo (2003).
Remark. Not every analytic function of (ρ, θA ) is an analytic function of theassociated Cartesian coordinates. The standard counterexample for this is theradial coordinate ρ as deﬁned by Equation (19.23), whose second derivative withrespect to the coordinates (xα ) is singular at i. To have analyticity with respect19.2 Analyticity at inﬁnity

515

to the coordinates (xα ) one needs the right combination of spherical harmonics
and powers of ρ.
A particular case of the above discussion concerns the conformal factor Ω.
From Equation (19.8) it follows thatΩ=ζ
(1 + ζ 1/2 W )2.

A direct computation taking into account the asymptotic conditions (19.11)
shows that while Ω is C 2 at i, it will fail to be of class C 3 unless W = 0 –which, in the present gauge, means that m = 0. Thus, in general, the conformal
factor Ω is not analytic in the harmonic coordinates (xα ) even if ζ is analytic. It
is, nevertheless, analytic in ρ.
19.2.1 A spacetime conformal completion of static solutions
Theorem 19.1 is a statement about the conformal structure of hypersurfaces of a
canonical foliation of a static spacetime. Thus, it is of natural interest to analyse
the consequences of this property from a spacetime perspective. Intuitively, one
expects that the nice conformal properties of the leaves of the foliation will leadto a good spacetime conformal behaviour. As the spatial conformal factor is notanalytic with respect to the harmonic coordinates x = (xα ), one cannot expect
analyticity of a spacetime conformal extension in terms of these coordinates.
Instead, one looks for extensions which are analytic in the associated radial
coordinate.
Following Remark (iii) after Theorem 19.1, it is assumed that the harmonic
coordinates x = (xα ) are h-normal and centred at i. Writing θ = (θA ), one hasthat for θ = θ ﬁxed and s ∈ [0, s ) for s ≥ 0 suitably small, xα (s) = sρα (θ )describes a geodesic passing through i. A function f : U → R evaluated along
one of these geodesics will be denoted by f (sρα ). From xα = ρ ρα it follows thatdxα = ρα dρ + dρα = ρα dρ + ρ∂A ρα dθA ,so that, using the normal coordinates condition hαβ xα = −δαβ xα , one concludesthath = −dρ ⊗ dρ + ρ2 k,(19.24)wherek ≡ hαβ dρα ⊗ dρβ = hαβ ∂A ρα ∂B ρβ dθA ⊗ dθB
corresponds to the metric of the two-dimensional surfaces of constant ρ. In
particular, one has that k|ρ=0 = −σ – the negative deﬁnite standard metricof S2 .
Putting together the discussion of the previous paragraph and recalling thath = Ω2 h̃, one ﬁnds that the static metric (19.1) can be rewritten asg̃ = v 2 dt ⊗ dt − Ω−2 dρ ⊗ dρ + Ω−2 ρ2 k.

(19.25)

516

Static solutions

The claim is now that the conformal metric g ≡ Ξ2 g̃ withΞ = Ω1/2
gives rise to a conformal extension of the static spacetime which is as regular
as one can possibly expect, that is, analytic in the coordinates (ρ, θA ). From
Equation (19.25) one has thatg = Ωv 2 dt ⊗ dt − Ω−1 dρ ⊗ dρ + ρ2 Ω−1 k.

(19.26)

Recalling that Ω = O(ρ2 ) and v = O(1), one ﬁnds that while the ﬁrst and thirdterms of the above metric are regular, the second one is singular. This singularity
is a coordinate artefact which can be removed by considering the null coordinate
u≡t+

 ρ
ρ

ds
,
v(sρα )Ω(sρα )for ﬁxed ρα and ρ > 0. Observe, in particular, that as a consequence of the
behaviour of Ω near i one has that u → −∞ as ρ → 0. The diﬀerential of thenull coordinate u is given by
du = dt −1
dρ + λ,
vΩ

where
A

λ ≡ λA dθ ,

λA ≡



 ρ
∂A
ρ

1
v(sρα )Ω(sρα )ds.

Substituting the above expressions into the conformal metric (19.26) yields
g = Ωv 2 du ⊗ du + v(du ⊗ dρ + dρ ⊗ du) − Ωv 2 (du ⊗ λ + λ ⊗ du)− v(λ ⊗ dρ + dρ ⊗ λ) + Ωv 2 λ ⊗ λ + ρ2 Ω−2 k,which is regular whenever Ω = 0. Moreover, following the discussion of Section
19.2, the various metric coeﬃcients are analytic in the coordinates (ρ, θA ). The
conformal representation of static spacetimes given above shows that static
spacetimes admit a smooth conformal extension which includes a portion of nullinﬁnity. However, this description is not suitable for a spacetime discussion of
spatial inﬁnity. This issue will be elaborated in Chapter 20. The discussion ofthis section can be extended to include stationary spacetimes; for a discussionof the required considerations, see Dain (2001b).

19.3 A regularity condition

517

19.3 A regularity condition
As an application of the results on the analyticity of solutions to the conformal
static equations at i, in this section a proof is given of a property of the conformal
structure of static solutions which plays a central role in the discussion ofChapter 20. The analysis of this section is best carried out in spinors and is
adapted from Beig (1991).
Before stating the main result of this section it is convenient to discuss someancillary consequences of the conformal static equations. In what follows, all
the spinors are expressed in terms of their components with respect to somespin dyad {A A } associated to the frame {ei } corresponding to the particularrealisation of harmonic h-normal coordinates at i.
Lemma 19.1 (behaviour of the symmetrised derivatives of ζ at i)
A solution to the conformal static equations satisﬁes
D(Ap B p DAp−1 B p−1 · · · DA1 )B 1 ζ(i) = 0.
Proof For the cases p = 0, 1, 2, the result follows from a direct computation
using the conditions in (19.11), observing that h(ABCD) = 0. For higher orderderivatives, the result follows by induction, using that the spinorial version of
the equation associated to the zero quantity Sij is given by
DAB DCD ζ = ςhABCD + (μ − 1)ζsABCD ,(19.27)and using that DEF hABCD = 0 and h(ABC)D = 0.
Remark. Lemma 19.1 implies, in particular, that
D(Ap B p DAp−1 B p−1 · · · DA1 B 1 ) ζ(i) = 0.
The main result of this section is the following:
Proposition 19.1 (behaviour of the derivatives of the Cotton spinor ati) A solution to the conformal static equations satisﬁes
D(Ap B p DAp−1 B p−1 · · · DA1 B 1 bABCD) (i) = 0,

p = 0, 1, 2, . . . .

(19.28)The original proofs of this result were given independently by Friedrich (1988)
and Beig (1991).
Proof
form

The proof of this result follows from considering Equation (19.18) in the(1 − μζ)bABCD = 2μsQ(ABC DD) Q ζ.

(19.29)

518

Static solutions

Using the conditions in (19.11) one obtains bABCD (i) = 0. Repeated diﬀerentiation and symmetrisation of Equation (19.29) yields(1 − μζ)D(Ap B p DAp−1 B p−1 · · · DA1 B 1 bABCD)− pμD(Ap B p ζDAp−1 B p−1 · · · DA1 B 1 bABCD)+ · · · − μD(Ap B p DAp−1 B p−1 · · · DA1 B 1 ζbABCD)= D(Ap B p · · · DA1 B 1 s|Q|ABC DD) Q ζ+ · · · + sQ(ABC DAp B p · · · DA1 B 1 DD) Q ζ.
Using Lemma 19.1 it follows that every term in the above expression, save forthe ﬁrst one in the left-hand side, vanishes when evaluated at i. This yields thedesired result.
Remark. Condition (19.28) has been called in Friedrich (1988), for reasons tobe elaborated in Chapter 20, the radiativity condition. In Friedrich (1998c) it
has been given the name regularity condition. In tensorial notation Equation(19.28) takes the form
D{αp Dαp−1 · · · Dα1 bβγ} (i) = 0

p = 0, 1, 2, . . . .

Conformal transformation properties
Let ω denote a smooth function deﬁned in a neighbourhood of i satisfying
ω(i) = 0. From the conformal transformation properties of the Cotton tensor –see Equation (5.19) – it follows that under the rescaling h → h = ω 2 h the
Cotton spinor satisﬁesbABCD = ω −1 bABCD .
Thus, bABCD (i) = 0 if bABCD (i) = 0. Using the transformation law ofb(i) = DA1 B 1 bABCD (i) as thethe connection one ﬁnds that DA
1 B 1 ABCD
correction terms associated to the transition tensor involve bABCD (i) = 0.
b(i) = 0. Proceeding inductively one concludes that
Hence, D(A1 B 1 ABCD)



D(A
DA
· · · DAb(i) = 0,p−1 B p−1
1 B 1 ABCD)p Bp

p = 0, 1, 2, . . . .

Consequently, the regularity condition (19.28) is conformally invariant. This
conformal invariance allows the following reading of Proposition 19.1: theconformal class of a 3-metric satisfying the static equations cannot be arbitrary.
More precisely, condition (19.28) is a necessary condition for a metric h to belong
to the conformal class of a static metric.
19.4 Multipole moments
In Newtonian gravity time-independent gravitational ﬁelds are characterised bya sequence of multipole moments. It is desirable to have a similar characterisation
for time-independent solutions to the Einstein ﬁeld equations describing isolatedbodies. One of the advantages of the conformal approach to static spacetimes

19.4 Multipole moments

519

is that it allows a geometric formulation of the notion of multipole moments.
Following the original
treatment in Geroch (1970a,b) one deﬁnes a sequence
of tensor ﬁelds P, Pi , Pi1 i2 , . . . in a neighbourhood U of i via the recursive
relationsP ≡ Ω−1/2 (1 − v),
Pi ≡ Di P,
1
Pi2 i1 ≡ D{i2 Pi1 } − P ri2 i1 ,2
1
Pip+1 ···i1 ≡ D{ip+1 Pip ···i1 } − p(2p − 1)P{ip+1 ···i3 ri2 i1 } ,2

p = 2, 3, . . . .

The particular form of the lower order correction terms in the deﬁnition of thetensors Pip+1 ···i1 has been chosen so as to ensure conformal invariance of thedeﬁnition of multipole moments to be given below – this observation follows froma tedious computation which will not be further elaborated here. The multipole
moments of a static solution are then obtained by evaluating the above tensors
at i. To this end, choose a smooth coordinate system x = (xα ) on U and denote
by Pα , Pα2 α1 , . . . the components of the tensors with respect to these coordinatesand deﬁne the multipole moments of the static solution with respect tothe coordinates x to be the sequence {m, mα , mα2 α1 , . . .} withm ≡ P (i),mαp ···α1 ≡ Pαp ···α1 (i),p = 1, 2, 3 . . . .

For a given p, the 2p quantities mαp ···α1 are called the 2p -poles. In particular, mis the monopole (the mass) and mα is the dipole moment. As the multipole
moments are expressed as the value of a tensor at a point, it follows that, undera coordinate transformation x = (xα (x)), the multipole moments transform asm = m,mα = Aα β mβ ,mαp ···α1 = Aαp βp · · · Aα1 β1 mβp ···β1 ,(19.30)where (Aα β ) are the components of 3×3 invertible real matrices; that is, (Aα β ) ∈GL(3, R). Observe that the monopole is invariant under a change of coordinates.
For the particular choice of the conformal factor given by Equation (19.7) one
has that P = m so that Di P = 0 and accordingly mα = 0; in other words, inthe conformal gauge determined by (19.7) one is automatically in the centre ofmass.
The properties of the multipole expansions in Newtonian gravity raise thequestion of to what extent the general relativistic multipole moments determine asolution to the static equations, and vice versa. The construction described in theprevious paragraph can be thought of as mapping a static solution (v, h̃) to thecollection of multipoles {m, mα , mα2 α1 , . . .}. Now, two collections of multipoles{m, mα , mα2 α1 , . . .} and {m , mα , mα2 α1 , . . .} are said to be equivalent if
there exists (Aα β ) ∈ GL(3, R) such that the relations in (19.30) hold. In Beig
and Simon (1980a) the following has been proved:

520

Static solutions


Theorem 19.2 (multipole theorem) If two static solutions (v, h̃) and (v  , h̃ )lead to multipole sequences which are equivalent, then the static solutions areisometric in a neighbourhood of i.
Although a detailed proof of the above theorem will not be provided, it isof interest to discuss the basic underlying ideas. The fundamental problem isthe following: given a sequence of multipoles {m, mα , mα2 α1 , . . .}, how can
one reconstruct the pair (v, h̃) solving the static equations? To answer thisquestion one ﬁrst employs an inductive argument which relies on the deﬁnitionof the multipole moments, the conformal static equations and the commutatorof covariant derivatives to show that the values of the ﬁelds ζ, s, sαβ and anyof their covariant derivatives at the point i can be expressed in terms of themultipole moments. Thus, one can compute the Taylor expansions (in harmonich-normal coordinates) of these ﬁelds around i. From the general theory of Taylor
expansions one knows that the expansions are unique. Moreover, it is a classicalresult of Riemannian geometry that the sequence{rαβγδ (i), Dη rαβγδ (i), Dη2 η1 rαβγδ (i), . . .}determines, in a unique way, the Taylor expansion of the components of themetric hαβ – again, in h-normal coordinates (xα ) centred at i; see, for example,
Günther (1975). A ﬁnal argument shows that applying the above procedure totwo equivalent sequences of multipoles leads to two metrics which are isometric.
Now, given any set of multipole moments subject to the appropriate convergence condition, it is natural to expect that there exists a static solution having
precisely those multipole moments. In other words, the sequence of multipolescharacterises (in a suitable) unique manner the static spacetime. As a result of
the analyses in Friedrich (2007) and Herberthson (2009) one has the following:
Theorem 19.3 (suﬃcient conditions on the sequence of multipoles
for the existence of a static solution) Let {m, mα , mα2 α1 , . . .} denote thecomponents of a sequence of real valued, totally symmetric trace-free tensors
at the origin of R3 expressed in terms of Cartesian coordinates x = (xα ). If
constants M, C > 0 can be found such that|mαp ···α1 | ≤

p!M
,
Cp

(19.31)then there exists a static, asymptotically ﬂat spacetime having the multipole
moments {m, mα , mα2 α1 , . . .}.
The proof of the above result goes beyond the scope of this book. Again, only
the basic underlying ideas are brieﬂy discussed. The starting point of the analysis
is to exploit the analyticity of the solutions to the conformal static equations
provided by Theorem 19.1 to implement a complex analytic extension of
the whole setting. More precisely, the ﬁelds hαβ , ζ, ς, sαβ can be extended

19.4 Multipole moments

521

near i by analyticity into the complex domain and regarded as holomorphic (i.e.
complex analytic) ﬁelds on a complex analytic manifold SC . Restricting SC to
be a suﬃciently small neighbourhood of i one can use similarly extended normal
coordinates x = (xα ) centred at i to deﬁne an analytic system of coordinates on
SC which identiﬁes the latter with an open neighbourhood of the origin in C3 .
The original manifold S is then a three-dimensional real analytic submanifold
of SC . Under the analytic extension the main diﬀerential geometric concepts
and formulas remain valid. In particular, the extended ﬁelds, to be denoted
again by hαβ , ζ, s, sαβ , satisfy the conformal static vacuum ﬁeld equations on
SC . In order to provide a geometric perspective of the problem, one considers
the function Γ ≡ δαβ xα xβ on S which extends to a holomorphic function on
SC satisfying the equation hαβ Dα ΓDβ Γ = −4Γ. While restricted to S, thefunction Γ vanishes only at i. On SC its set of zeros is a two-dimensional complex
submanifold of SC ,Ni ≡ {p ∈ SC | Γ(p) = 0},
the so-called complex null cone at i. This cone is generated by the complex
null geodesics through i; see Figure 19.1. The analogy between the (spacetime)
conformal ﬁeld equations and the conformal static ﬁeld equations discussed in
Section 19.1.1 suggests the formulation of a characteristic initial value problem
for the conformal static ﬁeld equations on the null cone Ni . The formulation ofthis characteristic initial value problem requires the determination of suitable
initial data. An argument involving the idea of exact sets of ﬁelds – see Penrose
and Rindler (1984) – allows one to show that the basic data for this characteristic
problem are is given by the sequence of ﬁeldssαβ (i), D{α1 sαβ} (i), . . . , D{αp ···α1 sαβ} (i), . . . .

Figure 19.1 Schematic representation of the complex null cone through i, Ni ,as described in the main text.

522

Static solutions

The above null data can be obtained by repeated diﬀerentiation along thedirection of the complex null generators of Ni of the components of sαβ ; see, forexample, the discussion in Friedrich (2004, 2007).
Given the analyticity of the setting described in the previous paragraphs, one
can make use of the Cauchy-Kowalewskaya theorem to discuss the existenceof analytic solutions to this characteristic problem and to provide convergenceconditions on the null data which ensure the existence of a solution; see the
Appendix to this chapter. The convergence conditions thus obtained are similar
to the ones in Equation (19.31) of Theorem 19.3 and, in particular, ensure theexistence of a real static solution. This is the main result of Friedrich (2007).
To obtain the convergence condition on the sequence of multipoles one needs
to analyse the relation between the null data and the sequence of multipoles.
Inspection shows that the null data and the multipole moments are in a one-toone correspondence. This correspondence, however, is non-linear and implicit.
The detailed analysis of this correspondence in Herberthson (2009) allows thetransformation of the convergence conditions for the null data into convergenceconditions for the sequence of multipoles.
19.5 Uniqueness of the conformal structure of static metrics
As a ﬁnal application of the conformal static equations, the extent to which theconformal class of the 3-metric h̃ determines a solution to the static equations
will be analysed. This question was ﬁrst analysed in Beig (1991) from where themain ideas of the analysis are adapted. An alternative discussion of some aspectsof this problem is given in Friedrich (2008a,b).
The multipole Theorem 19.2 shows that a static solution is determined by its
multipole moments. Thus, it is natural to try to relate the multipole moments
to the conformal class of the metric h. In what follows, for p = 1, 2, 3, . . . deﬁne
βAp B p ···A1 B 1 A0 B 0 ≡ DQ (B p DAp−1 B p−1 · · · DA1 B 1 bA0 B 0 Ap Q) (i).
Using an inductive argument similar (albeit lengthier!) to the one leading to
Proposition 19.1 one obtains the family of identities
βAp B p ···A1 B 1 A0 B 0 = 6μD(Ap B p · · · DA2 B 2 sA1 B 1 A0 B 0 ) (i),(19.32)for p = 1, 2, 3, . . . with μ = m2 /4; see Equation (19.12). The above identities
constitute the main tool for the reminder of the section. Observe that thetensorial counterpart of the symmetrised derivatives of the spinor enter directlyin the deﬁnition of the multipole moments. The quantities βAp B p ···A0 B 0 havegood conformal properties. Recalling that under the rescaling h = ω 2 h with
ω(i) = 0, one has the transformation rulesbABCD = ω −1 bABCD ,AB = ω −1 AB .

19.5 Uniqueness of the conformal structure of static metrics

523

It follows thatβA
= ω −2 (i)βAp B p ···A0 B 0 .
p B p ···A0 B 0Consider now two solutions (h, sABCD , ζ, ς) and (h , sABCD , ζ  , ς  ) to theconformal static equations such thath = ω 2 h,and consider the question of under which circumstances will the above twosolutions determine the same physical static solution (v, h̃) – modulo isometries.
The identities (19.32) show how the multipole moments of the two solutions
are connected to each other. One then needs further conditions that allow one
to constrain the relation between solutions further. In view of the conformal
nature of the problem, the natural object to look for those extra conditions
is the Cotton spinor. Proposition 19.1 and the identities (19.32) alreadyprovide information about some of the derivatives of bABCD at i. The only
derivatives which have not yet been considered are divergences of the form
DP Q D(P Q DAp B p · · · DA1 B 1 bABCD) . A direct computation using Equations(19.27) and (19.29) yields
DP Q bP QCD (i) = 0,

(19.33a)

D

PQ

D(P Q bABCD) (i) = 0,

(19.33b)

D

PQ

D(P Q DEF bABCD) (i) = 0.

(19.33c)

However, a lengthy computation reveals that
DP Q D(P Q DGH DEF bABCD) (i) = −24μD(GH sQ EF A (i)sBCD)Q (i).
Deﬁning, for convenience, OGHEF ABCD ≡ D(GH sQ EF A (i)sBCD)Q (i), a
computation using the deﬁnition of the quantities βAp B p ···A0 B 0 allows thereexpression of this quantity in the form
OGHEF ABCD =1 Q
β (GHF EA βBCD)Q .
36(19.34)The conformal transformation properties can be easily read from this last
expression. Namely, one has that

−5
OGHEF
(i)OGHEF ABCD .
ABCD = ωOn the other hand, it can be checked that




−3
DP Q D(P(i)DP Q D(P Q DGH DEF bABCD) (i).
Q DGH DEF bABCD) (i) = ω

From the above transformation rules, assuming that OGHEF ABCD = 0 one
concludes that
ω 2 (i) = μ /μ.

524

Static solutions

Observe that if OGHEF ABCD = 0, no conclusion can be extracted from the analysis. As a consequence of Equation (19.34), the requirement OGHEF ABCD = 0 is
a condition on the conformal structure of the static solutions under consideration.
If it holds, then using the identities (19.32) one concludes that the two solutions
to the conformal static equations will have the same multipole moments if they
have the same mass. Moreover, as a consequence of the multipole Theorem 19.2,
they are isometric. The analysis of this section is summarised in the following
theorem, ﬁrst proven in Beig (1991):
Theorem 19.4 (uniqueness of the conformal structure of static solutions) Two solutions to the conformal static equations with the same mass, lying
in the same conformal class and satisfying OGHEF ABCD = 0 are isometric.
The condition OGHEF ABCD = 0 can be seen to be violated if the two static
solutions are axially symmetric about a common axis; see, for example, Beig
(1991). As discussed in Friedrich (2008a) this is, in fact, the only possibility.
More precisely, a static solution which admits a non-trivial rescaling leading to
a new static solution must be axially symmetric and admit a conformal Killing
vector. There exists a three-parameter family of such solutions. These have been
explicitly found in Friedrich (2008b).
The Schwarzschild solution
A case of particular interest is when h̃ is conformally ﬂat. It then follows thatβAp B p ···A0 B 0 = 0 for all p, and the only non-vanishing mass multipole is themass m. Invoking, again, the multipole Theorem 19.2 it follows that for a given
value of m there exists only one solution, up to isometries, with this property –namely, the Schwarzschild solution. An alternative derivation of the uniqueness of
the Schwarzschild spacetime among the class of conformally ﬂat static solutions
which makes no use of the multipole theorem has been given in Friedrich (2004).
In this analysis the conformal static equations are explicitly integrated along
geodesics starting at i.
19.6 Characterisation of static initial data
An issue related to the questions discussed in the previous section concerns thecharacterisation of initial data for a static spacetime – this question will beof relevance in Chapter 20. More precisely, one is interested in the following
question: given a 3-metric h, under which circumstances does there exist in
the conformal class [h] another metric h̃ which, together with some scalar v,
constitutes a solution to the static equations?
As in the rest of the chapter, the above question is restricted to a suitableneighbourhood of inﬁnity. Proposition 19.1 shows that not every conformal classwill contain a static metric. In other words, condition (19.28) is a necessary

19.7 Further reading525

condition for a metric h to be conformal to a static metric. Now, condition(19.28) is not suﬃcient. The relations (19.33b) and (19.33c) show that there exist
further conditions (in fact, an inﬁnite hierarchy of them) on the conformal class,algebraically independent from (19.28), which need to be satisﬁed by a metric hin order to be conformal to a static metric. The gap between a conformal classof 3-metrics satisfying the regularity condition (19.28) and a conformal classcontaining a static metric has been analysed in detail in Friedrich (2013).
The level of detail required to discuss the main result of Friedrich (2013)
goes well beyond the scope of this chapter, and only the key ideas are
brieﬂy mentioned. If a metric h is conformal to a metric h solving theconformal static equations, then writing h = ω 2 h for some suitable conformal
factor ω, it is possible to rewrite the conformal static equations as a highly
overdetermined system of diﬀerential equations for ω. To analyse the solvabilityof the conditions one needs to consider the associated integrability conditions. As
already anticipated by (19.33b) and (19.33c), these integrability conditions giverise, in addition to (19.28), to restrictions on the conformal structure which takethe form of an inﬁnite hierarchy of diﬀerential conditions on the Cotton tensor
at i. These conditions can be expressed in terms of requirements on a covector
constructed from the 3-metric h. An interesting feature of the analysis is thatthe overdetermined system involving the conformal factor ω is highly singular
at i. For this system to have a solution, a hierarchy of regularity conditions
need to be imposed on the singular part of the equation so that it admits asmooth extension to a neighbourhood of i – this is reminiscent of a procedure
which arises in the construction of radiative initial data sets in Section 20.2.
Remarkably, the required regularity conditions turn out to be nothing else but
the conditions (19.28).
19.7 Further reading
A systematic analysis of time-independent solutions to the Einstein ﬁeld
equations is provided in Beig and Schmidt (2000). This reference provides an
excellent point of entry to the extensive literature on static and stationary
solutions in general relativity. A survey of the various approaches to deﬁne
multipole moments for time-independent solutions to the Einstein ﬁeld equations
can be found in Quevedo (1990). An analysis of global aspects of static andstationary spacetimes can be found in Anderson (2000).
Several of the results discussed in this chapter admit a generalisation to thecase of stationary solutions. The deﬁnitions of multipole moments given by
Geroch (1970a,b) have been extended to the stationary case in Hansen (1974).
The analyticity of solutions of the conformal stationary ﬁeld equations has been
analysed in Beig and Simon (1980b, 1981); see also Kundu (1981). However, in
this case the 3-metric h of a surface of constant time will not be analytic; seeDain (2001b). Instead, the analyticity refers to the 3-metric γ of the quotient
space obtained from identifying points on the spacetime lying on the same orbit

526

Static solutionsof the stationary Killing vector. The analysis of the convergence conditions fornull data of static solutions in Friedrich (2007) has been extended to the caseof stationary solutions in Aceña (2009). An alternative analysis of multipole
expansions of static solutions with the aim of obtaining convergence conditions
on sequences of multipoles has been given in Bäckdahl and Herberthson (2005a,b,
2006) and Bäckdahl (2007).
The analysis of the conformal static equations by means of the complex nullcone through i was ﬁrst introduced in Friedrich (1988). Further extensions of
this method have been given in Friedrich (2004, 2007, 2013).
Appendix 1: Hölder conditions
Given 0 < α ≤ 1, a real valued function f on an open set U ⊂ Rn is said to satisfy
the Hölder condition with exponent α on U if there exists a non-negative
constant C such that|f (x) − f (y)| ≤ C|x − y|α ,for all x, y ∈ U .

If the above is the case, one writes f ∈ C 0,α (U ). The Hölder condition is astronger notion of continuity; that is, a function satisfying the Hölder condition
is continuous, but not all continuous functions satisfy the Hölder condition forsome α. More generally, one says that f ∈ C k,α (U ) if all its derivatives up toorder k satisfy the Hölder condition for a given α. The Hölder condition plays animportant role in the regularity of solutions to elliptic PDEs; see, for example,
Evans (1998) for further details.
Appendix 2: the Cauchy-Kowalewskaya theorem
The Cauchy-Kowalewskaya theorem asserts the local existence, in a neighbourhood of t = 0, of a real analytic solution u(t, x) to the quasilinear ﬁrst-order
initial value problem∂t u = Aα (t, x, u)∂α u + B(t, x, u),u(0, x) = u (x),where Aα (t, x, u), B(t, x, u) and u (x) are real analytic functions of theirarguments; see, for example, Evans (1998) for further details. A discussion of thevarious approaches to prove this result can be found in Shinbrot and Welland(1976).

20
Spatial inﬁnity

This chapter discusses the properties of the conformal Einstein ﬁeld equations
and the behaviour of their solutions in a suitable neighbourhood of spatial inﬁnity. This analysis is key in any attempt to extend the semiglobal existence resultsfor Minkowski-like spacetimes of Chapter 16 to a truly global problem where
initial data is prescribed on a Cauchy hypersurface. An interesting feature of thesemiglobal existence Theorem 16.1 is that the location of the intersection of theinitial hyperboloid with null inﬁnity does not play any role in the formulationof the result. This observation suggests that the essential diﬃculty in formulatinga Cauchy problem is concentrated in an arbitrary (spacetime) neighbourhood of
spatial inﬁnity. The subject of this chapter can be regarded, in some sense, asa natural extension of the study of static spacetimes in Chapter 19 to dynamic
spacetimes – a considerable amount of the discussion of the present chapter
is devoted to understanding why this is indeed the case. A further objectiveof this chapter is to understand the close relation between the behaviour of thegravitational ﬁeld at spatial inﬁnity and the so-called peeling behaviour discussed
in Chapter 10. The main technical tool in this chapter is the construction of
the so-called cylinder at spatial inﬁnity – a conformal representation of spatial
inﬁnity allowing the formulation of a regular initial value problem by meansof which it is possible to relate properties of the initial data on a Cauchy
hypersurface with the behaviour of the gravitational ﬁeld at null inﬁnity.
Despite recent developments in the understanding of the behaviour of solutions
to the Einstein ﬁeld equations in a neighbourhood of spatial inﬁnity, several key
issues still remain open.
20.1 Cauchy data for the conformal ﬁeld equations
near spatial inﬁnity
To begin to understand the diﬃculties behind the formulation of a standard
initial value problem for a Minkowski-like spacetime, it is convenient to look at
the behaviour of Cauchy data for the conformal equations near spatial inﬁnity.

528

Spatial inﬁnity
20.1.1 General set up

In what follows, initial data sets (S̃, h̃, K̃) which are asymptotically Euclidean
and regular in the sense of Deﬁnition 11.2 will be considered. As the discussion
in this chapter will be mainly concerned with the behaviour of the data ina neighbourhood of spatial inﬁnity, it will be assumed, without any loss ofgenerality, that the manifold S̃ has only one asymptotic end. The basic aspectsof the analysis of spatial inﬁnity are already present in time-symmetric initial
data sets. Thus, attention is restricted to this type of conﬁguration. Finally, it
will be assumed, unless otherwise explicitly stated, that the conformal metric h
is analytic in a suitable neighbourhood of the point at inﬁnity i. This assumption
allows the simpliﬁcation of a number of arguments and calculations and allows
one to analyse the solutions to the Einstein ﬁeld equations under optimal
regularity assumptions of the initial data – it is, however, non-essential.
Remark. Static initial data sets satisfy the assumptions described in theprevious paragraph.
In Chapter 11 it has been seen that the conformal factor Ω linking a particular
choice of conformal metric h with the physical metric h̃ admits, in a suitableneighbourhood U of i and in terms of normal coordinates x = (xα ) centred at i,the decomposition
Ω=

|x|2
,
(U + |x|W )2|x|2 = δαβ xα xβ ,(20.1)

where U/|x|2 is the Green function of the Yamabe operator and describes the
local geometry around i, while W contains global information; see the discussion
in Section 11.6.4. In particular, one has that U = 1 + O(|x|2 ) is analytic if h
is analytic and, moreover, W (i) = m/2 where m denotes the Arnowitt-DeserMisner (ADM) mass of the initial data set.
There is a certain amount of freedom in the choice of the conformal scalingof the metric h. For the purposes of the present discussion, it is convenient toconsider the scaling introduced in Section 11.6.2 for whichhαβ = −δαβ + O(|x|3 ),(20.2)

so that the curvature tensor of h satisﬁes, in this gauge, rαβγδ (i) = 0. This gauge
construction is supplemented by an h-normal frame {ei } centred at i; that is,
one hashij ≡ h(ei , ej ) = −δij ,Dγ̇ ei = 0,

where γ̇ denotes the tangent vector to any geodesic passing through i; compare
the discussion in Section 18.4.1. Consistent with Equation (20.2), the framecoeﬃcients in ei = ei α ∂α satisfy
ei α = δi α + O(|x|3 ).

20.1 Cauchy data for the conformal ﬁeld equations near spatial inﬁnity

529

Moreover, the associated connection coeﬃcients are of the form
γi j k = O(|x|2 ),and one has thatrij = O(|x|),where rij ≡ rαβ ei α ej β are the components of the Ricci tensor of h with respect
to the frame {ei }.
20.1.2 The rescaled Weyl and Schouten tensors on U
For time-symmetric initial data, the components of the electric part of therescaled Weyl tensor, dij , and the Schouten tensor, Lij , with respect to theframe {ei } are given on U , respectively, by
dij =


1 
D{i Dj} Ω + Ωsij ,Ω21
1
Lij = − D{i Dj} Ω + rhij ;Ω
12

see Section 11.4.3. The spinorial version of the above expressions is readily foundto be given by
1 
D(AB DCD) Ω + ΩsABCD ,Ω21
1
LABCD = − D(AB DCD) Ω + rhABCD .
Ω12
φABCD =

(20.3a)(20.3b)

The ﬁrst of the above equations implies an expression for the Cotton spinor
bABCD ; see Equation (19.19). Rewriting Equation (20.3a) in the form
Ω2 φABCD = D(AB DCD) Ω + ΩsABCD ,taking the spinorial curl of the latter, commuting covariant derivatives in theterm with the triple derivatives of Ω and recalling that the Cotton spinor isgiven by bABCD = D(A Q sBCD)Q , one concludes thatbABCD = 2D(A Q ΩφBCD)Q + ΩD(A Q φBCD)Q .

(20.4)

Behaviour close to inﬁnity
As in the case of hyperboloidal data discussed in Section 11.7, the expressions(20.3a) and (20.3b) are formally singular whenever Ω = 0. Accordingly, thediscussion of the behaviour of dij and Lij close to i requires some care.
In view of the decomposition (20.1) it is convenient to deﬁne the masslesspart of the conformal factor as Ὼ ≡ |x|2 /U 2 . By construction one hasῺ(i) = 0,Di Ὼ(i) = 0,

Di Dj Ὼ(i) = −2δij ,(20.5)

530

Spatial inﬁnity

so that one obtains the expansionῺ = δαβ xα xβ + O(|x|3 ).
In particular, it is observed that D{i Dj} Ὼ = O(|x|3 ).
One can deﬁne a massive part of the conformal factor as Ω̌ ≡ Ω − Ὼ.
Rewriting Equation (20.1) as

−2
|x|W
Ω = Ὼ 1 +,U
and using that
Di |x| =xi+ O(|x|2 ),r

Di Dj |x| =1(|x|2 δij − xi xj ) + O(|x|),|x|3where xi ≡ δi β δαβ xα , one concludes, taking into account the boundary
conditions (20.5), thatD{i Dj} Ω = −3mx{i xj}
+ O(|x|2 ).
|x|

Finally, observing that, in the present gauge, sij = O(|x|) and r = O(|x|), one
ﬁnds
dij = −

3mx{i xj}
+ O(|x|−2 ),
|x|5

Lij =

3mx{i xj}
+ O(|x|0 ).
|x|3

Accordingly, one concludes that both dij and Lij are singular at i with
dij = O(|x|−3 ),Lij = O(|x|−1 ),as |x| → 0.

The analysis of the consequences of this singular behaviour and how to deal with
it will be the central subject of the remainder of this chapter.
Remark. Even if the massive part of the conformal factor vanishes, one still
has a potential source of singularities in the ﬁelds dij and Lij . This can be seenfrom computing the massless part of the electric part of the Weyl tensor
given by1
(D{i Dj} Ὼ + Ὼsij )d`ij ≡
Ὼ2
1  2
U D{i Dj} |x|2 − 4U D{i |x|2 Dj} U
=
|x|4


− 2|x|2 U D{i Dj} U + 6|x|2 D{i U Dj} U + |x|2 U 2 sij .

(20.6)

A similar expression holds for L̀ij – the massless part of the Schouten tensor. In
the next section, it will be seen that under suitable assumptions on the metrich, both d`ij and L̀ij extend to analytic ﬁelds in a neighbourhood of i.

20.2 Massless and purely radiative spacetimes

531

20.2 Massless and purely radiative spacetimes
Intuition on the behaviour of solutions to the conformal Einstein ﬁeld equations
in a neighbourhood of spatial inﬁnity can be obtained from the analysis ofmassless initial data sets, that is, data sets for which Ω = Ὼ. In view of themass positivity theorem – see Schoen and Yau (1979) – the idea of considering
initial data sets which are massless might at ﬁrst seem strange. The rigidity partof this theorem implies that if the ADM mass m of an initial data set vanishes,
then one has, in fact, initial data for the Minkowski spacetime or the initial data
set is singular somewhere. Since the present chapter is mainly focused on an
analysis local to i (i.e. in a suitably small neighbourhood of i) the presence of
singularities in the interior of the 3-manifold S can be disregarded.
20.2.1 Geometric setting
Given a massless initial data set for the conformal Einstein ﬁeld equations in aneighbourhood U of the point at inﬁnity i, the conformal evolution equations
determine a (future and past) development (M, g, Ξ) which is contained in
D(U ) = D+ (U ) ∪ D− (U ). Following the notation of Chapter 14, let I + (i) andI − (i) denote the timelike future and timelike past of i in (M, g) and by Ni + andNi − the null cones generated by the null geodesics passing through i. From theboundary conditions (20.5) satisﬁed by Ὼ it follows that the spacetime conformal
factor Ξ has a non-degenerate critical point at i which, for simplicity, is assumed
to be the only critical point of Ξ. The locus of points for which Ξ = 0 coincides
with Ni + ∪ Ni − ; see the discussion in Section 16.3.
As observed in Friedrich (1988), the development (M, g, Ξ) of the conformal
ﬁeld equations can be regarded from a dual perspective:(i) On the set Mci ≡ M \ I + (i) ∪ Ni + ∪ I − (i) ∪ Ni − , corresponding to theexterior of the null cones, the metric g̃ ≡ Ξ−2 g is a solution to the Einsteinﬁeld equations with vanishing mass for which i represents spatial inﬁnity i0 .
(ii) On I + (i) the metric g̃ represents a solution to the Einstein ﬁeld equations
for which the point i represents past timelike inﬁnity i− and the set I − ≡Ni + \ {i} past null inﬁnity. For suitably smooth initial data, the solution
thus obtained has a regular past timelike inﬁnity and provides an example
of purely radiative spacetimes; see the discussion in Section 18.4.
A schematic depiction of the above geometric setting can be found in
Figure 20.1.
20.2.2 A regularity condition at spatial inﬁnity
Not all initial data sets lead to developments (M, g, Ξ) such that I + (i) admits
a regular past timelike inﬁnity – as given in point (ii) of the previous section.
The purpose of this section is to identify the initial data sets with this property.

532

Spatial inﬁnity

Figure 20.1 Schematic depiction of the geometric set up
 for massless spacetimes as described in the main text. The set Mci ≡ M \ I + (i) ∪ Ni + ∪ I − (i) ∪Ni − contains a solution to the vacuum Einstein ﬁeld equations with vanishingmass for which i represents spatial inﬁnity i0 , while on I + (i) one obtains apurely radiative solution where Ni + represents past null inﬁnity I − and icorresponds to past timelike inﬁnity i− .

The arguments of this section are always carried out in a suitable neighbourhoodof the vertex i.
As already discussed, direct inspection of expression (20.6) shows that
although Ὼ = |x|2 /U 2 is a real analytic function in a suitable neighbourhood
U of i ﬁxed by the equation
12ῺΔh Ὼ = 3Di ῺDi Ὼ − Ὼ2 r[h]2and the boundary conditions (20.5), in general, the corresponding ﬁelds d`ij andL̀ij will not have the same degree of smoothness.
To identify conditions on h ensuring that the ﬁelds d`ij and L̀ij are also
analytic, it is convenient to consider a complex analytic extension of U
similar to the one discussed in Section 19.4. To this end, one allows the normal
coordinates x = (xα ) in U to take values in a neighbourhood UC of the origin of
C3 so that the original neighbourhood U is a real three-dimensional analytic
submanifold of UC . Accordingly, the ﬁelds Γ ≡ |x|2 , h, ei , Ὼ, sij and r[h]are extended by analyticity into the complex domain and are regarded asholomorphic ﬁelds over UC . Assuming that i = {p ∈ UC | xα (p) = 0} is the
only critical point of Ὼ in UC , the complex null cone generated by the complex
null geodesics through i is given by the two-dimensional complex submanifold
NC (i) ≡ {p ∈ UC | Γ(p) = 0}.
By construction, the set of points in UC where Ὼ vanishes coincides with NC (i).

20.2 Massless and purely radiative spacetimes

533

A ﬁrst criterion ensuring the analyticity of φ̀ABCD – or, equivalently, d`ij – isgiven by the following:
Proposition 20.1 (analyticity of the massless part of the Weyl spinor:
ﬁrst version) The analyticity of φ̀ABCD near i is equivalent to the requirement(20.7)
D(P p Qp · · · DP 1 Q1 ) DEF D(AB DCD) Ὼ + ῺsABCD (i) = 0
for p = 0, 1, 2, . . ..
Remark. As will be seen in the following, this condition is, in fact, a conditionon the conformal class of h. The conformal constraint equations imply that L̀ij
is analytic if d`ij is analytic.
Proof The proof of the lemma makes repeated use of a factorisation lemma forholomorphic functions, which is discussed in the Appendix to this chapter; see
Lemma 19.2. The analyticity of h implies that the ﬁeld φ̀ABCD on U extends
to a holomorphic ﬁeld on UC \ NC (i) which satisﬁesῺ2 φ̀ABCD = D(AB DCD) Ὼ + ῺsABCD .

(20.8)

Now, if φ̀ABCD is analytic at NC (i) one can take a derivative of the above
expression and evaluate on NC (i) to ﬁnd that
(20.9)
DEF D(AB DCD) Ὼ + ῺsABCD N (i) = 0.
C

Conversely, given condition (20.9), one would like to verify that φ̀ABCD is
analytic at NC (i). Using the factorisation Lemma 19.2 in the Appendix to thischapter, it follows that there is a holomorphic ﬁeld fABCDEF such that, in aneighbourhood of NC (i), one has(20.10)
DEF D(AB DCD) Ὼ + ῺsABCD = ῺfABCDEF .
Deﬁning ZABCD ≡ D(AB DCD) Ὼ + ῺsABCD , the last equation can be written
as DEF ZABCD = ῺfABCDEF . Moreover, transvecting Equation (20.10) with
DEF Ὼ one obtains
DEF ῺDEF ZABCD N (i) = 0,C

(20.11)

which can be read as an ordinary diﬀerential equation for ZABCD along thegenerators of NC (i). It follows that the ﬁeld ZABCD is constant along thegenerators, so that evaluating Equation (20.11) at the vertex one concludes that
D(AB DCD) Ὼ + ῺsABCD = 0

on NC (i).

Using again Lemma 19.2 again, one ﬁnds that there exists a further holomorphic
ﬁeld fABCD such that
D(AB DCD) Ὼ + ῺsABCD = ῺfABCD

in a neighbourhood of NC (i).

534

Spatial inﬁnity

Taking a derivative of this expression and comparing the result with Equation (20.10) it follows that
DEF ῺfABCD N (i) = 0.
C

One observes that DEF Ὼ = 0 on UC \ {i}. It follows that there exists a
holomorphic function gABCD such that fABCD = ῺgABCD so that
Ὼ2 gABCD = D(AB DCD) Ὼ + ῺsABCD .
Comparing the latter with Equation (20.8) one concludes that gABCD coincides
with φ̀ABCD on UC \ NC (i), and, thus, φ̀ABCD is analytic near i as required.
Having encoded the analyticity of φ̀ABCD in terms of the vanishing of aspinorial ﬁeld at NC (i), one makes use of Lemma 19.3 in the Appendix to thischapter to express the latter condition as an equivalent series of conditions atthe vertex i.
An alternative way of imposing conditions on the metric h ensuring thatφ̀ABCD is analytic at i can be obtained using expression (20.4) for the Cotton
spinor. One has the following:
Proposition 20.2 (analyticity of the massless part of the Weyl spinor:
second version) A necessary condition on the metric h for φ̀ABCD to beanalytic in a neighbourhood of i is given by the sequence of conditions
D(P p Qp · · · DP 1 Q1 b̀ABCD) (i) = 0,

p = 0, 1, 2, . . .

(20.12)

Proof As in the proof of Proposition 20.1, one considers an arbitrary nullgeodesic γ(s) ∈ NC (i), s ∈ C, such that γ(0) = i with tangent vector having
the spinorial counterpart κA κB with κA parallely propagated along γ(s). The
latter implies that
κA DAB Ὼ = 0.

(20.13)For Ω = Ὼ, relation (20.4) takes the formb̀ABCD = 2D(A Q Ὼφ̀BCD)Q + ῺD(A Q φ̀BCD)Q ,which can be extended to UC by analyticity. In particular, at NC (i), contracting
the previous expression four times with κA one obtains, in view of (20.13), that
 A B C Dκ κ κ κ b̀ABCD N (i) = 0.
C

PQ

Applying repeatedly κ κ DP Q one ﬁnds that
 Pp Qκ κ p · · · κP 1 κQ1 κA κB κC κD DP p Qp · · · DP 1 Q1 b̀ABCD N (i) = 0,C

for p = 0, 1, 2, . . . Restricting the previous expression to i and recalling that κA
is arbitrary, one obtains (20.12).

20.2 Massless and purely radiative spacetimes

535

Remark. In Chapter 19 it has been shown that the Cotton spinor of the 3metric of a static solution satisﬁes a condition that is identical to (20.12); see
Proposition 19.1. The role it plays in ensuring the analyticity of the rescaled
Weyl tensor at i motivates the alternative name regularity condition. From
the analysis of Section 19.3 it follows that the expression (20.12) is conformallyinvariant and, accordingly, is a condition on the conformal class [h].
In Friedrich (1998c) it has been proven that conditions (20.7) and (20.12) are,in fact, equivalent. The following proposition rounds out nicely the discussion ofthis section.
Proposition 20.3 (equivalence between the conditions ensuring analyticity of the Weyl spinor ) Conditions (20.7) and (20.12) are equivalent.
Consequently, a necessary and suﬃcient condition on the conformal class [h] toensure that the ﬁelds φ̀ABCC and L̀ABCD extend analytically to i is given byD(P p Qp · · · DP 1 Q1 b̀ABCD) (i) = 0,

p = 0, 1, 2, . . .

The proof of the equivalence between (20.7) and (20.12) involves a lengthy
computation that goes beyond the scope of this section. Interested readers arereferred to Friedrich (1998c), theorem 4.2 and its proof, for full details.
20.2.3 Construction of massless data
In Friedrich (1988) it has been observed that asymptotically initial data sets canbe used as seeds for the construction of massless initial data sets.
Let (h , sij , ζ, ς) denote an asymptotically Euclidean solution to the conformal
static Equations (19.17). The above ﬁelds are expressed in a conformal gauge forwhich r[h ] = 0. Moreover, the conformal factor linking the conformal metrich with the physical metric h̃ via h ≡ Ω2 h̃ satisﬁes
−1/2

Ω

1
= ζ −1/2 + m
2(20.14)

with m the ADM mass of the static solution; compare Equation (19.8). One canthen look for conformal factors Ω solving the conformal Hamiltonian constraint2ΩΔh Ω = 3|D  Ω|2– compare Equation (11.15a) – where D  and Δh denote, respectively, theLevi-Civita covariant derivative and Laplacian of the conformally static metrich . Making use of the ansatz Ω = Ω(ζ), observing that
Di Ω =

dΩ
Di ζ,
dζ

Di Dj Ω =d2 ΩdΩDi Dj ζ,Di ζDj ζ +dζ 2dζand taking into account the conformal static equations one arrives at the ordinary
diﬀerential equation

536

Spatial inﬁnity
dΩ
d2 Ω
= 3ζ
2ζΩ 2 + 3Ω
dζ
dζ



dΩ
dζ

2
.

The general solution to this equation is given by
Ω=

c1 ζ
√ ,
(1 + c2 ζ)2c1 , c2 constants.

The subclass of analytic solutions is given by c2 = 0, so that – up to a constantfactor – one hasΩ = ζ.

(20.15)

When c2 = 0, that is, in the case of a non-analytic solution, one has a nonvanishing mass. In hindsight, the solution (20.15) could have been guessed
directly from Equation (20.14) as ζ satisﬁes the boundary conditions (20.5);
compare also Equation (19.11). Summarising, one has:
Proposition 20.4 (massless initial data out of static data) Given asolution to the conformal static equations (h , sij , ζ, ς) in a neighbourhood Uof the point at inﬁnity i, the metrich̃ = ζ −2 h ,deﬁned in a suitable punctured neighbourhood of i, satisﬁes the time-symmetric
Hamiltonian constraint r[h̃] = 0 and has vanishing mass. Moreover, the rescaled
Weyl and Schouten spinors obtained from Equations (20.3a) and (20.3b) bysetting Ω = ζ are analytic at i.
The above result can be generalised to obtain massless initial data for theconformal Einstein-Maxwell ﬁeld equations; see Simon (1992).
20.2.4 Evolution of massless data
Proposition 20.4 can be combined with the conformal evolution equations to
obtain a development admitting the dual interpretation discussed in Section
20.1.1. The simplest way of implementing the construction is to make use of theextended conformal Einstein ﬁeld equations expressed in terms of a conformal
Gaussian gauge; see Section 13.4.
Initial data for the congruence of conformal geodesics (x(τ ), β̃(τ )) underlying
the conformal Gaussian gauge can be set by the conditions
τ = 0,ẋ ⊥ U ,Θ = ζ,Θ̇ = 0,d ≡ Θ β̃  = dζ,on U .

The conformal factor associated to the congruence of conformal geodesics – see
Proposition 5.1 – is then given by


ςτ 2
1
Θ=ζ 1+,
ς ≡ Δh ζ.
2ζ
3

20.2 Massless and purely radiative spacetimes

537

In the above expression ζ and ς are regarded as constant along a given conformal
geodesic. Now, one has that ς(i) = −2. Hence, by choosing U suﬃciently small
so thatς < 0 in this neighbourhood, one can ensure that Θ has real roots at
τ = ± 2ζ/|ς|. Observe, in addition, that Θ = 0 at i.
The existence of a development for the massless data provided by Proposition
20.4 is given by the following result:
Theorem 20.1 (existence of purely radiative spacetimes) Let u denote
initial data for the extended conformal Einstein ﬁeld equations on a neighbourhood U of i constructed from a pair (h , ζ) as given by Proposition 20.4. Then
there exists τ• > 0 ensuring the existence of a smooth solution u to the conformal
Einstein ﬁeld equations onMτ• ≡ D+ (U ) ∩ [0, τ• ) × U ,such that the restriction of u to U coincides with u . DeﬁneNτ• ≡ {p ∈ Mτ• | Θ(p) = 0},
and let g be the Lorentzian metric constructed from the solution u. For thissolution one has the following:(i) On Mτ• \ (Nτ• ∪ (I + (i) ∩ Mτ• )) the metric g̃ ≡ Θ−2 g is a solution to thevacuum Einstein ﬁeld equations with vanishing mass for which Nτ• \ {i}represents future null inﬁnity I + and the point i corresponds to spatial
inﬁnity i0 .
(ii) On I + (i) ∩ Mτ• the Lorentzian metric g̃ is a purely radiative solution to the
Einstein ﬁeld equations for which Nτ• \ {i} represents past null inﬁnity andthe point i corresponds to past timelike inﬁnity.
A schematic depiction of the spacetimes constructed via the above result isgiven in Figure 20.1.
Proof The local existence of smooth solutions follows from the hyperbolic formof the evolution equations given in Proposition 13.3 together with the local
existence for this type of evolution equations provided by Kato’s Theorem 12.2.
The existence of an actual solution to the full conformal Einstein ﬁeld equations
follows from the form of the associated subsidiary system – see Proposition 13.4 –while the existence of a solution to the Einstein ﬁeld equations is obtained from
Proposition 8.3 whenever Θ = 0. The interpretation of the solutions in theregions where Θ > 0 and Θ < 0 follows from the discussion in Section 20.2.1.
Remark. Although the result is, from the conformal perspective, purely local,from the physical point of view, it is nevertheless of a semi-global nature. It
follows from the smoothness of the solution u to the conformal Einstein ﬁeld
equations provided by Theorem 20.1 on Nτ• \ {i} and, in particular, of the ﬁeld

538

Spatial inﬁnity

φABCD , that the Weyl tensor of both of the spacetimes in (i) and (ii) satisfy
the Peeling behaviour ; see Theorem 10.4.
20.3 A regular initial value problem at spatial inﬁnity
The purpose of this section is to discuss the formulation of a regular asymptotic
initial value problem for the conformal evolution equations for data with nonvanishing mass.
Consider a suitable neighbourhood U of the point at inﬁnity i of the pointcompactiﬁcation (S, h, Ω) of an asymptotically Euclidean (time-symmetric)
Cauchy hypersurface of a vacuum spacetime (M̃, g̃). Let {ei } denote an
h-orthonormal frame and let {A A } denote an associated spin frame. The key
idea behind the formulation of a regular asymptotic initial value problem is basedon the observation that a conformal rescaling of the form
Ω → Ω ≡ κ−1 Ω(20.16)

induces a rescaling of the frame of the formei → ei ≡ κei ,A A → A A ≡ κ1/2 A A .

Accordingly, the components of the rescaled Weyl spinor with respect to the spinframe {A A } rescale as
φABCD → φABCD ≡ κ3 φABCD .
Now, if one considers the rescaling (20.16) with a function κ of the form
κ = |x|κ,with κ smooth such that κ(i) = 0,(20.17)

one ﬁnds that φABCD = O(1). Thus, the components of the Weyl spinor withrespect to the new frame are bounded at i.
20.3.1 Rescaling of the initial data for the conformal ﬁeld equations
The discussion of the previous paragraph suggests that the rescaling (20.16) with
κ given by (20.17) could be used to formulate a regular Cauchy problem on U .
Note, however, that while φABCD is bounded at i, there is no guarantee that
it will be smooth since |x| is not a smooth function of the normal coordinates
x = (xα ). Thus, one needs to resort to polar coordinates similar to the ones
used in Section 19.2.1 to analyse the spacetime conformal extensions of static
spacetimes. Letting
ρ2 ≡ δαβ xα xβ ,

ρα ≡

xα
,
|x|

and using some coordinates θ = (θA ) on S2 to parametrise the position vector
ρα , one has that the 3-metric h can be written as
h = −dρ ⊗ dρ + ρ2 k,k ≡ hαβ ∂A ρα ∂B ρβ dθA ⊗ dθB ,20.3 A regular initial value problem at spatial inﬁnity

539with k|ρ=0 = −σ; compare Equation (19.24). It is natural to consider an horthonormal frame {ei } with dual coframe {ω i } satisfying
ω 3 = dρ,

ρ2 k = −ω 1 ⊗ ω 1 − ω 2 ⊗ ω 2 .

The indexing of the basis vectors has been chosen so as to match that of thespatial Infeld-van der Waerden symbols; see Equations (4.11a) and (4.11b).
From the above expressions it follows, writing ω i = ω i α dxα , that ω 3 α = O(1),ω 1 α , ω 2 α = O(ρ), while for the frame coeﬃcients in ei = ei α ∂ α one has thate3 α = O(1), e1 α , e2 α = O(ρ−1 ).
Consistent with the rescaling (20.16), letei ≡ κei ,ω i ≡ κ−1 ω i ,(20.18)and set ei = ei α ∂α and ω i = ω i α dxα . It follows that if the function κ is chosen
as in Equation (20.17), thene3 α = O(ρ),ω 3 α = O(ρ−1 ),e1 α , e2 α = O(1),ω1 α , ω2 α = O(1),and, moreover, that1h ≡ κ−2 h = − 2 dρ ⊗ dρ + k.
ρ

(20.19)Thus, the coframe coeﬃcients and, consequently, also the metric coeﬃcientsare singular at ρ. This singular behaviour is, however, not an obstacle for theconstruction of a regular initial value problem as these objects do not explicitly
appear as unknowns in the spinorial conformal Einstein ﬁeld equations in either
their standard or their extended form. Introducing the coordinate r ≡ − log ρ sothat r → ∞ as ρ → 0 one obtains the line elementh = −dr ⊗ dr + k.
Hence, the locus of points for which ρ = 0 lies at inﬁnity with respect tothe metric h but has ﬁnite circumference – and is, in fact, a metric 2-sphere.
Accordingly, the rescaling (20.19) resolves (blows up) the point at inﬁnity into a
2-sphere which is described locally in terms of the coordinates θ = (θA ).
In the remainder of this chapter, for the blow up of i to S2 it will be
understood the pairC(U ), {ei }consisting of a 3-manifold C(U ) with boundary ∂C(U ) ≈ S2 such that C(U ) \∂C(U ) can be identiﬁed with U \ {i} and where the frame {ei } is given as inEquation (20.18) with a choice of κ as in Equation (20.17). The set I 0 ≡ ∂C(U )will be called the sphere at inﬁnity. Observe that the deﬁnition of a blow up
makes reference neither to the metric h nor to the coframe {ω i } which are
singular as ρ → 0.

540

Spatial inﬁnity

The previous deﬁnition of the blow up of i has the purpose of simplifying thediscussion in the remainder of the chapter. A precise and rigorous discussion ofthis construction requires the use of the language of ﬁbre bundles. The interested
reader is referred to Friedrich (1998c, 2004) for a detailed account; see also Aceñaand Valiente Kroon (2011).
The rescaling of the conformal ﬁelds
The eﬀects of the frame rescaling (20.18) on the connection coeﬃcients can beanalysed by means of the usual transformation formulae for the connection.
One hasγi j k = ω j k ei i Di ek k= ω j k ei i Di ek k − κ−1 ω j k ei i ek l Sil mk Dm κ= κγi j k − (δk j Di κ + δi j Dk κ + δik Dj κ);compare a similar computation in Section 15.1.2. The spinorial version of theabove expression is given by1= κγABCD − (AC DBD κ + BD DAC κ).
γABCD
2To complete the discussion of the connection, one needs to consider the rescalingof the covector f . From the transformation rules of solutions to the conformal
geodesics equations, Equation (5.40), it follows that f  = f +dκ. Thus, if f  = 0,
it follows thatfi = Di κ,fAB= DAB κ.

Finally, it follows from the transformation rules of the components of the
Schouten tensor under the transition to a Weyl connection thatL̂ij = κ2 Lij ,ΘABCD = κ2 ΘABCD .

Comparing the above expressions with (20.3b), it follows that if κ is chosen asin (20.17), then ΘABCD = O(1).
A closer look at the frame
It is convenient to have a more detailed expression for the frame {ei } or,alternatively, its frame spinorial index counterpart {eAB } – recall that eAB ≡σ i AB ei . Let {∂ + , ∂ − } denote a local basis of vectors on S2 with dual cobasis{α+ , α− } such that ∂ − = ∂ + and, furthermore,
σ = 2(α+ ⊗ α− + α− ⊗ α+ ),with σ denoting the standard metric of S2 . The vectors can be expressed in termsof the local coordinates θ = (θA ), but the explicit correspondence will not be
required. The vectors {∂ + , ∂ − } originally deﬁned on S2 can be extended to C(U )20.3 A regular initial value problem at spatial inﬁnity

541

by Lie propagation along the radial direction given by ∂ ρ ; that is, one requiresthat [∂ ρ , ∂ ± ] = 0. Using the vector ﬁelds {∂ ρ , ∂ ± } one can then locally writeeAB = eAB 3 ∂ ρ + eAB + ∂ + + eAB − ∂ − ,for suitable frame coeﬃcients eAB 3 and eAB ± . These coeﬃcients can be
expanded, in turn, in terms of the basic valence-2 symmetric spinorsxAB ≡√2(A 0 B) 1 ,1
yAB ≡ − √ A 1 B 1 ,2

1zAB ≡ √ A 0 B 0 ,2satisfyingxAB xAB = −1,
yAB y AB = 0,xAB y AB = 0,1
yAB z AB = − ,2xAB z AB = 0,

(20.20a)zAB z AB = 0.

(20.20b)

Expressing the spinorial basis {A A } in the form 0 A = oA , 1 A = ιA one ﬁnds
that the ﬁelds xAB , yAB and zAB are, up to a normalisation, the componentsof the pairs o(A ιB) , oA oB and ιA ιB with respect to the spin basis. Taking into
account the contractions (20.20a) and (20.20b) and the line element (20.19) one
ﬁnds the more detailed expressioneAB = xAB ∂ ρ + eAB + ∂ + + eAB − ∂ − ,e01 ± = 0.

20.3.2 The cylinder at spatial inﬁnity
After providing regular initial data for the conformal ﬁeld equations in theneighbourhood U of i, one can now specify in more detail the conformal Gaussian
system underlying the hyperbolic reduction of the conformal Einstein ﬁeld
equations.
In what follows, the initial data for the congruence of conformal geodesics will
be ﬁxed, so that for p ∈ U \ {i} one has:x ≡ x(0) = p,ẋ ≡ ẋ(0) = e0Θ g̃(ea , eb ) = ηab ,future directed and orthogonal to S̃,Θ > 0,β̃, ẋ = 0.
For the above data one further letsΘ = κ−1 Ω,β̃  = Ω−1 dΩin U \ {i},(20.21)

where, in a slight abuse of notation, β̃  denotes the pull-back of β̃ to U \ {i}.
While β̃  is singular at i, d ≡ Θ β̃  is regular under the present assumptions.
Using Proposition 5.1 in Chapter 5 it follows that
κ2 τ 22κΩτ −1,κdΩ,(20.22)da = −
Θ = κ−1 Ω 1 − 2 ,
ω
ω2

542

Spatial inﬁnity

where
ω≡

2Ω
|dΩ|2

.

Now, as Ω = O(ρ2 ), it follows that ω = O(ρ) so that, together with the choice(20.17) for κ one ﬁnds that κ/ω → 1. Moreover, both Θ and da can be seen tohave well-deﬁned limits as ρ → 0. Accordingly, the conformal Gaussian gauge
can be extended to the set
I 0 ≡ {p ∈ U | ρ(p) = 0} ≈ S2 ,despite the fact that the second prescription in (20.21) is singular at theabove set.
Assume now that the congruence of conformal geodesics underlying the gauge
has no conjugate points on D(U ). A point p ∈ D(U ) is described by coordinates(τ, x ) where x denotes the normal coordinates of the intersection of the uniqueconformal geodesic passing through p with U . Accordingly, a suitable region ofD(U ) close to U can be thought of as a subset of R × U . In the following it will
be convenient to consider the sets


ω(q)
ω(q)
≤τ ≤
,
(20.23a)M(U ) ≡ (τ, q) ∈ R × U −
κ(q)
κ(q)


(20.23b)I ≡ (τ, q) ∈ M(U ) | q ∈ I 0 , |τ | < 1 ,I ± ≡ (τ, q) ∈ M(U ) | q ∈ I 0 , τ ± 1 ,(20.23c)ω(q)I ± ≡ (τ, q) ∈ M(U ) τ = ±.
(20.23d)κ(q)If an existence result for solutions to the conformal evolution equations can beobtained, then the set M(U ) gives rise to an extension of the physical spacetime
manifold M̃ in a neighbourhood of spatial inﬁnity, while I ± describe the twocomponents of null inﬁnity. In this representation the point i0 is replaced by
an extended set I, the cylinder at spatial inﬁnity, with both spatial andtemporal dimensions. The sets I ± where “null inﬁnity touches spatial inﬁnity”
will be called, for reasons which will become clearer in the subsequent discussion,
the critical sets.
The set up discussed in the previous paragraphs is ﬁxed up to a speciﬁc choiceof the function κ in (20.17). A convenient and simple choice of this function
consists in setting κ = ρ so that κ = 1 – this choice will be called the basic
representation. A schematic depiction of the sets in (20.23a)–(20.23d) of thebasic representation is given in Figure 20.2. An alternative choice consists ofsetting κ = ω. In this case Θ vanishes at τ ± 1, and, accordingly, one calls
this construction the horizontal representation. A schematic depiction of
the sets in (20.23a)–(20.23d) of the horizontal representation is given in Figure
20.3.

20.3 A regular initial value problem at spatial inﬁnity

543

Figure 20.2 Schematic depiction of the basic representation of the set up of
the cylinder at spatial inﬁnity. Left, a three-dimensional diagram; right, a twodimensional longitudinal section. See the main text for further details. Notethat the diagram to the right is not a conformal diagram but a graph of thelocation of the conformal boundary in the conformal Gaussian coordinates.

Figure 20.3 Schematic depiction of the horizontal representation of the set
up of the cylinder at spatial inﬁnity. Left, a three-dimensional diagram; right,
a two-dimensional longitudinal section. See the main text for further details.
Note that the diagram to the right is not a conformal diagram but a graph ofthe location of the conformal boundary in the conformal Gaussian coordinates.

20.3.3 The cylinder at spatial inﬁnity for the Minkowski and
Schwarzschild spacetimes
A good way of obtaining intuition about the properties of the conformal evolution
equations in a neighbourhood of I is to consider the case of initial data for the
Schwarzschild spacetime. The discussion in this section follows that of section 6
in Friedrich (1998c).

544

Spatial inﬁnity

Time-symmetric initial data for the Schwarzschild spacetime has been discussed in Section 11.6. It has been shown that the hypersurface S̃ characterisedby the condition t = 0 is conformally ﬂat, so that setting ρ = 1/r̄, with r̄ the
Schwarzschild radial isotropic coordinate, one obtains the following conformal
metric and conformal factor:
h = −dρ ⊗ dρ − ρ2 σ,

Ω= 

ρ2

2 .
1 + 12 mρ

(20.24)

A comparison with the split (20.1) shows that, close to the point at inﬁnity i(i.e. for ρ close to 0), one has
U = 1,W =m.
2

The basic data (20.24) allow one to compute the data for the conformal
evolution equations. Following the discussion of Section 20.3.1 and setting κ = ρ(i.e. using the standard representation) one ﬁnds thateAB 0 = 0,eAB 1 = ρxAB ,fAB = xAB ,eAB + = zAB ,ξABCD = 0,6mρ2
2 ABCD ,1
1 + 2 mρ

ΘABCD = eAB − = yAB ,χ(AB)CD = 0,

φABCD = −6m2ABCD ,(20.25a)(20.25b)(20.25c)where 2ABCD ≡ (A 0 B 0 C 1 D) 1 . In addition, the functions associated to theconformal Gaussian gauge can be computed to be


τ2
2τ ρ
ρ
Θ̇ = − Θ= 
2 1 − 
2 ,
4 ,
1
1
1 + 2 mρ1 + 2 mρ1 + 12 mρ
dAB = 

2ρxAB

3 .
1 + 12 mρ

The simple form of the initial data (20.25a)–(20.25c) suggests that thediscussion of the conformal evolution equations can be simpliﬁed by consideringa speciﬁc ansatz for the solutions. Some experimentation shows that a consistentansatz is given by
eAB 0 = e0 xAB ,eAB 1 = e1 xAB ,eAB + = e+ zAB ,eAB − = e− yAB ,1
ξABCD = √ ξ(AC xBD + BD xAC ),2
1
χ(AB)CD = χ2 2ABCD + χhABCD ,3
1
1
ΘABCD = θ2 2ABCD + θh hABCD + √ θx AB xCD ,32

fAB = f xAB ,φABCD = φ2ABCD ,20.3 A regular initial value problem at spatial inﬁnity

545

where the components of the vector-valued unknownu = (e0 , e1 , e+ , e− , f, ξ, χ2 , χ, θ2 , θh , θx , φ)are assumed to be real-valued functions of (τ, ρ). The ansatz allows one to reduce
the spinorial evolution equations to a system of scalar equations. A lengthy
computation renders1(χ2 − χ)e0 − f,31∂τ e± = − (χ2 + 2χ)e± ,61∂τ f = (χ2 − χ)f + θx ,3
1
1
∂τ χ = − χ22 − χ2 − θh ,63
11
∂τ θh = − χ2 θ2 − χθh ,631(χ2 − χ)e1 ,3
11∂τ ξ = − (χ2 + 2χ)ξ − χ2 f − θx ,
6
2
1 2 2
∂τ χ2 = χ2 − χ2 χ − θ2 + Θφ,
63
1
1∂τ θ2 = χ2 θ2 − (χ2 θh + χθ2 ) − Θ̇φ,
63
12ρ∂τ θx = (χ2 − χ)θx − 3 φ,
33 1 + 1 mρ∂ τ e0 =∂ τ e1 =2

1∂τ φ = − (χ2 + 2χ)φ.
2Initial data for these components can be obtained from a comparison of theansatz with Equations (20.25a)–(20.25c). One concludes thate0 = 0,e1 = ρ,θ2 = 

e+ = 1,6mρ

e− = 1,2 ,
1 + 12 mρ

f = 1,

θh = 0,ξ = 0,

θx = 0,χ2 = 0,χ = 0,

φ = −6m.

The symmetry reduced system and associated initial data can be written ina schematic form as∂τ u = F (u, τ, ρ; m),u(0, ρ; m) = u (ρ; m),(20.26)

where F and u are analytic functions of their arguments.
The m = 0 case
The particular case m = 0 – that is, the Minkowski spacetime – can be solved
explicitly with the only non-vanishing geometric ﬁelds given by
e0 = −τ,e1 = ρ,e± = 1,

f = 1,(20.27)

while the ﬁelds associated to the conformal gauge areΘ = ρ(1 − τ 2 ),dAB = 2ρxAB .

Consequently, this solution exists for all τ, ρ ∈ R. From the expressions in (20.27)
one ﬁnds that


√

τ
1
AA
= 2 dτ + dρ , ω AB = − xAB dρ − 2y AB α+ − 2z AB α− .
ω ≡ τAA ω
ρ
ρ

546

Spatial inﬁnity

Using the above covectors one can recover the metric associated to theconformal representation of the Minkowski spacetime under consideration. From
Equation (4.14) one ﬁnds
g=


1 2
ρ dτ ⊗ dτ + τ ρ(dτ ⊗ dρ + dρ ⊗ dτ ) − (1 − τ 2 )dρ ⊗ dρ − σ .
ρ2Consistent with the discussion of the previous sections, this metric is singular atρ = 0. Now, as
f = fAB ω AB =1dρρ

ˆ associated to thisis a closed form, it follows that the Weyl connection ∇representation is, in fact, the Levi-Civita connection of the metric ρ2 g. The
standard Minkowski metric can be recovered by setting x0 = τ ρ so that
 0

1
0
2
g̃ = Θ−1 g = 
2 dx ⊗ dx − dρ ⊗ dρ − ρ σ ,ρ2 − (x0 )21
ημν dxμ ⊗ dxν .
=(xλ xλ )2
Performing the inversion xμ → −xμ /(xλ xλ ) in the last line element one obtains
the standard Minkowski metric; compare the discussion in Section 6.2.2.
Null geodesics orthogonal to the spheres of constant ρ are given bys,ρ = ρ (1 ± s),
θ = (θA ) = (θA ),(20.28)τ=
1±s
for constant ρ , θA and s an aﬃne parameter. A direct computation shows thatoutgoing null geodesics intersecting future null inﬁnity I + correspond to the
choice of the minus sign in Equations (20.28) – the intersection occurring at
s = 12 so that ρ = 12 ρ . These outgoing null geodesics do not intersect pastnull inﬁnity I − for a ﬁnite value of s. As ρ → 0, the outgoing null geodesics
approach in a non-uniform manner the set I − ∪ I ∪ I + ∪ I − . An analogous
discussion applies to the incoming null geodesics obtained from taking the plus
sign in (20.28). Accordingly, the cylinder at spatial inﬁnity can be regarded as alimit set of outgoing and incoming null geodesics; see Figure 20.4.
The m = 0 caseNow, returning to the case m = 0, it follows from the Cauchy stability of ordinary
diﬀerential equations – see, for example, Hartman (1987) – that given τ• > 1
there exist m• > 0, ρ• > 0 such that the solution u(τ, ρ; m) is analytic in all
variables and exists for|τ | ≤ τ• ,ρ ≤ ρ• ,|m| ≤ m• .

By choosing τ• suﬃciently large and observing the properties of thereference m = 0 solution, one can ensure that for each conformal geodesic with

20.3 A regular initial value problem at spatial inﬁnity

547

Figure 20.4 Schematic depiction of the null geodesics close to the cylinder at
spatial inﬁnity for the Minkowski spacetime as discussed in the main text; see
the parametric equations in (20.28). The curves intersecting I + are outgoinggeodesics, while the ones intersecting I − are incoming. The cylinder I canbe seen as a limit set of the two classes of geodesics.

0 < ρ < ρ• there exists a τI < τ• such that Θ|±τI = 0, dΘ|±τI = 0. To obtaina statement that is valid for any value of m, it is observed that the symmetryreduced evolution equations and the associated data are invariant under the
rescaling
m →

1
m,


ρ → ρ,

φ →

1
φ,


e1 → e1 ,

Θ → Θ,for  > 0. Consequently, for any arbitrary m it is always possible to obtain asolution to the symmetry-reduced system (20.26) reaching null inﬁnity if ρ is
suﬃciently small. Moreover, if ρ is suﬃciently small, the underlying congruenceof conformal geodesics is free of conjugate points on M(U ). Null geodesics in
the Schwarzschild spacetime behave more and more like null geodesics in the
Minkowski spacetime as ρ → 0. Numerically constructed solutions of the reduced
spherically symmetric evolution system for the Schwarzschild spacetime can befound in Zenginoglu (2006, 2007).
20.3.4 Structural properties of the conformal evolution equations
near the cylinder at spatial inﬁnity
Having brieﬂy analysed the regular initial value problem at spatial inﬁnity for
the Minkowski and the Schwarzschild spacetimes, one is now in the position of
making some general remarks about this type of initial value problem.
The cylinder at spatial inﬁnity as a total characteristic
Following Proposition 13.3, the hyperbolic reduction of the extended conformal
Einstein ﬁeld equations by means of a conformal Gaussian system leads to an
evolution system which can be written as548

Spatial inﬁnity∂τ υ̂ = Kυ̂ + Q(Γ̂)υ̂ + L(x)φ,
I + A0 (e) ∂τ φ + Aα (e)∂α φ = B(Γ̂)φ.

(20.29a)(20.29b)

For convenience, it is assumed that Equation (20.29b) corresponds to theboundary adapted Bianchi system; see Section 13.4.4. Despite the fact that thecylinder I is, from the point of view of the metric g, a singular hypersurface, it
is regular from the point of view of Equations (20.29a) and (20.29b) and its data(υ̂  , φ ); see Section 20.3.1.
Inspection of the explicit form of the conformal evolution equations reveals
that L(x) = 0 whenever the conformal factor Θ and the covector d vanish. It
follows that, on I, the conformal evolution equations decouple and one has[0]∂τ υ̂ [0] = Kυ̂ [0] + Q(Γ̂ )υ̂ [0] ,υ̂ [0] ≡ υ̂|I ,

Γ̂

[0]

≡ Γ̂|I .

These transport equations can be integrated along the cylinder I from theobservation that, irrespective of the particular choice of κ, the restriction of
the initial data υ̂  to I 0 coincides with the restriction of initial data for the
Minkowski spacetime as given in Section 20.3.3. Accordingly, the solution one
obtains must also coincide with the Minkowskian one – namely,(e1AB )[0] = 0,[0](e+
= yAB ,
AB )

[0]
(e−
= zAB ,AB )(χ(AB)CD )[0] = 0,

(fAB )[0] = 0,

(ΘABCD )[0] = 0.

(e0AB )[0] = −τ xAB ,(ξABCD )[0] = 0,

Substituting the above values in the restriction to I of the partial diﬀerential
equation (PDE) (20.29b) one ﬁnds that the normal matrix A3 satisﬁes
A3 |I = 0.

(20.30)Hence, on I the restriction of Equation (20.29b) acquires the simpliﬁed form
[0]
I + A0 (e[0] ) ∂τ φ[0] + A+ (e[0] )∂ + φ[0] + A− (e[0] )∂ − φ[0] = B(Γ̂ )φ[0](20.31)

where φ[0] ≡ φ|I ; that is, one obtains an interior system. It follows that thecylinder at spatial inﬁnity I is a total characteristic of the conformal evolution
Equations (20.29a) and (20.29b) and the restriction to I of all the conformalﬁelds can be obtained from the restriction of the initial data to I 0 by solving theresulting system of transport equations. Thus, although at ﬁrst sight it seems thatthe construction of the cylinder at spatial inﬁnity is introducing a set on whichboundary data must be prescribed, the structural properties of the equations
do not allow this: no boundary conditions can be prescribed on I; compare thediscussion in Section 12.4.
The solution to the interior equations for the Weyl tensor, Equation (20.31),
can be obtained by observing that the restriction of the initial data for the Weyltensor coincides with that of Schwarzschildean data so that the solution must be20.3 A regular initial value problem at spatial inﬁnity

549

the Schwarzschild spacetime. From the symmetry-reduced conformal evolution
equations it follows that φABCD is constant along I. Accordingly, one ﬁnds that(φABCD )[0] = −6m2ABCD .
The conformal evolution system and the critical sets
The analysis of the transport equations on I provides valuable insights into thehyperbolicity of the conformal evolution system (20.29a) and (20.29b). Observing
that (e0AB )[0] = −τ xAB it follows that⎛⎞1−τ 0 0 0
0
⎜ 0
1 0 0
0 ⎟
⎜
⎟


I + A0 (e) |I = ⎜
0
0
1
0
0 ⎟
⎜
⎟;
⎝ 0
0 0 1
0 ⎠
0
0 0 0 1+τcompare Equation (13.61). Accordingly, the matrix A0 loses rank at the criticalsets I ± and is no longer positive deﬁnite. Thus, the standard theory of hyperbolic
PDEs as discussed in Chapter 12 cannot be employed to make assertions about
the existence and uniqueness of solutions of the evolution system (20.29a) and(20.29b) up to I ± . This degeneracy of the conformal evolution system is theessential source of diﬃculties in the analysis of the so-called problem of spatial
inﬁnity and requires the development of tailor-made techniques in order for one
to be able to make assertions about the behaviour of its solutions.
Expansions in a neighbourhood of the cylinder at spatial inﬁnity
On an intuitive level, one would expect the degeneracy of the conformal evolution
system at the critical sets I ± to manifest itself through a loss of smoothness of
its solutions. The discussion of Section 20.3.3 shows that this potential loss ofregularity does not occur for all initial data. This observation hints that the
precise algebraic structure of the evolution equations plays a decisive role in thenature of the solutions. In Friedrich (1998c) a procedure to analyse in detailthe properties of the solutions to the conformal evolution equations has been
put forward. Exploiting the total characteristic nature of the cylinder at spatial
inﬁnity one can repeatedly diﬀerentiate the evolution Equations (20.29a) and(20.29b) with respect to ∂ρ and then evaluate on I. In view of condition (20.30)
one obtains a hierarchy of transport equations for the ﬁeldsυ̂ [p] ≡ ∂ρp υ̂|I ,φ[p] ≡ ∂ρp φ|I ,p = 1, 2, 3, . . . ,of the form
[0]

[p]

∂τ υ̂ [p] = Kυ̂ [p] + Q(Γ̂ )υ̂ [p] + Q(Γ̂ )υ̂ [0]p−1  p Q(Γ̂[j] )υ̂ [p−j] + L[j] φ[p−j] + L[p] φ[0] ,
+
j
j=1

550

Spatial inﬁnity
I + A0 (e[0] ) ∂τ φ[p] + A+ (e[0] )∂ + φ[p] + A− (e[0] )∂ − φ[p]
p  
p 
[p]
[0]
B(Γ̂[j] )φ[p−j] − A+ (e[j] )∂ + φ[p−j]= B(Γ̂ )φ +
j
j=1

− [j]
− A (e )∂ − φ[p−j] .

The above equations will be called the transport equations of order p. The
non-homogeneous terms depend on υ̂ [j] and φ[j] for 0 ≤ j < p. Thus, if theselower order solutions are known, the above pair of equations constitutes aninterior system for υ̂ [p] and φ[p] on I. The principal part of these equations
is universal – in the sense that it is independent of the value of p. Initial data forthese transport equations can be obtained from repeated ρ-diﬀerentiation andevaluation on I 0 of the initial data υ̂  , φ on U . The coeﬃcients obtained fromthis integration can, in turn, be collected in formal expansions of the form∞

1 [p] pυ̂ ρ ,
υ̂ =
p!
p=0

∞

φ̂ =

1 [p] p
φ ρ .
p!
p=0

(20.32)At the time of writing, the analysis of the convergence of these formal expansions
and the way they relate to actual solutions to the conformal Einstein ﬁeld
equations is an outstanding open aspect in the understanding of the problem of
spatial inﬁnity. Some ideas on how this problem could be addressed can be foundin, for example, Friedrich (2003b) and Valiente Kroon (2009).
The structure of the hierarchy of transport equations for υ̂ [p] and φ[p] makes
them amenable to a treatment by means of computer algebra methods. Thisapproach has been pursued in Valiente Kroon (2004a,b) where solutions up toorder p = 8 have been obtained. As is to be expected, the algebraic complexityof the solutions increases as p increases, eventually making the evaluation offurther orders in the expansion no longer feasible due to computer limitations.
The solutions to the transport equations obtained in this manner provide a
valuable insight into the behaviour of the conformal ﬁeld equations at spatial
inﬁnity.
As ﬁrst observed in Friedrich (1998c), quite remarkably, there is a non-trivial
relation between the regularity condition for the Cotton tensor, Equation (20.12),
and the smoothness of the solutions to the transport equations:
Theorem 20.2 (necessary conditions for the regularity of solutions
to the conformal ﬁeld equations at the critical sets) Given a vacuum
time-symmetric initial data set with a conformal metric which is analytic in aneighbourhood of inﬁnity, the solution to the regular ﬁnite initial value problem
at spatial inﬁnity is smooth through I ± only if the regularity condition
D(E p F p · · · DE 1 F 1 bABCD) (i) = 0(20.33)holds for p = 0, 1, 2, . . . If this condition is violated at some order p , the solutions
to the transport equations at order p will develop logarithmic singularities at I ± .

20.3 A regular initial value problem at spatial inﬁnity

551

The analysis leading to the above result requires only the homogeneous partof the transport equations.
A toy model: the spin-2 massless ﬁeld
A way of gaining insight into the behaviour of the solutions to the conformal
evolution equations on the cylinder I is to consider an analogous discussion fora test spin-2 massless ﬁeld on the Minkowski spacetime. Accordingly,
let ζABCD denote the components of a totally symmetric rank-4 spinorial ﬁeld
satisfying the equation∇Q A ζABCQ = 0.

(20.34)The principal part of the evolution equations implied by (20.34) along thecylinder I is identical to that of the Bianchi evolution equations. Several aspectsof this toy model have been considered in Valiente Kroon (2002), Friedrich(2003b) and Beyer et al. (2012), and the following discussion is adapted fromvarious parts of these references.
The background Minkowski geometry has already been obtained in Section
20.3.3; see Equation (20.27). From these expressions, a computation shows thatthe evolution equations implied by the spin-2 massless ﬁeld equation can be
explicitly written as(1 − τ )∂τ ζ0 + ρ∂ρ ζ0 − ðζ1 − 2ζ0 = 0,1
∂τ ζ1 − (ðζ2 + ð̄ζ0 ) − ζ1 = 0,21
∂τ ζ2 − (ðζ3 + ð̄ζ1 ) = 0,21
∂τ ζ3 − (ðζ4 + ð̄ζ2 ) + ζ3 = 0,2
(1 + τ )∂τ ζ4 − ρ∂ρ ζ4 − ð̄ζ3 + 2ζ4 = 0,

(20.35a)(20.35b)(20.35c)(20.35d)(20.35e)where ζ0 ≡ ζ0000 , ζ1 ≡ ζ0001 , . . ., and where for convenience of the subsequentdiscussion, the connection coeﬃcients associated to S2 (i.e. Γ00CD and Γ11CD )have been absorbed in the diﬀerential operators ð and ð̄; see the Appendix to
Chapter 10. The subsequent analysis will also require the constraint equations
implied by Equation (20.34). These are given by1τ ∂τ ζ1 − ρ∂ρ ζ1 − (ðζ0 − ð̄ζ2 ) = 0,21τ ∂τ ζ2 − ρ∂ρ ζ2 − (ðζ1 − ð̄ζ2 ) = 0,21τ ∂τ ζ3 − ρ∂ρ ζ3 − (ðζ2 − ð̄ζ4 ) = 0.
2

(20.36a)(20.36b)(20.36c)

Diﬀerentiating Equations (20.35a)–(20.35e) and (20.36a)–(20.36c) repeatedly
with respect to ∂ρ and evaluating at I one obtains the transport equations

552

Spatial inﬁnity
[p]

[p]

[p]

(1 − τ )∂τ ζ0 − ð̄ζ1 + (p − 2)ζ0 = 0,
1
[p]
[p]
[p]
[p]
∂τ ζ1 − (ðζ0 + ð̄ζ2 ) − ζ1 = 0,
2
1
[p]
[p]
[p]
∂τ ζ2 − (ðζ1 + ð̄ζ3 ) = 0,
2
1
[p]
[p]
[p]
[p]
∂τ ζ3 − (ðζ2 + ð̄ζ4 ) + ζ3 = 0,
2
[p]
[p]
[p]
(1 + τ )∂τ ζ4 − ðζ3 − (p − 2)ζ4 = 0,

(20.37a)(20.37b)(20.37c)(20.37d)(20.37e)

and
1
[p]
[p]
[p]
[p]
τ ∂τ ζ1 − (ðζ4 + ð̄ζ2 ) − pζ1 = 0,
2
1
[p]
[p]
[p]
[p]
τ ∂τ ζ2 − (ðζ3 − ð̄ζ1 ) − pζ2 = 0,
2
1
[p]
[p]
[p]
[p]
τ ∂τ ζ3 − (ðζ2 − ð̄ζ0 ) − pζ3 = 0.
2

(20.38a)(20.38b)(20.38c)

The linearity of the above equations suggests eliminating the angular dependence of the solutions through an expansion in terms of spin-weighted sphericalharmonics. Consistent with the spin weight of the various components of ζABCD
one considers the ansatz
p

l

[p]

ζk =

zk,p;l,m (τ ) k−2 Ylm .
l=|k−2| m=−lObserve, in particular, that the number of l-modes is bounded by the diﬀerentiation order p. This ansatz can be shown to be the most general possible. Taking
into account the action of the operators ð and ð̄ on the spin-weighted sphericalharmonics, a calculation combining Equations (20.37a)–(20.37e) and (20.38a)–(20.38c) shows that the coeﬃcients zk,p;l,m (τ ) satisfy the Jacobi ordinary
diﬀerential equation
(1 − τ 2 )z̈k,p;l,m + 2(k − 2) + 2(p − 1)τ żk,p;l,m + l(l + 1) − p(p − 1) zk,p;l,m = 0,
where ˙ denotes diﬀerentiation with respect to τ . The solutions to this equation
are well understood; see, for example, Szegö (1978). For |k − 2| ≤ l < p thesolutions are a linear combination of the polynomialsp+k−21−τ(−p−6+k,−p+k−2)(p+k−2,−p+k−2)
(τ ),
Pl−2
(τ ),
Pp−l−12(α,β)

where Pn(τ ) denotes the Jacobi polynomial of degree n with integer
parameters (α, β) given byn−s 
sn n+α n+βτ −1τ +1.
Pn(α,β) (τ ) ≡
s
n−s
2
2
s=0

20.3 A regular initial value problem at spatial inﬁnity

553

The case l = p is the one of most interest as the general solution can be foundto be a linear combination ofp+k−2 p+2−k
1+τ
1−τ
,
2
2

p+k−2 p+2−k  τ
1+τ
1−τ
ds
.
p−1+k (1 + s)p+3−k
2
2
(1
−
s)
0
Using partial fractions one ﬁnds that the integral in the second solution can be
expressed as τdsap+2−ka1= a• ln(1 + τ ) ++ ··· +
p−1+k
p+3−k
p+2−k
(1 + s)(1 + τ )1+τ
0 (1 − s)
bp−2+kb1+ b0 ,+ ··· ++ b• ln(1 − τ ) +(1 − τ )p−2+k1−τwhere a• , b• , ap+2−k , . . . , a1 , bp−2+k , . . . , b0 are some constants. Thus, generically, the solutions for the l = p modes will be non-smooth and develop
logarithmic singularities at the critical sets I ± even in the case where the initial
data is as smooth as it can be. Direct inspection of the above expressions showsthat at τ = 1 the most singular component of ζABCD is ζ0 , while at τ = −1 itis ζ4 .
The singular behaviour can be avoided if the initial data is ﬁne tuned. Indeed,a lengthy analysis renders the following (see Valiente Kroon (2002)):
Proposition 20.5 (regularity of solutions to the massless spin-2 ﬁeld
equations at the critical sets) The solutions to the transport equations
implied on the cylinder at spatial inﬁnity I of the Minkowski spacetime by the
spin-2 massless ﬁeld Equation (20.34) extends analytically to the critical sets I ±if and only if the regularity condition
D(E p F p · · · DE 1 F 1 b̆ABCD) (i) = 0,

p = 0, 1, 2, . . . ,whereb̆ABCD ≡ 2DP (A ΩζBCD) P + ΩDP (A ζBCD) P
denotes the linearisation of the Cotton spinor around Minkowski data.
This result is the spin-2 ﬁeld version of Theorem 20.2 for the full conformal
Einstein ﬁeld equations.
20.3.5 The cylinder at spatial inﬁnity and static solutions
The analysis of static solutions provides deeper insights into the behaviour ofthe solutions to the transport equations on I. The discussion in Section 19.2.1
shows that static solutions admit a smooth conformal completion at null inﬁnity.
Thus, it is natural to conjecture that they also extend smoothly through the554

Spatial inﬁnity

critical sets I ± . The analysis of the conformal evolutions for the Schwarzschildspacetime provides further support to this idea – this evidence, however, must
be taken with care as the spherical symmetry of the spacetime gives rise to anumber of non-generic simpliﬁcations.
A lengthy computation which combines the ideas of Sections 19.2.1 and 20.3.3
yields the following satisfactory result:
Proposition 20.6 (regularity of static solutions at the critical sets) The
solutions to the transport equations at I for static data extend smoothly (and, infact, analytically) through the critical sets I ± .
A proof of this result can be found in Friedrich (2004). A generalisation of theanalysis to the stationary case is given in Aceña and Valiente Kroon (2011).
20.4 Spatial inﬁnity and peelingAt the time of writing, one of the outstanding challenges in the analysis of
the problem of spatial inﬁnity is to obtain a satisfactory understanding of theconnection between the solutions to the transport equations on I and the peeling(or lack thereof) of the Weyl tensor at I .
The key hypothesis in the peeling theorem, Theorem 10.4, is the smoothnessof the rescaled Weyl tensor on null inﬁnity. Direct inspection allows one to relax
this assumption to a certain minimum regularity threshold. Now, it has been
seen in the previous section that generic solutions to the transport equations
on the cylinder I have logarithmic singularities at the critical sets I ± . In viewof the hyperbolic character of the conformal evolution equations, it is to be
expected that this singular behaviour will spread along the conformal boundary,thus destroying the smoothness of the rescaled Weyl tensor along the conformalboundary. These singularities may lead to a restricted peeling behaviour ; see,for example, Chruściel et al. (1995) and Valiente Kroon (1998, 1999a,b) for a
discussion of more general types of peeling. A detailed and rigorous treatmentof these ideas is not yet available; some heuristic discussions can be found inValiente Kroon (2002, 2003, 2005, 2007a).
The most promising avenue to obtain a link between the generic singularbehaviour at the critical sets and the peeling behaviour at null inﬁnity consistsof computing the formal expansions (20.32) up to a certain order N . Letting υ̂and φ denote the actual solutions (if any) to the conformal evolution equations
one deﬁnes the remaindersN

RN [υ̂] ≡ υ̂ −1 [p] pυ̂ ρ ,
p!
p=0

N

RN [φ] ≡ φ −1 [p] p
φ ρ .
p!
p=0

If the expansion order N is suﬃciently high, it may be possible to use theconformal evolution equations to obtain estimates on the remainders RN [υ̂]and RN [φ]. The idea behind this approach is that the expansion terms should

20.5 Existence of asymptotically simple spacetimes

555

contain the most singular part of the solution, thus leaving a remainder which
is more regular and, accordingly, more amenable to an analytic treatment. This
strategy has been implemented with success for the model problems of the spin2 massless ﬁeld in the Minkowski spacetime in Friedrich (2003b) and for the
spinorial Maxwell equations (i.e. the spin-1 massless ﬁeld) on the Schwarzschildspacetime in Valiente Kroon (2009).
20.5 Existence of asymptotically simple spacetimes
The regularity of static solutions at spatial inﬁnity provides a procedure toconstruct a wide class of asymptotically simple solutions to the Einstein ﬁeld
equations from a Cauchy initial value problem: the so-called Cutler-WaldChruściel-Delay construction; see Cutler and Wald (1989); Chruściel and
Delay (2002) and Corvino (2007). The key idea behind this construction is toconsider time-symmetric initial data sets (S̃, h̃) for the Einstein ﬁeld equations
which are exactly Schwarzschildean in a suitable exterior region Ẽ of theasymptotic end but otherwise arbitrary in a compact region B in the interior.
The existence of such initial data sets is ensured by the exterior asymptotic
gluing construction; see Theorems 11.3 and 11.4. Denote by (S, h) a suitable
point compactiﬁcation of the data (S̃, h̃) and let E denote the neighbourhood of icorresponding to the exterior region Ẽ. As a consequence of the causal propertiesof general relativity, the development of (S̃, h̃) is such that D+ (E) coincides witha suitable spacetime neighbourhood of the spatial inﬁnity of S. In a slight abuseof terminology one can say that these data have compact support. Accordingly,
D+ (S) will contain hyperboloidal hypersurfaces H which coincide with S \ E on
D+ (S \ E). On H ∩ D+ (E), the initial data for the conformal ﬁeld equations
implied by the development on H will be Schwarzschildean hyperboloidal data –and, thus, smooth at I ∩H. An important technical aspect of this construction isto ensure that the gluing region does not drift away into the asymptotic region as
one considers a sequence of data tending to data for the Minkowski spacetime.
This is ensured by Theorem 11.4. Now, if the data on B are suﬃciently close
to data for the Minkowski spacetime, one can apply the semi-global existence
Theorem 16.1 to the data on H to obtain a development D+ (H) which is
asymptotically simple. As the Schwarzschild spacetime is asymptotically simple,
one concludes that D+ (S) is asymptotically simple; see Figure 20.5. In viewof the time symmetry of the initial data, one, in fact, obtains a spacetime
where the two components of null inﬁnity I − and I + are complete. While thedevelopment D+ (S) is static in D+ (E), in general, radiation will be registeredon I + ∩ J + (S \ E) and I − ∩ J − (S \ E).
For further details on the construction described in the previous paragraph,see Chruściel and Delay (2002).
Remark. The original version of the above construction was carried out forsolutions to the Einstein-Maxwell equations. Remarkably, it is possible to556

Spatial inﬁnity

Figure 20.5 Schematic depiction of the Cutler-Wald-Chruściel-Delay construction. The spacetime is asymptotically simple and coincides with the
Schwarzschild spacetime on D+ (H). Generically, radiation is registered on
I + ∩ J + (S \ E).

construct initial data with compact support for the Einstein-Maxwell equations
without the need of a gluing construction; see Cutler and Wald (1989).
20.6 Obstructions to the smoothness of null inﬁnity
The spacetimes obtained from the Cutler-Wald-Chruściel-Delay construction
are very special. Thus, it is natural to ask whether it is possible to constructasymptotically simple spacetimes which do not have such a rigid behaviour ina neighbourhood of spatial inﬁnity. Insight into this question can be obtained
from the analysis of the transport equations on the cylinder at spatial inﬁnity.
The systematic analysis of the transport equations on I has shown that two
diﬀerent types of obstructions to the smoothness of null inﬁnity arisein the development of time-symmetric data (S̃, h̃) admitting a smooth pointcompactiﬁcation (S, h) at spatial inﬁnity. These are brieﬂy discussed in thefollowing.
Obstructions associated to the conformal class [h̃]As already discussed, obstructions to the smoothness of null inﬁnity associatedto the conformal class [h̃] can be removed by requiring that the Cotton tensor
of the conformal metric h satisﬁes the regularity condition (20.33).
Obstructions associated to the scaling of the conformal metric
To discuss the obstructions to the smoothness of null inﬁnity associated to theparticular scaling of the conformal metric, suppose that the conformal metric h
is a solution to the conformal static equations with associated conformal factor
Ω; see Chapter 19. Now, restricting the subsequent considerations to a suitable20.7 Further reading

557

small neighbourhood U of i consider another conformal factor Ω satisfying theboundary conditions of a point compactiﬁcation and such that the metric h̃ ≡Ω−2 h satisﬁes the time-symmetric Hamiltonian constraint on Ũ ≡ U \{i}; that
is, r[h̃ ] = 0. It follows that there exists ϑ ∈ C 2 (U ) ∩ C ∞ (Ũ ) such that Ω ≡ ϑΩ.
Now, assume that ϑ(i) = 1, dϑ(i) = 0 and Hess ϑ(i) = 0 so that the metricsh̃ and h̃ = ϑ−2 h̃ have the same mass. As the conformal metric h is static it
satisﬁes the regularity condition (20.33). Moreover, as h = ϑ2 h ∈ [h ] it alsosatisﬁes the regularity condition. After a lengthy inductive argument one obtains
the following:
Theorem 20.3 (obstructions to the smoothness of null inﬁnity associated to the scaling of the conformal metric) Given time-symmetric initial
data with an analytic conformal metric h, the solution to the regular ﬁnite
initial value problem at spatial inﬁnity for the conformal Einstein ﬁeld equations
is smooth through the critical sets I ± (and, in particular, free of logarithmic
singularities) if and only if ϑ − 1 vanishes at i at all orders.
The proof of this result can be found in Valiente Kroon (2010, 2011). The
analysis leading to the above theorem assumed the analyticity of the metricin U . However, the result also holds if one assumes smoothness. This result
provides strong indication that static initial data play a privileged role among
the class of time-symmetric data which extend smoothly through the criticalsets. A precise clariﬁcation of this role is one of the outstanding challenges in
the analysis. Despite the insights obtained so far, at the time of writing, itcannot be excluded that there exist data which are not asymptotically static at
i and for which the solutions to the transport equations on I extend smoothly
through the critical sets. To address this point, it is necessary to identify the gap
between initial data satisfying the regularity condition (20.33) and static data.
The further conditions required to single out static data have been analysed inFriedrich (2013). It has been found that a suﬃcient condition for the staticityof the data satisfying condition (20.33) and the non-degeneracy requirement
associated to the hypothesis of Theorem 19.4 (concerning the uniqueness ofthe conformal structure of a static solution) can be expressed in terms of a
covector with conformally invariant diﬀerential. The challenge is now to analyse
whether data violating this suﬃcient condition develop singularities at thecritical sets.
20.7 Further reading
Although it has long been recognised that, for spacetimes with a non-vanishingmass, spatial inﬁnity is a singular point of the conformal structure – see,for example, Penrose (1963, 1965) – systematic attempts to understand the
behaviour of the geometry of spacetime in a neighbourhood of this pointin the light of the Einstein ﬁeld equations took time to get started. Early

558

Spatial inﬁnity

analyses of the behaviour of the Einstein ﬁeld equations in a neighbourhoodof a suitable representation of spatial inﬁnity have been given in Schmidt (1981),
Beig and Schmidt (1982), Beig (1984) and Schmidt (1987). The approach tothe analysis of spatial inﬁnity discussed in this chapter started in Friedrich(1988). The construction of the cylinder at spatial inﬁnity was presented inFriedrich (1998c) which to date remains the most comprehensive reference inthe matter. A useful discussion which overlaps with the previous reference but
also expands in certain aspects not covered in the original work is given inFriedrich (2004); this reference provides, in particular, a detailed discussionof the construction of the cylinder at spatial inﬁnity for static solutions. The
extension of the later analysis to stationary solutions has been carried out inAceña and Valiente Kroon (2011). A programme to analyse the solutions to
the transport equations on I was started in Friedrich and Kánnár (2000a);
see also Friedrich and Kánnár (2000b). Expansions to a suﬃciently high orderto observe the ﬁrst obstructions to the smoothness of null inﬁnity have been
carried out in Valiente Kroon (2004a,b,c, 2005). General results concerningthese expansions showing the special role played by static solutions (in a timesymmetric setting) are given in Valiente Kroon (2010, 2011). An account of
the state of the art concerning the problem of spatial inﬁnity is provided inFriedrich (2013) where the gap between data satisfying the regularity conditionon the Cotton tensor and static data is analysed in detail. A discussion of general
aspects of the behaviour of the massless spin-2 ﬁeld in a neighbourhood of spatial
inﬁnity of the Minkowski spacetime can be found in Valiente Kroon (2002); seealso Beyer et al. (2012). A method for the construction of estimates for the
massless spin-2 ﬁeld which remain regular at the critical sets of the Minkowski
spacetime has been provided in Friedrich (2003b). These ideas have been
adapted to the case of the Maxwell equations on a Schwarzschild background inValiente Kroon (2007b, 2009).

Appendix: properties of functions on the complex null cone
The following result of complex analysis is used repeatedly in the main text of
this chapter.
Lemma 19.2 (factorisation lemma) Let f denote a holomorphic function
on a neighbourhood UC of the origin of C3 , and let NC (0) denote the complex
null cone through the origin. If f |NC (0) = 0, then there exists a holomorphic
function g deﬁned on a neighbourhood of the origin of C3 such that f = Γgwhere Γ = |x|2 .
The proof of this result can be found in Kodaira (1986). Recall that NC (0)coincides with the locus of points in C3 for which Γ vanishes. One also has thefollowing:20.7 Further reading

559

Lemma 19.3 (characterisation of functions vanishing on the null cone)
A holomorphic spinorial ﬁeld ζA···D in some neighbourhood UC of the origin in
C3 vanishes on NC (0) if and only if it satisﬁes the sequence of conditions
D(P p Qp · · · DP 1 Q1 ) ζA···D (0) = 0,

p = 0, 1, 2, . . .

(20.39)The proof of this result is based on the observation that the conditions (20.39)
can be used to construct a Taylor-like expansion of the ﬁeld ζA···D of the form∞

ζA···D (γ(s)) =1 p P p Qp
s κ κ · · · κP 1 κQ1 DP p Qp · · · DP 1 Q1 ζA···D (0)p!
p=0along the generators γ(s) of NC (0) for s an aﬃne parameter suﬃciently close to 0.
As a consequence of the analyticity of the set up, the above expansion uniquelydetermines the function ζA···D in a neighbourhood of 0 on NC (0). A more detailed
discussion of the proof can be found in Friedrich (2013), lemma 6.1.

21
Perspectives

And it seemed as though in a little while the solution would be found, and thena new and splendid life would begin; and it was clear to both of them that they
had still a long, long road before them, and that the most complicated anddiﬃcult part was only just beginning.
– A. Chekhov, The lady with the dog
Conformal notions provide valuable tools for the analysis of global properties ofspacetimes. In Part IV of this book it has been shown how a conformal point ofview leads to proofs of the global existence and non-linear stability of de Sitterlike spacetimes, of the semiglobal existence and non-linear stability of Minkowskilike spacetimes, and how they provide a systematic procedure for the constructionof anti-de Sitter-like spacetimes. Moreover, conformal methods provide a robust
framework for the analysis of the gravitational ﬁeld of isolated systems in aneighbourhood of both null and spatial inﬁnity.
The application of conformal methods in general relativity is a mature area of
research with a considerable number of open problems. Several of these have been
discussed in various places of this book. Unavoidably, there are other problems
and aspects of the subject which, for reasons of space, could not be covered in
the main text. This last chapter presents a list of ideas and problems which,in the opinion of the author, may play a role in the future development of thesubject.
21.1 Stability of cosmological models
The global non-linear stability of the de Sitter spacetime was discussed in
Chapter 15. This exact solution can be regarded as a basic cosmological model.
The analysis of Chapter 15 can be extended to include a non-vacuum matter
content with good conformal properties: for example, the Maxwell, Yang-Millsand conformally coupled scalar ﬁeld; see Friedrich (1991) and Lübbe and Valiente

21.1 Stability of cosmological models

561

Kroon (2012). More recently, the ideas behind these proofs have been adaptedin Lübbe and Valiente Kroon (2013b) to provide an analysis of the futurestability of Friedman-Robertson-Walker cosmological models with a perfect ﬂuid
with the equation of state of radiation; see Section 9.4. A natural question iswhether conformal methods could be adapted to more general matter models,that is, matter models with a non-vanishing trace. That this may be possible issuggested by the analysis in Friedrich (2015b) where it is shown that massive
scalar ﬁelds for which the mass parameter is related to the cosmological constantby the condition 3m2 = −2λ give rise to a set of regular conformal evolution
equations for the Einstein-massive scalar ﬁeld system. A further indicationthat conformal methods may be applicable to more general matter models isprovided by the observation that for a large class of equations of state, perfect
ﬂuid Friedman-Lemaı̂tre-Robertson-Walker (FLRW) cosmological models can besmoothly conformally compactiﬁed – see, for example, Griﬃths and Podolský
(2009), section 6.4 – this despite the fact that the “natural” conformal evolution
equations for these models are not conformally regular.
An important motivation behind the analysis of the future non-linear stabilityof cosmological models is the so-called cosmic no-hair conjecture – theexpectation that for a large class of models the late-time evolution approximates,in some sense, a de Sitter state; see, for example, Wald (1983). As the analysis
of Lübbe and Valiente Kroon (2013b) exempliﬁes, conformal methods provide aconvenient setting for this discussion – at least for some suitable matter models.
Conformal methods provide a natural tool for the analysis of so-called
isotropic cosmological singularities. These are singularities of the physicalspacetime that can be removed by means of a conformal rescaling of the metric.
The singularity of the rescaled metric is assumed to occur on a spacelike surface.
Accordingly, the conformal structure can be extended through the singularityand one can study the Cauchy problem for the cosmological model with data at
the location of the singularity; see, for example, Tod (2002) for an introduction
into the subject and Anguige and Tod (1999a,b) and Tod (2003) for further
details. The Big Bang singularity in FLRW models provides the prototypicalexample of this type of singularity: as these spacetimes are conformally ﬂat, any
curvature singularity must be restricted to the (physical) Ricci tensor. In viewof the highly symmetric nature of FLRW spacetimes, the Ricci tensor has only
one essential component; combining this observation with the fact that underconformal rescalings the Ricci tensor satisﬁes a transformation law which is nonhomogeneous, one can then see that in FLRW spacetimes the conformal factor
can be chosen so as to absorb the singular behaviour of the curvature.
In the analysis of isotropic singularities, the role of the conformal factor is
diﬀerent from the one in the study of asymptotics: the conformal factor diverges
at the singularity rather than going to zero – thus, it “blows up” the shrinking
physical metric to make it ﬁnite. This type of behaviour is not expected to bea general feature of cosmological solutions to the Einstein ﬁeld equations. This

562

Perspectives

observation is related to Penrose’s Weyl curvature hypothesis: the idea thatthe early geometry of the universe should be such that the Weyl tensor vanishes,
singling out a state of low gravitational entropy; see Penrose (1979).
In the discussion of isotropic cosmological singularities one pursues conformal
rescalings of the form g̃ = 02 g where g̃ denotes the physical metric, while theunphysical metric g extends the conformal structure through the singularity
characterised by the condition 0 → 0. Under these conventions the Einsteinﬁeld equations, written in a suitable gauge, lead to conformal evolution equations
having a well-understood singular behaviour at the Big Bang. These evolution
equations are an example of so-called Fuchsian diﬀerential equations – aclass of equations with a well-deﬁned theory. Using this theory, a number ofstatements concerning isotropic singularities can be obtained; see again Tod(2002) and references within for further details. More recently, it has been shownthat a duality property of the conformally coupled scalar ﬁeld equation allows
one to analyse isotropic singularities in a framework involving the conformal
Einstein ﬁeld equations; see Lübbe (2014). It is of interest to see whether theseideas can be pursued further and extended to more general contexts.
21.2 Stability of black hole spacetimes
One of the outstanding open problems in mathematical general relativity is thequestion of the non-linear stability of the Kerr spacetime; see, for example,
Dafermos and Rodnianski (2010) for an entry point into the literature of thesubject. The expectation associated with this question is that perturbations ofa Kerr metric should dynamically approach a member of the Kerr family ofsolutions in the exterior of the black hole region. This problem involves bothan orbital and an asymptotic stability analysis; see the discussion in Section
14.4. The non-linear stability of the Kerr spacetime poses both technical andconceptual challenges. On the technical side, it requires the development ofrobust partial diﬀerential equation (PDE) techniques to control the behaviourof the Einstein ﬁeld equations in the strong gravitational ﬁeld regime of a black
hole. Current eﬀorts in this direction have involved a detailed analysis of linear
wave equations whose solutions propagate on the domain of outer communicationof a Kerr background. This analysis makes systematic use of so-called vector
ﬁeld methods. The expectation is that these wave equations provide a suitablemodel for the Einstein equations written, say, in harmonic coordinates; seeagain Dafermos and Rodnianski (2010) for an account of this approach. On theconceptual side, the problem needs a detailed speciﬁcation, compatible with theneeds of PDE theory, of what is meant by the statement that a given spacetime isclose to the Kerr spacetime; some ideas on how to address this issue are discussed
in Bäckdahl and Valiente Kroon (2010a,b).
Given that conformal methods, as discussed in this book, provide a tool for the
analysis of the non-linear stability of asymptotically simple spacetimes – compare

21.2 Stability of black hole spacetimes

563

Chapters 15 and 16 – it is natural to wonder whether they could also provide
an avenue for the analysis of the non-linear stability of black hole spacetimes.
The stability proofs discussed in this book start from the premise that a detailed
understanding of the conformal geometry of a background solution is key to theanalysis. Once this has been achieved, the existence and stability results follow
by means of general results of the theory of PDEs – namely, the Cauchy stability
guaranteed by Theorem 12.2. From the perspective of conformal geometry, theessential diﬀerence between the basic asymptotically simple spacetimes and theexact solutions describing black hole spacetimes is that while the former are
conformally regular, the latter have a conformal structure with singular regions.
This observation rules out the possibility of directly using arguments based
solely on the notion of Cauchy stability to prove global existence and stability of
black hole spacetimes. In order to go any further, it seems necessary to analyse
the structure of the singularities in the conformal structure of the backgroundsolutions so as to obtain, if possible, conformal representations of the black holespacetimes for which the conformal Einstein ﬁeld equations acquire a form whichis amenable to a PDE analysis. An example of the regularisation of singularities
in the conformal structure is provided by the analysis of the problem of spatial
inﬁnity in Section 20.3 where a detailed knowledge of the singular behaviour ofthe various conformal ﬁelds led to the construction of a regular Cauchy problem
for the conformal ﬁeld equations. It is possible that some singular regions in theconformal structure of black hole spacetimes – such as neighbourhoods of i± inthe extreme Reissner-Nordström and extreme Kerr spacetimes – are amenable
to an analogous discussion; see, for example, Lübbe and Valiente Kroon (2014).
A systematic approach to the analysis of the conformal structure of black
holes is through the study of suitable congruences of conformal geodesics. In
Friedrich (2003a) it is shown that it is possible to construct a non-intersecting
congruence of conformal geodesics that covers the whole of the Kruskal extensionof the Schwarzschild spacetime. This congruence is prescribed by initial data on
the time symmetric hypersurface of the spacetime, and it provides a preferredconformal representation of the spacetime as well as a global conformal Gaussiangauge system from which, say, a global numerical evaluation of the spacetime
can be undertaken; see the discussion in the next section. In addition, this typeof construction sheds some light on the singular behaviour of the conformal
structure at the timelike inﬁnity; see Friedrich (2002), section 1.4.4. A similar
analysis has been carried out in the Reissner-Nordström spacetime (including
the extremal case) using so-called conformal curves in Lübbe and Valiente
Kroon (2013a) and the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter
spacetime with conformal geodesics in Garcı́a-Parrado et al. (2014). It would beof great interest extend this type of analysis to stationary black holes, that is,
the Kerr spacetime.
The expectation driving the constructions described in the previous paragraphis that they will lead to a suitable conformal representation of the background564

Perspectives

black hole spacetimes which, in turn, lends itself to the formulation of an
initial value problem allowing the analysis of the non-linear stability of blackhole spacetimes. Nevertheless, the presence of singular points of the conformal
structure of the background solution will require considerations of asymptotic
stability – rather than just orbital stability as in the case of the proofs of stabilityof the de Sitter and Minkowski spacetimes given in Chapters 15 and 16. The
development of methods that allow this type of analysis for the conformal ﬁeld
equations is an interesting and challenging problem.
Finally, it should be mentioned that the notion of conformal compactiﬁcationof spacetimes, as introduced in Chapter 7, has been used as the starting point ofa programme to construct a theory of peeling and scattering of ﬁelds (including
gravity) on black hole spacetimes; see Nicolas (2015) and references within. It
would be of great interest to combine this approach to the asymptotic analysis ofspacetimes with the methods for the conformal Einstein ﬁeld equations developedin this book.
21.3 Conformal methods and numerics
Numerical relativity, the study of the Einstein ﬁeld equations by means of
numerical methods, has undergone a great development in recent years. Extended
numerical simulations of coalescing black holes have become almost routine; see,for example, Alcubierre (2008), Pretorious (2009) and Baumgarte and Shapiro(2010). To a great extent, these numerical simulations have been concerned withastrophysical aspects of black holes – most notably the extraction of gravitational
wave forms; see, for example, Lehner and Pretorious (2014). In addition to thisimportant application aimed at the detection of gravitational waves, numerical
relativity oﬀers a powerful tool in mathematical investigations of the equationsof general relativity. Some promising areas for this type of interaction have been
described in, for example, Jaramillo et al. (2008); for an alternative perspective,
see Andersson (2006).
The conformal ﬁeld equations suggest the possibility of performing global
numerical evaluations of spacetimes, that is, evaluations which are not limitedin their spatial and temporal dimensions by the ﬁniteness of the computational
grids. In addition, one would expect such evaluations to be free, to some extent, ofthe problems posed by the presence of unphysical boundary conditions required
to obtain a discretisation in a ﬁnite grid without periodic boundary conditions.
There have been a number of eﬀorts geared towards the construction of global
numerical evaluations of spacetimes using the conformal Einstein ﬁeld equations.
An early implementation of these ideas for the spherically symmetric Einsteinconformally invariant scalar ﬁeld system can be found in Hübner (1995). A
programme to construct a computer code for numerical simulations using themetric vacuum conformal ﬁeld equations has been reported in Hübner (1999a,b,
2001b) and culminated in Hübner (2001a) where a numerical demonstration of
the semi-global existence result for hyperboloidal data discussed in Chapter 16

21.3 Conformal methods and numerics

565

Figure 21.1 Focusing of the generators of null inﬁnity in a numerical evaluationof hyperboloidal initial data close to Minkowski initial data. From Figure
2 in P. Hübner (2001), From now to timelike inﬁnity in a ﬁnite grid,Class. Quantum Grav. 18, 1871–1884. © IOP Publishing. Reproduced bypermission of IOP Publishing. All rights reserved.

has been provided. Remarkably, the numerical simulations obtained by meansof this code show how the generators of null inﬁnity intersect, up to numericalprecision at a single point, future timelike inﬁnity i+ ; see Figure 21.1. An alternative approach based on the frame version of the standard vacuum conformal ﬁeld
equations has been described in Frauendiener (1998a,b, 2002) and implemented
in Frauendiener and Hein (2002); see also the review by Frauendiener (2004).
A critical discussion of the numerical implementation of the standard conformal
Einstein ﬁeld equation can be found in Husa (2002).
Conformal Gaussian gauge systems provide an alternative approach to thenumerical implementation of the conformal Einstein ﬁeld equations. As shownin Chapter 13, the evolution equations implied by the extended conformal
Einstein equations in this type of gauge splits into a subsystem of transport
equations for the components of the frame, connection and Schouten tensor
and a symmetric hyperbolic system for the components of the rescaled Weyltensor. This remarkable structure, highlighting the special role of the Weyltensor as describing the free gravitational ﬁeld, may facilitate the numerical
implementation of the system. An added advantage of this formulation of theconformal ﬁeld equations is, in the vacuum case, the a priori knowledge of theconformal factor linking the unphysical spacetime with the physical spacetime
for which the Einstein equations hold.
A programme to analyse the global dynamics of cosmological spacetimes
by numerical methods using the extended conformal ﬁeld equations has been
pursued in Beyer (2007, 2008, 2009a,b). This work has provided valuable insights

566

Perspectives

into the cosmic no-hair conjecture – see Section 21.1 – and the role of the socalled Nariai solution. Cosmological spacetimes provide a convenient testbed forthe numerical implementation of the conformal ﬁeld equations as they allow theuse of compact spatial domains – say, the 3-sphere S3 , the 3-torus S × S × S or
the 3-handle S2 × S – so that no boundary conditions in the spatial domain are
required. In addition, compact spatial sections are naturally amenable to the useof spectral methods; see, for example, Beyer (2009c).
A further application of the extended conformal Einstein ﬁeld equations
is the global numerical evaluation of spherically symmetric static black holespacetimes. This idea was ﬁrst investigated in Zenginoglu (2006, 2007) for the
Schwarzschild spacetime and later extended to the electrovacuum case (i.e. theReissner-Nordström spacetime) in Valiente Kroon (2012). The assumption of
spherical symmetry implies a great simpliﬁcation in the equations so as to rendera reduced evolution system consisting of transport equations solely. Notice,
however, that the conformal gauge in terms of which the evolution equations
are expressed is not adapted to the orbits of the static Killing vectors, and, thus,
one has non-trivial gauge dynamics. An important property of these sphericallysymmetric reduced equations is that their essential dynamics is governed by acore system consisting of three equations in the vacuum case (for a componentof the connection, a component of the Schouten tensor and the non-vanishingcomponent of the rescaled Weyl tensor) and four equations in the electrovacuum
case (connection, Schouten tensor, rescaled Weyl tensor and the single nonvanishing component of the Faraday tensor). These equations can be easily
implemented and numerically solved with present-day desktop computers andallow the global computation of a privileged conformal representation of theblack hole spacetime from an initial hypersurface to either the singularity or nullinﬁnity and beyond; see Figure 21.2. These small-scale numerical simulations
could be used, in the future, as the ﬁrst step in the global numerical evaluationof dynamic, non-spherically symmetric spacetimes.
More recently, there have been eﬀorts aimed at the numerical implementationof the construction of the cylinder at spatial inﬁnity described in Section
20.3.2. The ultimate goal of this programme is the numerical computation of
hyperboloidal data from Cauchy data and to obtain insight into the numericalconsequences of the obstructions to the smoothness of null inﬁnity discussed
in the later sections of Chapter 20. At the time of writing, the analysis has
been restricted to the analysis of the spin-2 ﬁeld equation on a Minkowski
background – in the spirit of Section 20.3.4 – with the expectation that thissituation contains the essential diﬃculties in the implementation; see Beyer et al.
(2012) and Frauendiener and Hennig (2014).
Foliations of spacetimes by means of hyperboloidal hypersurfaces have beenused in numerical simulations aimed at analysing radiative processes in gravitational collapse and perturbations of black hole spacetimes; see, for example,
Zenginoglu (2008, 2011a,b), Rinne (2010, 2014), Zenginoglu and Kidder (2010)

21.4 Computer algebra

567

Figure 21.2 The Schwarzschild spacetime in a conformal Gaussian gaugesystem; see the discussion in the main text and compare also Section 20.3.3.
From Figure 3.3, page 57, of A. Zenginoglu, A conformal approach to numericalcomputations of asymptotically simple spacetimes, PhD thesis, University of
Postdam (2006). Reproduced courtesy of the author.

and Rinne and Moncrief (2013). These numerical investigations make use offormulations of the Einstein ﬁeld equations alternative to the ones discussed
in this book. Finally, hyperboloidal foliations have also been used in theimplementation of fully spectral (i.e. in time and space) evolution schemes forvarious ﬁelds; see Hennig and Ansorg (2009) and Macedo and Ansorg (2014).

21.4 Computer algebra
The analysis of hyperbolic reductions in Chapter 13 shows that despite theirelegance and appealing geometric nature, the study of the consequences of
the conformal ﬁeld equations requires a considerable amount of algebraic
manipulations. Modern computer algebra systems provide a natural way of
performing these manipulations in an eﬀective and eﬃcient way. At the timeof writing, the suite of packages xAct for Mathematica provides a robust
and versatile framework for the type of tensorial and spinorial manipulations
discussed in this book; see Martı́n-Garcı́a (2014). The packages in the suite xAct
allow one to perform tensorial and spinorial abstract index computations on
generic tensors as well as explicit component computations for a given metric.
At the core of xAct is a canonicalisation routine which allows one to simplify
large tensorial and spinorial expressions by identifying “dummy” indices andexploiting the symmetries of the various objects involved. In addition, xAct
allows one to carry out cumbersome operations such as the decomposition of
spinors into irreducible parts. An additional appeal of this system is that it

568

Perspectives

provides its output in standard index notation. The system also provides facilities
to produce Latex output of the calculations
The capabilities of modern computers have reached the point that, for example, using xAct, it is possible to perform certain types of analyses which would
have been impractically long otherwise. As an example, the study of asymptotic
expansions using the framework of the cylinder at spatial inﬁnity describedin Chapter 20 and reported in Valiente Kroon (2004a,b,c, 2005) depended, ina crucial manner, on computer algebra calculations. These calculations werecarried out with purpose-built routines in the computer algebra system Maple V.
21.5 Concluding remarks
This book has discussed a particular approach to the use of conformal methods inmathematical general relativity. Clearly, the approach presented is not the only
one possible nor are the potential applications restricted to the ones discussed
in these pages. It constitutes a body of work extending over a period of morethan 30 years starting with the work of H. Friedrich in the early 1980s – or
50 years if one considers the seminal work by R. Penrose in the 1960s. Thisextended period of time is proof of the vitality of the subject. Nevertheless,a more exacting assessment of its vitality and relevance should come from its
inﬂuence in the whole of mathematical general relativity and its ability to foster
new ideas and research problems. Time will be the ultimate judge on this matter.
This book is an attempt to bring to the fore the relevance of conformal methods
in modern research in general relativity and to make the subject as accessible
as possible to those interested in using these ideas in their own research. The
reader is left to decide whether this goal has been achieved.

References

Abraham, R., and Robbin, J. 1967. Transversal mapping and ﬂows. W. A. Benjamin.
Aceña, A. 2009. Convergent null data expansions at spacelike inﬁnity of stationary
vacuum solutions. Ann. Henri Poincaré, 10, 275.
Aceña, A., and Valiente Kroon, J. A. 2011. Conformal extensions for stationary
spacetimes. Class. Quantum Grav., 28, 225023.
Alcubierre, M. 2008. Introduction to 3+1 numerical relativity. Oxford University Press.
Ambrosetti, A., and Prodi, G. 1995. A primer of nonlinear analysis. Cambridge
University Press.
Anderson, M. T. 2000. On stationary vacuum solutions to the Einstein equations. Ann.
Henri Poincaré 1, 977.
Anderson, M. T. 2005a. Existence and stability of even dimensional asymptotically deSitter spaces. Ann. Henri Poincaré, 6, 801.
Anderson, M. T. 2005b. Geometric aspects of the AdS/CFT correspondence. Page 1 of:
Biquard, O. (ed), AdS/CFT correspondence: Einstein metrics and their conformalboundaries. Euro. Math. Soc. Zürich.
Anderson, M. T. 2006. On the uniqueness and global dynamics of AdS spacetimes.
Class. Quantum Grav., 23, 6935.
Anderson, M. T., and Chruściel, P. T. 2005. Asymptotically simple solutions of thevacuum Einstein equations in even dimensions. Comm. Math. Phys., 260, 557.
Anderson, M. T., Chruściel, P. T., and Delay, E. 2002. Non-trivial static, geodesicallycomplete vacuum spacetimes with a negative cosmological constant. JHEP, 10, 6.
Andersson, L. 2006. On the relation between mathematical and numerical relativity.
Class. Quantum Grav., 23, S307.
Andersson, L., and Chruściel, P. T. 1993. Hyperboloidal Cauchy data for vacuum
Einstein equations and obstructions to smoothness of null inﬁnity. Phys. Rev. Lett.,
70, 2829.
Andersson, L., and Chruściel, P. T. 1994. On “hyperboloidal” Cauchy data for vacuum
Einstein equations and obstructions to smoothness of scri. Comm. Math. Phys.,
161, 533.
Andersson, L., Chruściel, P. T., and Friedrich, H. 1992. On the regularity of solutions
to the Yamabe equation and the existence of smooth hyperboloidal initial data for
Einstein’s ﬁeld equations. Comm. Math. Phys., 149, 587.
Anguige, K., and Tod, K. P. 1999a. Isotropic cosmological singularities I. Polytropic
perfect ﬂuid spacetimes. Ann. Phys., 276, 257.
Anguige, K., and Tod, K. P. 1999b. Isotropic cosmological singularities II. The EinsteinVlasov system. Ann. Phys., 276, 294.
Appel, W. 2007. Mathematics for physics and physicists. Princeton University Press.
Arnowitt, R., Deser, S., and Misner, C. W. 1962. The dynamics of general relativity.
Page 227 of: Witten, L. (ed), Gravitation: an introduction to current research. John
Wiley & Sons.

570

References

Arnowitt, R., Deser, S., and Misner, C. 2008. Reprint of: The dynamics of generalrelativity. Gen. Rel. Grav., 40, 1997.
Ashtekar, A. 1980. Asymptotic structure of the gravitational ﬁeld at spatial inﬁnity.
Page 37 of: Held, A. (ed), General relativity and gravitation: one hundred years
after the birth of Albert Einstein, vol. 2. Plenum Press.
Ashtekar, A. 1984. Asymptotic properties of isolated systems: recent developments.
Page 37 of: Bertotti, B., de Felice, F., and Pascolini, A. (eds), General relativity
and gravitation. D. Reidel Publishing Company.
Ashtekar, A. 1987. Asymptotic quantization. Bibliopolis.
Ashtekar, A. 1991. Lectures on non-perturbative canonical gravity. World Scientiﬁc.
Ashtekar, A. 2014. Geometry and physis of null inﬁnity. Page 99 of: Bieri, L., andYau, S.-T. (eds), One hundred years of general relativity. Surveys in Diﬀerential
Geometry, vol. 20. International Press.
Ashtekar, A., and Das, S. 2000. Asymptotically anti-de Sitter spacetimes: conserved
quantities. Class. Quantum Grav., 17, L17.
Ashtekar, A., and Dray, T. 1981. On the existence of solutions to Einstein’s ﬁeld
equations with non-zero Bondi news. Comm. Math. Phys., 79, 581.
Ashtekar, A., and Hansen, R. O. 1978. A uniﬁed treatment of null and spatial inﬁnity
in general relativity. I. Universal structure, asymptotic symmetries, and conserved
quantities at spatial inﬁnity. J. Math. Phys., 19, 1542.
Ashtekar, A., and Magnon, A. 1984. Asymptotically anti-de Sitter space-times. Class.
Quantum Grav., 1, L39.
Ashtekar, A., and Schmidt, B. G. 1980. Null inﬁnity and Killing ﬁelds. J. Math. Phys.,21, 862.
Ashtekar, A., and Xanthopoulos, B. C. 1978. Isometries compatible with asymptotic
ﬂatness at null inﬁnity: a complete description. J. Math. Phys., 19, 2216.
Ashtekar, A., Horowitz, G. T., and Magnon-Ashtekar, A. 1982. A generalization of
tensor calculus and its applications to physics. Gen. Rel. Grav., 14, 411.
Aubin, T. 1976. Équations diﬀérentielles non linéaires et le problème de Yamabe
concernant la courbure scalaire. J. Math. Pures Appl., 55, 269.
Bäckdahl, T. 2007. Axisymmetric stationary solutions with arbitrary multipole
moments. Class. Quantum Grav., 24, 2205.
Bäckdahl, T. 2009. Relating the Newman-Penrose constants to the Geroch-Hansen
multipole moments. Class. Quantum Grav., 26, 231102.
Bäckdahl, T., and Herberthson, M. 2005a. Explicit multipole moments of stationary
axisymmetric spacetimes. Class. Quantum Grav., 22, 3585.
Bäckdahl, T., and Herberthson, M. 2005b. Static axisymmetric space-times withprescribed multipole moments. Class. Quantum Grav., 22, 1607.
Bäckdahl, T., and Herberthson, M. 2006. Calculations of, and bounds for, the multipole
moments of stationary spacetimes. Class. Quantum Grav., 23, 5997.
Bäckdahl, T., and Valiente Kroon, J. A. 2010a. Geometric invariant measuring thedeviation from Kerr data. Phys. Rev. Lett., 104, 231102.
Bäckdahl, T., and Valiente Kroon, J. A. 2010b. On the construction of a geometricinvariant measuring the deviation from Kerr data. Ann. Henri Poincaré, 11, 1225.
Bartnik, R. 1986. The mass of an asymptotically ﬂat manifold. Comm. Pure Appl.
Math., 661.
Bartnik, R., and Isenberg, J. 2004. The constraint equations. Page 1 of: Chruściel,
P. T., and Friedrich, H. (eds), The Einstein equations and the large scale behaviourof gravitational ﬁelds. Birkhauser.
Baston, R. J., and Mason, L. J. 1987. Conformal gravity, the Einstein equation andspaces of complex null geodesics. Class. Quantum Grav., 4, 815.

References

571

Baumgarte, T. W., and Shapiro, S. L. 2010. Numerical relativity: solving Einstein’s
equations on the computer. Cambridge University Press.
Beig, R. 1984. Integration of Einstein’s equations near spatial inﬁnity. Proc. Roy. Soc.
Lond. A, 391, 295.
Beig, R. 1985. A remarkable property of spherical harmonics. J. Math. Phys., 26, 769.
Beig, R. 1991. Conformal properties of static spacetimes. Class. Quantum Grav., 8, 263.
Beig, R., and Heinzle, J. M. 2005. CMC-slicings of Kottler-Schwarzschild-de Sitter
Cosmologies. Comm. Math. Phys., 260, 673.
Beig, R., and Husa, S. 1994. Initial data for general relativity with toroidal conformal
symmetry. Phys. Rev. D, 50, R7116.
Beig, R., and O’Murchadha, N. 1991. Trapped surfaces due to concentration ofgravitational radiation. Phys. Rev. Lett., 66, 2421.
Beig, R., and O’Murchadha, N. 1994. Trapped surfaces in vacuum spacetimes. Class.
Quantum Grav., 11, 419.
Beig, R., and O’Murchadha, N. 1996. The momentum constraints of general relativity
and spatial conformal isometries. Comm. Math. Phys., 176, 723.
Beig, R., and O’Murchadha, N. 1998. Late time behaviour of the maximal slicing of
the Schwarzschild black hole. Phys. Rev. D, 57, 4728.
Beig, R., and Schmidt, B. G. 1982. Einstein’s equation near spatial inﬁnity. Comm.
Math. Phys., 87, 65.
Beig, R., and Schmidt, B. G. 2000. Time-independent gravitational ﬁelds. Page 325
of: Schmidt, B. G. (ed), Einstein’s ﬁeld equations and their physical implications.
Springer.
Beig, R., and Simon, W. 1980a. Proof of a multipole conjecture due to Geroch. Comm.
Math. Phys., 75, 79.
Beig, R., and Simon, W. 1980b. The stationary gravitational ﬁeld near spatial inﬁnity.
Gen. Rel. Grav., 12, 1003.
Beig, R., and Simon, W. 1981. On the multipole expansions for stationary space-times.
Proc. Roy. Soc. Lond. A, 376, 333.
Beig, R., and Szabados, L. B. 1997. On a global invariant of initial data sets. Class.
Quantum Grav., 14, 3091.
Bekenstein, J. 1974. Exact solutions of Einstein-conformal scalar equations. Ann. Phys.,
82, 535.
Bernal, A. N., and Sánchez, M. 2007. Globally hyperbolic spacetimes can be deﬁned as‘causal’ instead of strongly causal. Class. Quantum Grav., 24, 745.
Besse, A. L. 2008. Einstein manifolds. Springer Verlag.
Beyer, F. 2007. Asymptotics and singularities in cosmological models with positive
cosmological constant. PhD thesis, University of Potsdam.
Beyer, F. 2008. Investigations of solutions of Einstein’s ﬁeld equations close to lambdaTaub-NUT. Class. Quantum Grav., 25, 235005.
Beyer, F. 2009a. Non-genericity of the Nariai solutions: I. Asymptotics and spatiallyhomogeneous perturbations. Class. Quantum Grav., 26, 235015.
Beyer, F. 2009b. Non-genericity of the Nariai solutions: II. Investigations within the
Gowdy class. Class. Quantum Grav., 26, 235016.
Beyer, F. 2009c. A spectral solver for evolution problems with spatial S3-topology.
J. Comput. Phys., 228, 6496.
Beyer, F., Doulis, G., Frauendiener, J., and Whale, B. 2012. Numerical space-times
near space-like and null inﬁnity. The spin-2 system on Minkowski space. Class.
Quantum Grav., 29, 245013.

572

ReferencesBičák, J. 2000. Selected solutions of Einstein’s ﬁeld equations: their role in generalrelativity and astrophysics. In: Schmidt, B. G. (ed), Einstein ﬁeld equations andtheir physical implications (selected essays in honour of Juergen Ehlers). Springer
Verlag.
Bičák, J., and Krtouš, P. 2001. Accelerated sources in de Sitter spacetime and the
insuﬃciency of retarded ﬁelds. Phys. Rev. D, 64, 124020.
Bičák, J., and Krtouš, P. 2002. The ﬁelds of uniformly accelerated charges in de Sitter
spacetime. Phys. Rev. Lett., 88, 211101.
Bičák, J., and Schmidt, B. G. 1989. Asymptotically ﬂat radiative space-times withboost-rotation symmetry: the general structure. Phys. Rev. D, 40, 1827.
Bičák, J., Scholtz, M., and Tod, K. P. 2010. On asymptotically ﬂat solutions ofEinstein’s equations periodic in time: II. Spacetimes with scalar-ﬁeld sources. Class.
Quantum Grav., 27, 175011.
Bizon, P. 2013. Is AdS stable? Gen. Rel. Grav., 46, 1724.
Bizon, P., and Friedrich, H. 2012. A remark about wave equations on the extreme
Reissner-Nordström black hole exterior. Class. Quantum Grav., 30, 065001.
Bizon, P., and Rostworowski, A. 2011. Weakly turbulent instability of anti-de Sitter
spacetime. Phys. Rev. Lett., 107, 031102.
Bondi, H., van der Burg, M. G. J., and Metzner, A. W. K. 1962. Gravitational waves
in general relativity VII. Waves from axi-symmetric isolated systems. Proc. Roy.
Soc. Lond. A, 269, 21.
Brännlund, J. 2004. Conformal isometry of the Reissner-Nordström-de Sitter Black
Hole. Gen. Rel. Grav., 36, 883.
Branson, T., Eastwood, M., and Wang, M. 2004. Conformal geometry. In
www.birs.ca/workshops/2004/04w5006/report04w5006.pdf.
Brill, D.R., and Hayward, S. A. 1994. Global structure of a black hole cosmos and its
extremes. Class. Quantum Grav., 11, 359.
Buchdahl, H. A. 1959. Reciprocal static metrics and scalar ﬁelds in the general theoryof relativity. Phys. Rev., 115, 1325.
Butscher, A. 2002. Exploring the conformal constraint equations. Page 195 of:Frauendiener, J., and Friedrich, H. (eds), The conformal structure of spacetime:geometry, analysis, numerics. Lect. Notes. Phys. Springer.
Butscher, A. 2007. Perturbative solutions of the extended constraint equations ingeneral relativity. Comm. Math. Phys., 272, 1.
Cagnac, F. 1981. Problème de Cauchy sur un conoı̈de caractéristique pour des equations
quasi-lineaires. Ann. Mat. Pure Appl., 129, 13.
Carmeli, M. 1977. Group theory and general relativity: representations of the Lorentz
group and their applications to the gravitational ﬁeld. McGraw-Hill.
Carter, B. 1966a. The complete analytic extension of the Reissner-Nordström metricin the special case e2 = m2 . Phys. Rev. Lett., 21, 423.
Carter, B. 1966b. Complete analytic extension of the symmetry axis of Kerr’s solutionof Einstein’s equations. Phys. Rev., 141, 1242.
Carter, B. 1968. Global structure of the Kerr family of gravitational ﬁelds. Phys. Rev.,
174, 1559.
Carter, B. 1971. Causal structure in space-time. Gen. Rel. Grav., 1, 349.
Carter, B. 1973. Black hole equilibrium states. Page 61 of: DeWitt, C., and DeWitt,
B.S. (eds), Black holes: les astres occlus. Gordon and Breach.
Chaljub-Simon, A. 1982. Decomposition of the space of covariant two-tensors on R3 .
Gen. Rel. Grav., 14, 743.

References

573

Chaljub-Simon, A., and Choquet-Bruhat, Y. 1980. Global solutions of the Licnerowicz
equation in general relativity on an asymptotically Euclidean manifold. Gen. Rel.
Grav., 12, 175.
Chang, S.-Y. A., Qing, J., and Yang, P. 2007. Some progress in conformal geometry.
Sigma, 3, 122.
Choptuik, M. W. 1993. Universality and scaling in gravitational collapse of a masslessscalar ﬁeld. Phys. Rev. Lett., 70, 9.
Choquet-Bruhat, I., Isenberg, J., and York, J. W. Jr. 2000. Einstein constraints on
asymptotically Euclidean manifolds. Phys. Rev. D, 61, 084034.
Choquet-Bruhat, Y. 2007. Results and open problems in mathematical generalrelativity. Milan J. Math., 75, 273.
Choquet-Bruhat, Y. 2008. General relativity and the Einstein equations. Oxford
University Press.
Choquet-Bruhat, Y., and Christodoulou, D. 1981. Existence of global solutions of theYang-Mills, Higgs and spinor ﬁeld equations in 3+1 dimensions. Ann. Sci. l’E.N.S.,
14, 481.
Choquet-Bruhat, Y., and Geroch, R. 1969. Global aspects of the Cauchy problem ingeneral relativity. Comm. Math. Phys., 14, 329.
Choquet-Bruhat, Y., and York, J. W. Jr. 1980. The Cauchy problem. In: Held, A. (ed),General relativity and gravitation, Vol. I. Plenum Press.
Choquet-Bruhat, Y., Dewitt-Morette, C., and Dillard-Bleck, M. 1982. Analysis,
manifolds and physics: part I. North Holland Publishing Company.
Choquet-Bruhat, Y., Chruściel, P. T., and Martı́n-Garcı́a, J. M. 2011. The Cauchy
problem on a characteristic cone for the Einstein equations in arbitrary dimensions.
Ann. Henri Poincaré, 12, 419.
Christodoulou, D. 1986. The problem of a self-gravitating scalar ﬁeld. Comm. Math.
Phys., 105, 337.
Christodoulou, D., and Klainerman, S. 1993. The global nonlinear stability of the
Minkowski space. Princeton University Press.
Christodoulou, D., and O’Murchadha, N. 1981. The boost problem in general relativity.
Comm. Math. Phys., 80, 271.
Chruściel, P. T. 1991. On the uniqueness in the large of solutions of Einstein’s equations(“Strong Cosmic Censorship”). Centre for Mathematics and Its Applications,
Australian National University.
Chruściel, P. T., and Delay, E. 2002. Existence of non-trivial, vacuum, asymptoticallysimple spacetimes. Class. Quantum Grav., 19, L71.
Chruściel, P. T., and Delay, E. 2003. On mapping properties of the general relativistic
constraint operator in weighted function spaces, with applications. Mem. Soc.
Math. France, 94, 1.
Chruściel, P. T., and Delay, E. 2009. Gluing constructions for asymptotically hyperbolic
manifolds with constant scalar curvature. Comm. Anal. Geom., 17, 343.
Chruściel, P. T., and Paetz, T.-T. 2012. The many ways of the characteristic Cauchy
problem. Class. Quantum Grav., 29, 145006.
Chruściel, P. T., and Paetz, T.-T. 2013. Solutions of the vacuum Einstein equations
with initial data on past null inﬁnity. Class. Quantum Grav., 30, 235037.
Chruściel, P. T., MacCallum, M. A. H., and Singleton, D. B. 1995. Gravitational waves
in general relativity XIV. Bondi expansions and the “polyhomogeneity” of I . Phil.
Trans. Roy. Soc. Lond. A, 350, 113.
Chruściel, P. T., Ölz, C. R., and Szybka, S. J. 2012a. Space-time diagrammatics. Phys.
Rev. D, 86, 124041.

574

References

Chruściel, P. T., and Costa, J. L., and Heusler, M. 2012b. Stationary black holes:
uniqueness and beyond. Living Rev. Relativity 15, 7. URL (cited on 24 May 2016):
www.livingreviews.org/lrr-2012-7.
Cocke, W. J. 1989. Table for constructing spin coeﬃcients in general relativity. Phys.
Rev. D, 40, 650.
Cook, G. B. 2000. Initial data for numerical relativity. Living Rev. Relativity, 3, 5. URL(cited on 24 May 2016): www.livingreviews.org/lrr-2000-5.
Corvino, J. 2000. Scalar curvature deformations and a gluing construction for the
Einstein constraint equations. Comm. Math. Phys., 214, 137.
Corvino, J. 2007. On the existence and stability of the Penrose compactiﬁcation. Ann.
Henri Poincaré, 8, 597.
Corvino, J., and Schoen, R. 2006. On the asymptotics for the Einstein constraint
vacuum equations. J. Diﬀ. Geom., 73, 185.
Cotton, É. 1899. Sur les variétés à trois dimensions. Ann. Fac. Sci. Toulouse 2e série,
1, 385.
Couch, W. E., and Torrence, R. J. 1984. Conformal invariance under spatial inversion
of extreme Reissner-Nordström black holes. Gen. Rel. Grav., 16, 789.
Courant, R., and Hilbert, D. 1962. Methods of mathematical physics. Vol. II. John
Wiley & Sons.
Courant, R., and John, F. 1989. Introduction to calculus and analysis II/1. Springer.
Cutler, C. 1989. Properties of spacetimes that are asymptotically ﬂat at timelike
inﬁnity. Class. Quantum Grav., 6, 1075.
Cutler, C., and Wald, R. M. 1989. Existence of radiating Einstein-Maxwell solutions
which are C ∞ on all of I + and I − . Class. Quantum Grav., 6, 453.
Dafermos, M. 2003. Stability and instability of the Cauchy horizon for the sphericallysymmetric Einstein-Maxwell-scalar ﬁeld equations. Ann. Math., 158, 875.
Dafermos, M. 2005. The interior of charged black holes and the problem of uniquenessin general relativity. Comm. Pure Appl. Math., 58, 0445.
Dafermos, M., and Rodnianski, I. 2005. A proof of Price’s law for the collapse of aself-gravitating scalar ﬁeld. Invent. Math., 162, 381.
Dafermos, M., and Rodnianski, I. 2010. Lectures on black holes and linear waves.
Page 97 of: Ellwood, D., Rodnianski, I., Staﬃlani, G., and Wunsch, J. (eds), Evolution equations. Clay Mathematics Proceedings, Vol. 17. American MathematicalSociety-Clay Mathematics Institute.
Dain, S. 2001a. Initial data for a head-on collision of two Kerr-like black holes withclose limit. Phys. Rev. D, 64, 124002.
Dain, S. 2001b. Initial data for stationary spacetimes near spacelike inﬁnity. Class.
Quantum Grav., 18, 4329.
Dain, S. 2001c. Initial data for two Kerr-like black holes. Phys. Rev. Lett., 87, 121102.
Dain, S. 2006. Elliptic systems. Page 117 of: Frauendiener, J., Giulini, D., andPerlick, V. (eds), Analytical and numerical approaches to general relativity. Lect.
Notes. Phys., vol. 692. Springer Verlag.
Dain, S., and Friedrich, H. 2001. Asymptotically ﬂat initial data with prescribed
regularity at inﬁnity. Comm. Math. Phys., 222, 569.
Dain, S., and Gabach-Clement, M. E. 2011. Small deformations of extreme Kerr blackhole initial data. Class. Quantum Grav., 28, 075003.
Damour, T., and Schmidt, B. 1990. Reliability of perturbation theory in generalrelativity. J. Math. Phys., 31, 2441.
Dossa, M. 1986. Solution globale d’un probléme de Cauchy caractéristique non linéare.
Comptes Rendus de l’Academie des Sciences. Series I, 303, 795.

References

575

Dossa, M. 1997. Espaces de Sobolev non isotropes à poids et problémes de Cauchy
quasi-linéaires sur un conoı̈de caractéristique. Ann. Inst. H. Poincaré Phys. Théor.,
1, 37.
Dossa, M. 2002. Solutions C ∞ d’une classe de problèmes de Cauchy quasi-linéaires
hyperboliques du second ordre sur un conoı̈de caractéristique. Annales de la faculté
des sciences de Toulouse Sér. 6, 11, 351.
Eastwood, M. 1996. Notes on conformal diﬀerential geometry. Page 57 of: Slovák, J.
(ed), Proceedings of the 15th Winter School “Geometry and Physics”. Rendicontidel Circolo Matematico de Palermo, Serie II, vol. Supplemento No. 43. Circolo
Matematico di Palermo.
Ehlers, J. 1973. Spherically symmetric spacetimes. Page 114 of: Israel, W. (ed),Relativity, astrophysics and cosmology. D. Reidel Publishing Company.
Ellis, G. F. R. 1984. Relativistic cosmology: its nature, aims and problems. In:
Bertotti, B., de Felice, F., and Pascolini, A. (eds), General relativity andgravitation. D. Reidel Publishing Company.
Ellis, G. F. R. 2002. The state of cosmology 2001: two views and a middle way. In:
Bishop, N. T., and Maharaj, S. D. (eds), General relativity and gravitation. World
Scientiﬁc.
Ellis, G. F. R., and van Elst, H. 1998. Cosmological models: Cargese lectures 1998.
NATO Adv. Study Inst. Ser. C. Math. Phys. Sci., 541, 1.
Ellis, G. F. R., Maartens, R., and MacCallum, M. A. H. 2012. Relativistic cosmology.
Cambridge University Press.
Estabrook, F., Wahlquist, H., Christensen, S., DeWitt, B., Smarr, L., and Tsiang, E.
1973. Maximally slicing a black hole. Phys. Rev. D, 7, 2814.
Evans, L. C. 1998. Partial diﬀerential equations. American Mathematical Society.
Exton, A. R., Newman, E. T., and Penrose, R. 1969. Conserved quantites in the
Einstein-Maxwell theory. J. Math. Phys., 10, 1566.
Feﬀerman, C., and Graham, C. R. 1985. Conformal invariants. Page 95 of: Élie Cartan
et les mathématiques d’aujord’hui. The mathematical heritage of Élie Cartan,Sémin. Lyon 1984. Astérisque, No. Hors Sér.
Feﬀerman, C., and Graham, C. R. 2012. The ambient metric. Annals of Mathematical
Studies, vol. 178. Princeton University Press.
Fischer, A. E., and Marsden, J. E. 1972. The Einstein evolution equations as a ﬁrstorder quasi-linear symmetric hyperbolic system. I. Comm. Math. Phys., 28, 1.
Fourès-Bruhat, Y. 1952. Théorème d’existence pour certains systèmes d’équations aux
derivées partielles non linéaires. Acta Mathematica, 88, 141.
Frances, C. 2005. The conformal boundary of anti de Sitter spacetimes. Page 205 of:Biquard, O. (ed), AdS/CFT correspondence: Einstein metrics and their conformalboundaries. Euro. Math. Soc. Zürich.
Frankel, T. 2003. The geometry of physics. Cambridge University Press.
Frauendiener, J. 1998a. Numerical treatment of the hyperboloidal initial value problem
for the vacuum Einstein equations. I. The conformal ﬁeld equations. Phys. Rev. D,
58, 064002.
Frauendiener, J. 1998b. Numerical treatment of the hyperboloidal initial value problem
for the vacuum Einstein equations. II. The evolution equations. Phys. Rev. D, 58,
064003.
Frauendiener, J. 2002. Some aspects of the numerical treatment of the conformal ﬁeld
equations. Page 261 of: Frauendiener, J., and Friedrich, H. (eds), The conformal
structure of space-time: geometry, analysis, numerics. Springer.
Frauendiener, J. 2004. Conformal inﬁnity. Living Rev. Relativity, 7, 1. URL (cited on
24 May 2016): www.livingreviews.org/lrr-2004-1.

576

References

Frauendiener, J., and Hein, M. 2002. Numerical evolution of axisymmetric, isolated
systems in general relativity. Phys. Rev. D, 66, 124004.
Frauendiener, J., and Hennig, J. 2014. Fully pseudospectral solution of the conformallyinvariant wave equation near the cylinder at spacelike inﬁnity. Class. Quantum
Grav., 31, 085010.
Frauendiener, J., and Sparling, G. A. 2000. Local twistors and the conformal ﬁeld
equations. J. Math. Phys., 41, 437.
Frauendiener, J., and Szabados, L. B. 2001. The kernel of the edth operators on highergenus spacelike 2-surfaces. Class. Quantum Grav., 18, 1003.
Friedrich, H. 1981a. The asymptotic characteristic initial value problem for Einstein’s
vacuum ﬁeld equations as an initial value problem for a ﬁrst-order quasilinear
symmetric hyperbolic system. Proc. Roy. Soc. Lond. A, 378, 401.
Friedrich, H. 1981b. On the regular and the asymptotic characteristic initial value
problem for Einstein’s vacuum ﬁeld equations. Proc. Roy. Soc. Lond. A, 375, 169.
Friedrich, H. 1982. On the existence of analytic null asymptotically ﬂat solutions ofEinstein’s vacuum ﬁeld equations. Proc. Roy. Soc. Lond. A, 381, 361.
Friedrich, H. 1983. Cauchy problems for the conformal vacuum ﬁeld equations inGeneral Relativity. Comm. Math. Phys., 91, 445.
Friedrich, H. 1984. Some (con-)formal properties of Einstein’s ﬁeld equations andconsequences. In: Flaherty, F. J. (ed), Asymptotic behaviour of mass and spacetime
geometry. Lecture notes in physics 202. Springer Verlag.
Friedrich, H. 1985. On the hyperbolicity of Einstein’s and other gauge ﬁeld equations.
Comm. Math. Phys., 100, 525.
Friedrich, H. 1986a. Existence and structure of past asymptotically simple solutionsof Einstein’s ﬁeld equations with positive cosmological constant. J. Geom. Phys.,
3, 101.
Friedrich, H. 1986b. On the existence of n-geodesically complete or future complete
solutions of Einstein’s ﬁeld equations with smooth asymptotic structure. Comm.
Math. Phys., 107, 587.
Friedrich, H. 1986c. On purely radiative space-times. Comm. Math. Phys., 103, 35.
Friedrich, H. 1988. On static and radiative space-times. Comm. Math. Phys., 119, 51.
Friedrich, H. 1991. On the global existence and the asymptotic behaviour of solutions
to the Einstein-Maxwell-Yang-Mills equations. J. Diﬀ. Geom., 34, 275.
Friedrich, H. 1992. Asymptotic structure of space-time. Page 147 of: Janis, A. I., andPorter, J. R. (eds), Recent advances in general relativity. Einstein Studies, vol. 4.
Birkhauser.
Friedrich, H. 1995. Einstein equations and conformal structure: existence of anti-deSitter-type space-times. J. Geom. Phys., 17, 125.
Friedrich, H. 1996. Hyperbolic reductions for Einstein’s equations. Class. Quantum
Grav., 13, 1451.
Friedrich, H. 1998a. Einstein’s equation and geometric asymptotics. Page 153 of:Dadhich, N., and Narlinkar, J. (eds), Proceedings of the GR-15 conference.
Inter-University Centre for Astronomy and Astrophysics.
Friedrich, H. 1998b. Evolution equations for gravitating ideal ﬂuid bodies in generalrelativity. Phys. Rev. D, 57, 2317.
Friedrich, H. 1998c. Gravitational ﬁelds near space-like and null inﬁnity. J. Geom.
Phys., 24, 83.
Friedrich, H. 1999. Einstein’s equation and conformal structure. Page 81 of: Huggett,
S. A., Mason, L. J., Tod, K. P., Tsou, S. T., and Woodhouse, N. M. J. (eds),
The geometric universe: science, geometry and the work of Roger Penrose. Oxford
University Press.

References

577

Friedrich, H. 2002. Conformal Einstein evolution. Page 1 of: Frauendiener, J., andFriedrich, H. (eds), The conformal structure of spacetime: geometry, analysis,numerics. Lecture Notes in Physics. Springer.
Friedrich, H. 2003a. Conformal geodesics on vacuum spacetimes. Comm. Math. Phys.,
235, 513.
Friedrich, H. 2003b. Spin-2 ﬁelds on Minkowski space near space-like and null inﬁnity.
Class. Quantum Grav., 20, 101.
Friedrich, H. 2004. Smoothness at null inﬁnity and the structure of initial data. In:
Chruściel, P. T., and Friedrich, H. (eds), 50 years of the Cauchy problem in generalrelativity. Birkhausser.
Friedrich, H. 2005. On the non-linearity of the subsidiary systems. Class. Quantum
Grav., 22, L77.
Friedrich, H. 2007. Static vacuum solutions from convergent null data expansions atspace-like inﬁnity. Ann. Henri Poincaré, 8, 817.
Friedrich, H. 2008a. Conformal classes of asymptotically ﬂat, static vacuum data. Class.
Quantum Grav., 25, 065012.
Friedrich, H. 2008b. One-parameter families of conformally related asymptotically ﬂat,
static vacuum data. Class. Quantum Grav., 25, 135012.
Friedrich, H. 2009. Initial boundary value problems for Einstein’s ﬁeld equations andgeometric uniqueness. Gen. Rel. Grav., 41, 1947.
Friedrich, H. 2011. Yamabe numbers and the Brill-Cantor criterion. Ann. Inst. H.
Poincaré, 12, 1019.
Friedrich, H. 2013. Conformal structure of static vacuum data. Comm. Math. Phys.,321, 419.
Friedrich, H. 2014a. On the AdS stability problem. Class. Quantum Grav., 31, 105001.
Friedrich, H. 2014b. The Taylor expansion at past time-like inﬁnity. Comm. Math.
Phys., 324, 263.
Friedrich, H. 2015a. Geometric asymptotics and beyond. Page 37 of: Bieri, L., andYau, S.-T. (eds), One hundred years of general relativity. Surveys in Diﬀerential
Geometry, vol. 20. International Press.
Friedrich, H. 2015b. Smooth non-zero rest-mass evolution across time-like inﬁnity. Ann.
Henri Poincaré, 16, 2215.
Friedrich, H., and Kánnár, J. 2000a. Bondi-type systems near space-like inﬁnity andthe calculation of the NP-constants. J. Math. Phys., 41, 2195.
Friedrich, H., and Kánnár, J. 2000b. Calculating asymptotic quantities near space-like
inﬁnity and null inﬁnity from Cauchy data. Annalen Phys., 9, 321.
Friedrich, H., and Nagy, G. 1999. The initial boundary value problem for Einstein’s
vacuum ﬁeld equation. Comm. Math. Phys., 201, 619.
Friedrich, H., and Rendall, A. D. 2000. The Cauchy problem for the Einstein equations.
Lect. Notes. Phys., 540, 127.
Friedrich, H., and Schmidt, B.G. 1987. Conformal geodesics in general relativity. Proc.
Roy. Soc. Lond. A, 414, 171.
Friedrich, H., and Stewart, J. 1983. Characteristic initial data and wavefront singularities in general relativity. Proc. Roy. Soc. Lond. A, 385, 345.
Garabedian, P. R. 1986. Partial diﬀerential equations. AMS Chelsea Publishing.
Garcı́a, A. A., Hehl, F. W., Heinicke, C., and Macias, A. 2004. The Cotton tensor in
Riemannian spacetimes. Class. Quantum Grav., 21, 1099.
Garcı́a-Parrado, A., Gasperı́n, E., and Valiente Kroon, J.A. 2014. Conformal geodesics
in the Schwarzshild-de Sitter and Schwarzschild anti-de Sitter spacetimes. In
preparation.

578

References

Gasperı́n, E., and Valiente Kroon, J. A. 2015. Spinorial wave equations and the stabilityof the Milne universe. Class. Quantum Grav., 32, 185021.
Geroch, R. 1968. Spinor structure of spacetimes in general relativity I. J. Math. Phys.,
9, 1739.
Geroch, R. 1970a. Multipole moments. I. Flat space. J. Math. Phys., 11, 1955.
Geroch, R. 1970b. Multipole moments. II. Curved space. J. Math. Phys., 11, 2580.
Geroch, R. 1970c. Spinor structure of spacetimes in general relativity II. J. Math. Phys.,
11, 343.
Geroch, R. 1971a. A method for generating new solutions of Einstein’s equations. J.
Math. Phys., 12, 918.
Geroch, R. 1971b. Space-time structure from a global view point. Page 71 of: Sachs,
R. K. (ed), General relativity and cosmology. Proceedings of the International
School in Physics “Enrico Fermi”, Course 48. Academic Press.
Geroch, R. 1972a. A method for generating new solutions of Einstein’s equations. II.
J. Math. Phys., 13, 394.
Geroch, R. 1972b. Structure of the gravitational ﬁeld at spatial inﬁnity. J. Math. Phys.,
13, 956.
Geroch, R. 1976. Asymptotic structure of space-time. In: Esposito, E. P., and Witten, L.
(eds), Asymptotic structure of spacetime. Plenum Press.
Geroch, R., Held, A., and Penrose, R. 1973. A space-time calculus based on pairs ofnull directions. J. Math. Phys., 14, 874.
Gourgoulhon, E. 2012. 3 + 1 Formalism in general relativity: bases of numerical
relativity. Lect. Notes. Phys. 846. Springer Verlag.
Gowers, T. (ed). 2008. The Princeton companion to mathematics. Princeton University
Press.
Graham, C. R., and Hirachi, K. 2005. The ambient obstruction tensor and Q-curvature.
Page 59 of: AdS/CFT correspondence: Einstein metrics and their conformalboundaries. IRMA Lect. Math. Theor. Phys., vol. 8. Eur. Math. Soc. Zürich.
Griﬃths, J. B., and Podolský, J. 2009. Exact space-times in Einstein’s general relativity.
Cambridge University Press.
Guès, O. 1990. Problème mixte hyperbolique quasi-linéaire caracteristique. Comm.
Part. Diﬀ. Eqns., 15, 595.
Gundlach, C., and Martı́n-Garcı́a, J. M. 2007. Critical phenomena in gravitational collapse. Living Rev. Relativity, 10, 5. URL (cited on 24 May 2016):
www.livingreviews.org/lrr-2007-5.
Günther, P. 1975. Spinorkalkül und Normalkoordinaten. Z. Angew. Math. Mech.,55, 205.
Guven, J., and O’Murchadha, N. 1995. Constraints in spherically symmetric classicalgeneral relativity. I. Optical scalars, foliations, bounds on the conﬁguration spacevariables, and the positivity of the quasilocal mass. Phys. Rev. D, 52, 758.
Hamilton, R. 1982. The inverse function theorem of Nash and Moser. Bull. Am. Math.
Soc., 7, 65.
Hansen, R. O. 1974. Multipole moments of stationary spacetimes. J. Math. Phys.,
15, 46.
Hartman, P. 1987. Ordinary diﬀerential equations. SIAM.
Hawking, S. W., and Ellis, G. F. R. 1973. The large scale structure of space-time.
Cambridge University Press.
Henneaux, M., and Teitelboim, C. 1985. Asymptotically anti-de Sitter spaces. Comm.
Math. Phys., 98, 391.
Hennig, J., and Ansorg, M. 2009. A fully pseudospectral scheme for solving singular
hyperbolic equations. J. Hyp. Diﬀ. Eqns., 6, 161.

References

579

Herberthson, H. 2009. Static spacetimes with prescribed multipole moments: a proofof a conjecture by Geroch. Class. Quantum Grav., 26, 215009.
Herberthson, M., and Ludwig, G. 1994. Time-like inﬁnity and direction-dependent
metrics. Class. Quantum Grav., 11, 187.
Holst, M., Nagy, G., and Tsogtgerel, G. 2008a. Far-from-constant mean curvature
solutions of Einstein’s contraint equations with positive Yamabe metrics. Phys.
Rev. Lett., 100, 161101.
Holst, M., Nagy, G., and Tsogtgerel, G. 2008b. Rough solutions of the Einsteinconstraints on closed manifolds without near-CMC conditions. Comm. Math.
Phys., 288, 547.
Hübner, P. 1995. General relativistic scalar-ﬁeld models and asymptotic ﬂatness. Class.
Quantum Grav., 12, 791.
Hübner, P. 1999a. How to avoid artiﬁcial boundaries in the numerical calculation of
black hole spacetimes. Class. Quantum Grav., 16, 2145.
Hübner, P. 1999b. A scheme to numerically evolve data for the conformal Einsteinequation. Class. Quantum Grav., 16, 2823.
Hübner, P. 2001a. From now to timelike inﬁnity on a ﬁnite grid. Class. Quantum Grav.,
18, 1871.
Hübner, P. 2001b. Numerical calculation of conformally smooth hyperboloidal data.
Class. Quantum Grav., 18, 1421.
Hughes, T. J. R., Kato, T., and Marsden, J. E. 1977. Well-posed quasi-linear secondorder hyperbolic systems with applications to nonlinear elastodynamics andgeneral relativity. Arch. Ration. Mech. Anal., 63, 273.
Husa, S. 2002. Problems and successes in the numerical approach to the conformal ﬁeld
equations. Page 239 of: Frauendiener, J., and Friedrich, H. (eds), The conformal
structure of space-time: geometry, analysis, numerics. Springer.
Ionescu, A. D., and Klainerman, S. 2009a. On the uniqueness of smooth, stationary
black holes in vacuum. Inventiones mathematicae, 175, 35.
Ionescu, A. D., and Klainerman, S. 2009b. Uniqueness results for ill-posed characteristic
problems in curved spacetimes. Comm. Math. Phys., 285, 873.
Isenberg, J. 1995. Constant mean curvature solutions of the Einstein constraint
equations on closed manifolds. Class. Quantum Grav., 12, 2249.
Isenberg, J. 2013. Initial value problem in general relativity. In: Ashtekar, A., andPetkov, V. (eds), The Springer handbook of spacetime. Springer Verlag.
Ishibashi, A., and Wald, R. M. 2004. Dynamics in non-globally-hyperbolic spacetimes
III: anti de Sitter spacetime. Class. Quantum Grav., 21, 2981.
Jaramillo, J. L., Valiente Kroon, J.A., and Gourgoulhon, E. 2008. From geometry to
numerics: interdisciplinary aspects in mathematical and numerical relativity. Class.
Quantum Grav., 25, 093001.
Kánnár, J. 1996a. Hyperboloidal initial data for the vacuum ﬁeld equations withcosmological constant. Class. Quantum Grav., 13, 3075.
Kánnár, J. 1996b. On the existence of C∞ solutions to the asymptotic characteristic
initial value problem in general relativity. Proc. Roy. Soc. Lond. A, 452, 945.
Kato, T. 1975a. The Cauchy problem for quasi-linear symmetric hyperbolic systems.
Arch. Ration. Mech. Anal., 58, 181.
Kato, T. 1975b. Quasi-linear equations of evolution, with applications to partial
diﬀerential equations. Lect. Notes Math., 448, 25.
Kenneﬁck, D. 2007. Traveling at the speed of thought: Einstein and the quest for
gravitational waves. Princeton University Press.
Kenneﬁck, D., and O’Murchadha, N. 1995. Weakly decaying asymptotically ﬂat staticand stationary solutions to the Einstein equations. Class. Quantum Grav., 12, 149.

580

References

Klainerman, S. 1984. The null condition and global existence to nonlinear wave
equations. Lect. Appl. Math. 23, 293.
Klainerman, S. 2008. Partial diﬀerential equations. Page 455 of: Gowers, T., BarrowGreen, J., and Leader, I. (eds), The Princeton companion to mathematics.
Princeton University Press.
Kobayashi, S. 1995. Transformation groups in diﬀerential geometry. Springer Verlag.
Kobayashi, S., and Nomizu, K. 2009. Foundations of diﬀerential geometry. Vol. 1. Wiley.
Kodaira, K. 1986. Complex manifolds and deformation of complex structures. Springer
Verlag.
Kozameh, C., Newman, E. T., and Tod, K. P. 1985. Conformal Einstein spaces. Gen.
Rel. Grav., 17, 343.
Krantz, S. G. 2006. Geometric function theory. Birkhäuser.
Kreiss, H.-O., and Lorenz, J. 1998. Stability for time-dependent diﬀerential equations.
Acta Numerica, 7(203).
Kreiss, H.-O., Reula, O., Sarbach, O., and Winicour, J. 2009. Boundary conditions forcoupled quasilinear wave equations with applications to isolated systems. Comm.
Math. Phys., 289, 1099.
Kulkarni, R. S., and Pinkall, U. (eds). 1988. Conformal geometry. Aspects ofMathematics. Friedrich Vieweg & Sohn.
Kundu, P. 1981. On the analyticity of stationary gravitational ﬁelds at spacial inﬁnity.
J. Math. Phys., 22, 2006.
Künzle, H. P. 1967. Construction of singularity free spherically symmetric spacetime
manifolds. Proc. Roy. Soc. Lond. A, 297, 244.
Lee, J. M. 1997. Riemannian manifolds: an introduction to curvature. Springer Verlag.
Lee, J. M. 2000. Introduction to topological manifolds. Springer Verlag.
Lee, J. M. 2002. Introduction to smooth manifolds. Springer Verlag.
Lee, J. M., and Parker, T. H. 1987. The Yamabe problem. Bull. Am. Math. Soc., 17, 37.
Lehner, L., and Pretorious, F. 2014. Numerical relativity and astrophysics. Ann. Rev.
Astron. Astrophys., 52, 661.
Lübbe, C. 2014. Conformal scalar ﬁelds, isotropic singularities and conformal cyclic
cosmologies. In arXIv:1312.2059.
Lübbe, C., and Valiente Kroon, J. A. 2009. On de Sitter-like and Minkowski-likespacetimes. Class. Quantum Grav., 26, 145012.
Lübbe, C., and Valiente Kroon, J. A. 2010. A stability result for purely radiative
spacetimes. J. Hyp. Diﬀ. Eqns., 7, 545.
Lübbe, C., and Valiente Kroon, J. A. 2012. The extended conformal Einstein ﬁeld
equations with matter: the Einstein-Maxwell system. J. Geom. Phys., 62, 1548.
Lübbe, C., and Valiente Kroon, J. A. 2013a. A class of conformal curves in the ReissnerNordström spacetime. Ann. Henri Poincaré, 15, 1327.
Lübbe, C., and Valiente Kroon, J. A. 2013b. A conformal approach for the analysis of
the non-linear stability of pure radiation cosmologies. Ann. Phys., 328, 1.
Lübbe, C., and Valiente Kroon, J. A. 2014. On the conformal structure of the extremal
Reissner-Nordström spacetime. Class. Quantum Grav., 31, 175015.
Ludvigsen, M., and Vickers, J. A. G. 1981. The positivity of the Bondi mass. J. Phys. A,
14, L389.
Ludvigsen, M., and Vickers, J. A. G. 1982. A simple proof of the positivity of the Bondi
mass. J. Phys. A, 15, L67.
Macedo, R. P., and Ansorg, M. 2014. Axisymmetric fully spectral code for hyperbolic
equations. In arXiv:1402.7343.
Machado, M. P., and Vickers, J. A. G. 1995. A space-time calculus invariant under nullrotations. Proc. Roy. Soc. Lond. A, 450, 1.

References

581

Machado, M. P., and Vickers, J. A. G. 1996. Invariant diﬀerential operators and the
Karlhede classiﬁcation of type N vacuum solutions. Class. Quantum Grav., 13,1589.
Maldacena, J. 1998. The large N limit of superconformal ﬁeld theories and supergravity.
Adv. Theor. Math. Phys., 2, 231.
Maliborski, M., and Rostworowski, A. 2013. Lecture notes on turbulent instability ofanti-de Sitter spacetime. J. Mod. Phys. A, 28, 1340020.
Martı́n-Garcı́a, J. M. 2014. www.xact.es.
Mason, L. J. 1986. The conformal Einstein equations. Twistor Newsletter, 22, 41.
Mason, L. J. 1995. The vacuum and Bach equations in terms of light cone cuts. J.
Math. Phys., 36, 3704.
Miao, P. 2003. On the existence of static metric extensions in general relativity. Comm.
Math. Phys., 241, 27.
Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1973. Gravitation. W. H. Freeman.
Morrey, C. B. 1958. On the analyticity of the solutions of analytic non-linear elliptic
systems of partial diﬀerential equations. Am. J. Math., 80, 198.
Morris, M. S., and Thorne, K. S. 1988. Wormholes in spacetime and their use forinterstellar travel: a tool for teaching general relativity. Am. J. Phys., 56, 395.
Müller zu Hagen, H. 1970. On the analyticity of stationary vacuum solutions ofEinstein’s equation. Proc. Camb. Phil. Soc., 68, 199.
Müller zu Hagen, H., and Seifert, H.-J. 1977. On characteristic initial-value and mixed
problems. Gen. Rel. Grav., 8, 259.
Newman, E. T., and Penrose, R. 1962. An approach to gravitational radiation by amethod of spin coeﬃcients. J. Math. Phys., 3, 566.
Newman, E. T., and Penrose, R. 1963. Errata: an approach to gravitational radiation
by a method of spin coeﬃcients. J. Math. Phys., 4, 998.
Newman, E. T., and Penrose, R. 1965. 10 exact gravitationally-conserved quantities.
Phys. Rev. Lett., 15, 231.
Newman, E. T., and Penrose, R. 1966. Note on the Bondi-Metzner-Sachs group. J.
Math. Phys., 7, 863.
Newman, E. T., and Penrose, R. 1968. New conservation laws for zero rest-mass ﬁelds
in asymptotically ﬂat space-time. Proc. Roy. Soc. Lond. A, 305, 175.
Newman, E. T., and Tod, K. P. 1980. Asymptotically ﬂat space-times. In: Held, A.
(ed), General relativity and gravitation: one hundred years after the birth of Albert
Einstein. Plenum.
Newman, R. P. A. C. 1989. The global structure of simple space-times. Comm. Math.
Phys., 123, 17.
Nicolas, J.-P. 2015. The conformal approach to asymptotic analysis. In
arXiv:1508.02592.
O’Donnell, P. 2003. Introduction to 2-spinors in general relativity. World Scientiﬁc.
Ogiue, K. 1967. Theory of conformal connections. Kodai Math. Sem. Rep., 19, 193.
O’Murchadha, N. 1988. The Yamabe problem and general relativity. Proc. Centre Math.
Anal. (A.N.U.), 19(137).
O’Neill, B. 1983. Semi-Riemannian geometry with applications to relativity. Academic
Press.
O’Neill, B. 1995. The geometry of Kerr black holes. A. K. Peters.
Paetz, T.-T. 2015. Conformally covariant systems of wave equations and theirequivalence to Einstein’s ﬁeld equations. Ann. Henri Poincaré, 16, 2059.
Penrose, R. 1960. A spinor approach to general relativity. Ann. Phys. (New York),
10, 171.

582

References

Penrose, R. 1963. Asymptotic properties of ﬁelds and space-times. Phys. Rev. Lett.,
10, 66.
Penrose, R. 1964. Conformal approach to inﬁnity. In: DeWitt, B. S., and DeWitt, C. M.
(eds), Relativity, groups and topology: the 1963 Les Houches lectures. Gordon andBreach.
Penrose, R. 1965. Zero rest-mass ﬁelds including gravitation: asymptotic behaviour.
Proc. Roy. Soc. Lond. A, 284, 159.
Penrose, R. 1967. Structure of space-time. Page 121 of: DeWitt, C. M., and Wheeler,
J. A. (eds), Battelle rencontres: 1967 lectures in mathematics and physics. W. A.
Benjamin.
Penrose, R. 1969. Gravitational collapse: the role of general relativity. Rev. Nuovo
Cimento, I, 257.
Penrose, R. 1979. Singularities and time asymmetry. Page 581 of: Hawking, S. W.,
and Israel, W. (eds), General relativity: an Einstein centenary survey. Cambridge
University Press.
Penrose, R. 1980. Null hypersurface initial data for classical ﬁelds of arbitrary spin andfor general relativity. Gen. Rel. Grav., 12, 225.
Penrose, R. 1983. Spinors and torsion in general relativity. Found. Phys., 13, 325.
Penrose, R. 2002. Gravitational collapse: the role of general relativity. Gen. Rel. Grav.,
34, 1141.
Penrose, R. 2011. Reprint of: Conformal treatment of inﬁnity. Gen. Rel. Grav., 43, 901.
Penrose, R., and Rindler, W. 1984. Spinors and space-time. Vol. 1: Two-spinor calculus
and relativistic ﬁelds. Cambridge University Press.
Penrose, R., and Rindler, W. 1986. Spinors and space-time. Vol. 2: Spinor and twistor
methods in space-time geometry. Cambridge University Press.
Persides, S. 1979. A deﬁnition of asymptotically Minkowskian space-times. J. Math.
Phys., 20, 1731.
Persides, S. 1980. Structure of the gravitational ﬁeld at spatial inﬁnity. II. Asymptotically Minkowski spaces. J. Math. Phys., 21, 142.
Persides, S. 1982a. Timelike inﬁnity. J. Math. Phys., 23, 283.
Persides, S. 1982b. A uniﬁed formulation of timelike, null and spatial inﬁnity. J. Math.
Phys., 23, 289.
Petersen, P. 1991. Riemannian geometry. Graduate Texts in Mathematics, vol. 171.
Springer.
Poisson, E., and Will, C. M. 2015. Gravity: Newtonian, post-Newtonian, relativistic.
Cambridge University Press.
Porrill, J. 1982. The structure of timelike inﬁnity for isolated systems. Proc. Roy. Soc.
Lond. A, 381, 323.
Pretorious, F. 2009. Binary black hole coalescence. Page 305 of: Colpi, M., Casella, P.,
Gorini, V., Moschella, U., and Possenti, A. (eds), Physics of relativistic objects incompact binaries: from birth to coalescence. Springer.
Pugliese, D., and Valiente Kroon, J. A. 2012. On the evolution equations for ideal
magnetohydrodynamics in curved spacetime. Gen. Rel. Grav., 44, 2785.
Pugliese, D., and Valiente Kroon, J. A. 2013. On the evolution equations for a selfgravitating charged scalar ﬁeld. Gen. Rel. Grav., 45, 1247.
Quevedo, H. 1990. Multipole moments in general relativity: static and stationary
vacuum solutions. Fortsch. der Physik, 38, 733.
Reinhart, B. L. 1973. Maximal foliations of extended Schwarzschild space. J. Math.
Phys., 14, 719.
Reiris, M. 2014a. Stationary solutions and asymptotic ﬂatness I. Class. Quantum Grav.,
31, 155012.

References

583

Reiris, M. 2014b. Stationary solutions and asymptotic ﬂatness II. Class. Quantum
Grav., 31, 155013.
Rendall, A. D. 1990. Reduction of the characteristic initial value problem to the Cauchy
problem and its application to the Einstein equations. Proc. Roy. Soc. Lond. A,
427, 221.
Rendall, A. D. 2005. Theorems on existence and global dynamics for the Einstein equations. Living Rev. Relativity, 8, 6. URL (cited on 24 May 2016):
www.livingreviews.org/lrr-2005-6.
Rendall, A. D. 2008. Partial diﬀerential equations in general relativity. Oxford
University Press.
Reula, O. 1989. On existence and behaviour of asymptotically ﬂat solutions to thestationary Einstein equations. Comm. Math. Phys., 122, 615.
Reula, O. 1998. Hyperbolic methods for Einstein’s equations. Living Rev. Rel., 3, 1.
Ringström, H. 2009. The Cauchy problem in general relativity. Eur. Math. Soc. Zürich.
Rinne, O. 2010. An axisymmetric evolution code for the Einstein equations on
hyperboloidal slices. Class. Quantum Grav., 27, 035014.
Rinne, O. 2014. Formation and decay of Einstein-Yang-Mills black holes. Phys. Rev. D,
90, 124084.
Rinne, O., and Moncrief, V. 2013. Hyperboloidal Einstein-matter evolution and tails
for scalar and Yang–Mills ﬁelds. Class. Quantum Grav., 30, 095009.
Rodnianski, I., and Speck, J. 2013. The nonlinear future-stability of the FLRW familyof solutions to the Euler-Einstein system with a positive cosmological constant. J.
Eur. Math. Soc., 15, 2369.
Sachs, R. K. 1962a. Asymptotic symmetries in gravitational theory. Phys. Rev., 128,
2851.
Sachs, R. K. 1962b. Gravitational waves in general relativity VIII. Waves inasymptotically ﬂat space-time. Proc. Roy. Soc. Lond. A, 270, 103.
Sachs, R. K. 1962c. On the characteristic initial value problem in gravitational theory.
J. Math. Phys., 3, 908.
Sbierski, J. 2013. On the existence of a maximal Cauchy development for the Einsteinequations: a dezorniﬁcation. Ann. Henri Poincaré: Online First.
Schmidt, B., Walker, M., and Sommers, P. 1975. A characterization of the BondiMetzner-Sachs group. Gen. Rel. Grav., 5, 489.
Schmidt, B. G. 1978. Asymptotic structure of isolated systems. Page 11 of: Ehlers, J.
(ed), Isolated systems in general relativity, Proceedings of the International Schoolof Physics “Enrico Fermi”, Course 67. North Holland Publishing Company.
Schmidt, B. G. 1981. The decay of the gravitational ﬁeld. Comm. Math. Phys., 78, 447.
Schmidt, B. G. 1986. Conformal geodesics. Lect. Notes. Phys., 261, 135.
Schmidt, B. G. 1987. Gravitational radiation near spatial and null inﬁnity. Proc. Roy.
Soc. Lond. A, 410, 201.
Schmidt, B. G. 1996. Vacuum space-times with toroidal null inﬁnities. Class. Quantum
Grav., 13, 2811.
Schmidt, B. G., and Walker, M. 1983. Analytic conformal extensions of asymptoticallyﬂat spacetimes. J. Phys. A: Math. Gen., 16, 2187.
Schoen, R. 1984. Conformal deformation of a Riemannian metric to constant scalar
curvature. J. Diﬀ. Geom., 20, 479.
Schoen, R., and Yau, S. T. 1979. On the proof of the positive mass conjecture in generalrelativity. Comm. Math. Phys., 65, 45.
Schouten, J. A. 1921. Über die konforme Abbildung n-dimensionaler Mannigfaltigkeiten
mit quadratischer Massbestimmung auf eine Mannigfaltigkeit mit euklidischer
Massbestimmung. Math. Zeitschrift, 11, 58.

584

References

Schwarzschild, K. 1916. Über das Gravitationsfeld eines Massenpunktes nach derEinsteinschen Theorie. Sitz. Preuss. Akad. Wiss. Berlin, 7, 189–196.
Schwarzschild, K. 2003. “Golden oldie”: on the gravitational ﬁeld of a mass pointaccording to Einstein’s theory. Gen. Rel. Grav., 35, 951.
Sen, A. 1981. On the existence of neutrino “zero-modes” in vacuum spacetimes. J.
Math. Phys., 22, 1781.
Sexl, R. U., and Urbantke, H. K. 2000. Relativity, groups, particles: special relativity
and relativistic symmetry in ﬁeld and particle physics. Springer.
Shapiro, I. 1999. A century of relativity. Rev. Mod. Phys., 71, S41.
Shinbrot, M., and Welland, R. R. 1976. The Cauchy-Kowaleskaya. J. Math. An. App.,55, 757.
Simon, W. 1992. Radiative Einstein-Maxwell spacetimes and “no-hair” theorems. Class.
Quantum Grav., 9, 241.
Sommers, P. 1980. Space spinors. J. Math. Phys., 21, 2567.
Speck, J. 2012. The nonlinear future-stability of the FLRW family of solutions to theEuler-Einstein system with a positive cosmological constant. Selecta Mathematica,
18, 633.
Spivak, M. 1970. A comprehensive introduction to diﬀerential geometry. Vol. I. Publish
or Perish.
Stephani, H., Kramer, D., MacCallum, M. A. H., Hoenselaers, C., and Herlt, E. 2003.
Exact solutions of Einstein’s ﬁeld equations, 2nd edition. Cambridge University
Press.
Stewart, J. 1991. Advanced general relativity. Cambridge University Press.
Synge, J. L. 1960. Relativity: the general theory. North Holland Publishing Company.
Szabados, L. B. 1994. Two-dimensional Sen connections in general relativity. Class.
Quantum Grav., 11, 1833.
Szabados, L. B. 2009. Quasi-local energy-momentum and angular momentum inGeneral Relativity. Living Rev. Relativity, 12, 4. URL (cited on 24 May 2016):
www.livingreviews.org/lrr-2009-4.
Szegö, G. 1978. Orthogonal polynomials. AMS Colloq. Pub., vol. 23. AMS.
Szekeres, P. 1965. The gravitational compass. J. Math. Phys., 6, 1387.
Tataru, D. 2004. The wave maps equation. Bull. Am. Math. Soc., 41, 185.
Taubes, C. H. 2011. Diﬀerential geometry. Oxford University Press.
Taylor, M. E. 1996a. Partial diﬀerential equations I: basic theory. Springer Verlag.
Taylor, M. E. 1996b. Partial diﬀerential equations II: qualitative studies of linear
equations. Springer Verlag.
Taylor, M. E. 1996c. Partial diﬀerential equations III: nonlinear equations. Springer
Verlag.
Tod, K. P. 1984. Three-surface twistors and conformal embedding. Gen. Rel. Grav.,
16, 435.
Tod, K. P. 2002. Isotropic cosmological singularities. Page 123 of: Frauendiener, J.,
and Friedrich, H. (eds), The conformal structure of space-time: geometry, analysis,numerics. Lect. Notes. Phys. 604 Springer.
Tod, K. P. 2003. Isotropic cosmological singularities: other matter models. Class.
Quantum Grav., 20, 251.
Tod, K. P. 2012. Some examples of the behaviour of conformal geodesics. J. Geom.
Phys., 62, 1778.
Torres del Castillo, G. F. 2003. 3-D spinors, spin-weighted functions and theirapplications. Birkhäuser.
Trautman, A. 1958. Radiation and boundary conditions in the theory of gravitation.
Bull. Academ. Pol. Sciences: Series des Sciences Math. Astron. Phys., 6, 407.

References

585

Trudinger, N. 1968. Remarks concerning the conformal deformation of Riemannian
structures on compact manifolds. Ann. Sc. Norm. Sup. Pisa, 22, 265.
Valiente Kroon, J. A. 1998. Conserved quantities for polyhomogeneous spacetimes.
Class. Quantum Grav., 15, 2479.
Valiente Kroon, J. A. 1999a. A comment on the outgoing radiation condition and thepeeling theorem. Gen. Rel. Grav., 31, 1219.
Valiente Kroon, J. A. 1999b. Logarithmic Newman-Penrose constants for arbitrary
polyhomogeneous spacetimes. Class. Quantum Grav., 16, 1653.
Valiente Kroon, J. A. 2002. Polyhomogeneous expansions close to null and spatial
inﬁnity. Page 135 of: Frauendiner, J., and Friedrich, H. (eds), The conformal
structure of spacetimes: geometry, numerics, analysis. Lecture Notes in Physics.
Springer.
Valiente Kroon, J. A. 2003. Early radiative properties of the developments of timesymmetric conformally ﬂat initial data. Class. Quantum Grav., 20, L53.
Valiente Kroon, J. A. 2004a. Does asymptotic simplicity allow for radiation near spatial
inﬁnity? Comm. Math. Phys., 251, 211.
Valiente Kroon, J. A. 2004b. A new class of obstructions to the smoothness of nullinﬁnity. Comm. Math. Phys., 244, 133.
Valiente Kroon, J. A. 2004c. Time asymmetric spacetimes near null and spatial inﬁnity.
I. Expansions of developments of conformally ﬂat data. Class. Quantum Grav., 23,
5457.
Valiente Kroon, J. A. 2005. Time asymmetric spacetimes near null and spatial inﬁnity.
II. Expansions of developments of initial data sets with non-smooth conformal
metrics. Class. Quantum Grav., 22, 1683.
Valiente Kroon, J. A. 2007a. Asymptotic properties of the development of conformallyﬂat data near spatial inﬁnity. Class. Quantum Grav., 24, 3037.
Valiente Kroon, J. A. 2007b. The Maxwell ﬁeld on the Schwarzschild spacetime:
behaviour near spatial inﬁnity. Proc. Roy. Soc. Lond. A, 463, 2609.
Valiente Kroon, J. A. 2009. Estimates for the Maxwell ﬁeld near the spatial and nullinﬁnity of the Schwarzschild spacetime. J. Hyp. Diﬀ. Eqns., 6, 229.
Valiente Kroon, J. A. 2010. A rigidity property of asymptotically simple spacetimes
arising from conformally ﬂat data. Comm. Math. Phys., 298, 673.
Valiente Kroon, J. A. 2011. Asymptotic simplicity and static data. Ann. Henri
Poincaré, 13, 363.
Valiente Kroon, J. A. 2012. Global evaluations of static black hole spacetimes. In
preparation.
Wald, R. M. 1983. Asymptotic behaviour of homogeneous cosmological models in thepresence of a positive cosmological constant. Phys. Rev. D, 28, 2118.
Wald, R. M. 1984. General relativity. University of Chicago Press.
Walker, M. 1970. Block diagrams and the extension of timelike two-surfaces. J. Math.
Phys., 11, 2280.
Weyl, H. 1918. Reine Inﬁnitesimalgeometrie. Math. Zeitschrift, 2, 404.
Weyl, H. 1968. Zur Inﬁnitesimal Geometrie: Einordnung der projektiven und derkonformen Auﬀassung. Page 195 of: Gesammelte Abhandlungen., vol. II. Springer
Verlag.
Will, C. M. 2014. The confrontation between general relativity and experiment. Living.
Rev. Relativity, 17.
Willmore, T. J. 1993. Riemannian geometry. Oxford University Press.
Winicour, J. 2012. Characteristic evolution and matching. Living. Rev. Relativity, 14.
Witten, E. 1998. Anti de Sitter space and holography. Adv. Theor. Math. Phys., 2, 253.

586

References

Witten, E., and Yau, S.-T. 1999. Connectedness of the boundary in the AdS/CFT
correspondence. Adv. Theor. Math. Phys., 1635.
Wong, W. W.-Y. 2013. A comment on the construction of the maximal globally
hyperbolic Cauchy development. J. Math. Phys., 54, 113511.
Yamabe, H. 1960. On a deformation of Riemannian structures on compact manifolds.
Osaka J. Mathematics, 12, 6126.
York, J. W. Jr. 1971. Gravitational degrees of freedom and the initial-value problem.
Phys. Rev. Lett., 26, 1656.
York, J. W. Jr. 1972. Role of conformal three-geometry in the dynamics of gravitation.
Phys. Rev. Lett., 28, 1082.
York, J. W. Jr. 1973. Conformally covariant orthogonal decomposition of symmetrictensor on Riemannian manifolds and the initial value problem of general relativity.
J. Math. Phys., 14, 456.
Zenginoglu, A. 2006. A conformal approach to numerical calculations of asymptoticallyﬂat spacetimes. PhD thesis, Max-Planck Institute for Gravitational Physics (AEI)and University of Potsdam.
Zenginoglu, A. 2007. Numerical calculations near spatial inﬁnity. J. Phys. Conf. Ser.,66, 012027.
Zenginoglu, A. 2008. A hyperboloidal study of tail decay rates for scalar and Yang-Mills
ﬁelds. Class. Quantum Grav., 25, 195025.
Zenginoglu, A. 2011a. A geometric framework for black hole perturbations. Phys.
Rev. D, 83, 127502.
Zenginoglu, A. 2011b. Hyperboloidal layers for hyperbolic equations on unboundeddomains. J. Comput. Phys., 230, 2286.
Zenginoglu, A., and Kidder, L. E. 2010. Hyperboloidal evolution of test ﬁelds in threespatial dimensions. Phys. Rev. D, 81, 124010.
Zhiren, J. 1988. A counterexample to the Yamabe problem for complete non-compact
manifolds. Lect. Notes Math., 1306, 93.

IndexAB -operator, 89
2-sphere, standard metric, 142
3-sphere, 142
North and South Poles, 142
acceleration vector, 106
achronal set, 392
ADM-angular momentum, 273
ADM-linear momentum, 273
ADM-mass, 273
algebraic curvature, 195, 203, 264
spinorial counterpart, 198, 207
anti-de Sitter spacetime, 159
behaviour of geodesics, 160
conformal geodesics, 162
instability, 475
anti-de Sitter-like spacetimes, 17, 223, 454
boundary adapted gauge, 460, 463
conformal geodesics, 457
covariant formulation of boundary data,
468
formulation of an initial boundary value
problem, 460
identiﬁcation of boundary conditions, 466
initial data, 286
local existence, 455, 467, 468
asymptopia, 12, 452
asymptotic characteristic initial value
problems, see characteristic initialvalue problems
asymptotic distance functions, 274
asymptotic Einstein condition, 224
asymptotic end, 271
asymptotic initial value problem, 419
asymptotic Killing vectors, 239
asymptotic simplicity, 178, see
asymptotically simple spacetimes, 222
asymptotic symmetries, 239
asymptotically Cartesian coordinates, 271
asymptotically empty and simple spacetimes,
179
asymptotically Euclidean manifolds, 271
asymptotically Euclidean and regularmanifolds, 274, 507
conformal gauge freedom, 274
asymptotically simple spacetimes, 13, 178

existence, 555
global hyperbolicity of, 394
related deﬁnitions, 181
remarks concerning, 179
atlas, 28
maximal, 28
Bach tensor, 473
Banach spaces, 306
basis
dual, 31orientation, 32
orthonormal, 45
spin, 66
Bianchi identity
boundary adapted evolution system, 375,
548
constraint equations, 351
electric-magnetic parts decomposition, 370
ﬁrst, 41
hyperbolic reduction, 350
second, 42
Weyl connection, 120
standard evolution system, 351, 373
subsidiary evolution equations, 362, 379
third, 473
Birkhoﬀ theorem, 163
black holes, 163
uniqueness theorems, 163
BMS group, 238
Bondi coordinates, 236
Bondi mass, 238, 438
boost-rotation symmetric spacetimes, 181
boundary, of a manifold, 29
boundary deﬁning function, 285
bundlescotangent, 34
tangent, 34
tensor, 34
Cartan’s formula, 355, 389
Cauchy development, 6, 393, 398
maximal, 7
Cauchy horizon, 394, 448
Cauchy hypersurface, 7, 393
Cauchy-Kowalewskaya theorem, 522, 526

588

Index

Cauchy stability, 16, 309
causal future/past, 391
causality, 8
for symmetric hyperbolic systems, 296
Causality Principle, 4caustic, see conformal geodesic
characteristic initial value problems, 16, 319
asymptotic, 16, 477
on a cone, 496
local existence, 491, 501
standard, 478
constraint equations, 478, 486
freely speciﬁable data, 478, 490, 500
interior equations, 321, 478, 485, 486, 489
reduced conformal ﬁeld equations, 490
reduction to a Cauchy problem, 324
transport equations, 478
transverse equations, 485
well- and ill-posed problems, 320
characteristic polynomial, 297
characteristic set, 297
characteristics, 9, 297
charts, 27chronological future/past, 391
classical determinism, 7
Codazzi-Mainardi equation, 62, 254
relation to the conformal constraints, 262
commutation coeﬃcients, 339
commutator, 34
Jacobi identity, 35
compact 3-manifolds, 268
complex analytic extensions, 520, 532
complex null cone, 521
computer algebra, 567
concomitants, 48
cone past boundary, 497
conformal boundary, 10
conformal constraints on, 266
conformal class, 11, 113conformal compactiﬁcation, 114
conformal covariance, 11
conformal decompactiﬁcation, 280
conformal Einstein constraint equations, 259
basic properties, 263
relation to the Einstein constraints, 265
conformal Einstein ﬁeld equations, 13
frame formulation, 195
gauge freedom, 193, 200
metric formulation, 185, 190
relation to the Einstein ﬁeld equations,
191, 196, 199
solutions to, 191, 196
spinorial formulation, 198
vacuum, 191
as wave equations, 384
conformal extension, 9, 114
conformal factor
evolution equations for concomitants, 352
massive part, 530

massless part, 530
conformal ﬂatness, 118conformal gauge source functions, see gaugesource functions
conformal Gaussian coordinates, 367
conformal Gaussian gauge systems, 331, 366
model equations, 368
conformal geodesic, 14
canonical factor associated to, 132
caustics, 136
as conformal invariant, 128
deﬁnition, 127g̃-adapted equations, 134
physical acceleration, 134
reparametrisation, 129
spiralling, 135transformation under change of
connection, 127conformal geodesic deviation equation, 135g̃-adapted form, 137conformal geometry, 10
conformal group, 114
conformal inﬁnity, 179
conformal invariance, 11
conformal Killing operator, 257
conformal Killing vectors, 239
conformal normal gauge, 276
conformal rescaling, 9, 112conformal static ﬁeld equations, 507, 511
conformal structure, see conformal classconformal transformation, 113conformally invariant scalar ﬁeld, see scalarﬁeld
conformomorphism, see conformaltransformation
congruence, 55
acceleration, 57
expansion, 57
twist, 57
connection, 38
coeﬃcients, 51
spatial, 109
coeﬃcients for a metric connection, 52
Levi-Civita, 46
metric, 46
Weyl, 120constraint equations, see Einstein constraint
equationscontracted Christoﬀel symbols, 387, 513
coordinate gauge source functions, see gaugesource functions
corner conditions, 316, 467
cosmic censorship, 7, 400
cosmic no-hair conjecture, 561, 566
cosmological models, 561
cosmology, 2
Cotton spinor, 512
Cotton tensor, 116, 510
rescaled, see rescaled Cotton tensor

Index
Cotton-York tensor, 118, 188, 471
3-dimensional, 118
covariant derivative, 38
commutator, 39
modiﬁed commutator, 40
covariant directional derivative, 51
covector, 31
critical sets, 542
curvature, 40
3-dimensional Riemann tensor, 60
Riemann tensor, 40
frame components, 53
spinorial counterpart, 84
transformation under conformal
rescalings, 124
curvecomplete integral, 55
deﬁnition, 30
extendible/inextendible, 30
integral, 55
proper time parametrisation, 45
cuts, of null inﬁnity, 225
cylinder at spatial inﬁnity, 541
basic representation, 542
horizontal representation, 542
Minkowski spacetime, 543
Schwarzschild spacetime, 543
static solutions, 553
structural properties, 547
transport equations, 548, 550
cylindrical asymptotic end, 273
D’Alambertian operator, 89
de Sitter spacetime, 155
behaviour of geodesics, 157
compactiﬁcation into the Einstein
cylinder, 157
in conformal Gaussian gauge, 412
conformal geodesics, 158
gauge source functions, 411
Gaussian coordinates, 409
as a solution to the conformal ﬁeld
equations, 408
de Sitter-like spacetimes, 16, 223
a priori structure of the conformalboundary, 426
asymptotic initial data, 419
evolution equations, 421, 429
global existence and stability results, 407,
425, 430
standard Cauchy initial data, 416
derivations, 30
deviation covector, 135g̃-adapted form, 137
deviation vector, 43, 135g̃-adapted form, 137
diﬀeomorphism, 27
diﬀerential, 31
diﬀerential manifolds, see manifolds589

diﬀerential operators, 252
Dirac delta, 279, 293
direct sum, 56
directional covariant derivative, 82
space spinors, 109
distribution, 55
divergence theorem, 278, 293
domain, 28
of dependence, 304, 393
dual space, 31edge, 313, 392
Einstein constraint equations, 5, 254
on compact manifolds, 268
conformal gauge freedom, 258
Einstein cosmos, 144conformal geodesics, 144
Einstein cylinder, see Einstein Cosmos
Einstein ﬁeld equations, 1
vacuum, 2
Einstein spaces, 2
Einstein static universe, see Einstein Cosmos
Einstein tensor, 49
Einstein-Rosen bridge, 271
Einstein’s summation convention, 31, 32
electric-magnetic parts decomposition, 249
frame expressions, 251
electromagnetic ﬁeld, see Maxwell ﬁeld
ellipticity, 252
overdetermined, 252
undetermined, 252
embedding, 54, 247
energy density, 254
unphysical, 255
energy ﬂux vector, 254
unphysical, 255
energy-momentum tensor
physical, 186, 211
unphysical, 186, 211
equivalent multipole sequences, 519
Euclidean metric, deﬁnition, 47
exact solutions, 3
expanding Einstein cylinder, 145, 438
exponential map, 44, 275
extended conformal Einstein ﬁeld equations,14, 201
conformal covariance, 205
evolution system, 370
frame formulation, 204
relation to the Einstein ﬁeld equations, 208
spinor formulation, 208
subsidiary equations, 376
extended Einstein constraint equations, 286
extension, of smooth functions, 292, 307
exterior asymptotic gluing, 288, 452, 556
extrinsic curvature, 61
conformal transformation formulae, 248

590

Index

Faraday tensor, 213
auxiliary ﬁelds, 214dual, 213
self-dual, 213
spinorial counterpart, 215
unphysical, 214
ﬁrst fundamental form, see intrinsic metricfoliation, 54
formal adjoint of a diﬀerential operator, 253
formally regular equations, 185
frame gauge source functions, see gaugesource functions
Frobenius’s theorem, 55, 119
Fuchsian equations, 562
Fundamental Theorem of Riemannian
Geometry, 46
future null cone, 497
gauge freedom, 4, 14
gauge source functions, 331, 336
conformal, 193, 347, 483
coordinates, 338, 387
frames, 344
a model equation, 337
Gauss-Codazzi equation, 62, 254
relation to the conformal constraints, 262
generalised wave coordinates, 339, 386
generators, of null inﬁnity, 225, 448
geodesic balls, 275
geodesic deviation equation, 43
geodesic distance, 276
geodesics, 43
as conformal geodesics, 130congruence of, 43transformation under conformal rescaling,126
geometric curvature, 194, 203, 264
spinorial counterpart, 197, 206geometric uniqueness, 6, 400, 475
global hyperbolicity, 7
gravitational radiation, 237
Hadamard’s parametrix construction, 509
Hamiltonian constraint, 254conformal, 255
harmonic coordinates, 287, 387, 512
Hermitian conjugation, 96
Hermitian inner product, 94, 295
Hodge dual, 50, 250
left, right, 50
Hölder condition, 512, 526
hyperbolic equations, 4
hyperbolic reductions, 4, 331
hyperboloidal initial data set
deﬁnition, 284
existence, 285
hyperboloidal manifolds, 284
hyperboloids, 18, 154
hyperplane, 55
hypersurface, 54

incoming radiation, 237
indices
abstract index notation, 35
index-free notation, 35
lowering and raising, 44
spacetime frame, 32
Infeld-van der Waerden symbols, 74
spatial, 99
initial boundary value problems, 313
existence, 318
geometric uniqueness, 475
uniqueness, 316
initial data sets, 255
initial value problems, 4, 296
Cauchy, 4, 298
characteristic, 298
regular at spatial inﬁnity, 538
interior equations, 297
intrinsic metric, 57
isolated systems, 2, 182
isometries, 44
isotropic coordinates, 273
isotropic cosmological singularities, 561
Jacobi identity, 68
Jacobi ordinary diﬀerential equation, 552
Jacobi polynomials, 552
Jacobian, 28
Kerr spacetime, non-linear stability, 562
Killing horizons, 167
Killing vector, 504
Koszul formula, 46
Kronecker’s delta, 31
2-dimensional, 68
Laplace operator, see Laplacian
Laplacian, 253
lapse, 55
leave, 54
lens-shaped domain, 301
Licnerowicz equation, 256
Licnerowicz’s theorem, 507
Licnerowicz-York conformal method,255
Lie derivative, 37
linear connection, see connection
linearised Ricci operator, 289
local coordinates, 28
Lorentz group, relation to spinors, 73
Lorentz transformations, 80, 114
Lorentzian causality, 8, 390, 448
manifolds
with boundary, 29
deﬁnition, 27
dimension, 28
Husdorﬀ, 29
integrable, 55

Indexorientable, 29paracompact, 29
map, 36mass positivity theorem, 531
massless initial data sets, 531
construction, 535
evolution, 536
maximal analytic extension, 4, 400
maximal Cauchy development, 400
maximal hypersurfaces, 249
maximally dissipative boundary conditions,
315, 466
maximum principle, 268, 283
Maxwell equations, 213
behaviour under conformal rescaling, 215
hyperbolic reductions, 332
propagation of the constraints, 335
spinorial constraint equations, 333
spinorial evolution equations, 333
spinorial version, 215
as a symmetric hyperbolic system, 333
Maxwell ﬁeld, 213
energy-momentum tensor, 213, 215
Maxwell spinor, 215
mean curvature, 249
mean value theorem, 302, 329
metric, 44
conformal/conformally related, 113conformally ﬂat, 113, 118
Lorentzian, 44
Riemannian, 44
negative-deﬁnite, 44
signature, 44
metric tensor, see metric
Minkoswki metric, deﬁnition, 47
Minkowski spacetime, 145
behaviour of geodesics, 149
compactiﬁcation into the Einstein
cylinder, 147conformal geodesics, 151
in the Einstein cylinder, 153
future and past null inﬁnity, 148
future and past timelike inﬁnity, 148
hyperboloidal data, 442
null coordinates, 146
Penrose diagram, 148semiglobal existence and stability, 437, 450
spatial inﬁnity, 148
standard hyperboloids, 154, 440
standard metric, 145translated hyperboloids, 155
Minkowski-like spacetimes, 13, 18, 223
a priori structure of the conformalboundary, 444
general properties of null inﬁnity, 225
hyperboloidal initial data, 442
timelike inﬁnity, 447
momentum constraint, 254conformal, 255

591

Monge cone, see characteristic set
multipole moments, 518
multipole theorem, 519
Nariai spacetime, 181, 566
Newman-Penrose constants, 240
Newman-Penrose formalism, 91
Newman-Penrose gauge, 231
news function, 237
non-trivial topology, 271
normal, 54, 248
of a foliation, 55
normal coordinates, 275, 499
normal frame, 499
normal gauge, 499
normal matrix, 314, 466, 548
normal neighbourhoods, 275
normal vector, 54
null cone, 45, 496
null inﬁnity, 10
adapted frames, 232
future, 225
general properties, 225
obstructions to the smoothness, 556
past, 225
null tetrads, 77
null vectors
future pointing, 76
past pointing, 76
numerical relativity, 564
O(3) transformation, 100, 114
order symbols, 223, 291
orthogonal transformation, see O(3)transformation
orthogonal vectors, 44
outgoing radiation, see radiation ﬁeld
parallel propagation, 43
Pauli matrices, 75
peeling behaviour, 180, 452, 554
peeling theorem, 230
Penrose diagrams, 10
Eddington-Finkelstein coordinates, 170
elementary blocks, 167
ﬂipping blocks, 169
gluing blocks, 169
Kruskal-like coordinates, 171
seams, 167
Penrose’s proposal, 13, 182
perfect ﬂuids, 219
barotropic equations of state, 220
density, 220energy-momentum tensor, 220
equation of energy conservation, 220
equations of motion, 220
pressure, 220
radiation equation of state, 220

592

Index

perturbations, 401
Petrov type, 230
physical spacetime, 184
point compactiﬁcation, 143, 274
points at inﬁnity, 274
regularity, 277
polar coordinates, 514, 538
positive deﬁnite matrices, 308
principal part, of a diﬀerential operator, 252,
295
projector, 55, 249
spinorial counterpart, 103
propagation, of the constraints, 6, 331, 354
pull-backs, 37
pure radiation problem, 497
purely radiative spacetimes, 453, 496, 531,
536
push-forwards, 36
quasilinear diﬀerential operators, 253
quasilinear evolution equations, 294
radiation ﬁeld, 237, 500
radiativity condition, see regularity condition
rays, see generators, of null inﬁnity
reduced Einstein ﬁeld equations, 388
reﬂective boundary conditions, 473
regularity condition, 518, 533–535
Reissner-Nordström spacetime
extremal, 172, 174
non-extremal, 173
rescaled Cotton tensor, 189, 212
spinorial decomposition, 198
rescaled Weyl spinor, 198
rescaled Weyl tensor, 11, 188
Riccati system, 487
Ricci scalar, 48, 110
Ricci tensor, 48
3-dimensional, 60
spinorial counterpart, 88
Riemann mapping theorem, 226
Riemann tensor, see curvaturespinorial counterpart, 84
right-handedness, 77, 78
sameness map, 67
scalar ﬁeld, 30, 216
auxiliary ﬁelds, 218
conformally invariant, 216
energy-momentum tensor, 217
massive, 561
relation to other wave equations, 219
spinorial description, 217
unphysical, 216
Schouten tensor
3-dimensional, 60
4-dimensional, 48
of a Weyl connection, 122

Schwarzschild spacetime, 163
conformal extension, 164
Eddington-Finkelstein coordinates, 163
isotropic radial coordinate, 270
Kruskal-Székeres extension, 164
maximal conformal extensions, 176
negative mass case, 176
Penrose diagram, 165
standard coordinates, 163
time symmetric initial data, 270
Schwarzschild-de Sitter and -anti de Sitter
spacetimes, 175
second fundamental form, see extrinsic
curvaturesee-saw rule, 68
semiglobal existence, 437
semilinear equations, 295
Sen connection, 105
shear, 225
slice, see leavesmooth tensor ﬁeld, see tensor ﬁeldsmoothness, 27
Sobolev embedding theorems, 307
Sobolev norm, 306
Sobolev spaces, 306
soldering forms, 80
spacetimecausal, 391
with complete past null cone, 497
deﬁnition, 1, 45
globally hyperbolic, 391
strongly causal, 392
time oriented, 45
spatial inﬁnity, 19
problem of, 438, 527, 549, 563, 566
sphere, at inﬁnity, 539
spherical coordinates, 142
spherical symmetry, 141
centre of symmetry, 141
quotient manifold, 141
quotient metric, 141
warped product form, 142
spin algebra, 67
spin connection coeﬃcients, 82, 197, 206
spatial, 109transformation under conformal rescaling,123
Weyl connection, 124
spin-2 massless ﬁeld, 551
spin-boosts, 241
spin-weight, 242
spin-weighted spherical harmonics, 240, 242
spinor covariant derivative, 81
space spinors, 108
spinor ﬁelds, 81
spinorial divergence equation, 298
spinors, 15
1 + 1 + 2 formalism, 464
abstract index notation for, 66

Indexcomplex conjugation, 72
components with respect to a basis, 71
contravariant, 67
covariant, 67
decomposition in irreducible components,
70
ﬂag, 77
Hermitian, 73
higher rank, 67
imaginary, 97
real, 97
space spinor formalism, 97
valence 1, 67
stability, 401standard Euclidean metric, 142
star shaped neighbourhood, 275
static equations, 506
static solutions, 506
cylinder at spatial inﬁnity, 553
static spacetimes, 504
conformal completions, 515
static spherically symmetric spacetimes, 165
conformal extension, 165
strictly globally static spacetimes, 475
structure equations, 472
SU (2, C) transformation, 100
submanifolds, 54
subsidiary evolution system, 331, 354, 388
model equations, 354
supertranslations, 239
supremum norm, 292
symbol, of a diﬀerential operator, 252, 295
symmetric hyperbolic systems, 295
characteristic initial value problems,319
constraint equations, 297
existence of solutions, 312
ﬁnite speed of propagation, 304
global existence, 328
initial boundary value problems, 313
initial data, 296
local existence, 308
localisability of solutions, 304, 310
null directions, 297
patching of solutions, 311
stability, 309, 312
uniqueness of solutions, 301
symmetry reduced evolution system, 545
symplectic transformation, 66
tangent space, 31
Taylor expansions, 292
tensor algebra, 33
tensor ﬁeld, 34
tensors, 32
antisymmetric, 33
antisymmetric part, 36
contraction, 35
contravariant, 32

593

covariant, 32
mixed, 32
product, 33
space of, 33
space spinor counterpart, 97
spatial, 250
spinorial counterpart, 76
symmetric, 33
symmetric part, 36
temporal and spatial components, 53
trace-free, 47transverse and longitudinal components,
56
time function, 54, 394
time independent solutions, 19time reﬂection symmetric initial data sets,
255
topological spaces, 27
torsion, 39
frame components, 53
tensor, 39
total characteristics, 297, 548
trace, 47
trace-free Ricci tensor, 48
spinorial counterpart, 110
transition tensor, 42translations, 239
transport equations, 368, 462, 548
transverse equations, 297
unit normal vector, see normal vector
unit position vector, 514
unphysical spacetime, 184
vector spaces
complex, 65
complex conjugate, 65
simplectic, 65
vectors, 31
causal, 391
future directed, 45
future/past pointing, 391
future pointing, 78
past directed, 45
tangent, 30
timelike, null, spacelike, 45
Weyl propagated, see Weyl propagation
volume form, 49
spatial, 249
spatial, spinorial counterpart, 105
spinorial counterpart, 78
wave coordinates, 387
wave equation
existence theory, 328
as a symmetric hyperbolic system, 300
weakly asymptotically simple spacetimes,
181
Weingarten map, 56

594
Weingarten tensor, 57
well-posedness, 6, 298
Weyl candidates, 250
Weyl connection, see connection
Weyl curvature hypothesis, 561
Weyl propagation, 121
Weyl tensor, 11, 48
rescaled, 11, 188
spinorial counterpart, 87
Weyl-Schouten theorem, 118
Whitney’s extension theorem, 324, 330
world tensor, 73
world-vector, 74

Index
Yamabe classes, 258
positive, 269
Yamabe equation, 278
Yamabe invariant/number, 280
Yamabe operator, 256, 508
Yamabe problem, 258, 269
York splitting, 257
zero-quantities, 195, 198, 199, 203, 207
-spinor, 67
ð and ð̄ operators, 228, 242
X-linearity, 34