INTRODUCTION TO GENERAL RELATIVITY, BLACK HOLES, AND COSMOLOGYIntroduction to General Relativity,
Black Holes, and Cosmology

Yvonne Choquet-Bruhat
French Academy of Sciences, Paris,and the American Academy of Arts and Science

With a foreword by

Thibault Damour
Permanent Professor, Institut des Hautes Études Scientiﬁque
Member of the French Academy of Sciences, Paris

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Foreword
Yvonne Choquet-Bruhat has made many deep and lasting contributions
to mathematical and physical aspects of General Relativity, startingwith her epoch-making 1952 proof of the well-posedness of the Cauchy
problem for Einstein’s equations. We are all very fortunate that she
has undertaken to present, in terms accessible to all, a comprehensive
account of all the aspects of General Relativity. Indeed, this beautiful
book is quite unique both in the range of topics it covers and in the way
each topic is treated.
First, the range of topics presented by Yvonne Choquet-Bruhat istruly remarkable. She covers successively the basics of Riemannian
geometry (Chapter I) and of Special Relativity (Chapter II); the mathematical and physical deﬁnition of General Relativity (Chapter III);the main properties and consequences of Einstein’s ﬁeld equations(Chapter IV); the mathematics and physics of the Schwarzschildspacetime (Chapter V); a mathematically rigourous discussion of black
holes (Chapter VI); a brief, but up to date, discussion of relativistic
cosmology (Chapter VII); a thorough presentation of the Cauchy
problem (Chapter VIII); and last, but not least, a detailed discussionof two of the most important phenomenological matter models, namely
relativistic ﬂuids (Chapter IX) and the relativistic kinetic theory of
N -particle systems (Chapter X).
But, most importantly, the treatment of each one of these broad topics
is both very comprehensive and remarkably concise. She has succeeded inreaching a Landau- and Lifshitz-like ideal of covering all the crucial issues
in the most concise way, while expounding each topic in a mathematically rigourous way. This rare combination of qualities makes this bookparticularly valuable. For instance, her discussion of the Schwarzschildspacetime covers, in one go: (i) a derivation of the solution that includes
a proof of Birkhoﬀ’s theorem; (ii) the form of the solution in ﬁve diﬀerent
types of coordinates; (iii) a preliminary discussion of the event horizon(which is developed in the following chapter); (iv) the motion of planetsand of light, and their comparison with the most recent observations; (v)the stability of circular orbits; (vi) a presentation of Fermat’s principlethat includes its little-known generalization to arbitrary spacetimes due
to G. Ferrarese; (vii) the redshift and time-delay eﬀects; (viii) a discussion of spherically symmetric interior solutions that includes several
theorems about their general properties; (ix) the Reissner-Nordström solution; (x) the Schwarzschild solution in any dimension; and (xi) a precise
but concise account of the results of Gu Chao Hao, Christodoulou, andvi

Forewordothers on spherically symmetric gravitational collapse. The chapter isthen nicely capped by some problems, followed by their solutions.
I am sure that this remarkably concise and complete book by Yvonne
Choquet-Bruhat will allow many readers to achieve a deep understanding of General Relativity through her unique mathematico-physicalknowledge of one of the fundamental pillars of modern physics. Thankyou, Yvonne for sharing with us the deep wisdom you have acquired
during your lifelong exploration of the beautiful theoretical landscape
opened, nearly a century ago, by Albert Einstein.
Thibault Damour
Institut des Hautes Études ScientiﬁquesPreface
Special Relativity (1905) and General Relativity (1915), both due toEinstein’s genius, are fundamental pillars of modern physics. They have
revolutionized the scientiﬁc concepts of space and time, formerly due
to everyday experience at a human scale, and also to previous scientiﬁc work and experiments made with clocks, very precise and reliable
at the considered scale. These concepts of absolute space and absolute
time had become ingrained in the minds of common folk as well as
scientists, and were diﬃcult to overthrow for a long time.1 Nowadays,
the world explored by humanity, at microscopic as well as cosmologicalscales, has become very much greater, and old concepts have had to be
replaced by more general ones. Spectacular improvements in technology
have changed the lives of a large part of humanity, and new information
processes have permitted easier access to scientiﬁc knowledge and theacceptance of new concepts.
Modern physical theories have a mathematical formulation, often
geometrical, with consequences deduced from mathematical theorems.
The validity of a physical theory results from the veriﬁcation of its
consequences by observations or experiments.
General Relativity is a beautiful geometric theory, simple in its general mathematical formulation, which has numerous consequences withstriking physical interpretations: gravitational waves, black holes, cosmological models. Several of these consequences have been veriﬁed with agreat accuracy. The Einstein equations present a wide variety of new andinteresting mathematical problems with possible physical interpretation.
The aim of this book is to present with precision, but as simply as possible, the foundations and main consequences of General Relativity. It
is written for an audience of mathematics students interested in physics
and physics students interested in exact mathematical formulations—or indeed for anyone with a scientiﬁc mind and curious to know more of
the world we live in. The mathematical level of the ﬁrst seven chaptersis that of undergraduates specializing in mathematics or physics; thesechapters could be the basis for a course on General Relativity. The next
three chapters are more advanced, though not requiring very sophisticated mathematics; they are aimed at graduate students, lecturers, andresearchers. No a priori specialized physics knowledge is required. These
chapters could serve as the text for a course for graduate students.
I wish to everyone a good trip in this strange but fascinating world.
Yvonne Choquet-Bruhat

1 For instance, the fact that two twins

who live diﬀerent lives age diﬀerently
was called the ‘twin paradox’, although
in fact there is no paradox there, except
in the human-built deﬁnition of time.

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PrefaceAcknowledgements
My gratitude goes to IHES, its scientiﬁc director Jean Pierre Bourguignon, and its permanent professor Thibault Damour, for inducing
Pierre Cartier, who kindly accepted, to give me a desk in his oﬃce. I havethus beneﬁted for the last ten years from a pleasant environment andworking facilities together with instructive seminars and conversations
that have been intellectually stimulating as well as friendly.
I am very fortunate that I have been able to meet regularly at IHES
Thibault Damour, who has always been ready to answer my questions
and allow me to beneﬁt from his great and wide knowledge of relativistic
physics and cosmology, as well as of the varied elaborate mathematical
tools used in this ﬁeld—from the oldest to the most recent. He was kindenough to read the whole of a preliminary version of this book. The
present version has been very much improved through his informationand his constructive criticism. Thank you Thibault—this book would
never have been completed without your help and encouragement.
I also thank Richard Kerner, who spent time reading my text, pointing
out useful corrections.
I thank François Bachelier for his pertinent help with tex.
Cover painting: Cosmos Birth by Geneviève Choquet-Kastylevsky.

Notation
The sign ≡ denotes an identity and the sign := a deﬁnition, althoughsometimes when there is no confusion possible I denote by the simple
sign = one or the other of these.
In the text, I usually denote by a single letter a geometric object, for example X or T for a vector or tensor. In equations, I
prefer to use indices, since this makes operations simpler to writeand understand—without speaking of possible numerical or algebraic
computing applications.
The spacetime metric signature is − + + . . . + . In the Lorentzian case,the spacetime indices are Greek letter. When speciﬁed, a time index isdenoted by zero, and the space indices by Latin letters from the middleof the alphabet, i, j, . . . .
Deﬁnitions of new notions are in bold characters and are assembled
in the index.

Note on references
I have not tried to ﬁnd for each result an original source. This would
have been a diﬃcult and sometimes controversial task. In the side notes,
I give references, assembled at the end of the book, to papers I have used,
where the interested reader can ﬁnd relevant details, complements, andreferences to previous papers about the subjects treated.
• The reference YCB-OUP2009 is to Yvonne Choquet-Bruhat General
Relativity and the Einstein Equations, published by Oxford University
Press in 2009.
• The references CB-DMI and II are to Y. Choquet-Bruhat and
C. DeWitt-Morette, Analysis, Manifolds and Physics, Parts I and II,
published by North-Holland in 1982 and 2000, respectively.

Contents
Part AI

Fundamentals

Riemannian and Lorentzian geometry
I.1 Introduction
I.2 Diﬀerentiable manifolds and mappings
I.2.1 Diﬀerentiable manifolds
I.2.2 Diﬀerentiable mappings
I.2.3 Submanifolds
I.2.4 Tangent and cotangent spaces
I.2.5 Vector ﬁelds and 1-forms
I.2.6 Moving frames
I.3 Tensors and tensor ﬁelds
I.3.1 Tensors, products and contraction
I.3.2 Tensor ﬁelds. Pullback and Lie derivative
I.3.3 Exterior forms
I.4 Structure coeﬃcients of moving frames
I.5 Pseudo-Riemannian metrics
I.5.1 General properties
I.5.2 Riemannian metrics
I.5.3 Lorentzian metrics
I.6 Causality
I.6.1 Causal and null cones
I.6.2 Future and past
I.6.3 Spacelike submanifolds
I.6.4 Length and geodesics
I.7 Connections
I.7.1 Linear connection
I.7.2 Riemannian connection
I.8 Geodesics, another deﬁnition
I.8.1 Pseudo-Riemannian manifolds
I.8.2 Riemannian manifolds
I.8.3 Lorentzian manifolds
I.9 Curvature
I.9.1 Deﬁnitions
I.9.2 Symmetries and antisymmetries
I.9.3 Diﬀerential Bianchi identity and contractions
I.10 Geodesic deviation
I.11 Linearized Ricci tensor
I.11.1 Linearized Bianchi identities

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I.12
I.13
I.14

II

Physical comment
Solutions of selected exercises
Problems
I.14.1 Liouville theorem
I.14.2 Codiﬀerential δ and Laplacianof an exterior form
I.14.3 Geodesic normal coordinates
I.14.4 Cases d = 1, 2, and 3
I.14.5 Wave equation satisﬁed by the
Riemann tensor
I.14.6 The Bel–Robinson tensor
I.14.7 The Weyl tensor
I.14.8 The Cotton–York tensor
I.14.9 Linearization of the Riemann tensor
I.14.10 Second derivative of the Ricci tensor

Special relativity
II.1
Introduction
II.2
Newtonian mechanics
II.2.1The Galileo–Newton Spacetime
II.2.2
Newtonian dynamics. Galileo group
II.2.3
Physical comment
II.2.4The Maxwell equations in
Galileo–Newton spacetime
II.3
The Lorentz and Poincaré groups
II.4
Lorentz contraction and dilation
II.5
Electromagnetic ﬁeld and Maxwell equationsin Minkowski spacetime M 4
II.6
Maxwell equations in arbitrary dimensions
II.7
Special Relativity
II.7.1
Proper timeII.7.2
Proper frame and relative velocities
II.8
Some physical comments
II.9
Dynamics of a pointlike mass
II.9.1
Newtonian law
II.9.2
Relativistic law
II.9.3
Newtonian approximation of therelativistic equation
II.9.4Equivalence of mass and energy
II.9.5
Particles with zero rest mass
II.10 Continuous matterII.10.1 Case of dust (incoherent matter),massive particles
II.10.2 Perfect ﬂuids
II.10.3 Yang–Mills ﬁelds
II.11 Problems
II.11.1 Lorentz transformation of theMaxwell equations
II.11.2 The relativistic Doppler–Fizeau eﬀect

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III

IV

General Relativity
III.1 Introduction
III.2 Principle of general covariance
III.3 The Galileo–Newton equivalence principle
III.4 General Relativity
III.4.1 Einstein equivalence principles
III.4.2 Conclusion
III.5 Constants and units of measurement
III.6 Classical ﬁelds in General Relativity
III.6.1 Perfect ﬂuid
III.6.2 Electromagnetic ﬁeld
III.6.3 Charged ﬂuid
III.7 Gravitation and curvature
III.8 Observations and experiments
III.8.1 The Einstein equivalence principle
III.8.2 Deviation of light rays
III.8.3 Proper time, gravitational time delay
III.8.4 Conclusion
III.9 Problems
III.9.1 Newtonian gravitation theory inabsolute space and time E n × R
III.9.2 Mass in length units (case n = 3)
III.9.3 Planck units

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The Einstein equations
IV.1 Introduction
IV.2 The Einstein equations
IV.2.1 The Einstein equations in vacuum
IV.2.2 Equations with sources
IV.2.3 Matter sources
IV.2.4 Field sources
IV.3 The cosmological constant
IV.4 General Einsteinian spacetimes
IV.4.1 Regularity
IV.4.2 Boundary conditions
IV.4.3 Physical comment
IV.5 Newtonian approximation
IV.5.1 Determination of GE
IV.5.2 Equations of motion
IV.5.3 Post-Newtonian approximation
IV.6 Minkowskian approximation
IV.6.1 Linearized equations at η
IV.6.2 Plane gravitational waves
IV.6.3 Further results on gravitational waves
IV.6.4 Tidal force
IV.6.5 Gravitational radiation
IV.7 Strong high-frequency waves
IV.7.1 Introduction
IV.7.2 Phase and polarization

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IV.7.3
Propagation and backreaction
IV.7.4
Observable displacements
IV.8
Stationary spacetimes
IV.8.1Deﬁnition
IV.8.2EquationsIV.8.3Non-existence of gravitational solitons
IV.8.4Gauss’s law
IV.9
Lagrangians
IV.9.1
Einstein–Hilbert Lagrangian in vacuo
IV.9.2
Lagrangians for Einstein equations
with sources
IV.10 Observations and experiments
IV.11 Problems
IV.11.1
The Einstein cylinder
IV.11.2
de Sitter spacetime
IV.11.3
Anti-de Sitter spacetime
IV.11.4
Taub–NUT spacetime
IV.11.5
The quadrupole formula
IV.11.6
Gravitational waves
IV.11.7
Landau–Lifshitz pseudotensor
IV.11.8
High-frequency waves from aspherically symmetric star
IV.11.9
Static solutions with compact
spacelike sections
IV.11.10 Mass of an asymptoticallyEuclidean spacetime
IV.11.11 Taub Lagrangian
V

The Schwarzschild spacetime
V.1
Introduction
V.2
Spherically symmetric spacetimes
V.3
Schwarzschild metric
V.4
Other coordinates
V.4.1
Isotropic coordinates
V.4.2
Wave (also called harmonic) coordinates
V.4.3
Painlevé–Gullstrand-like coordinates
V.4.4
Regge–Wheeler coordinates
V.5
Schwarzschild spacetime and event horizon
V.6
The motion of the planets and perihelion precession
V.6.1
Equations
V.6.2
Results of observations
V.6.3
Escape velocity
V.7
Stability of circular orbits
V.8
Deﬂection of light rays
V.8.1
Theoretical prediction
V.8.2
Fermat’s principle and light travel
parameter time
V.8.3
Results of observation

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V.9

Redshift and time dilation
V.9.1
Redshift
V.9.2
Time dilation
V.10 Spherically symmetric interior solutions
V.10.1 Static solutions. Upper limit on mass
V.10.2 Matching with an exterior solution
V.10.3 Non-static interior solutions
V.11 Spherically symmetric gravitational collapse
V.11.1 Tolman, Gu, Hu, and Claudel–
Newman metrics
V.11.2 Monotonically decreasing density
V.12 Problems
V.12.1 Relativistic and Newtonian
gravitational masses
V.12.2 The Reissner–Nordström solution
V.12.3 Schwarzschild spacetime indimension n + 1
V.12.4 Schwarzschild metric in isotropic
coordinates, n = 3
V.12.5 Wave coordinates for the
Schwarzschild metric in dimension
n+1
VIBlack holes
VI.1 Introduction
VI.2 The Schwarzschild black hole
VI.3 Eddington–Finkelstein extensions
VI.3.1 Eddington–Finkelstein white hole
VI.3.2 Kruskal spacetime
VI.4 Stationary black holes
VI.4.1 Axisymmetric and stationary spacetimes
VI.5 The Kerr spacetime and black hole
VI.5.1 Boyer–Lindquist coordinates
VI.5.2 The Kerr–Schild spacetime
VI.5.3 Essential singularity
VI.5.4 Horizons
VI.5.5 Limit of stationarity. The ergosphere
VI.5.6 Extended Kerr spacetime
VI.5.7 Absence of realistic interior solutions
or models of collapse
VI.6 Uniqueness theorems for stationary black holes
VI.6.1 The Israel uniqueness theorem
VI.6.2 Uniqueness of the Kerr black holeVI.6.3 Stability of the Kerr black hole
VI.7 General black holes
VI.7.1 Deﬁnitions
VI.7.2 Weak cosmic censorship conjecture
VI.7.3 Thermodynamics of black holes

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VI.8 Conclusions
VI.8.1 Observations
VI.8.2 The interiors of black holes
VI.9 Solution of Exercise VI.3.1
VI.10 Problems
VI.10.1 Lemaı̂tre coordinates
VI.10.2 Reissner–Nordström black hole
VI.10.3 Kerr–Newman metric
VI.10.4 Irreducible mass (Christodoulou–Ruﬃni)
VI.10.5 The Riemannian Penrose inequality
VII Introduction to cosmology
VII.1 Introduction
VII.2 The ﬁrst cosmological models
VII.2.1 Einstein static universe
VII.2.2 de Sitter spacetime
VII.2.3 General models
VII.3 Cosmological principle
VII.3.1 Assumptions
VII.3.2 Observational support
VII.4 Robertson–Walker spacetimes
VII.4.1 Robertson–Walker universes, metricat given t
VII.4.2 Robertson–Walker cosmologies
VII.5 General properties of Robertson–Walker spacetimes
VII.5.1 Cosmological redshift
VII.5.2 The Hubble law
VII.5.3 Deceleration parameter
VII.5.4 Age and future of the universe
VII.6 Friedmann–Lemaı̂tre universes
VII.6.1 Equations
VII.6.2 Density parameter
VII.6.3 Einstein–de Sitter universe
VII.6.4 General models with p = 0
VII.6.5 ΛCDM cosmological model
VII.7 Primordial cosmology
VII.8 Solution of Exercises VII.6.1 and VII.6.2
VII.9 Problems
VII.9.1 Isotropic and homogeneous
Riemannian manifolds
VII.9.2 Age of the universe
VII.9.3 Classical Friedmann–Lemaı̂tre universes
VII.9.4 Milne universe

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Part B
VIII

Advanced topics

General Einsteinian spacetimes. The
Cauchy problem
VIII.1 Introduction
VIII.2 Wave coordinates
VIII.2.1
Generalized wave coordinates
VIII.2.2
Damped wave coordinates
VIII.3 Evolution in wave gauge
VIII.3.1
Solution of the reduced
equations in vacuum
VIII.3.2
Equations with sources
VIII.4 Preservation of the wave gauges
VIII.4.1
Wave gauge constraints
VIII.5 Local existence and uniqueness
VIII.6 Solution of the wave gauge constraints
VIII.6.1
Asymptotically Euclidean manifolds
VIII.6.2
Compact manifolds
VIII.7 Geometric n + 1 splitting
VIII.7.1
Adapted frame and coframe
VIII.7.2
Dynamical system withconstraints for ḡ and K
VIII.7.3
Geometric Cauchy problem.
Regularity assumptions
VIII.8 Solution of the constraints
by the conformal method
VIII.8.1
Conformally formulated (CF)constraints
VIII.8.2
Elliptic system
VIII.8.3
Physical comment
VIII.9 Motion of a system of compact bodies
VIII.9.1
Eﬀective one-body (EOB) method
VIII.9.2
Numerical Relativity
VIII.10 Global properties
VIII.10.1 Global hyperbolicity andglobal uniqueness
VIII.10.2 Global existence
VIII.11 Singularities and cosmic censorship conjectures
VIII.11.1 Strong cosmic censorship conjecture
VIII.11.2 Weak cosmic censorship conjecture
VIII.12 Problems
VIII.12.1 Symmetric hyperbolic systems
VIII.12.2 The wave equation as asymmetric hyperbolic system
VIII.12.3 The evolution set of Maxwell
equations as a ﬁrst-ordersymmetric hyperbolic system

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VIII.12.4

Conformal transformation of
the CF constraints
VIII.12.5
Einstein equations in dimension
2+1
VIII.12.6
Electrovac Einsteinian
spacetimes, constraints
VIII.12.7
Electrovac Einsteinian
spacetimes, Lorenz gauge
VIII.12.8
Wave equation for F
VIII.12.9
Wave equation for the Riemann tensor
VIII.12.10 First-order symmetric
hyperbolic system for the
Riemann tensor, Bel–Robinson energy
VIII.12.11 Schwarzschild trapped surface
IX

Relativistic ﬂuids
IX.1
Introduction
IX.2
Case of dust
IX.3
Charged dust
IX.4
Perfect ﬂuid
IX.4.1Stress–energy tensor
IX.4.2Euler equations
IX.5
Thermodynamics
IX.5.1Conservation of rest mass
IX.5.2Deﬁnitions. Conservation of entropy
IX.5.3Equations of state (n = 3)
IX.6
Wave fronts and propagation speeds
IX.6.1
Characteristic determinantIX.6.2
Wave front propagation speed
IX.6.3
Case of perfect ﬂuids
IX.7
Cauchy problem for the Euler and entropy system
IX.7.1The Euler and entropy equations
as a Leray hyperbolic system
IX.7.2First-order symmetric hyperbolic systems
IX.8
Coupled Einstein–Euler–entropy system
IX.8.1
Initial data
IX.8.2Evolution
IX.9
Dynamical velocity
IX.9.1
Fluid index and Euler equations
IX.9.2
Vorticity tensor and Helmholtz equations
IX.9.3General perfect ﬂuid enthalpy h
IX.10 Irrotational ﬂows
IX.10.1 Deﬁnition and properties
IX.10.2 Coupling with the Einstein equations
IX.11 Equations in a ﬂow-adapted frame
IX.12 Shocks
IX.13 Charged ﬂuids
IX.13.1 Equations

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IX.14

IX.15
IX.16

IX.17
IX.18
IX.19
IX.20

X

IX.13.2 Fluids with zero conductivity
IX.13.3 Fluids with ﬁnite conductivity
Magnetohydrodynamics
IX.14.1 Equations
IX.14.2 Wave fronts
Yang–Mills ﬂuids (quark–gluon plasmas)
Viscous ﬂuids
IX.16.1 Generalized Navier–Stokes equations
IX.16.2 A Leray–Ohya hyperbolic systemfor viscous ﬂuids
The heat equation
Conclusion
Solution of Exercise IX.6.2
Problems
IX.20.1 Speciﬁc volume
IX.20.2 Motion of isolated bodies
IX.20.3 Euler equations for the dynamic velocity
IX.20.4 Hyperbolic Leray system for the
dynamical velocity
IX.20.5 Geodesics of conformal metric
IX.20.6 Cosmological equation of statep = (γ − 1)μRelativistic kinetic theory
X.1
Introduction
X.2
Distribution function
X.2.1
Deﬁnition
X.2.2
Interpretation
X.2.3
Moments of the distribution function
X.2.4
Particles of a given rest mass
X.3
Vlasov equations
X.3.1
General relativistic (GR)–Vlasov equation
X.3.2
EM–GR–Vlasov equation
X.3.3
Yang–Mills plasmas
X.4
Solution of a Vlasov equation
X.4.1
Construction
X.4.2
Global existence theorem
X.4.3
Stress–energy tensor
X.5
The Einstein–Vlasov system
X.5.1
Equations
X.5.2
Conservation law
X.6
The Cauchy problem
X.6.1
Cauchy data and constraints
X.6.2
Evolution
X.6.3
Local existence and uniqueness theorem
X.6.4
Global theorems
X.7
The Maxwell–Einstein–Vlasov system
X.7.1
Particles with given rest mass and charge
X.7.2
Particles with random masses and charges
X.8
Boltzmann equation. Deﬁnitions

227
228
229
229
230
231
231
232
233
234
235
235
235
235
235
236
237
239
240
241
241
242
242
242
243
245
246
246
247
248
249
249
250
250
251
251
251
252
252
253
253
253
254
254
255
256

xix

xx

Contents

X.9
X.10
X.11

X.12
X.13
X.14

Moments and conservation laws
Einstein–Boltzmann system
Thermodynamics
X.11.1 Entropy and the H theorem
X.11.2 Maxwell–Jüttner equilibrium distribution
X.11.3 Dissipative ﬂuids
Extended thermodynamics
Solutions of selected exercises
Problems
X.14.1 Liouville’s theorem and generalization
X.14.2 Vlasov equation for particles withrandom charges
X.14.3 Distribution function on a
Robertson–Walker spacetime with
Vlasov source

257
258
259
259
261
262
262
262
264
264
264

265

References

267

Index

273

Part A
Fundamentals
This part contains the mathematical deﬁnitions and physical interpretations necessary for a basic understanding of Einstein’s General
Relativity. The basic deﬁnitions and properties of Riemannian andLorentzian geometry are given. Special Relativity is introduced, withdiscussion of proper time, the equivalence of mass and energy, and physical observations. The spacetimes of General Relativity are deﬁned, andtheir basic mathematical properties and physical interpretations are described, along with comments on experimental results. The theoreticalfoundations of the Einstein equations are given and these equations arederived. The Newtonian and Minkowskian approximations of their solutions are described. Gravitational waves and gravitational radiation
are introduced. The spherically symmetric Schwarzschild spacetime isdeduced from the Einstein equations in vacuum, its mathematical consequences are computed, and their observational conﬁrmation is described.
Black holes, a phenomenon unknown to Newtonian mechanics, but
predicted by the Einstein equations, are discussed. Both sphericallysymmetric and axisymmetric (Kerr) black holes are constructed andstudied. Some mathematical results and conjectures are introduced, andan introduction to the thermodynamics of black holes and Hawking radiation is provided. Relevant astrophysical observations are described.
The construction of the ﬁrst cosmological models—Einstein, de Sitter,and anti-de Sitter—is given. General Robertson–Walker spacetimes, thecosmological redshift, and the Hubble law are discussed. Friedman–
Lemaı̂tre universes and the presently accepted ΛCDM model withaccelerated expansion are described, as is the current majority view on
the content of the universe, including the mysterious dark matter anddark energy. A brief account of primordial cosmology is given.

I

Riemannian andLorentzian geometry
I.1

Introduction

We give in this chapter a survey of the basic deﬁnitions of Riemannian and Lorentzian diﬀerential geometry1 that are necessary in General
Relativity.

I.2

Diﬀerentiable manifolds and mappings

I.2.1

Diﬀerentiable manifolds

The fundamental arena of diﬀerential geometry is a diﬀerentiable manifold. For the physicist, the most concrete and useful deﬁnition makes
apparent the local identiﬁcation of a manifold with Rn , the space of setsof n real numbers with its usual topology.2 The deﬁnition proceeds asfollows.
A chart on a set X is a pair (U, φ), with U a subset of X, called
the domain of the chart, and φ a bijection from U onto an open set uof Rn , i.e. a one-to-one invertible mapping φ : U → u by x → φ(x) ≡(x1 , . . . , xn ). The numbers xi , i = 1, . . . , n, are called local coordinatesof the point x ∈ X.
An atlas on X is a collection of charts (UI , φI ), with {I} an arbitrary
set of indices, whose domains cover X.
A mapping f between open sets of Rn , f : u → v, is called a homeomorphism if it is bijective and if f and its inverse mapping f −1 arecontinuous. A bijective mapping is a C k diﬀeomorphism if f and f −1
are of class C k .
Exercise I.2.1 Prove that a C 1 diﬀeomorphism with f of class C k isa C k diﬀeomorphism.
Hint: ∂(f f −1 )/∂xi ≡ 0.
An atlas endows X with the structure of a topological manifold, of
are homeomorphisms between
dimension n, if the mappings φI ◦ φ−1J
the open sets of Rn , φJ (UJ ∩ UI ) and φI (UJ ∩ UI ). The mappings
φI ◦ φ−1
J deﬁne changes of local coordinates in the intersection UJ ∩ UI ,(x1 , . . . , xn ) → (y 1 , . . . , y n ). The manifold is a diﬀerentiable manifold of class C k if these mappings are C k diﬀeomorphisms; that is, if

I.1 Introduction
I.2 Diﬀerentiable manifolds
and mappings

33

I.3 Tensors and tensor ﬁelds 7
I.4 Structure coeﬃcientsof moving frames10
I.5 Pseudo-Riemannian
metrics
I.6 Causality

11
14

I.7 Connections
I.8 Geodesics, anotherdeﬁnition

19

I.9 Curvature
I.10 Geodesic deviation
I.11 Linearized Ricci tensor

20
22
23

I.12
I.13

24

Physical comment
Solutions of selected
exercises

I.14 Problems

16

24
28

1 Morecomplete treatments can befound in many textbooks, in particularin CB-DMI.
2 A basis of open sets in R comprises thesets determined by the order relationa < x < b. The topology of Rn is the
direct product of n copies of R.

4

Riemannian and Lorentzian geometry

the functions y i (x1 , . . . , xn ) are of class C k and the Jacobian determinant D(y)/D(x) with elements the partial derivatives ∂y i /∂xj is diﬀerent
from zero. Henceforth, by smooth we shall mean of class C k with k large
enough for the statement under consideration to be true; a particularcase is k = ∞.
Two C k atlases on X are said to be equivalent if their union is again
a C k atlas. We consider them to deﬁne the same C k manifold. It is
known that any C 1 manifold can be endowed with a C ∞ structure.
Unless otherwise stated, all manifolds considered here are assumed tobe smooth.
The given deﬁnition of a manifold does not imply that it is a Hausdorﬀ
topological space; that is, it admits the possibility for two points not tohave non-intersecting neighbourhoods. In the following, we will, unlessotherwise stated, only consider Hausdorﬀ manifolds and call themsimply manifolds.
Examples of manifolds are the sphere and the torus.
An open set of Rn is oriented by the order of the coordinates
1(x , . . . , xn ). A diﬀerentiable manifold is said to be orientable (andoriented by the choice of coordinates) if its deﬁning atlas is such that
D(y)/D(x) > 0 in all intersections of domains of charts.
Unless otherwise stated, the manifolds considered here will be C ∞ ,connected and oriented.

I.2.2

Diﬀerentiable mappings

A function f on an n-dimensional manifold V n is a mapping V n → R
by x → f (x). Its representative in local coordinates of the chart (U, φ)
is a function on an open set of Rn , fφ := f ◦ φ−1 : (xi ) → f (φ−1 (xi )).
The function f is of class C k at x if fφ is of class C k at φ−1 (x). Thisdeﬁnition is chart-independent if V n is smooth. The gradient, also called
diﬀerential, of f is represented in a chart by the partial derivatives offφ . If (U, φ) and (U  , φ ) are two charts containing x, it holds that at x
(calculus relations)∂fφ∂fφ ∂xj=.
∂xi∂xj  ∂xi

(I.2.1)This equivalence relation entitles the diﬀerential of f to be called acovariant vector (see below). A covariant vector is a geometric object,independent of a particular of coordinates choice.
A diﬀerentiable mapping f between diﬀerentiable manifolds anddiﬀeomorphisms can be deﬁned analogously. The diﬀerential of f at
x ∈ W n , the source, is represented in a chart around x and a chart at
f (x) ∈ V p , the target, by a linear mapping from Rn into Rp ; that is, ann × p matrix with elements ∂f α /∂xi , α = 1, . . . , p, i = 1, . . . , n.

I.2

I.2.3

Diﬀerentiable manifolds and mappings

Submanifolds

An embedded submanifold of V is the image by an injective
diﬀerentiable mapping of a smooth manifold W.

I.2.4

Tangent and cotangent spaces

A tangent vector v to a diﬀerentiable manifold V at a point x ∈ V
is a geometric object. It can be deﬁned through local coordinates as
an equivalence class of triplets (UI , φI , vφI ), where (UI , φI ) are charts
containing x, while vφI = (vφi I ), i = 1, . . . , n, are vectors in Rn . The
equivalence relation is given by
vφi I = vφj J

∂xiI

∂xjJ

,

(I.2.2)where xiI and xiJ are respectively local coordinates in the charts (UI , φI )and (UJ , φJ ). The vector vφ ∈ Rn is the representative of the vector
v in the chart (U, φ). The vector v is attached to the manifold by theassumption, compatible with the equivalence relation, that the numbers
vφi are the components of vφ in the frame of Rn deﬁned by the tangentsto the coordinate curves, curves in Rn where only one coordinate varies.
Tangent vectors at x constitute a vector space, the tangent space
to V n at x, which is denoted Tx V n . The set of pairs (x, vx ), x ∈ V n ,vx ∈ Tx V n , denoted by T V n , is called the tangent bundle to V n .
An arbitary set of n linearly independent tangent vectors at x constitute a frame at x. The natural frame associated to a chart (U, φ)is the set of n vectors e(i) , i = 1, . . . , n, such that ej(i),φ = δij ; they are
represented in local coordinates by tangent vectors to the coordinates
curves of the chart. The numbers vφi I are the components of the vector
v in the natural frame. Traditionally, indices of components of vectors
are written upstairs.
The cotangent space Tx∗ to V is the dual of Tx , that is, the spaceof 1-forms on Tx , which are also called covariant vectors at x ∈ V,
the tangent vectors being called contravariant. The components of acovariant vector at x in a chart (U, φ) containing x, are a set of n numbers
vi , i = 1, . . . , n, with indices traditionally written downstairs. Under achange of chart from (U, φ) to (U, φ ), it holds thatvi = vj

∂xj
.
∂xi

(I.2.3)

Covariant vectors can be deﬁned by this equivalence relation. By (2.1),
the diﬀerential at x of a diﬀerentiable function f is a covariant vector,
called the gradient of f. The natural coframe is the set of diﬀerentials
dxi of the coordinate functions x → xi .

5

6

Riemannian and Lorentzian geometry

I.2.5

Vector ﬁelds and 1-forms

A vector ﬁeld on V assigns a tangent vector at x ∈ V to each point x.
The relations (2.1) and (2.2) show that, given a diﬀerentiable function
f on V, the quantity v(f ) deﬁned for points x in the domain U of thechart (U, φ) byv(f ) := vφi

3 Evenlinear, that is also such thatv(λf ) = λv(f ), with λ a constant in
R, a property included in the Leibniz
rule.

∂fφ
,
∂xiφ

fφ := f ◦ φ−1(I.2.4)is chart-independent: v deﬁnes a mapping between diﬀerentiable functions. It is easy to check that this mapping is additive,3v(f + g) = v(f ) + v(g),(I.2.5)and satisﬁes the Leibniz rule,v(f g) = f v(g) + gv(f ).

(I.2.6)

The properties (2.5) and (2.6) characterize a derivation operator,and v(f ) is called the derivative of f along v.
If we take for v a vector of a natural frame eφ,(i) , that is, v j = δij ,thenv(f ) ≡ eφ,(i) (f ) =4 See, for instance, CB-DMI, III B 1.

(I.2.7)

Alternatively, a tangent vector at a point x on a diﬀerentiable manifold can be deﬁned, without ﬁrst introducing its representatives and theequivalence relation, as the value at x of a linear ﬁrst-order derivation
operator acting on diﬀerentiable functions deﬁned in a neighbourhoodof x.4 The natural frame can thus be deﬁned as the set of operators∂/∂xi , i = 1, . . . , n.
A covariant vector ﬁeld on V assigns a covariant vector at x to eachpoint x ∈ V . It is called a 1-form on V.

I.2.6

5 It has been proved that all orientablediﬀerentiable 3-manifolds are parallelizable; this is not true in all otherdimensions.

∂(f ◦ φ−1 ).
∂xiφMoving frames

A moving frame, often called simply a frame in what follows, in a
subset U of a diﬀerentiable n-dimensional manifold V is a set of n vectorﬁelds on U that are linearly independent in the tangent space Tx V ateach point x ∈ U. A manifold that admits global frames (not necessarily
global coordinates) is called parallelizable.5
A coframe on U is a set of n 1-forms θi that are linearly independent at
each x ∈ U in the dual space Tx∗ V . In the domain U of a coordinate chart,a coframe is speciﬁed by n linearly independent diﬀerential 1-forms
θi ≡ aij dxj ,with aij functions on U.

(I.2.8)

I.3

The diﬀerential of a diﬀerentiable function f is a covariant vector ﬁeld
called an exact 1-form
df :=

∂f i
dx ≡ ∂i f θi .
∂xi

Its components ∂i f in the coframe θi are called Pfaﬀ derivatives.
The dual frame to a coframe θi is the set of vector ﬁelds such thatθi (ej ) = δji .

(I.2.9)The natural frame ∂/∂xi and the natural coframe dxi are dual.

I.3

Tensors and tensor ﬁelds

I.3.1

Tensors, products and contraction

A covariant p-tensor at a point x ∈ V can be deﬁned as a p-multilinear
form on p direct products of the tangent space Tx V with itself. Contravariant tensors can be deﬁned as multilinear forms on direct productsof the cotangent space Tx∗ V.
Tensors, contravariant or covariant, can also be deﬁned, like vectors, through equivalence relations between their components in various
charts. For instance a 1-contravariant 2-covariant tensor T at x ∈ V isan equivalence class of triplets (UI , φI , TφI ,ij h ), i, j, h = 1, . . . , n, withthe equivalence relation being the law of change of components of T bychange of local coordinates from (U, φ) to (U, φ ) given byTi j h∂xk ∂xl ∂xh=
Tkl m .
∂xi ∂xj  ∂xm(I.3.1)The space of tensors of a given type is a vector space, in which summation of elements and their multiplication by scalars are deﬁned in termsof these operations acting on components.
The tensor product S ⊗ T of a p-tensor S and a q-tensor T is intrinsically deﬁned: it is the (p + q)-tensor with components deﬁned byproducts of components. For example, the tensor product of a covariant
vector ω with a contravariant 2-tensor T is the mixed 3-tensor ω ⊗ Twith components(ω ⊗ T )i jk = ωi T jk .

(I.3.2)

Although products of components are commutative, tensor products are
non-commutative: ω ⊗ T and T ⊗ ω of the previous example do notbelong to the same vector space. A p-covariant and q-covariant tensor isan element of the tensor product of p copies of Tx and q copies of Tx∗ ,but diﬀerent orders of these products give diﬀerent spaces of tensors:
Tx∗ ⊗ Tx = Tx ⊗ Tx∗ .
A basis of the vector space of tensors of a given type is obtained bytensor products of vectors and covectors of bases. For example, a natural

Tensors and tensor ﬁelds

7

8

Riemannian and Lorentzian geometrybasis of the space of covariant 2-tensors at x is dxi ⊗ dxj :T = Tij dxi ⊗ dxj .

(I.3.3)The symmetry [respectively antisymmetry] properties, for example Tij = Tji [respectively Tij = −Tji ], are intrinsic properties (i.e.
independent of coordinates).
Exercise I.3.1 Check these properties.
The contracted product of a p-contravariant tensor and a qcovariant tensor is a tensor of order p + q − 2 whose components are
obtained by summing over a repeated index appearing once upstairs
and once downstairs.

I.3.2

Tensor ﬁelds. Pullback and Lie derivative

Tensor ﬁelds are assignments of a tensor at x to each point x of the
manifold V. Diﬀerentiability can be deﬁned in charts; the notion of a C ktensor ﬁeld is chart-independent if the manifold is of class C k+1 .
The image of a contravariant tensor ﬁeld on V under a diﬀerentiablemapping between diﬀerentiable manifolds f : V → W is not necessarily
a tensor ﬁeld on W unless f is a diﬀeomorphism.
Exercise I.3.2 Prove this statement in the case of a vector ﬁeld.
The pullback f ∗ on V of a covariant tensor ﬁeld on W is a covariant
tensor ﬁeld. For instance, for a covariant vector ω and a mapping f given
in local charts by the functions y α = f α (xi ), with y α coordinates on W
and xi coordinates on V, the pullback is deﬁned by(f ∗ ω)i (x) =∂y α
ωα (y(x).
∂xi(I.3.4)

Exercise I.3.3 Extend the deﬁnition and property to covariant
tensors.

6 See, for instance, CB-DMI, III C.

The Lie derivative of a tensor ﬁeld T with respect to a vector ﬁeld
X is a derivation operator from p-tensors into p-tensors. If ft is the oneparameter local group of diﬀeomorphisms6 generated by X, then the Liederivative at x ∈ V of a contravariant tensor T is deﬁned by(LX T )(x) := lim[(ft−1 ) T (ft (x)) − T (x)].
t=0(I.3.5)For example, for a contravariant 2-tensor, it is computed to be(LX T )jk = X i

∂T jk
∂X j
∂X k
− T ik− T ji
.
i
i
∂x
∂x
∂xi

(I.3.6)

The Lie derivative of a vector ﬁeld X in the direction of the vector ﬁeld
Y is called the Lie bracket of X and Y ; it has components(LX Y )j ≡ [X, Y ]j ≡ X i ∂i Y j − Y i ∂i X j .

I.3

Tensors and tensor ﬁelds

9

An analogous deﬁnition gives Lie derivatives of covariant tensors. For
instance, for a covariant 2-tensor,(LX T )jk = X i

∂Tjk
∂X i
∂X i
+
T
+
T
.
ik
ji
∂xi
∂xj
∂xk

(I.3.7)

Exercise I.3.4 Check that the quantities (3.6) [respectively (3.7)]transform as the components of a 2-contravariant tensor [respectively
a 2-covariant tensor] under a change of coordinates.

I.3.3

Exterior forms

A totally antisymmetric covariant p-tensor ﬁeld is also called an exterior p-form. A natural basis of the space of p-forms is obtained byantisymmetrization. We call the p-antisymmetrization operator themixed tensor with components ε12...pi1 ...ip = 1 if i1 . . . ip is an even per-mutation of 12 . . . p and ε12...pi1 ...ip = −1 if it is an odd permutation The
components of a p-form can be written
ωi1 ...ip = ε12...pi1 ...ip ω12...pA p-form is determined by the data of its component with indices ofincreasing order.
We deﬁne the exterior product of exterior forms by antisymmetrization of tensor products. A basic example is the exterior product of two
1-forms given by7
1
dx ∧ dx = −dx ∧ dx := (dxi ⊗ dxj − dxj ⊗ dxi ).
2
i

j

j

i

(I.3.8)

7 Note that there is an alternative def-

inition of the exterior product in which
the factor of 12 is omitted: dxi ∧ dxj =dxi ⊗ dxj − dxj ⊗ dxi .

Using exterior products, a p-form reads in a natural coframeω=1
ωi ...i dxi1 ∧ . . . ∧ dxip .
p! 1 p(I.3.9)The exterior derivative of a p-form is a (p + 1)-form given in local
coordinates by
dω :=

1
1 ∂
dωi1 ...ip dxi1 ∧ . . . ∧ dxip ≡ωi ...i dxj ∧ dxi1 ∧ . . . ∧ dxip .
p!
p! ∂xj 1 p(I.3.10)

A form whose diﬀerential is zero is called a closed form. A form thatis the diﬀerential of an exterior form is called an exact form. An exact form is a closed form. The reciprocal of this property is true on
manifolds diﬀeomorphic to Rn (Poincaré lemma), but not on generalmanifolds. A necessary and suﬃcient condition is given by the de Rhamtheorem.8
Example I.3.1 Any n-form on the compact n-manifold S n is closed,
but is an exact diﬀerential only if its integral on S n vanishes.

8 The suﬃcient condition is that the in-tegral of the p-form is zero on p-cycles.

10

Riemannian and Lorentzian geometry

The interior product of a p-form with a vector ﬁeld X is the(p − 1)-form given in local coordinates by the formula(iX ω)i1 ...ip−1 :=1X h ωhi1...ip−1 .
(p − 1)!

Lemma I.3.1 The Lie derivative of an exterior form ω withrespect to a vector ﬁeld X is given by the formula
LX ω ≡ iX dω + d(iX ω).
Exercise I.3.5 Prove this formula for 2-forms by using local coordinates.

I.4

Structure coeﬃcientsof moving frames

For the natural coordinates xi in a chart of a manifold, which are functions x → xi , it holds that ∂xi /∂xj = δji , d2 xi ≡ 0. However, for amoving coframe in a domain U , the diﬀerentials of the 1-forms θi do notvanish in general: they are given by 2-forms
1 i jθ ∧ θh .
dθi ≡ − Cjh2(I.4.1)i
on U, antisymmetric in j and k, are called the
The functions Cjh
structure coeﬃcients of the frame.
The Pfaﬃan derivatives ∂i in the coframe θi of a function f on U aredeﬁned by df ≡ ∂i f θi . The identity d2 f ≡ 0 implies that1h(∂i ∂j f − ∂j ∂i f − Cij∂h f )θi ∧ θj ≡ 0;
(I.4.2)2
hence, Pfaﬃan derivatives in contrast to ordinary partial derivatives, do
not generally commute,d2 f ≡h∂h f.
(∂i ∂j − ∂j ∂i )f ≡ Cij(I.4.3)

Exercise I.4.1 Show that the structure coeﬃcients of a coframe
θi := aij dxj are given byiChk≡ Ajk ∂h aij − Ajh ∂h aij

where A is the matrix inverse of the matrix a.
We note that the basis ei dual to θi satisﬁes the commutation
conditions
1 h[ei , ej ] = Cijeh ,2
where [., ] denotes the Lie bracket of vector ﬁelds:[v, w] := Lv w = −Lw v.

I.5

I.5

Pseudo-Riemannian metrics

I.5.1

General properties

A pseudo-Riemannian metric on a manifold V is a symmetric covariant 2-tensor ﬁeld g such that the quadratic form it deﬁnes on contravariant vectors, g(X, X), given in local charts by gij X i X j , is non-degenerate;that is, the determinant Det(g) with elements gij does not vanish in any
chart; this property is independent of the choice of the charts because,
under a change of local coordinates (xi ) → (xi ), it holds that
2
D(x )
Det(g) = Det(g ).
(I.5.1)D(x)The inverse of the matrix (gij ) is denoted by (g ij ).
Exercise I.5.1 Show that the Kronecker symbol δij , δij = 1 for i = j,δij = 0 for i = j, is invariant under change of frame.
Show that the g ij are the components of a contravariant symmetric
2-tensor.
A metric is traditionally written in a moving frame:g = gij θi θj ,i.e.,

in a natural frame, g = gij dxi dxj .

(I.5.2)

It is known from algebra that any non-degenerate quadratic form over
the reals can be written as a sum

gij dxi dxj ≡εi (θi )2 with εi = ±1,(I.5.3)iwhere the θi are independent real linear forms of the dxi , that is, amoving frame. Given g, this decomposition can be done in many ways,
but the number of εi that are equal to +1 and the number that are equalto −1 are independent of the decomposition; this is called the signatureof the quadratic form. A moving frame where a metric takes the form
(5.3) is called an orthonormal frame, whatever its signature.
A pseudo-Riemannian metric g on V deﬁnes at each point x ∈ V
a scalar product, that is, a bilinear non-degenerate function on thetangent space Tx V :(v, w) := gx (v, w) ≡ gij (x)v i wj ,v, w ∈ Tx V.

Two vectors of a tangent space Tx V are said to be orthogonal if theirscalar product vanishes.
Through contracted products with the metric or its contravariant
counterpart, one associates canonically contravariant and covariant tensors, for example
Tij = gih gjk T hk .

(I.5.4)

An isometry of the pseudo-Riemannian manifold (V, g) is a diﬀeomorphism f that leaves g invariant; that is,(f ∗ g)(x) = g(x)at each point x ∈ V.

(I.5.5)

Pseudo-Riemannian metrics

11

12

Riemannian and Lorentzian geometry

If a metric g is invariant under the action of a one-parameter group
generated by the vector ﬁeld X, then its Lie derivative with respect to
X vanishes; the converse is locally true. The vector ﬁeld X is then calleda Killing vector of g.
Exercise I.5.2 Show that the Lie bracket of two Killing vectors is alsoa Killing vector.
A pseudo-Riemannian manifold of dimension n admits at mostn(n + 1)/2 Killing vectors.
Two pseudo-Riemannian manifolds (V, g) and (V  , g  ) are called locally isometric if there exists a diﬀerentiable mapping f such that any
point x ∈ V admits a neighbourhood U, and f (x) a neighbourhood U  ,with (U, g) and (U  , g  ) isometric. Locally isometric manifolds have thesame dimension but can have diﬀerent topologies (for example, a plane
and a cylinder).
A pseudo-Riemannian manifold is called a ﬂat space if it is isometricwith a pseudo-Euclidean space, that is, Rn with metric
gij dxi dxj ≡εi (dxi )2 with εi = ±1.
i

It is called locally ﬂat if it is locally isometric to a ﬂat space.
The volume form of the metric g is the exterior n-form that reads
in local coordinates1ωg = |Detg| 2 dx1 ∧ . . . ∧ dxn .

(I.5.6)This exterior form induces, on domains oriented by the order x1 , . . . , xn ,a volume element on a pseudo-Riemannian manifold, often denotedby the same symbol ωg , although the volume element concerns onlyoriented manifolds. It reads,
1μg = |Detg| 2 dx1 . . . dxn ,9 For integration of forms on manifolds,(I.5.7)the order of the diﬀerential dxi is irrelevant in integration (Fubini’s
theorem).9

see for instance CB-DMI, IV B 1.

I.5.2
10 Thatis, εi = +1 for all i in theformula (5.3).

Riemannian metrics

A metric is called Riemannian (or, for emphasis, properly Riemannian)
if the quadratic form deﬁned by g is positive-deﬁnite.10
If v is a tangent vector at x to a Riemannian manifold (V, g), thenon-negative scalar
11gx (v, v) 2 := [gij (x)v i v j ] 2(I.5.8)is the norm (in g) of v; it vanishes only if v is the zero vector, whose
components are v i = 0.

I.5

Pseudo-Riemannian metrics

13

The length of a parametrized curve λ → x(λ) joining two points of
V with parameters λ = a and λ = b is
1 b
dxi dxj 2:=
dλ.
(I.5.9)gij (x(λ))
dλ dλaElementary analysis shows that the deﬁnition is independent of the parametrization. The distance between these two points is the lower boundof the length of arcs joining them. A geodesic of a properly Riemannian
manifold is a curve for which this bound is attained.
Distance endows a properly Riemannian manifold with the structureof a metric space with a topology that coincides with its topology asa manifold. Riemannian manifolds and their topologies have been extensively studied by diﬀerential geometers, and their metrics are closely
related to their topological properties. They are of interest to General
Relativists as possible properties of space (see Chapter VII).

I.5.3

Lorentzian metrics

A pseudo-Riemannian metric g is called a Lorentzian metric if thesignature of the quadratic form deﬁned by g is (− + + · · · +). In the caseof a manifold with a Lorentzian metric, we denote its dimension by n + 1
and use Greek indices α, β, . . . = 0, 1, 2, . . . , n for local coordinates andlocal frames. A Lorentzian metric is a quadratic formg ≡ gαβ dxα dxβ(I.5.10)that admits a decomposition as a sum of squares of 1-forms θα with thefollowing signs, independent of the choice of these 1-forms:(θi )2 , θα = aα β dxβ .
(I.5.11)g ≡ −(θ0 )2 +i=1,...nRemark I.5.1 We adopt the ‘mostly plus’ MTW convention.11 Someauthors prefer the opposite convention, giving to Lorentzian metrics thesignature (+−− . . . −). Each of these conventions has its advantages anddisavantages, but of course they give equivalent geometrical results (note,however, that, surprisingly, this is not always true on non-orientablemanifolds12 ).
A metric of fundamental physical importance in the case n = 3 is the
Minkowski metric. For a general n, it is the ﬂat metric on Rn+1 thatreadsg ≡ −(dx0 )2 +(dxi )2 .
(I.5.12)i=1,2,...,nThe group of linear maps of Rn+1 ,αθα = Lαα θ ,(I.5.13)that preserves orthonormal frames (i.e. frames θα where the metric takes
the form (5.12)) is called the Lorentz group Ln+1 .

11 Misner–Thorne–Wheeler.

12 See CB-DMII Chapter I Section 7,
Pin and spin groups.

14

Riemannian and Lorentzian geometry

I.6

Causality

I.6.1

Causal and null cones

At each point x of a Lorentzian manifold V, one deﬁnes in Tx V, thetangent space to V at x, a double cone Cx called the causal cone, by
the inequality
gx (v, v) ≤ 0,v ∈ Tx V.

(I.6.1)The boundary of Cx in Tx V is the double cone Γx , called the null coneor light cone,gx (v, v) = 0.

(I.6.2)

A vector v ∈ Tx v is called causal if
gx (v, v) ≤ 0.

(I.6.3)

It is called timelike [respectively spacelike] if
gx (v, v) < 0 [respectively gx (v, v) > 0].

(I.6.4)

It is called a null vector, not to be confused with the zero vector, if
gx (v, v) = 0

I.6.2(I.6.5)

Future and past

The causal cone Cx splits into two convex cones, Cx+ and Cx− , characterized in an orthonormal Lorentzian frame by the properties v ∈ Cx
andCx+ : θ0 (v) ≡ v 0 > 0,

Cx− : v 0 < 0.

(I.6.6)

If it is possible to choose this splitting continuously on V , the manifold
is said to be time-orientable. It is time-oriented by the choice. Unlessotherwise stated, the manifolds called Lorentzian manifolds are oriented
manifolds with time-oriented Lorentzian metrics.
A causal vector is future- [respectively past-] oriented if it is suchthat v ∈ Cx+ [respectively v ∈ Cx− ].
A curve γ joining xa to xb in a manifold V is the image in V of asegment of R, λ → γ(λ) ∈ V, a ≤ λ ≤ b. The tangent to γ at a pointγ(λ) is the derivative dγ/dλ, a vector in Tγ(λ) with components dγ α /dλin local coordinates.
In a Lorentzian manifold (V, g), a future causal curve is a curve with
future causal tangent vectors, and a future timelike curve has future
timelike tangent vectors. The future of a point x is the set of pointsy ∈ V that can be reached from x by a future timelike curve. The futureof a subset is the union of the future of its points.

I.6

I.6.3

Spacelike submanifolds

Let M be an n-dimensional diﬀerentiable submanifold of the (n + 1)dimensional diﬀerentiable manifold V. Assume that f is deﬁned by anequation f = 0, with f a diﬀerentiable function with non-vanishinggradient. Then, we denote by ν the contravariant vector associated in
the metric g to the gradient of f at x ∈ V.
Exercise I.6.1 Show that ν is orthogonal in g to vectors v in Tx M.
The submanifold M is called spacelike [respectively, timelike, null] if
its normal ν is timelike [respectively, spacelike, null].
Exercise I.6.2 Show that in a small enough neighbourhood, local coordinates of a Lorentzian manifold can always be chosen such that thetime lines are orthogonal to spacelike submanifolds; that is, the metriccan be written
ds2 := −N 2 (dx0 )2 + gij dxi dxj .

I.6.4

Length and geodesics

The length of a causal curve γ joining xa to xb is
1
 bdγ α dγ β 2
:=
dλ.
−gαβ
dλ dλ
a

(I.6.7)Null curves have zero length.
To deﬁne geodesics on a Lorentzian manifold, we replace the integral (6.7), which causes problem at points where the curve has a nulltangent vector, by the integral, which is no longer independent of theparameter λ,
 b
dxα
.
(I.6.8)L dλ, with L:=gαβ (x(λ))ẋα ẋβ ,ẋα :=dλa
A geodesic joining x(a) and x(b) is deﬁned as a critical point of this
Lagrangian, that is, a solution of the Euler equations
d ∂L
∂L
−
= 0.
dλ ∂ ẋα∂xαThese equations read explicitly


d2 xβ
∂gαβ
∂gβγ dxβ dxγ
= 0;
2gαβ
+ 2
−
dλ2
∂xγ
∂xα
dλ dλ

(I.6.9)that is, because of the symmetry in the indices β and γ,


β
γ
d2 xρ
1 αρ ∂gαβ∂gαγ∂gβγ
ρ dx dxρ= 0, Γβγ := g
+ Γβγ
+
−
.
dλ2
dλ dλ
2
∂xγ
∂xβ
∂xα
(I.6.10)
Lemma I.6.1 On a curve solution of the equations (6.10), it holds thatgαβ (x(λ))ẋα ẋβ = constant(I.6.11)

Causality15

16

Riemannian and Lorentzian geometry

Proof. Equation (6.11), as can easily be checked directly, is a special
case of the energy equality satisﬁed by solutions of the Euler equationsof a Lagrangian:∂L αẋ = constant.
∂ ẋα

L−

(I.6.12)The above lemma shows that the deﬁnition (6.10) of geodesics deﬁnes
not only geometric curves but parametrized ones. It implies that thecritical points of the functional (6.10) are also critical points of (6.7),
but the converse is not true.
Exercise I.6.3 Prove this statement.
It can be proved that a geodesic between two points that are causally
related realizes, in the Lorentzian case, a local (i.e. among nearby such
curves) maximum of distance.
Exercise I.6.4 Check this statement by drawing in a plane with Cartesian coordinates (x, t) a broken causal line joining the origin (0, 0) toa point (0, a) and comparing the Minkowskian lengths of this line andthe straight line joining the same points.

I.7

Connections

I.7.1

Linear connection

Partial derivatives of components of tensor ﬁelds are not componentsof tensor ﬁelds – they do not transform as such under a change of coordinates. Geometric derivation operators are deﬁned by endowing the
manifold with a new structure called a connection. These operators arecalled covariant derivatives and are usually denoted by ∇. They map
diﬀerentiable vector ﬁelds into 2-tensor ﬁelds and obey the following
laws:∇(v + w) = ∇v + ∇w

additivity,∇(f v) = f ∇v + df ⊗ v(I.7.1)

Leibnitz rule for a product with a function.
(I.7.2)

When acting on a scalar function, they coincide with the diﬀerentiation
operator; that is,∇f := df,

a covariant vector.

Consider in the domain U of a chart a vector ﬁeld v = v i ei in anarbitrary frame. By the previous rules, one has∇v = v i ∇ei + dv i ⊗ ei ,(I.7.3)

I.7

Connections

17and hence ∇v is determined in U by the covariant derivatives of thebasis vectors ei , 2-tensor ﬁelds that we write in the chosen frame as
h j∇ei = ωjiθ ⊗ eh .

(I.7.4)hThe ωij
are functions of the local frame on U. They are called connection coeﬃcients. The covariant derivative of a vector reads, by (7.3)
and (7.4),∇v = (∇j v i )θj ⊗ ei ,withi∇j v i = ∂j v i + ωjhvh .

(I.7.5)h
The connection coeﬃcients ωji
are not components of a tensor;13 theirtransformation law under a change of local frame results from thetensorial character of ∇v and is found to be








h
ωji
= Ahh ∂j Ahi + Ahh Ajj Aii ωjh i ,(I.7.6)

where A is the change-of-frame matrix. In the case of a change of natural
frame, it holds thatAii =∂xi.
∂xi

Exercise I.7.1 Prove the connection frame-change formula (7.6).
Show that the diﬀerence of two connections on a manifold V is a tensor
on V.
The covariant derivative of a covariant vector is deﬁned so that thecovariant derivative of the scalar v i ui is the ordinary derivative andobeys the Leibniz rule:∇j (v i ui ) =∂j (v i ui ) ≡ (∂j v i )ui + v i (∂j ui ) and∇j (v i ui ) =(∇j v i )ui + v i (∇j ui ).
It then follows from (7.5) thath∇j ui := ∂j ui − ωjiuh .

(I.7.7)The covariant derivatives of a (contravariant) vector v or a covariant
vector u in the direction of another vector w are vectors of the same
type ∇w v and ∇w u with components
wi ∇i v jandwi ∇i uj .

Remark I.7.1 The set of numbers ∇i v j or ∇i uj deﬁned as components of 2-tensors can also be interpreted for any given i as thecomponents of the vector or covector that is the derivative of v or u
in the direction of the frame vector e(i) .
The property that the covariant derivative in the direction of a vector
w of a contravariant or covariant vector is also a contravariant or covariant vector, respectively, allows us to deﬁne the covariant derivatives oftensor ﬁelds by assuming that, as is usual for derivation operators, theysatisfy the additivity condition and also obey the Leibniz rule, but here

13 They deﬁne a matrix valued 1-form
i given by ω h := ω h θ j .
ωh
i
ji

18

Riemannian and Lorentzian geometry

14 Onecannot apply the Leibniz rule
directly to tensor ﬁelds, because of the
non-commutativity of the tensor product: ∇u ⊗ v and u ⊗ ∇v do not belong
to the same vector space.

only for derivation in the direction of a given vector,14 equivalently on
representatives (i.e. components). We have, for two arbitrary tensors,∇v (S + T ) = ∇v S + ∇v T

linearity,∇v (S ⊗ T ) = ∇v S ⊗ T + S ⊗ ∇v T

Leibniz rule for tensor productand the derivation ∇v .

These general deﬁnitions give, for example for the 2-covariant, 2contravariant tensor
T := Tj hk l θj ⊗ eh ⊗ ek ⊗ θl ,∇i T := ∇ei T

the components of its covariant derivative, a 3-covariant, 2-contravariant
tensormhk
Tm hk l + ωim
Tj mk l + ωim
Tj hm l − ωilm Tj hk m .
∇i Tj hk l = ∂i Tj hk l − ωijRemark I.7.2 Lie derivatives obey the Leibniz rule
LX (S ⊗ T ) ≡ LX S ⊗ T + S ⊗ LX T.

I.7.2

Riemannian connection

The Riemannian connection of the pseudo-Riemannian metric g is
the linear connection ω such that
• The covariant derivative of the metric is zero; that is,λλgβλ − ωαβgλγ = 0.
∂α gβγ − ωαγ(I.7.8)• The second covariant derivatives of scalar functions commute:∇α ∂β f − ∇β ∂α f ≡ 0.

(I.7.9)One says that the connection has vanishing torsion.
Theorem I.7.1 The conditions (7.8) and (7.9) determine one andonly one Riemannian connection, given byβ≡ Γβαγ + g βμ ω̃αγ,μ ,ωαγ(I.7.10)1λλλ(gμλ Cαγ− gλγ Cαμ
− gαλ Cγμ),2(I.7.11)withω̃αγ,μ ≡λare the structure coeﬃcients of the moving frame,
where the Cαγvanishing in a natural frame, while

Γβαγ ≡1 βμg (∂α gγμ + ∂γ gαμ − ∂μ gαγ ).
2(I.7.12)The quantities Γ are called the Christoﬀel symbols of the metric g;they are zero for an orthonormal frame and, more generally, when theI.8

gαβ are constant. In a natural frame, the connection coeﬃcients reduceto the Christoﬀel symbols, which then read


∂gγμ
1
∂gαμ
∂gαγ
Γβαγ ≡ g βμ+−.
(I.7.13)
2
∂xα
∂xγ
∂xμ


Applying the general formula (7.6) to a change (xα ) → (xα ) of local
coordinates in a Riemannian connection gives
 δ


∂x
∂xβ ∂xα ∂xγ β
∂xβ ∂
β 
Γα γ  =.
(I.7.14) Γαγ +

β
α
γ
δ
α
∂x ∂x ∂x
∂x ∂x
∂xγ 
Proposition I.7.1 By a suitable choice of local coordinates, the connection coeﬃcients can be made to vanish at any one given point, andeven along any one given curve.
Exercise I.7.2 Show that these properties hold.
This proposition is very useful to remember when computing tensorial
expressions containing covariant derivatives.
Exercise I.7.3 Prove that a Killing vector ﬁeld satisﬁes the equations∇i Xj + ∇j Xi = 0.

I.8

Geodesics, another deﬁnition

I.8.1

Pseudo-Riemannian manifolds

Parallel transport: a vector ﬁeld v is said to be parallely transported
along a curve λ → x(λ) with tangent vector u if it satisﬁes along thiscurve the diﬀerential equation
dxα.
(I.8.1)dλA diﬀerentiable curve is called a geodesic if its tangent vector isparallely transported:uα ∇α v β = 0,uα =uα ∇α uβ = 0;
that is, in local coordinates, ifdxα ∂uβββ+ Γαλ u=0dλ ∂xα(I.8.2)(I.8.3)or, equivalently,
α
γ
d2 xβ
β dx dx
= 0.
+
Γ
αγ
dλ2
dλ dλ
Lemma I.8.1 The scalar(I.8.4)dxα dxβdλ dλis constant along a geodesic with parallely transported tangent uα .
uα uα ≡ gαβ

Geodesics, another deﬁnition

19

20

Riemannian and Lorentzian geometry

Proof. Along a geodesic, it holds thatuα uβ ∇α uβ ≡ uα ∇α (uβ uβ ) ≡d(uβ uβ ) = 0.
dλIt should be noted that the given deﬁnition of geodesics concerns parametrized curves—the property (8.2) is not conserved under a changeof parameter; indeed, if λ = f (λ), uα := dxα /dλ and uα := dxα /dλ ,
then

2
d2 xα
duα
d2 xα dλdxα d2 λ
≡
=
+
,
2
dλ
dλ2
dλ
dλ (dλ)2(dλ )and (8.4) becomes  2  2 α

α
γ
d x
dλ
dxα d2 λβ dx dx
+
Γ
= 0,
+
αγ
dλ
(dλ )2dλ dλdλ (dλ)2(I.8.5)or, equivalently,uα ∇α uβ + c(λ)uα = 0withc(λ) :=

d2 λ 
(dλ)2



dλ
dλ

2
.

Therefore, uα is parallely transported only if d2 λ /(dλ)2 = 0, i.e. λ =aλ + b; that is, λ and λ are linked by an aﬃne transformation. The
parameter λ on a geodesic, with uα ≡ dxα /dλ parallely transported,is called a canonical aﬃne parameter. The geodesics considered in
Section I.6.4 are canonically parametrized.

I.8.2

Riemannian manifolds

In a Riemannian manifold, a geodesic joining two points attains theminimum length among neighbouring curves joining these points.

I.8.3

Lorentzian manifolds

Since uα uα is constant along a geodesic, the timelike, null or spacelike
character of a geodesic is the same along the whole curve. The canonicalparameter λ is proportionnal to arc length if the curve is timelike.
In contrast to the Riemannian case, we have the following theorem.
Theorem I.8.1 In a Lorentzian manifold, a timelike geodesic joining two points attains the maximum length among neighbouring curves
joining these points.

I.9

Curvature

I.9.1

Deﬁnitions

The non-commutativity of covariant derivatives is a geometrical property of the metric, manifesting itself via its Riemannian connection.

I.9

Curvature21

The commutation (∇α ∇β − ∇α ∇β )v λ of two covariant derivatives ofa vector v is a 2-covariant, 1-contravariant tensor that is found by calculus to depend linearly on v, and not on its derivatives; that is, there
exist coeﬃcients Rαβ λ μ such that(∇α ∇β − ∇β ∇α )v λ ≡ Rαβ λ μ v μ .

(I.9.1)

15

These coeﬃcients are the components of a tensor, Riem(g), antisymmetric in its ﬁrst two indices, called the Riemann curvature tensor.16
The identity (9.1) is called the Ricci identity.
It can be proved17 that a manifold with vanishing Riemann curvature
is locally ﬂat.
Straightforward calculus using the expressions for the components innatural coordinates of the covariant derivative of a vector shows thatthe components of the Riemann tensor in natural coordinates are

15 Because the left-hand side is a tensor

and v is an arbitrary vector.
16 Note that the sign convention for the
Riemannian curvature tensor in (I.9.1)
is that used in CB-DMII; an alterative
convention is used in CB-DMI.
17 See CB-DMI, V B 2.

∂ λ
∂ λ
Γβμ −
Γ + Γλαρ Γρβμ − Γλβρ Γραμ .
(I.9.2)α∂x∂xβ αμThe Ricci tensor Ricci(g) is deﬁned by the contraction α = λ of the
Riemann tensor and has components
Rαβ λ μ ≡

∂ α
∂ α
ρ
λ
Γβμ −
Γ + Γρβμ Γααρ − Γβρ Γμλ .
α∂x∂xβ αμThe scalar curvature is
Rβμ ≡R := g αβ Rαβ .
18

The Einstein tensor(I.9.3)(I.9.4)18 Denoted G
αβ by some authors.

is1
Sαβ := Rαβ − gαβ R.
2Exercise I.9.1 Show that in an arbitrary moving frame,
ρ
ρ
α
α
α ρ
α
α
− ∂μ ωλβ+ ωλρ
ωμβ − ωμρ
ωλβ
− ωρβ
Cλμ
.
Rλμ α β ≡ ∂λ ωμβ(I.9.5)(I.9.6)

Remark I.9.1 The Riemann curvature tensor is an exterior 2-formtaking values in the set of linear maps from the tangent plane to itself.

I.9.2

Symmetries and antisymmetries

The Riemann tensor is obviously antisymmetric in its ﬁrst pair of indices.
It is straightforward to show that when written in full covariant form
Rαβ,γμ := gγλ Rαβ λ μ ,it is also antisymmetric in its last pair of indices and satisﬁes the so-called
algebraic Bianchi identity
Rαβ,λμ + Rβλ,αμ + Rλα,βμ ≡ 0.

(I.9.7)This identity can be used to show that the Riemann tensor is invariant
under the interchange of these two pairs. One says that it is a symmetricdouble 2-form.
The Ricci and Einstein tensors are symmetric 2-tensors.

22

Riemannian and Lorentzian geometry

I.9.3

Diﬀerential Bianchi identity and contractions

The deﬁnition of the Riemann tensor implies the diﬀerential Bianchi
identity∇α Rβγ,λμ + ∇β Rγα,λμ + ∇γ Rαβ,λμ ≡ 0.

(I.9.8)Contracting this identity gives the identity∇α Rβγ α μ − ∇β Rγμ + ∇γ Rβμ ≡ 0,

(I.9.9)and a further contraction gives the following identity satisﬁed by the
Einstein tensor:∇α S αβ ≡ 0.

(I.9.10)

This identity, called the conservation identity, plays a fundamentalrole in the choice of the Einstein equations.

I.10

Geodesic deviation

An important phenomenon that signals to observers the curvature ofspacetime is its inﬂuence on the distance between nearby geodesics.
Let Cσ be a one-parameter family of geodesics with canonical parameter denoted by s, Cσ (s) := ψ(σ, s). Denote by v = ∂ψ/∂s thetangent vector to Cσ and by X = ∂ψ/∂σ the vector that characterizes
the inﬁnitesimal displacement of Cσ .
Lemma I.10.1 These two vector ﬁelds commute; that is,v α ∇α X β − X α ∇α v β = 0(I.10.1)

Proof. The relation is tensorial and pointlike. It is suﬃcient to prove it
at an arbitrary point in particular coordinates. Choose coordinates suchthat at that point the Christoﬀel symbols vanish. Thenv α ∇α X β − X α ∇α v β =∂ 2 ψβ∂ 2 ψβ
−
= 0.
∂s∂σ
∂σ∂s



Theorem I.10.1 The rate of acceleration of distance between nearby
geodesics is linked with the curvature by the equation

19 Onealso says that the right-handside is the relativistic tidal force. Thisterminology is inspired by the fact thattides on Earth are due to the diﬀerential of the gravitational attraction (theHessian of the gravitational potential in
the Newtonian approximation) mainlyof the Moon, on the seas.

∇2v2 X β ≡D2 β
X = X α v λ v μ Rλα β μ ,Ds2(I.10.2)

called the equation of geodesic deviation.19
Proof. Diﬀerentiating the relation (10.1) in the direction of v gives, using
the parallel transport of v,
v λ v α ∇λ ∇α X β − v λ ∇λ X α ∇α v β − X α v λ ∇λ ∇α v β = 0.

(I.10.3)

I.11

Linearized Ricci tensor

23

Hence, using the Ricci identity,v λ v α ∇λ ∇α X β − v λ ∇λ X α ∇α v β − X α v λ (∇α ∇λ v β + Rλα β μ v μ ) = 0.
(I.10.4)
and hence, by the parallel transport of v,
v λ v α ∇λ ∇α X β − v λ ∇λ X α ∇α v β + X α ∇α v λ ∇λ v β − X α v λ Rλα β μ v μ = 0.
(I.10.5)
The commutation (10.1) of v and X shows that this equation simpliﬁes
tov λ v α ∇λ ∇α X β = X α v λ v μ Rλα β μ ,(I.10.6)

which can also be written as∇2v2 X β ≡D2 β
X = X α v λ v μ Rλα β μ .
Ds2(I.10.7)


I.11

Linearized Ricci tensor

The linearization,20 also called ﬁrst variation, of an operator
P : u → P (u) between open sets of normed vector spaces E1 and E2 ,at a point u ∈ E1 , is a linear operator acting on vectors δu ∈ E2 givenby the (Fréchet) derivative Pu of P at u, that is, such thatδP := Pu δu,with

P (u + δu) − P (u) = Pu (u)δu + o(|δu|). (I.11.1)

We consider a pseudo-Riemannian metric, with components gαβ inlocal coordinates. The relation between gαβ and the inverse matrix g αβ ,g αλ gαβ = δβλ ,the Kronecker symbol,(I.11.2)

where we have set

hλμ := δgλμ .
(I.11.3)

implies thatδg αβ = −hλμ := −g αλ g βμ hλμ ,The deﬁnition and the above relation applied to the Christoﬀel symbols and the Ricci tensor gives by straightforward computation thatδΓλαβ is the following 3-tensor (indices raised with g λμ ):1
(I.11.4)∇α hλβ + ∇β hλα − ∇λ hαβ .
2From this formula and the expression for the Ricci tensor, it follows that
δRicci(g) is the linear operator on h := δg given in local coordinates byδΓλαβ ≡11∇λ ∇α hλβ + ∇λ ∇β hλα − ∇α ∇β hλλ . (I.11.5)
δRαβ ≡ − ∇λ ∇λ hαβ +22From this identity, there results the linearization of the scalar curvature:δR ≡ g αβ δRαβ + Rαβ δg αβ ,20 See, for instance, CB-DMI, II A, in
particular Problem 1.

24

Riemannian and Lorentzian geometry

where g αβ δRαβ is the divergence of a vector, namely,λα
g αβ δRαβ ≡ −∇λ ∇λ hα.
α − ∇α hI.11.1(I.11.6)

Linearized Bianchi identities

The contracted Bianchi identities∇α S αβ ≡ 0
implyδ(∇α S αβ ) ≡ 0;that is,∇α (δS αβ ) + S αλ δΓβαλ + S λβ δΓααλ ≡ 0,with
1
δS λμ ≡ δRλμ − (g λμ δR + hλμ R) ≡ 0.
2

I.12

Physical comment

We shall see in the following chapters that in Relativity (and alreadyin Newton’s theory) a model for reality at a macroscopic scale is basedon diﬀerentiable manifolds, geometric objects whose elements are called
points. Each point of, let us say, a 3-manifold is represented by a familyof sets of three numbers, each set being the coordinates of the point in areference frame; diﬀerent elements of the family are linked by an equivalence relation between reference frames. The physically realistic problem
is to link the abstract reference frames with a concrete observable one.
We will return to this subject in the following chapters.

I.13

Solutions of selected exercises

Exercise I.3.2 Image of a vector ﬁeld
Let f : V → W by x → y := f (x), and denote by (y α ) coordinates on
W and by (xi ) coordinates on V. The diﬀerential of f at x is a linear
map between Tx V and Ty W , denoted by f  , such that(f  v)α (y) ≡∂f i iv (x(y)).
∂xj

It is not a vector ﬁeld on W if the inverse mapping of f, f −1 : y → x(y),is not deﬁned on W.

I.13

Exercise I.3.4 Components of Lie derivatives
For a covariant 2-tensor, the components of the Lie derivative are(LX T )jh = X i

∂Tjh
∂X i
∂X i
+
T
+
T
.
ih
ji
∂xi
∂xj
∂xh

(I.13.1)



Under the change of coordinates xi = φi (y j ), we have X i = Aij  X j ,with
Aij  :=


∂
∂xi
∂
= Aji
,
 and
j
i
∂y
∂x
∂y j 

with (Aji ) the inverse matrix of (Aij  ). Straightforward computation
gives




(LX T )jh = Aim X m Ani

im∂(Apj Aqh Tp q ) ∂(A  X)m
+ Ani Aqh Tn q Apj


n
p
∂y
∂y


i
m
)
p n
h ∂(Am X+ Aj Ai Tp n Aq
.
q

∂y





Since Aji Aih = δhj  , the terms containing no derivatives of the A are
easily seen to reduce to
Apj Aqh



∂X m
∂X m
m ∂Tp q + Tm q+ Tp m
X
∂y m
∂y p
∂y q





≡ Apj Aqh (LX T )p q .
(I.13.2)

The terms containing derivatives of the A are








i
∂(Apj Aqh ) ∂A 
m
X Tp q+ Ani Aqh Tn q Apj X m

m
∂y
∂y p
i



 ∂A 
m+ Apj Ani Tp n Aqh X m
,
∂y q
m

which can be writteni∂(Apj Aqh )p m ∂Amn q
 q A X
X Tp q
+
A
A
T
n

i
j
h
∂y m
∂y p
i



 ∂A 
m+ Apj Ani Tp n Aqh X m
.
∂y q
m

Renaming indices then givesTp q X

m



i
i
∂(Apj Aqh ) ∂A 
p q  n ∂Amm
+ Api Aqh Anj + Aj Ai Ah
m
n
n
∂y
∂y
∂y 

Since partial derivatives commute, it holds that∂Aim
∂ 2 xi
∂Ain
≡
=
,



∂y n
∂y n ∂y m
∂y m

≡ 0.

Solutions of selected exercises

25

26

Riemannian and Lorentzian geometry

and we can writeTp q 

X

m



i
i
∂(Apj Aqh )p q  n ∂Anp q  n ∂An
+
A
A
A
+
A
A
A


j
h
i
j
i
h
∂y m
∂y m
∂y m

≡ 0.



But, since Ain and Anj are elements, of inverse matrices,∂Anj= −Api Ain mi ∂A nApi Anj
m

∂y

∂y



∂Apj
= − m .
∂yAn analogous computation replacing j by h completes the proof that

(LX T )jh ≡ Apj Aqh (LX T )p q .

Exercise I.3.5 Lie derivative of exterior forms
On a two-dimensional manifold, the maximal order of an exterior formis 2:
1
ω := ωij dxi ∧ dxj ≡ ω12 dx1 ∧ dx2
2
The general formula for the Lie derivative of a covariant tensor gives for
LX ω the 2-form


∂X h
∂
h ∂
ω12 + ω12
(X h ω12 ),(I.13.3)(LX ω)12 = X
≡
h
h
∂x
∂x
∂xh
while
iX ω = ω12 (X 1 dx2 − X 2 dx1 ),dω ≡ 0.

Hence,d(iX ω) =∂(ω12 X 1 ) ∂(ω12 X 2 ) 1+
dx ∧ dx2
∂x1
∂x2

and
LX ω ≡ d(iX ω) ≡ d(iX ω) + iX dω.

Exercise I.5.1 Kronecker symbol and contravariant
components of g
A mixed tensor X with components Xji := δji in coordinates xi has thefollowing components in coordinates xi :Xji ≡∂xi ∂xj iδ .
∂xi ∂xi j

Hence,


Xji ≡∂xi ∂xi= δji  ,∂xi ∂xi

because ∂xi /∂xi and ∂xi /∂xi are elements of inverse matrices.

I.13

The elements g ih of the inverse of the metric gij are such thatg ih gjh = δji ,and in another coordinate system they are such that

g i h gj  h = δji ;that is,




gi h

∂xj ∂xh
gjh = δji .
∂xj  ∂xhHence,
g ih = g i h∂xj ∂xh,∂xj  ∂xhand the g ih are the components of a contravariant tensor.

Exercise I.7.1 Connection frame-change formulaLet

v i = Aii v i ,i∇j v i = ∂j v i + ωjh
vh ,








∇j  v i = ∂j  v i + ωji  h v h .

We have ∂j = Ajj ∂j  , with Ajj and Ajj  inverse matrices, and hence






i
∇j v i = Ajj ∂j  (Aii v i ) + ωjhAhh v h










i
= Ajj Aii ∂j  v i + (Ajj ∂j  Aii )v i + ωjhAhh v h .

But, since ∇v is a mixed tensor,












∇j v i = Aii Ajj ∇j  v i = Aii Ajj (∂j  v i + ωji  h v h ).
Identifying these two expressions for ∇j v i gives the connection changeof-frame formula and the tensorial law for the diﬀerence of twoconnections, because the v h are arbitrary numbers.

Exercise I.7.3 Killing equations
A Killing vector X of a metric g is such that
(LX g)hk ≡ X i ∂i ghk + gih ∂k X i + gki ∂h X i = 0.
Using ∇i ghk ≡ 0 gives the result by straightforward computation.

Solutions of selected exercises

27

28

Riemannian and Lorentzian geometry

I.14

Problems

I.14.1

Liouville theorem

1. Let (V, g) be a pseudo-Riemannian manifold with tangent space T V.
Prove (Liouville theorem) that the volume form θ in T V given byθ := |Detg|dx0 ∧ dx1 ∧ . . . ∧ dxn ∧ dp0 ∧ dp1 ∧ . . . ∧ dpn
is invariant under the geodesic ﬂow with tangent vector X; that is,with LX denoting the Lie derivative with respect to X, it holds that
LX θ = 0.

(I.14.1)

2. Show that iX θ is a closed form.

Solution
In local coordinates (xA ) := (xα , pβ ) on T V, the Lie derivative of theexterior 2(n + 1)-form θ with respect to the vector ﬁeld X = (p, G), thatis, (X A ) = (pα , Gᾱ ), reads, using the fact that the components of θ do
not depend on p,
∂
∂X A
θ
+
θ
01...n,n+1...2(n+1)
A1...2(n+1)
∂xα
∂x0
A
A
∂X
∂X
+ θ0A...2(n+1)+ . . . + θ01...(2n+1)A 2(n+1) .
∂x1∂x(I.14.2)(LX θ)01...n,n+1...2(n+1) = pαThe expression for θ gives that∂
θ01...nn+1...2(n+1) = g λμ
∂xα




∂
g
λμ θ01...nn+1...2(n+1) ;∂xα(I.14.3)

on the other hand, since θ is antisymmetric, the same index cannot
appear twice in its components; therefore, the second line of (I.14.2) isequal to (recall that the components pα of X do not depend on xα )θ01...2(n+1)

∂X A
∂Gα
= θ01...2(n+1) α .
A∂x∂p(I.14.4)

The expression for G gives that∂Gα= −2Γλλα pα .
∂pα(I.14.5)

The result LX θ = 0 follows from the expression for the Christoﬀelsymbols.
2. The 2(n + 1)-exterior form θ is closed (dθ ≡ 0) on T V, a 2(n + 1)dimensional manifold. The equality LX θ ≡ iX dθ + diX θ = 0 implies
that diX θ = 0.

I.14

I.14.2

Problems

29

Codiﬀerential δ and Laplacian of an exteriorform

The codiﬀerential, denoted by δ, of an exterior p-formω :=1
ωi ....i dxi1 ∧ . . . ∧ dxip
p! 1 pon a pseudo-Riemannian manifold is deﬁned to be the exterior (p − 1)
-form
−1
δω :=
∇i ωij1 ....jp−1 dxj1 ∧ . . . ∧ dxjp−1 .
(p − 1)!
1. Show that δ and the exterior diﬀerential d are adjoint operators inthe sense that for forms with compact support in an open set of Rn ,
θ of degree p + 1 and ω of degree p, it holds that(ω, δθ) = (δω, dθ),where the scalar product of p-forms in the metric g is deﬁned by(ω, θ) =ωi1 ....ip ψ i1 ...ip μg ,Rnwith μg the volume element of g.
2. Show that Δ := −(δd + dδ), called the Laplacian,21 is an operator
from p-forms into p-forms given by (a hat means that the index isabsent from the sequence)Rihq ωhi1 ...ı̂q ...ip(Δω)i1 ...ip ≡ g jh ∇j ∇h ωi1 ...ip −
−2



1≤q≤p
h

j

R iq ir ωjhi1 ..;ı̂r ...ı̂q ...ip .

r<q≤p

If g is Lorentzian, then δd + dδ is usually denoted by g and called
the wave operator of g (the d’Alembertian if g is the Minkowski
metric η).

Solution
Take a 1 form ω = ωα dxα ; thendω =1(∇β ωα − ∇α ωβ )dxβ ∧ dxα , δω = −∇α ω α ≡ −∇α ωα2(δdω)β := −∇α (∇α ωβ − ∇β ωα ), (dδω)β = ∇β ∇α ωα .
Therefore, the components of the Laplacian of ω, a 1-form like ω, are(Δω)β ≡ ∇α ∇α ωβ − ∇α ∇β ωα + ∇β ∇α ωα ;that is, using the Ricci formula for commutation of covariant derivatives,(Δω)β ≡ ∇α ∇α ωβ − Rα βα λ ωλ ≡ ∇α ∇α ωβ − Rβ λ ωλ .
The general result is obtained similarly by straightforward computation.

21 Note that the Laplacian is sometimes (for instance in CB-DMI) deﬁned
without the minus sign:  := δd + dδ.

30

Riemannian and Lorentzian geometry

I.14.3

Geodesic normal coordinates

The geodesics arcs on a pseudo-Riemannian manifolds (V, g) are, in thetangent bundle T V , solutions of a ﬁrst-order diﬀerential system that
reads in local coordinates (xα , v α )
dxα
dv α
β γ
= vα ,= −Γαβγ v v .
dλdλ(I.14.6)

Classical theorems on such diﬀerential systems say that if the coeﬃcientsare Lipschitzian, i.e. if the metric is C 1,1 , then this system has one andonly one solution, deﬁned in a neighbourhood of λ = 0; taking givenααvalues for λ = 0, xα (0) = xα
0 , v (0) = a .
1. Show that if the functions
xα := φα (λ, aα )satisfy the system (14.6) with φα (0, aα ) = 0, ∂φα /∂λ(0, aα ) = aα ,
then


aα
α
α
α
x = xk := φ kλ,,k a real number.
k
Hint: Show that xα and xα
k satisfy the same diﬀerential system andtake the same initial value.
2. Deduce from the result of Part 1 that there exists a neighbourhoodof xα
0 such that the functions
xα = φα (1, y α )deﬁne an admissible change of coordinates (xα ) → (y α ). Show
that in the coordinates y α , called normal geodesic coordinates,
the geodesic arcs issuing from x0 are represented by straight lines,
λ → y α = aα λ. Show in particular that in these coordinates theChristoﬀel symbols vanish at x0 .

I.14.4

Cases d = 1, 2, and 3

One-dimensional manifolds are locally isometric with straight lines.
For two-dimensional manifolds, the Riemann tensor has only (up tosign) one non-zero component, R12,12 . What are the Ricci tensor andthe scalar curvature?
Show that in dimension d = 3, the Riemann tensor is linear in the
Ricci tensor (see Problem I.14.7 on the Weyl tensor).

I.14.522 Bel (1958).

Wave equation satisﬁed by the Riemann tensor

Deduce from the Bianchi identities a system of semilinear wave equations
satisﬁed by the Riemann tensor22 when the Ricci tensor is equal to agiven tensor ρ.

I.14

Problems

31

Solution
The Riemann tensor satisﬁes the Bianchi identities∇α Rβγ,λμ + ∇γ Rαβ,λμ + ∇β Rγα,λμ ≡ 0.

(I.14.7)

These identities together with the Ricci identity imply that∇α ∇α Rβγ,λμ + ∇γ ∇α Rαβ,λμ + ∇β ∇α Rγα,λμ + Sβγ,λμ ≡ 0,

(I.14.8)

where Sβγ,λμ is a homogeneous quadratic form in the Riemann tensor:
Sβγ,λμ ≡ {Rγ ρ Rρβ,λμ + Rα γβ ρ Rαρλμ + [(Rα γλ ρ Rαβ,ρμ )− (λ → μ)]} − {β → γ}.

(I.14.9)On the other hand, the Bianchi identities imply by contraction∇α Rβγ,αμ + ∇γ Rαβ,α μ + ∇β Rγα,αμ ≡ 0,

(I.14.10)

which can be written, using the symmetry Rαβ,λμ ≡ Rλμ,αβ , as∇α Rα β,λμ + ∇μ Rλβ − ∇λ Rμβ ≡ 0,

(I.14.11)

and the identities become∇α ∇α Rβγ,λμ + Sβγ,λμ + {∇γ (∇μ Rλβ − ∇λ Rμβ ) − (β → γ)} ≡ 0,
(I.14.12)
If the Ricci tensor Rαβ satisﬁes the Einstein equations
Rαβ = ραβ ,(I.14.13)

then the previous identities become the following quasidiagonal systemof semilinear wave equations23 for the Riemann tensor:∇α ∇α Rβγ,λμ + Sβγ,λμ + Jβγ,λμ = 0,

(I.14.14)

23 These equations are analogous to theMaxwell equations for the electromagnetic 2-form F .

where Jβγ,λμ depends on the sources ραβ and is zero in vacuum:Jβγ,λμ ≡ ∇γ (∇μ ρλβ − ∇λ ρμβ ) − (β → γ).

I.14.6(I.14.15)

The Bel–Robinson tensor

The Bel–Robinson tensor associated with the Riemann tensor ona four-dimensional pseudo-Riemannian manifold was deﬁned by Bel24
through the use of the left and right adjoints of the Riemann tensor
deﬁned by(∗ Ri)αβ,λμ ≡1ρσ
ηαβρσ Rλμ,2(R∗ )αβ,λμ ≡1ρσ
ηρσρσ Rαβ,2with ηαβλμ the volume form of the spacetime metric.

(I.14.16)

24 Bel (1959).

32

Riemannian and Lorentzian geometry

1. Prove that the left and right adjoints of the Riemann tensor are equal.
2. Prove the Lanczos identity(Ri +∗ Ri∗ )αβ,λμ ≡ Sαλ gβμ + Sβμ gαλ − Sαμ gβλ − Sβλ gαμ , (I.14.17)
with S the Einstein tensor
1Sαλ ≡ Rαλ − gαλ R.
(I.14.18)
2
3. The Bel–Robinson tensor Q is the fourth-order tensor given by1  αρ,λσ β μRRρ σ + 2(∗ R)αρ,λσ (∗ R)βρ σμ
Qαβλμ ≡
4+(∗ R∗ )αρ,λσ (∗ R∗ )βρ σμShow that it is symmetric in its two ﬁrst and two last indices, andunder commutation of these pairs of indices.
4. Show that the Bel–Robinson tensor of a vacuum Einstein spacetime,possibly with cosmological constant (Sαβ = Λgαβ ), is conservative;
namely, that it satisﬁes the equation∇α Qαβλμ = 0.

(I.14.19)

It can be proved that the contraction of the Bel–Robinson tensor withtimelike vectors is positive, thus giving the deﬁnition of a positive energy
density for the Riemann tensor. This is used in some existence proofs(see Chapter VIII).

Solution
Parts 1–3 follow by straightforward computation (easier to do in an
orthonormal frame).
Part 4 uses the Bianchi identities.

I.14.7

The Weyl tensor

Two metrics g and g̃ on a manifold V are called conformal if there is apositive scalar function Ω ≡ e2Φ on V such thatg̃ ≡ e2Φ g, i.e. g̃αβ = e2Φ gαβ ,g̃ αβ = e−2Φ g αβ .

(I.14.20)

Two conformal Lorentzian metrics have the same light cones and hencedeﬁne the same causal structures.
1. Show that the Christoﬀel symbols of two conformal metrics are linked
by the relationsααΓ̃αβγ = Γβγ + Sβγ ,(I.14.21)

where S is the tensorλΓ̃λβμ − Γλβμ := Sβμ≡ δμλ ∂β Φ + δβλ ∂μ Φ − gβμ ∂ λ Φ.

(I.14.22)

I.14

Problems

33

2. Show that the Riemann tensors of g and g̃ are linked byR̃αβ λ μ − Rαβ λ μ ≡ δβλ Φαμ − δαλ Φβμ + gαμ Φβ λ − gβμ Φα λ+ (δβλ gαμ − δαλ gβμ )∂ρ Φ∂ ρ Φ,withΦαβ := ∇α ∂β Φ − ∂α Φ∂β Φ.
3. Show that the Ricci tensors Rβμ and R̃βμ on a d-dimensional manifold
are linked byR̃βμ − Rβμ = − gβμ ∇α ∂α Φ − (d − 2)∇μ ∂β Φ + (d − 2)× ∂μ Φ∂β Φ − gβμ ∂ λ Φ∂λ Φ(I.14.23)

and the scalar curvatures by
e2Φ R̃ − R = −2(d − 1)∇α ∂α Φ − (d − 2)(d − 1)∂ λ Φ∂λ Φ.

(I.14.24)

4. Deﬁne the Weyl tensor25 by the relation1(gβδ Rαγ − gαγ Rβδ + gβγ Rαδ − gαδ Rβγ )d−2− (gβγ gαδ − gαγ gβδl )R.

Wαβ,γδ = Rαβ,γδ −Check that the Weyl tensor has the same symmetries as the
Riemann tensor and has zero trace.
Show that the components Wαβ λ μ are the same for two conformal
metrics.
Since W is obviously zero for a ﬂat metric, it is also zero if themetric is conformal to a ﬂat metric.26
5. Show that the Weyl tensor of a three-dimensional pseudo-Riemannian
manifold (M, g) is identically zero. Show that a three-dimensional
pseudo-Riemannian manifold with Ricci tensor identically zero is
locally ﬂat.

I.14.8The Cotton–York tensor

It can be proved that a 3-manifold is locally conformally ﬂat if and only if
its Cotton tensor vanishes. This tensor is a 3-tensor with components
given in terms of the Ricci tensor and the scalar curvature of a metricgij by11
Ckij: := ∇i Rjk − gjk R − ∇j Rik − gik R .
(I.14.25)
44Deﬁne a 2-tensor, called the Cotton–York tensor, by
Y ij := −εikl g mj Cmkl .
Show that Y is symmetric, traceless and transverse, that is, such that∇i Y ij = 0.

25 It is equal to the Riemann tensor ina Ricci-ﬂat space (Rαβ ≡ 0). It is considered that the Weyl tensor embodies
in some sense the non-Newtonian properties of the gravitational ﬁeld, in particular its radiation properties. This
point of view is supported by the factthat the equations for massless ﬁelds,
at least in four spacetime dimensions,are conformally invariant.

26 It can be proved that if d > 3, thenthe identical vanishing of the Weyl tensor implies that the metric is locallyconformally ﬂat.

34

Riemannian and Lorentzian geometry

I.14.9Linearization of the Riemann tensor

Write the linearization of the Riemann tensor as a geometric secondorder operator on hαβ := δgαβ .

Solution
∂ λ
∂ λ
Γ −
Γ + Γλαρ Γρβμ − Γλβρ Γραμ ,(I.14.26)∂xα βμ ∂xβ αμ∂∂δΓλ + Γλαρ δΓρβμ + (δΓλαρ )ΓρβμδRαβ λ μ ≡ α δΓλβμ −∂x∂xβ αμ− δΓλβρ Γραμ − Γλβρ (δΓραμ ),(I.14.27)

δRαβ λ μ ≡where δΓλαβ is the tensorλXαβ:= δΓλαβ ≡1
∇α hλβ + ∇β hλα − ∇λ hαβ .
2(I.14.28)

Therefore, δRαβ λ μ is the tensorλλδRαβ λ μ ≡ ∇α Xβμ− ∇β Xαμ.

I.14.10

Second derivative of the Ricci tensor

The second (Fréchet) derivative, also called the second variation, of
an operator P at u ∈ E1 is, if E1 is a Banach algebra, a quadratic form
Pu2 in δu such that
δ 2 P := Pu2 (δu, δu),(I.14.29)

with1
P (u + δu) − P (u) = Pu (u)δu + Pu2 (u)(δu, δu) + o(|δu|2 ). (I.14.30)
2Compute the second variation of the Ricci tensor at a metric g as aquadratic form in h := δg :
δ 2 Rαβ := Rαβ(g)(h, h).

(I.14.31)

Solution
27 Choquet-Bruhat (2000).

Straightforward though lengthy computation gives27
δ 2 Rαβ ≡ −hλμ [∇λ (∇α hβμ + ∇β hαμ − ∇μ hαβ ) − ∇α ∇β hλμ ]1−∇λ hλμ (∇α hβμ + ∇β hαμ − ∇μ hαβ ) + ∇β hλμ ∇α hλμ21+ ∇λ hρρ (∇α hβλ + ∇β hαλ − ∇λ hαβ )2+∇λ hμα ∇λ hβμ − ∇λ hμα ∇μ hλβ .
(I.14.32)

II

Special relativity
II.1

Introduction

Special Relativity, formulated by Enstein in 1905, revolutionized ourconceptions of time and space.1

II.2

Newtonian mechanics

II.2.1The Galileo–Newton Spacetime

In the spacetime of Galileo and Newton, the simultaneity of two events
is a notion independent of observers. Space and time are absolute objects,2 which exist independently of matter and events that may happen
in them. The mathematical model of Newton spacetime is the direct
product of R, where time varies, and space, which is assumed to be a
Euclidean space E 3 , that is, R3 with the Euclidean metric, which reads,
in natural frames that mathematicians call ‘Cartesian’ and physicists‘inertial’,(dxi )2 .
(II.2.1)
ds2 =II.1 Introduction
II.2 Newtonian mechanics

35
35

II.3 The Lorentz andPoincaré groups38
II.4 Lorentz contraction anddilation39
II.5 Electromagnetic ﬁeld
and Maxwell equationsin Minkowski
spacetime M 440
II.6 Maxwell equations inarbitrary dimensions
42
II.7 Special Relativity
II.8 Some physicalcomments

43
45

II.9 Dynamics of a pointlikemass
46
II.10 Continuous matter
49
II.11 Problems
54
1 See

e.g. the historical discussion in
Damour (2006).

i=1,2,3To link this mathematical model with observations, one has to identify an inertial frame of this Euclidean space with observed objects (seeSection II.2.3).
The isometry group of the Euclidean model of space permits the existence of solid bodies remaining isometric to themselves under motions.
In Galileo–Newton spacetime, lengths can be measured by comparison
with a standard metre, a piece of metal deposited in Sèvres. For distances that are too large to be compared directly with a copy of thestandard metre, one can use properties of Euclidean geometry, such as
triangulation. The existence of clocks measuring absolute time is predicted by the periodic phenomena resulting from Newtonian dynamics.3

2 Newton’scontemporary Leibniz wasin disagreement with this postulate.

3 The small oscillations of a pendulum

are approximately such a phenomenon.

II.2.2

Newtonian dynamics. Galileo group

The fundamental law of Newtonian dynamics for a particle assumed to
be pointlike is
F = mγ,(II.2.2)

36

Special relativity

where m is a constant, the mass of the particle (called inertial mass), γ
is its acceleration in absolute time with respect to absolute space, andF is a phenomenological vector, the applied force. The components of γ
in Cartesian coordinates xi for the particle in the absolute space E 3 are
the second partial derivatives γ i = d2 xi /dt2 .
Newton’s law (2.2) is invariant under the following time-dependentchange of coordinates in Newton’s absolute space E 3 :xi = xi + v i t + xi0 ,(II.2.3)

where the v i and xi0 are constants. The corresponding t-dependent Cartesian coordinates system in E 3 is in uniform translation with respect tothe original absolute space. All such reference frames are inertial frames.
The set of transformations (2.3) forms a group, called the Galileo
group. It had already been remarked by Galileo that a uniform-in-timetranslation of a boat cannot be detected by observers in the hold. Moregenerally, all the physical laws of Newtonian mechanics are assumed to
be invariant under the Galileo group, in the sense that they admit thesame formulation in all inertial frames.

II.2.34 Mach’s considerations partly inspired
Einstein’s General Relativity.
5 In Paris, 31 h 50 m, as can be veriﬁed
by a visit to the pendulum exhibited at
the Musée des arts et métiers or probably soon to one suspended beneath thedome of the Panthéon (its original location). This dependence results fromthe fact that the rotation of axes linked
with the Earth’s rotation with respect
to the Copernican inertial system leads
to what is called in Newtonian mechanics a complementary inertial force.
The appearance of sin λ, equal to 1 at
the North Pole, is due to the valueof the vertical (i.e. orthogonal to the
Earth’s surface) component of the rotation vector of the Earth, parallel tothe line between the poles. The horizontal component does not contributeto the complementary inertial force, because it is parallel to the velocity ofthe pendulum (see e.g. Bruhat, 1934,
p. 134).

Physical comment

Inertial frames are a physical reality. Someone seating on a disk turningwith respect to the Earth feels a ‘centrifugal force’ due the fact thataxes ﬁxed on the disk are not inertial for the Galileo–Newton spacetime.
More generally, non-inertial coordinates manifest themselves throughnon-vanishing Christoﬀel symbols of the Euclidean metric of the physical space E 3 in these axes. These non-zero Christoﬀel symbols cause
the appearance of what in Newtonian mechanics are called Coriolis orinertial forces. So the problem is posed, what is the physical origin of
the inertial frames? The answer appeared simple for Newton, namely
the a priori given absolute space and time, but the problem of identiﬁcation with observed reality was there all the same. The answer for
Mach was that ‘absolute’ space was determined by the matter content of
the universe.4 In practice, Mach and Newton agree, in an approximate
way. Experiments on Earth have shown that Newton’s laws of dynamics
written in his absolute spacetime are satisﬁed to a good approximation
if one assumes an inertial frame linked with the Earth, and to a better
approximation if the inertial frame, called a Copernican frame, has its
origin at the center of the Sun and with axes directed to speciﬁc stars.
This was observed by Foucault by recording the oscillations of a pendulum on Earth in a vertical plane that remains approximately ﬁxed
for a short time, but rotates through 360◦ in a time that depends on
the latitude λ, namely5 1 day/sin λ because of the centrifugal force dueto the rotation of the Earth. When the Solar System was known to berotating in the Milky Way, another inertial frame was considered, which
led to veriﬁcation of the existence of a further very small correction forexperiments made in the Solar System. Astronomers now know that theII.2

Milky Way itself does not stay still in the universe, but it is impossible
to determine an inertial frame for the universe if such a thing exists.
Finally, the physicists of the nineteenth century introduced a mysterious medium, called aether, invisible and intangible, though with all theproperties of a solid.6

II.2.4The Maxwell equations in Galileo–Newton
spacetime

Newtonian mechanics

37

6 Perhaps

physics is coming back to
the idea of an absolute spacetime determined by its content, everywherepresent pairs of virtual particles, but
perhaps spacetime is not a diﬀerentiable manifold, either below the Planck
length or at cosmological scales.

The Maxwell equations unify the various classical physical laws thatgovern the electric and magnetic ﬁelds (E, H). These equations in noninductive media read, in 3-vector notation on R3 ,div E = q,
div H = 0,

Coulomb’s law,(II.2.4)
Gauss’s law, non-existence of magnetic charges. (II.2.5)

The scalar q is the charge density. These equations are ‘constraints’, i.e.
they do not contain time derivatives. The following are evolution equations, with c, the speed of light, a dimensionless constant that depends
on the units of space and time (and can be made equal to 1 by choice ofa relation between these units7 ):
1 ∂H
,
c ∂t
1 ∂E
,
curl H = j +c ∂tcurl E = −Faraday’s law,(II.2.6)

Ampère–Maxwell law.

(II.2.7)

The vector j is the electric current. The term ∂E/∂t, called the ‘displacement current’, was introduced by Maxwell and led to the consideration of
the electric and magnetic ﬁelds as the splitting, in an observer-dependent
fashion, of one entity called the electromagnetic ﬁeld, into these electricand magnetic ﬁelds. The set of equations (2.4)–(2.7), still valid today,
are the shining success of nineteenth-century theoretical physics.
The various laws written above were interpreted before Einstein witht the absolute time and E 3 the absolute space determined by the
mysterious medium of the aether. The Maxwell equations imply thatelectric and magnetic ﬁelds—and also light,8 which is an electromagnetic phenomenon—propagate in vacuum with the constant velocity c,
independent of the time t and the location in this absolute space.9 The
number c can be made equal to 1 by appropriate choice of space andtime units.
The problem, for Newton’s mechanics, is that the Maxwell equations
are not invariant under the Galileo group.
It was the mathematical genius of Lorentz and Poincaré to discover thegroup that leaves invariant the Maxwell equations, but neither of them
discarded Newton’s absolute time, nor the aether ﬁlling the absolute
space.

7 See Section III.5 in Chapter III.

8 It was already known that the speed oflight is ﬁnite, and it had been measured
by Römer in 1675 from observations of
the eclipses of Jupiter’s Moons.
9 In the absence of electric current andcharge, this set of equations implies
that E and H satisfy the wave equationin the Minkowski metric
1 ∂2H
− 2
+ ΔH = 0,
c ∂t2
1 ∂2E
+ ΔE = 0,
− 2
c ∂t2
Δ :=



∂2
.
(∂ti )2
ı̂=1,2,,338Special relativity

II.3

The Lorentz and Poincaré groups

The (n + 1)-dimensional Lorentz group is the group of linear maps of
Rn+1 ,αX α = Lαα X ,

that preserves the quadratic form−(X 0 )2 +n

(X i )2 ;i=1that is, the elements Lαα of matrices L that represent the Lorentz group
n+1
α
n+1
are such that for each set {X } ∈ R, it holds that
− (L0α X α )2 +(Liα X α )2 ≡ −(X 0 )2 +(X i )2 . (II.3.1)

on R

i=1,...,ni=1,...,nThis identity with X 0 = 1, X i = 0 implies that
(L00 )2 = 1 +(Li0 )2 ≥ 1.

(II.3.2)

i=1,...n

The orthochronous Lorentz group is the subgroup that preserves
the orientation of R deﬁned by the element L00 , i.e. such that
L00 > 0.

(II.3.3)

The elements with L00 < 0 reverse the orientation of R. They do notconstitute a group.
Equation (3.1) implies that the determinant of L is equal to +1 or−1. Proper Lorentz transformations are those that preserve thespacetime orientation; i.e. they are such that
Det(L) = 1.

(II.3.4)

Particular Lorentz transformations are the space rotations(Lij  X j )2 ≡(X i )2 .
L0i = Li0 = 0,i=1,...,ni=1,...,nStandard geometrical considerations show that every proper Lorentz
transformation can be written
L = R1 LS R2 ,(II.3.5)

where R1 and R2 are space rotations and LS is a so-called special Lorentz transformation, acting only in the time and one space directions,
namely such that (3.1) reduce to(L00 )2 − (L10 )2 = 1,

(II.3.6)(L11 )2 − (L01 )2 = 1,
L00 L01 − L11 L10 = 0.

(II.3.7)(II.3.8)

II.4

Exercise II.3.1 Show that the general solution of these equations is,for an orthochronous transformation,
L00 = L11 = cosh ϕ,

L10 = L01 = sinh ϕ.

(II.3.9)

The (n + 1)-Poincaré group is the group of isometries of a ﬂatLorentzian (n + 1)-manifold, the Minkowski spacetime M n+1 . It is thesemidirect product of the translation group Rn+1 of Rn+1 and theLorentz group Ln+1 .
See Problem II.1 for the statement and proof of the invariance of theMaxwell equations written for a pair of vectors E, H.

II.4

Lorentz contraction and dilation

Before the new physics introduced by Einstein—that is, redeﬁnitionof time and space as observable reality instead of abstract a prioriconcepts—the results obtained by Lorentz and Poincaré led to controversial studies on the dynamics of charged bodies. We give some of theirresults, obtained by considering a special Poincaré transformation.
We√
set x0 = t, ξ 0√= τ , and V := tanh ϕ, and hence sinh ϕ = V / 1 − V 2 , andcosh ϕ = 1/ 1−V 2 (note that V ≤ 1). We obtain the transformation
law
t−τ =

t − τ  + V (x1 − ξ 1 )
√
,
1−V2

x1 − ξ 1 =x1 − ξ 1 + V (t − τ  )√,1−V2(II.4.1)

where (x1 , t) and (ξ 1 , τ ) are the coordinates of two points in the (x1 , t)plane in an inertial system, and the primed quantities are the coordinatesof these points in another inertial system in uniform translation along
the x1 axis with respect to the ﬁrst one. With classical interpretationof time and space, the relative velocity of this translation is dx1 /dt forﬁxed x1 , while the relative velocity of the unprimed frame with respect
to the primed one is dx1 /dt for ﬁxed x1 . These velocities resulting fromthe above formulas are found to beVdx1|x1 =const = √

dt
1−V2

and

dx1
−V
|x1 =const = √
.
dt
1−V2

(II.4.2)• Consider two events simultaneous in the primed frame: t = τ  . The
formulas (4.2) give thatx1 − ξ 1
x1 − ξ 1 = √,1−V2(II.4.3)

and hence the spatial distance observed in the primed frame issmaller than that observed in the unprimed one. This is the Lorentz
contraction.

Lorentz contraction and dilation

39

40

Special relativity• Consider two events with the same spatial location in the primed
frame: x1 = ξ 1 . Thent − τ ≥ t − τ  .

(II.4.4)

This is the Lorentz dilation.
The Lorentz contraction and dilation are not intrinsic phenomena.
They are relative to the observers, linked with their reference frames, as
is obvious from the fact that they are reversed by exchanging the rolesof these frames when deﬁning simultaneity or spatial coincidence.

II.5

10 It

was ﬁrst introduced as a technical tool (in the Euclideanized version
with x4 = ix0 ) by Poincaré in 1905.
Later, in 1908, Minkowski realized the
deeper mathematical–physical importance of the four-dimensional structuresη and F .

11 As

in Chapter I, we have adopted
the MTW convention, although thesignature (+ − −−), which is more convenient for some problems, is used bymany authors.

Electromagnetic ﬁeld and Maxwell
equations in Minkowski
spacetime M 4

Recall that the Poincaré–Minkowski spacetime10 M 4 is the (3 + 1)dimensional manifold R4 endowed with a Lorentzian, ﬂat metric that
reads,11 in so-called inertial coordinates,η = −(dx0 )2 +(dxi )2 .
(II.5.1)
i=1,2,3

We deﬁne the electromagnetic 2-form F on M 4 ,F ≡1
Fαβ dxα ∧ dxβ ,2(II.5.2)

by its components in inertial coordinates deduced from the electric andmagnetic vectors E and H previously considered on E 3 . We set
Fi0 = Ei ,F23 = H1 , F31 = H2 , F12 = H3 .

The electric current is deﬁned as the vector J α on M 4 with components
J i = −j i , J 0 = q.
Theorem II.5.1 Giving to the velocity of light its geometric valuec = 1, the Maxwell equations (2.4)–(2.7) can be written as a pair of equations for the electromagnetic 2-form F on Minkowski spacetime M 4 . In
arbitrary coordinates, the equations are, with ∇ the covariant derivativein the Minkowski metric in these coordinates,
dF = 0, i.e. ∇α Fβγ + ∇γ Fαβ + ∇β Fγα = 0,

(II.5.3)

and∇ · F = J,i.e.

∇α F αβ = J β ,(II.5.4)

where indices are raised with the Minkowski metric.
Equation (5.4) implies the conservation of the electric current J:∇α J α = 0.

(II.5.5)

II.5

Electromagnetic ﬁeld and Maxwell equations in Minkowski spacetime M 4

Proof. It is straightforward to check in inertial coordinates that equations (2.4) and (2.5) are equivalent12 to dF = 0 and that equations (2.6)
and (2.7) are equivalent to ∇ · F = J. This equation implies ∇ · J = 0(see Chapter I), which could have been deduced directly from equations(2.4) and (2.5) in inertial coordinates. Since d and ∇· are geometric operators, the Maxwell equations for F considered as an exterior 2-form
are valid in any frame of M 4 .
Since the operator d does not depend on the spacetime metric, and∇ · F is invariant under the group leaving the metric invariant, theMaxwell equations are invariant under the isometry group of Minkowski
spacetime.
Conversely, given on spacetime an exterior 2-form, we can deﬁne fora given observer a corresponding electric ﬁeld as follows. We consider alocal coordinate system for which the observer is at rest, i.e. the observer
describes a time line with unit tangent vector u with uα = δ0α . Then,the observer’s electric vector ﬁeld, orthogonal to u, is E β = F βα uα , i.e.,
in his proper frame,13 E i = F 0i .
The magnetic 2-form of an observer is the trace in his proper spaceof the electromagnetic 2-form. A magnetic space vector ﬁeld is deﬁned
only if this space is three-dimensional. In this case the magnetic ﬁeld
is the adjoint of the magnetic 2-form, given by the contracted productwith the volume form of the three-dimensional Euclidean metric:
H h :=1 ijh
η Fij .
2

Note that H so deﬁned (for n = 3) is a vector on the space manifold,but it is not a spacetime vector.
Under a change of observer, i.e. of proper frame, the space and timecomponents of the electromagnetic ﬁeld are mixed. This fact has beenchecked over and over in laboratories.
Exercise II.5.1 Write transformation of the electric and magnetic
ﬁelds under a change of Lorentzian frame with only non-zero componentsLαα with α and α equal to zero or one.
∂xα ∂xββ
Fα β  = Lα
α Lβ Fα β  .
α
β
∂x ∂x
Maxwell tensor
Hint: Use Fαβ ≡The Maxwell tensor τ is the symmetric 2-tensor given on Minkowski
spacetime by1ταβ := Fα λ Fβλ − ηαβ F λμ Fλμ .
4(II.5.6)

Exercise II.5.2 Show that the Maxwell tensor is traceless in spacetime
dimension 4.
The deﬁnitions of E and H in space dimension 3 show that thecomponents τ0α of τ in inertial coordinates read as follows, with εijl

41

12 Recall that in inertial coordinates on
M 4 with c = 1, it holds that F ij = Fij ,Fi0 = −F i0 , and ∇α = ∂/∂xα .

13 The proper frame of an observer isthe orthonormal frame with timelike
vector the unit tangent vector u toits trajectory. Its proper space is thehyperplane orthogonal to u.

42Special relativity

totally antisymmetric and ε123 = 1 :
τ00 =

1 2
(E + H 2 ),2

τ0i = −εijl E j H l .

(II.5.7)

The component τ00 = τ 00 is the energy density of the electromagnetic
ﬁeld, and the τ 0i = −τ0i , i = 1, 2, 3, are the components of the Poynting
energy ﬂux vector P, given in vector product notation on R3 by
P = E ∧ H.

(II.5.8)

Lemma II.5.1 Modulo the Maxwell equations, it holds that∇α τ αβ = J λ Fβλ .

(II.5.9)

Proof. Straightforward calculation gives1∇α τβα ≡ (∇α F αλ )Fβλ + F αλ ∇α Fβλ − F λμ ∇β Fλμ .
2Hence, changing names of indices and using antisymmetries,1∇α τβα ≡ (∇α F αλ )Fβλ + F αλ (∇α Fβλ + ∇λ Fαβ + ∇β Fλα ). (II.5.10)
2
The Maxwell equations (5.3) and (5.5) imply the result.



The vector J λ Fβλ is called the Lorentz force.
When the Lorentz force is identically zero, the equations ∇α τ αβ = 0
written in inertial coordinates are the usual equations of conservation ofenergy and momentum.

II.6

Maxwell equations in arbitrary
dimensions

The Minkowski spacetime M n+1 is the manifold Rn+1 endowed with a
ﬂat metric that reads in inertial coordinates(dxi )2 .
(II.6.1)
η = −(dx0 )2 +i=1,...,nIn a Minkowski spacetime of dimension n + 1, it is natural to deﬁne anelectromagnetic ﬁeld as an exterior 2-form F that satisﬁes the Maxwell
equationsdF = 0,∇ · F = J,

with

∇ · J = 0.

Let u be a unit timelike vector ﬁeld. The electric ﬁeld relative to u isthe vector ﬁeld uα Fαβ ; it is orthogonal to u. Choose for u the time
vector ∂/∂x0 of a system of inertial coordinates. The electric ﬁeld isthen a space vector with components Ei := Fi0 , while the components
Fij deﬁne on each submanifold M0 := { x0 = const} an exterior 2-form,
which can be called the magnetic 2-form. As mentioned previously
the magnetic vector ﬁeld is deﬁned only in space dimension 3.

II.7

Special Relativity

43

The deﬁnition (5.6) of the Maxwell stress energy tensor, as well as its
conservation law (5.9), extend to arbitrary dimension.

II.7

Special Relativity

II.7.1

Proper time

Lorentz and Poincaré had made mathematical studies, but it was Einstein who made the conceptual jump to discard Newton’s absolute timeand space as physical realities.14 Einstein, in Special Relativity, replaces
the direct product E 3 × R by the Minkowski spacetime M 4 . An event is
now a point in spacetime. Its history is a timelike trajectory in M 4 . An
Einsteinian revolution was to discover that the parameter t appearing
in the Minkowski metric has no physical meaning: the quantity measurable by well-deﬁned clocks (mechanical, atomic, or biological) is thelength of their timelike trajectories in spacetime; it is called the proper
time. More precisely, if C : t → C(t), t ∈ [t1 , t2 ] is a future causal curveparametrized by t joining two points of M4 , then the proper time alongthat curve is the parameter-independent quantity
 12
 t2  
dC dC
,
dt.
(II.7.1)
−η
dt dt
t1
Relativity postulates the existence of universal clocks deﬁned by speciﬁc physical phenomenon that measure the proper time. Such clocks
are nowadays obtained by using the frequency of radiation emitted by
speciﬁc atomic transitions, predicted by quantum theory to have a constant universal value. The actually adopted standard clock is the caesium
atom, which exhibits a particularly stable (of the order of 10−16 ) microwave transition between two particular energy levels. The second is now
deﬁned through the time measured by the caesium clock.15
Einstein’s concept of proper time has been proved valid in all
experiments performed to test it.
In a Lorentzian manifold, the length of timelike curves joining two
points has a local maximum for a timelike geodesic. Geodesics of
Minkowski spacetime are represented by straight lines in inertial coordinates. In particular, the line where only the parameter t of inertial
coordinates varies is a timelike geodesic. Therefore, the time measured
by an observer at rest in some inertial coordinate system is greater than
the time measured by a traveller which does not follow such a straightline between their separation and their reunion. This so-called ‘twin
paradox’ (see Fig. II.1)16 has been long veriﬁed for elementary particles in modern accelerators. The reality of proper time has also beenchecked in 1971 in an airplane ﬂight of a caesium clock which was latewith respect to a similar clock that remained on ground.17 In long spacetravels it could be veriﬁed with the human biological clock.
The case of spatial distances is more delicate to treat because of thelack of an absolute notion of simultaneity.

14 See Damour (2013a).

t
T

T
2

dx
= –v
dt

dx
=v
dt
v=

t

0
ds2 = –dt2 + dx2
T

T1 =

0

x

1 – v2 dt < T

Fig. II.1 The twin paradox.

15 It is likely, however, that in the future, the caesium clock will be replaced
as a standard by an optical clock.

16 There is actually no paradox, because the two twins do not have thesame history: one describes a geodesicin Minkowski spacetime, and the other
does not—in fact, he has to use a motor to follow his trajectory. This physical eﬀect is not to be confused withthe apparent time dilation, which is areciprocal eﬀect.
17 It is pleasant to know that the traveller ages less than the one who stays
put.

44

Special relativity

II.7.2

Proper frame and relative velocities

A velocity, even in Galileo–Newton mechanics, is always deﬁned with respect to some observer. The problem is more complex in Relativity, since
there is neither absolute time nor absolute simultaneity. In Relativity,
an observer is deﬁned by its world line, a timelike curve. The norm (inthe Minkowski metric) of its timelike tangent vector is dependent on the
choice of the parameter on the curve; this norm has no physical meaning
if the parameter is not speciﬁed. We label as a ‘unit tangent vector’ u
this tangent vector normalized by
η(u, u) = −1.

18 This deﬁnition comes from associat-

ing with an observer at some point of
spacetime the inertial system in which
he is momentarily at rest.

(II.7.2)

We call the proper frame of the observer (not necessarily following a
geodesic) at some point of spacetime an orthonormal Lorentzian framewith u as its timelike axis. Consider an object following a causal linewith tangent v at the considered point. If in the proper frame the vectorv has time component v 0 and space components v i , we say18 that theobject has velocity V with respect to this observer, where V is a space
vector, orthogonal to u, with components
Vi =

vi
.
v0

(II.7.3)

Since v is causal, it holds that
|V | ≡



1
2(V i )2≤ 1,(II.7.4)iwith |V | = 1, the speed of light, if and only if v is a null vector.
In particular, consider two observers at a point of spacetime with unit
velocities u and u . Choose their proper frames such that they are linked
by a special Lorentz transformation, i.e. e2 = e2 , e3 = e3 . The velocityof the primed observer with respect to the unprimed observer is thenalong the axis e1 and is given by, with ũα the components of u in theunprimed frame,
V =

L10
ũ1
=
,
ũ0
L00

(II.7.5)

since the components of u in the primed frame are u1 = 0, u0 = 1.
Addition of velocities
As foreseen from the overturning of the notion of absolute time andsimultaneity, relativistic relative velocities do not add simply as in
Newtonian mechanics.
Consider two systems of inertial coordinates (t, x1 , xa ) and (t , x1 , xa ).
They deﬁne at each point of spacetime two Lorentz frames, linked by aspecial Lorentz transformation, in relative motion with velocity V. The
velocity of a given point particle is dx1 /dt with respect to the unprimed

II.8

Some physical comments

45

frame and dx1 /dt with respect to the primed frame. The formula (4.1)
implies thatdx1
dx1 + V dt= ,dt
dt + V dx1(II.7.6)

and, therefore, setting dx1 /dt = U, dx1 /dt = U  ,U=U + V.
1 + V U(II.7.7)

This coincides in a ﬁrst approximation with the classical formula for
addition of velocities when U  V is small with respect to 1, the velocity
of light.
The formula (7.7) implies that U = 1 when U  = 1: the speed oflight is independent of the relative velocity V of diﬀerent observers. This
property, very surprising in the framework of Galilean kinematics, butalready implicit in the Maxwell equations, was found experimentally byMichelson19 and veriﬁed more accurately by Michelson and Morley.
They compared the speed of light in the direction of the orbital velocityof the Earth and in a transverse direction. To the surprise of the scientiﬁcworld, they found the same value in both cases to an accuracy higher
than possible experimental error. Since then, it has been veriﬁed withgreater and greater accuracy, in a medium that does not interfere withthis velocity, the vacuum. The most recent experiments give, with a 10−9
accuracy, 299 792.458 km/s. The Michelson–Morley experiment was inpart20 a source of inspiration to Einstein to discover Special Relativity.
Since the speed of light in vacuum is a universal constant, it canbe used to deﬁne the standards of length and time from one another.
Physics has shown that time can be measured more accurately, with acaesium clock, than length, for which the previous standard was related
to the wavelength of a krypton transition.21 It has been decided byscientiﬁc authorities that the metre is now oﬃcially deﬁned to be thedistance covered by light in (299 792 458)−1 seconds. The speed of lightin vacuum is therefore ﬁxed to be a universal constant. Unless otherwise
speciﬁed, we follow the mathematical usage of choosing units of lengthand time such that this constant is equal22 to 1 (its ‘geometric’ value). Of
course, in making comparison with observations or experiments, otherunits may be more appropriate.

II.8

Some physical comments

Like all mathematical models, Special Relativity aims at providing as
accurate as possible a picture of physical reality. No mathematical model
can replace reality, but the ﬁrst problem is to be able to compare theresults given by equations with observed facts. In Special Relativity,
one should physically identify spacetime inertial reference frames for
Minkowski spacetime. The choice posed a puzzle for Einstein, since
Minkowski spacetime considered as a global model is empty.

19 See footnote 8.

20 It seems that Einstein was more inspired by Faraday’s law of induction
than by the Michelson–Morley experiment.

21 The caesium clock deﬁnes the second as 9 192 631 770 periods of theradiation from a speciﬁc caesium-133
atomic transition. Optical clocks deﬁned by electromagnetic radiation areeven more precise. See, for instance,
Salomon (2013).
22 If the second is physically deﬁned (forinstance through the caesium clock),then unit of length for which the velocity of light is 1 is the distance coveredby light in 1 second, that is, 299 792 458
metres.

46

Special relativity

23 The rest mass is a number depending

only on the nature of the particle.

II.9

Dynamics of a pointlike mass

II.9.1

Newtonian law

The Newtonian equation of motion (2.5) of a particle with rest23 mass
m0 a constant, subjected to a force f , can be writtend(m0 v)= f,dt(II.9.1)

with v its velocity with respect to a Galilean inertial frame and t the
Newtonian absolute time. When f = 0, the particle is in uniform rectilinear motion in all Galilean inertial frames. In a non-Galilean frame, aterm, called the inertial force, must be added to f.
From (9.1), we obtain the Newtonian energy equationd 1( m0 v 2 ) = f · v,
dt 2

(II.9.2)

where the raised dot denotes the Euclidean scalar product.
If we want to write Newton’s law (9.1) in a non-Galilean frame, wemust add to f the so-called inertial forces due to the motion of theconsidered frame with respect to a Galilean one.

II.9.224 Recall our choice of signature− +. . . .+Relativistic law

The trajectory of a pointlike massive particle in Minkowski spacetime isa timelike curve. Since there is no absolute time to deﬁne its velocity,
we consider its unit velocity u, the tangent vector to its trajectory
parametrized by the proper time (the Minkowskian arc length s). The
components of u are, in arbitrary coordinates,24
uα =

dxα
,
ds

satisﬁng

uα uα = −1.

(II.9.3)

The acceleration of the particle is the derivative of u in the direction of itself, uα ∇α uβ . The time-dependent, coordinate-dependent,
Newtonian equations in space, (9.1) and (9.2), are replaced in relativistic
dynamics by a spacetime, coordinate-independent, equationuα ∇α (m0 uβ ) = F β ,(II.9.4)

where F is now a spacetime vector. Using uβ ∇α uβ = 0, a consequenceof uβ uβ = constant, (9.4) givesu α ∂ α m0 = u β F β .

(II.9.5)

II.9

Dynamics of a pointlike mass

47

This equation holds with m0 a constant25 on the trajectory if and only
if uβ F β = 0, i.e. if F is orthogonal (in the Minkowski metric) to the
trajectory,26 which we shall assume in what follows.

25 The rest mass of a molecule is modiﬁed by chemical reactions. The rest
mass of an atom is modiﬁed by its
absorption or emission of photons.

Remark II.9.1 Equation (9.4) reads (see Chapter I) in an arbitrary
frame
 2 β

α
λ
d x
β dx dxm0+ Γαλ(II.9.6)= F β.
ds2
ds ds

26 This is the case for the electromagnetic Lorentz force.

In a Minkowskian inertial reference frame, the connection coeﬃcientsvanish, and (9.6) with index i reduce to an analogue of Newton’s equation
in a Galilean frame. This remark leads to interpretation of the term
involving Γ as a kind of inertial force.

II.9.3

Newtonian approximation of the relativistic
equation

We denote by V as in Section II.7.2 the relative velocity of a massive particle with respect to some Minkowskian inertial coordinates xi ,xs0 = t, where its unit velocity is uα = dxα /ds; that is, we set
V i :=
Using (u0 )2 −
u0 =



ui
ds
dxi
= ui
= 0.
dt
dt
u

(II.9.7)

i 2
0 2
2
2
i (u ) = (u ) (1− |V | ) = 1 with |V | =

1
dt
=
ds
1 − |V |2

and

ui =



i (V

Vi
dxi
=
.
ds
1 − |V |2

i 2) gives(II.9.8)

These relations imply that the relativistic equations (9.4) can be written
in the inertial frame as
uα

∂
∂xα



m0 V i1 − |V |2= F i,

uα

∂
∂xα



m0
1 − |V |2= F 0.

On the trajectory, it holds that
uα

dt d
∂
d
=
,
=
α
∂x
ds
ds dt

and hence the above equations readd m0 V i
= F i 1 − |V |2 ,dt 1 − |V |2


m
d
 0
= F 0 1 − |V |2 .
dt 1 − |V |2(II.9.9)
The formulas with indices i look like the Newtonian ones if we replace
m0 by

m0
, and f i = F i 1 − |V |2 .
(II.9.10)m= 
2
1 − |V |48

Special relativity

For
i.e. for |U | small with respect to 1, we have small velocities,
1 − |U |2 =1
˜ − 12 |U |2 . Then,


1
m=m
˜ 0 1 + |U |2 .
(II.9.11)
2
Therefore, in this approximation, the mass m appearing in the relativistic equation is the sum of the rest mass of the particle and its kinetic
energy.
Remark
II.9.2 The force f i is replaced in the new notation byi
F 1 − |V |2 . The condition for the constancy of m0 reads
Vi F i − F 0=0uβ F β ≡ ui F i + u0 F 0 ≡ 
1 − |V |2
and hence is satisﬁed by the condition
F0 = 

Vi f i1 − |V |2

where Vi f i is in Newtonian mechanics the power of the force f.

II.9.427 It

is possible that all fundamental elementary particles have zero rest
mass and that the positive rest mass ofthe particles that appear to us as elementary is only an interaction energy.

28 After many years during which therest mass of the neutrino was believed
to be zero, it has now been establishedthat this rest mass is very small, but
non-zero.

Equivalence of mass and energy

The addition of the ‘kinetic energy’ and the energy of applied forces tothe rest mass m0 in the relativistic equation led Einstein to his famous
postulate of the equivalence of mass and energy,27 with a conversion
factor of the order of c2 , where the speed of light c is taken to be 1 inthe expression used for the Minkowski metric (geometrized units). The
equivalence of mass and energy has been veriﬁed in nuclear reactions
in a spectacular fashion. The energy can be created by the ﬁssion ofa uranium atom induced by a collision with a neutron. The sum of
the rest masses of the incoming particles is greater than that of the
post-ﬁssion particles, the diﬀerence being seen as kinetic (or radiated)energy. Nuclear fusion of particles into a single particle with rest mass
smaller than the sum of the rest masses of the incoming particles also
produces energy in accord with E = mc2 . Complex fusion reactions
involving hydrogen and its isotopes deuterium and tritium, leading inparticular to helium nuclei, is the main source of the heat producedby the Sun. Researchers are trying actively to reproduce it on Earth.
Constant rest masses are assigned to elementary particles, the photon(rest mass zero), electron, proton, neutrino,28 etc. The proton has now
been experimentally found to be composed of quarks and gluons.
Remark II.9.3 A useful physical quantity is the energy–momentum
P , a vector tangent to the trajectory and deﬁned for massive particles by
P α = muα ,hence

P α Pα = −m2 .

II.10Continuous matter49

In an arbitrary Lorentzian frame, the component P 0 is considered as theenergy of the particle with respect to that frame, while the components
P i deﬁne its momentum. This splitting is frame-dependent.

II.9.5

Particles with zero rest mass

An energy–momentum is also deﬁned for particles with zero rest mass.
Indeed, it was recognized by Einstein in 1905 that light can be described both by waves and by particles, the latter being called photons.
A particle with zero rest mass moves on a null geodesic, and the energy–
momentum of a photon is a null vector P tangent to this geodesic,
satisfying the equation
(P i )2 = 0.
(P 0 )2 −i

Its energy with respect to some Minkowskian reference frame is thecomponent P 0 .
To determine P , we consider that the wave associated to the considered photon is given by a scalar function eiϕ with hypersurfaces of
constant ϕ normal to the trajectory of the null particle and hence withgrad ϕ a null vector proportional to P. The wave has a frequency ν insome inertial coordinates if ∂ϕ/∂x0 = 2πν.
Einstein introduced in 1905 the postulate that the energy of a photonwith respect to some Minkowskian reference frame, i.e. the component
P 0 , is proportional to the frequency observed in that frame of the associated wave. The energy–momentum of a photon in some inertial
coordinates is, with h Planck’s constant (introduced by Planck in 1900
with a diﬀerent meaning),P =hgrad ϕ,2π

P 0 = hν.

The relation P 0 = hν introduced by Einstein to explain the photoelectric eﬀect is one of the key discoveries that led to quantum
mechanics.

II.10

Continuous matter

Our never-ending improvement in the exploration of reality has made us
see that matter is discontinuous at all scales: galaxies in the cosmos, stars
in galaxies, molecules in stars, atoms in molecules, ‘elementary’ particles
in atoms, quarks in protons, strings, . . . But, at some scales, which we call‘macroscopic’, we are not interested in the impossible task29 of following
the individual motions of constituents. We wish to describe the behaviour of volumes, small at the observed scale, but large at the scale of theconstituents, which we call particles, of the model that we are studying.
Particles eventually go in and out of such a volume. If it is possible to

29 Even without speaking of the intrusion of quantum mechanics!

50

Special relativity

deﬁne, pointwise on spacetime, measurable macroscopic quantities thatcharacterize the behaviour of matter at the scale of interest, we say that
we are dealing with continuous matter.
Real matter is much too complicated to be represented by a single
model. In fact, all models are only an approximation of a type of matter. Roughly speaking, ﬂuids are matter models where, in the absence of
external forces, the only action on an elementary volume is an interfacial
action with neighbouring elements that does not prevent them from slipping against each other. Perfect ﬂuids are those for which the antislipping
force has minimal action, that is, is orthogonal to the interface.
In Newtonian mechanics with absolute time and simultaneity, thestate of a ﬂuid is characterized, if one leaves aside thermodynamical considerations, by its density function μ and its ﬂow vector ﬁeld v. These areabsolute time-dependent quantities on Euclidean space E3 . They satisfy
on the one hand the conservation-of-matter equation∂μ+ ∂i (μv i ) = 0∂t(II.10.1)

and on the other hand the equations of motion obtained by passing to
the pointlike limit of the Newtonian dynamical laws applied to a small
volume in E3 . These equations ared(μv i )+ ∂k tik = 0,
dt

∂
∂xi
d
:=
+ vi
,
dt
∂t
∂x

(II.10.2)

where tik is the stress tensor, deduced in the case of ﬂuids from the force
f that the surrounding matter exerts at a point on an element of surface
with normal n by the linear relation
f i = tij nj .

30 Except for null ﬂuids, whereuα uα = 0.

(II.10.3)

The tensor t depends on the nature of the matter. It can be proved tobe symmetric when the matter has no intrinsic momentum density.
In relativistic dynamics, the fundamental macroscopic, spacetime
quantities characterizing a ﬂuid are the ﬂow vector ﬁeld u, a timelike unit vector,30 i.e. uα uα = −1, and the energy density function
μ. A Lorentzian frame with timelike vector u is called a comoving or
proper frame of the ﬂuid. The function μ on spacetime is the timecomponent in the proper frame of the energy–momentum vector ﬁeld
P = μu.
If there are no interactions (chemical, nuclear, inelastic shocks, etc.)
modifying the nature of the underlying particles, we can deﬁne for theﬂuid a rest mass density r, proportional to the particle number density if
all particles have the same rest mass. This scalar function on spacetime
satisﬁes a conservation equation that reads in arbitrary coordinates,∇α (ruα ) = 0.

(II.10.4)

II.10II.10.1

Continuous matter

51

Case of dust (incoherent matter),massive particles

The dust model of matter is a good approximation to other models,owing to the high conversion factor in usual units between rest massand other types of energy. A ﬂuid is called dust if neighbouring volume
elements exert no action on each other. It is then supposed that the ﬂow
lines are geodesics,uα ∇α uβ = 0.

(II.10.5)

The following lemma will be of fundamental importance in General
Relativity.
Lemma II.10.1 Equations (10.4) and (10.5) imply that the tensor
T αβ = ruα uβ(II.10.6)∇α T αβ = 0.

(II.10.7)

satisﬁes the conservation law

Proof. From uα uα =constant, it results that uβ ∇α uβ = 0. The proof isthen obtained by computing ﬁrst uβ ∇α T αβ .
Remark II.10.1 In a proper frame, the components of(10.6) are
T 00 = r,

II.10.2

T 0i = T i0 = 0,

T ij = 0.

the tensor(II.10.8)

Perfect ﬂuids

The equations of motion for Newtonian ﬂuids are generalized to relativistic ﬂuids by introducing a spacetime energy–momentum stress tensor
T , or stress–energy tensor for short, such that its components T 00 , T 0i
and T i0 in a proper frame of the ﬂuid represent energy, energy ﬂux andmomentum per unit volume31 relative to that frame, while the spacepart of this tensor is, in a ﬁrst consideration, identiﬁed with the stresstensor of Newtonian mechanics.
A relativistic ﬂuid is called a perfect ﬂuid if the components of T ina proper frame reduce to
T 00 = μ, energy density,
T
T

0i
i0

(II.10.9)= 0,

no energy ﬂux (no heat ﬂow),(II.10.10)= 0,zero momentum.

(II.10.11)

and, with p a scalar function called the pressure and e the Euclidean
metric,T ij = peij .

(II.10.12)

31 We do not give an axiomatic formulation of these physical notions: physics cannot be axiomatized (nor, by
the way, can mathematics at its very
foundations). Eventually, energy andmomentum are deﬁned as mathematical objects in elaborate theories. For
instance, energy appears as the conserved quantity associated to a timeindependent Hamiltonian.

52

Special relativity

32 The force acting on an element of the

The formula (10.12) coincides with the stress tensor of a Newtonian
perfect ﬂuid.32

boundary of an elementary volume isorthogonal to it: there is no antislipping
force.

Proposition II.10.1 The stress–energy tensor of a perfect ﬂuid in
Minkowski spacetime is, in arbitrary coordinates,
T αβ = μuα uβ + p(η αβ + uα uβ )(II.10.13)

Proof. One checks that in the proper frame the above tensor hascomponents given by (10.9)–(10.12).
Remark II.10.2 In the usual pressure and mass units, p is of theorder of c−2 with respect to μ; hence, at ordinary scales, p is very small
compared with μ.
The proper frame varies from point to point. It is not in general thenatural frame of an inertial coordinate system. It is therefore not legitimate to write the usual equations for separate conservation of energy
and momentum by using ordinary partial derivatives and the components (10.9)–(10.12) of the stress–energy tensor in a proper frame of the
ﬂuid. The dynamical equations of a perfect ﬂuid are postulated to be
spacetime tensorial equations that read, in arbitrary coordinates,∇α T αβ = 0.

(II.10.14)

These equations can be written in an arbitrary Lorentzian metric. We
will return to them after introducing General Relativity.

II.10.3

Yang–Mills ﬁelds

The electromagnetic ﬁeld, and the basics of Special Relativity have beentreated in Sections II.5–II.7. The deﬁnition of the electromagnetic potential has been omitted so far. It is a locally deﬁned 1-form A suchthatdA = F.
33 See,for instance, YCB-OUP2009,
Chapter III, Section 6.2.

Its existence, non-unique, results (Poincaré lemma) from the Maxwell
equation dF = 0.33 The non-uniqueness of the electromagnetic potential
A, the local character of its deﬁnition on a multiply connected manifold, and the physical properties of spinor ﬁelds in the presence of anelectromagnetic ﬁeld have led to the interpretation of A as the representative on spacetime of a U (1) connection on its tangent bundle, theelectromagnetic ﬁeld being the curvature of this connection.
After the discovery of other fundamental interactions, strong andweak, it was found by Yang and Mills that these interactions could also
be mathematically modelled by curvatures of connections on the tangent bundle of spacetime, but with higher-dimensional and non-Abeliangroups. The electromagnetic and weak interactions have been uniﬁed by
Weinberg and Salam as the curvature of an SU (2) × U (1) connection,with the addition of a scalar ﬁeld, the Higgs ﬁeld, made necessary by
the physical fact of the distinction (called symmetry breaking) of theseII.10interactions at lower energy scales. The strong interactions have now
been identiﬁed as a Yang–Mills ﬁeld, namely the curvature of a SU (3)connection. These connections together with associate spinor ﬁelds constitute what is now called the standard model.34 We recall brieﬂy35 thedescriptions of a classical (non-quantized) Yang–Mills ﬁeld, the Yang–
Mills equations and the conserved stress–energy tensor, which can betaken as a source36 for the Einstein equations.
Let G be a Lie group and G its Lie algebra. A representative on spacetime of a G-connection is a locally deﬁned 1-form with values in G; it iscalled a Yang–Mills potential. The curvature of this connection is represented by a G-valued 2-form, called a Yang–Mills ﬁeld, given in termsof A by (with [ , ] the Lie bracket in G)
F = dA + [A, A],i.e.

Fαβ = ∂α Aβ − ∂β Aα + [Aα , Aβ ].

Continuous matter

53

34 These Yang–Mills and spinor ﬁelds
appear in physics, up to now, only as
quantum ﬁelds.
35 See, for instance, CB-DMI Vbis
Problem 1.

36 Though, at a macroscopic scale, theYang–Mills ﬁelds are not directly observed.

(II.10.15)

The 2-form F satisﬁes the identityˆ ≡ 0,dFi.e

ˆ α Fβγ + ∇ˆ γ Fαβ + ∇ˆ β Fγα = 0,∇

(II.10.16)

ˆ is the metricwhere dˆ is the gauge-covariant exterior diﬀerential and ∇and gauge-covariant derivative:
ˆ α Fβγ := ∇α Fβγ + [Aα , Fβγ ],∇(II.10.17)

where ∇ is the covariant derivative associated with the spacetime metric. The Yang–Mills equations, which can be written in an arbitrary
Lorentzian metric, consist of the identity (10.16) and the following
equations that generalize the second set of Maxwell equations:ˆ α F αβ ≡ ∇α F αβ + [Aα , F αβ ] = 0.
∇

(II.10.18)

Note that (except for Abelian groups G) the Yang–Mills potential does
not disappear from these equations.
The Yang–Mills stress–energy tensor, a generalization of the Maxwell
tensor, is an ordinary tensor given by1
ταβ := Fα λ · Fβλ − gαβ F λμ · Fλμ .
4(II.10.19)

where a raised dot denotes the scalar product with respect to the Killingform37 of G. It is divergence-free if the Yang–Mills equations are satisﬁed:∇α τ αβ = 0.
Remark II.10.3 On a d-dimensional spacetime, the Einstein equations with Yang–Mills source are equivalent toRαβ = ραβ ≡ Fα λ · Fβλ −1gαβ F λμ · Fλμ .
2(d − 2)

(II.10.20)

The Yang–Mills stress–energy tensor has zero trace in four-dimensional
spacetimes, as the Maxwell tensor. In this case, ραβ = ταβ .

37 See, for instance, CB-DMI III D 6.

54Special relativity

II.11

Problems

II.11.1

Lorentz transformation of the Maxwell
equations

The Maxwell equations on R3 × R in non-inductive media read, withc = 1 the velocity of light, as follows:
1. The evolution equations are
∂E 1
∂H 3
∂E 2
∂H 2
∂H 1
∂E 3
=
=
−
− j,
−
,
(II.11.1)
2
3
3
∂t
∂x
∂x
∂t
∂x
∂x2
and equations obtained by circular permutation of the indices 1, 2, 3.
2. The constraint equations are
∂E 1
∂E 2
∂E 3
∂H 1
∂H 2
∂H 3
+
+
= q,
+
+
= 0. (II.11.2)
1
2
3
1
2
∂x
∂x
∂x
∂x
∂x
∂x3
The separation into evolution and constraints is coordinatedependent, as is the splitting of the electromagnetic ﬁeld in electricand magnetic parts. Consider the Lorentz transformationt + V x1
,
t= √
1−V2

x1 + V tx1 = √,
1−V2

x2 = x2 ,x3 = x3(II.11.3)

and determine E , H  , j  , q  depending on E, H, j, q, V and satisfying
the Maxwell equations.

II.11.2

The relativistic Doppler–Fizeau eﬀect

It has been well known since Doppler (in 1842) and experimentally veriﬁed with great accuracy (by Fizeau in 1848) that light appears reddened
when the source recedes from the observer, with the opposite occuringwhen the source approaches the observer. This fact is easily explainedby the longer time between two successive pulses needed to reach theobserver when the source recedes from him (or her). The classical formula results trivially from addition of velocities in the Newton–Galileo
kinematics. Give an expression for the relativistic Doppler–Fizeau eﬀectusing the addition-of-velocities formula in Special Relativity.

III

General Relativity
III.1

Introduction

Besides his desire to reconcile Special Relativity, valid when gravitational eﬀects are negligible, and Newton’s law of gravitation, Einstein
was led to his theory by physical facts and new ideas. We will not try inthis book to follow Einstein’s long path (1907–1915) of discovery of General Relativity and the Einstein equations. Their mathematics rely on
Lorentzian diﬀerential geometry, now a well-understood subject,1 withwhich we started this book.
The physical facts that inspired Einstein’s genius are the principle ofgeneral covariance and the Galileo–Newton equivalence principle.

III.2

55

III.3 The Galileo–Newton
equivalence principle
III.4 General Relativity

55
57

III.5 Constants and unitsof measurement
III.6 Classical ﬁelds inGeneral Relativity
III.7 Gravitation andcurvature
III.8 Observations andexperiments
III.9 Problems

Principle of general covariance

The principle of general covariance is an extension of the principle
called ‘material indiﬀerence’ in Newtonian mechanics, which essentiallysays that physical phenomena do not depend intrinsically on the reference frame in which we express their laws.2 The laws may look very
diﬀerent for diﬀerent observers, and hence in diﬀerent frames, but itshould be possible to ﬁnd frame-independent formulas for them. Tensor
ﬁelds are good candidates as objects for physical laws, being intrinsic
geometric objects on a manifold. Tensors can be represented by theircomponents, speciﬁc numbers attached to them by the choice of a particular reference frame but with general laws for passage from one frameto another.

III.3

III.1 Introduction
III.2 Principle of generalcovariance

55

59
60
62
62
65

1 In fact, General Relativity has given agreat impulse to the study of Riemannian and Lorentzian diﬀerential geometry.
2 A principle that seems trivial, if we be-lieve there exists a reality, although it
becomes debatable in quantum mechanics.

The Galileo–Newton equivalenceprinciple

The fundamental law of Newtonian dynamics,
F = mI γ,(III.3.1)

relates the acceleration γ of a test particle3 in Galileo–Newton absolute
spacetime E 3 × R to the force F acting on it and to its inertial mass
mI . In Newtonian mechanics, this inertial mass is a constant, depending
only on the nature of the particle.

3 Which is considered to be pointlike.

56

General Relativity

4 The story, perhaps not history, is that
Galileo veriﬁed this universality of fallby dropping objects from the Leaning Tower of Pisa. This fact has been
observed repeatedly down the years,
perhaps since Philiponos Eramnatikosin Alexandria around AD 300. Astronauts on the Moon veriﬁed it again bydropping at the same time a piece of
metal and a feather (which on Earthwould have been slowed down by theatmosphere).

5 Traditionnally, one introduces a passive and an active gravitational mass—
we bypass this distinction. By analogy
with the electric ﬁeld, we could havecalled the function ρG a gravitationalcharge density, but since it will be identiﬁed with the inertial mass density, avariable quantity in Special Relativity,
the analogy would be misleading.
6 The gravitational acceleration of apoint particle of inertial mass mI due
to ρG is the gradient of the potential
U, independent of mI . On the Earth’s
surface, the acceleration due to gravity
is approximately 9.81 cm s−2 .
7 Newton himself had some doubts
about this instantaneous “action at a
distance”, which contradicts our human experience.

8 Withproper units of mass relative
to the units of time and length, one
can set GN = 1, which is done bymany authors. Others prefer, as we doin this book because it is more convenient when working with the Einsteinequations, to choose units such that the
proportionality factor of the Einsteintensor and the stress–energy tensor ofsources is equal to 1.

In its primitive, fundamental form, the equivalence principle is theexpression of the fact that the acceleration γ due to gravity of a massive
body is independent of its mass.4
In Newtonian gravitation theory, the acceleration of a test particle in a gravitational ﬁeld depends only on its location in space(universality of free fall); it satisﬁes a diﬀerential equation that reads,
in inertial coordinates,
∂U
d2 xi
=
.
(III.3.2)2dt∂xi
The function U, called the gravitational potential, satisﬁes the
Poisson equation∂2
ΔU = −4πρG , Δ :=,(III.3.3)(∂xi )2i=1,2,3
where Δ is the Laplace operator on Euclidean space E 3 while the sourceρG is a, possibly time-dependent, positive scalar function on R3 equal
to the ‘gravitational mass density’5 of the sources at time t. We do notinclude in the deﬁnition of the potential of ρG the Newtonian gravitational constant GN . We will introduce GN later after proving the
proportionality of inertial mass and so-called gravitational mass.
Classical results on elliptic equations, in particular on the Laplace
operator, imply that the Newtonian gravitational potential U is given
at each instant of time and each point x ∈ E 3 by the space integral6
over E 3 at this time:7ρG (y) 3
d y, d3 y ≡ dy 1 dy 2 dy 3 .
U (x) =
(III.3.4)
R3 |x − y|The equivalence principle, the identity of the gravitational mass andthe inertial mass up to a choice of units, can now be proved throughthe Newtonian principle of equality of action and reaction, as follows.
Let the source of gravitation be a pointlike particle P located at theorigin O with gravitational mass mG ; that is, a density ρG := mG δ,
where δ is the Dirac density at O. This particle generates a potential
with value U = mG /r at a point x at distance r from O. The attractive
Newtonian force generated by P on a particle P  of inertial mass milocated at x has intensity mG mI /r2 . We can infer from the principleof action and reaction that mG mI = mG mI , and hence the quotientmG /mI is independent of the considered particle. Following the usual
notation, we denote the constant by GN , and call GN the Newtonian
gravitational constant, and we havemG = GN mI .

(III.3.5)

The value of this universal constant depends on the choice of units
for time, length and mass. It is approximately, in CGS units (seeSection III.5), the very small number8
GN =˜ 6, 67259 × 10−8 cm3 g−1 s−2 .

III.4General Relativity

57

The Newton equivalence principle (the equality mG = mI up to achoice of units of inertial and gravitational masses) has been veriﬁed byexperiments on the motion of pendulums by Galileo and Newton (1610–
1680), followed by several subsequent accurate experiments by Eötvös(1860–1908) and in modern times by Dicke, Braginski and especially
Aidelberger and his group, comparing with a torsion balance the gravitational force and the inertial, centrifugal, force due to rotation. There
have been also accurate tests of this equality from celestial mechanics,
starting from work of Newton and Laplace, up to modern tests based onlaser ranging to the Moon.

III.4

General Relativity

We have in Chapter II expressed the metric of Minkowski spacetime,
the arena of the dynamics of Special Relativity, in arbitrary coordinates. We have seen that in the equations of motion of test particles, theinertial forces appear through the Christoﬀel symbols of the metric.9
This fact, the Galileo–Newton equivalence principle and his genius inspired Einstein to replace the Minkowski spacetime of Special Relativity
by a general Lorentzian manifold (hence the name General Relativity).
The Newtonian gravitational potential had no inﬂuence on Newtonian
absolute spacetime structure. The new revolutionary idea due to Einsteinis that space and time are not a priori given structures but are united ina four-dimensional curved Lorentzian manifold whose metric g is linked
with the energy content of spacetime. This Lorentzian metric governs
the spacetime causality structure (see Chapter I). The length of atimelike curve measures the intrinsic, called proper, time along thatcurve: in Relativity, Special or General, the basic observable quantity
is proper time. It is assumed that quantum phenomena, namely vibrations of atoms, permit its measure, independently of everything else. Theoﬃcial clock is at present the caesium atom.
In General Relativity, as was already the case in Special Relativity,
there is no intrinsic notion of simultaneity, or, therefore, of spatial distances. Velocities are deﬁned mathematically in the tangent space atthe location of an observer at a point of the spacetime (see Chapter II).
Assuming that the speed of light is an absolute constant is physicallysensible because it is consistent with the experimental observation thatat any given point, the velocity of light is independent of direction, andwhen measured on Earth with standard of length a solid body, it appears to be constant with extremely great accuracy. Taking the velocity
c of light equal to 1 for any observer is no restriction—it is a deﬁnitionof the unit of length in terms of the unit of time.10

III.4.1

Einstein equivalence principles

Uniﬁcation of gravitation and inertia is at the root of Einstein General
Relativity.

9 Inertial

forces (for example Coriolis
forces) appear also in Newtonian mechanics through non-inertial (i.e. nonGalilean) frames.

10 Evaluating astronomical distances ina time unit, namely the light-year, andsaying that this is the time it takes
light to travel from one object to another is misleading since the proper
time is zero along a light ray. The distance is not the length of a spacelike
geodesic joining two points of spacetime, but is deﬁned in a more subtle
way and depends on the trajectoriesof the objects in the Lorentzian manifold that represents the spacetime (seeSection III.5).

58

General Relativity

In General Relativity, massive pointlike objects in free fall are assumed to follow timelike geodesics of the metric, and their equationsof motion are therefore independent of their mass, reading, in arbitrary
coordinates,
β
γ
d2 xα
α dx dx
= 0,
+
Γ
βγ
ds2
ds ds

with s the proper time (see Chapter I). The connection Γ of the metricrepresents in the coordinates xα both gravitation and inertia. This isoften called the weak Einstein equivalence principle. This priciple
is used in the formulation of equations of motion in a given gravitational ﬁeld of massive bodies small enough at the considered scale tobe taken as pointlike, with mass negligible compared with the masses
that produce the gravitational ﬁeld: this is the case for the motion of
the planet Mercury in the gravitational ﬁeld of the Sun. The advance ofthe perihelium of its elliptic orbit was the ﬁrst spectacular conﬁrmationof Einstein General Relativity (see Chapter V).
Light rays are null geodesics, trajectories of particles with zero rest
mass. Their deviation by gravitation was ﬁrst veriﬁed during the 1919
solar eclipse.
The weak Einstein equivalence principle can be shown to be a particular case, modulo the Einstein equations (see Chapter IV), of theEinstein equivalence principle, which says that in the dynamical equations previously expressed in Special Relativity, the spacetime
Minkowski metric η must be replaced, if the gravitatonal ﬁeld is not
negligible, by the spacetime Lorentzian metric, usually denoted by g.
One sometimes speaks of the strong Einstein equivalence principle whenself-gravitational phenomena are important.
We will treat in Chapter IV the coupling of gravitation and matter.

III.4.2

11 This is known under the name of the‘Anthropic principle’.

Conclusion

The Einstein equivalence principle has its root in Einstein’s basic ideaof identifying gravitation with inertia. In particle physics in Minkowski
spacetime, there appear phenomenological constants that are dimensionless, and hence cannot be given a geometric value by choice of units: theirvariation in spacetime would violate the Einstein equivalence principle.
No such violation has been found up to now. The ﬁne-structure constant α := e2 /hc has been estimated to have a fractional variation ofno more than order 10−7 from analysis of a natural ﬁssion reactor phenomenon that took place in Oklo, Gabon two billion years ago. Anotherdimensionless constant, the ratio μ = me /mp of the rest masses of electron and proton, has been estimated with a precision of 10−6 throughmeasurements of absorption lines in astronomical spectra. The value ofthese constants is puzzling. Some physicists have remarked that if theconstants of nature were not what they are, life as we know it would notexist and that this is enough of an explanation for our observations.11

III.5

III.5

Constants and units of measurement

59

Constants and units of measurement

This section recalls deﬁnitions, common to Newtonian and Einsteinianmechanics, that permit us to express quantitative facts, as required forthe confrontation of theory and reality. The three familiar items of everyday life are distance, mass and time. The fundamental requirement hereis the deﬁnition of units that are both common to a group of people
and reliable. The system of units used in nearly all countries in science and technology, and in most of them in everyday life, is the CGS
system.12 The Convention, an assembly of the French Republic, deﬁned
in 1795 the metre (100 centimetres) as the 1/(4 × 106 ) part of the meridian of the Earth and the gram as the mass of 1 cm3 of water. Standardsof mass and length made of platinum were deposited at an Institute inSèvres—a kilogram (1000 grams) for the mass, and were taken as thedeﬁnitions of the metre and kilogram, replacing the physical quantities
previously used to deﬁne them. The unit of time, the second, was deﬁned
ﬁrst as a fraction of the solar day, then as a fraction of a speciﬁc year
and then of a sideral day.13
Modern science has deﬁned fundamental universal dimensional14
constants. Two such constants can, in principle, be used to deﬁne twounits in terms of a third by giving an arbitrary ﬁxed value to these
constants. If this arbitrary value is 1, it is called the geometric value
and the units deduced from it are called geometric units. However, oneof the units must be deﬁned by an experimental device with veriﬁable
reproducibility.
An international committee ﬁxes the deﬁnition of the oﬃcially chosen
units, called SI for ‘Système International’. Of course, multiples of theseunits can equivalently be used when they lead to values that are easier tograsp at the relevant scale, as is the case with the mks (metre, kilograms,
second) and cgs systems.
The velocity of light, c, observed experimentally to be constant to very
great accuracy, is, in metres per second,˜ 3 × 108 .
cm s−1 = 299 792 458 =(III.5.1)

This equality is now taken as ﬁxing a relation between the units oflength and time. It is the unit of time that is actually chosen15 as basic
for formulation of experimental results; it is deﬁned at present by theperiod Tcaesium of a particular atomic transition of caesium, predicted tobe constant by quantum mechanics and observed experimentally with avery high accuracy to be the number
Tcaesium =1
second.
9 192 631 770

12 centimetre, gram, second.

13 Deﬁned by observation of the stars.
14 A constant is called dimensional if its
value depends on the units chosen. The
dimensional equation of a function depending on positive or negative powers
pL , pM and pT of length, mass and timeis

LpL M pM T pT .
For example, an acceleration has dimensional equation LT −2 , a force
M LT −2 . Under a change by a proportionality factor k of a unit with power p
in the dimensional equation, the function changes by a proportionality factor
k−p .
15 For a while, it was the unit of lengththat was chosen, namely the wavelength of a speciﬁed ray of the radiation
emitted by a speciﬁc transition of thekrypton atom, which replaced the definition of the metre by the standard inSèvres. But it was less precise and lessconvenient to use. Remember that astronomers have long been using time tomeasure distance, namely light-years.

(III.5.2)

The second is now deﬁned by this equality. In the future, it will probably
be deﬁned with still higher precision by an optical clock.
The velocity of light will be equal to 1 if we choose as unit of length
the distance16 covered by light in 1 second. We denote17 this distance

16 Note that, in Relativity, the deﬁnition of distance is ambiguous since
there is no absolute deﬁnition of simultaneity.
17 One could call it a light-second.

60

General Relativity

by Ls , and it holds thatcLs ,s−1 = 1.
The metre is deﬁned, using (5.1), by the formula
1 metre = (299 792 458)−1 Ls .
18 Thedimensions of GN can be deduced from the fact that GN m/r 2 isan acceleration, and hence has dimensions LT −2 . The dimensions of GN are
therefore L3 M −1 T −2 .

A unit of mass could be deﬁned through the previously chosen units by
taking the gravitational constant equal to 1; in cgs units, it is observed
to be,18
GN,cgs = 6.67259 × 10−8 .

19 This lack of accuracy is due to theextreme weakness of the gravitational
force between bodies of laboratory size.
Another diﬃculty appears with massesof planetary size: this is that only the
product of GN with their mass appears in their equations of motion, andthese masses must be determined fromthe planetary orbits—which requires
knowledge of the value of GN .

However, the lack of suﬃcient accuracy19 in the knowledge of GN has
so far prevented the use of this formula to deﬁne a unit of mass in termsof the length or time unit. It has recently been decided (July 2011) by
the BIPM (Bureau International des Poids et Mesures) that Planck’s
constant h, which is known with a very high accuracy, should be used
to oﬃcially deﬁne a unit of mass-energy. It is now h that is a geometricquantity.20

20 This oﬃcial deﬁnition does not pre-vent mathematicians from setting thegravitational constant equal to 1 in theequations on which they work. It must
only be remembered what this means
when one wants to make numericalestimates using various speciﬁed units.

III.6

Classical ﬁelds in General Relativity

We have written in Chapter II the expressions for the stress–energy
tensors of the usual and simplest forms of matter and ﬁelds in Special
Relativity. The Einstein equivalence principle leads immediately to theirexpressions in General Relativity. We give them in a general Lorentzian
metric g on a manifold V of arbitrary dimension n + 1, although thephysical case on a macroscopic scale is n = 3.

III.6.1

Perfect ﬂuid

The tensor representing the energy, stress and momentum of a perfect
ﬂuid is, in arbitrary coordinates, the following symmetric 2-tensor on V :Tαβ = μuα uβ + p(gαβ + uα uβ )(III.6.1)

where u is the unit timelike ﬂow vector, i.e. such thatgαβ uα uβ = −1,
while μ and p are scalar functions, respectively the pointwise energy andpressure density of the ﬂuid, both of which are non-negative in classicalsituations.
Exercise III.6.1 Show that in a proper frame of the ﬂuid, i.e. an
orthonormal frame for the metric g with timelike vector u, the components of the tensor T are respectively the ﬂuid pointwise densities
T00 = μ(energy), T0i = 0(momentum), Tij = pδij (stresses).
In the case of dust, p ≡ 0.

III.6Classical ﬁelds in General Relativity

The Einstein equivalence principle implies that a relativistic perfect
ﬂuid satisﬁes the conservation equations∇α T αβ = 0,
which have been found in Special Relativity as consequence of theclassical conservation laws of energy and momentum.

III.6.2

Electromagnetic ﬁeld

The electromagnetic ﬁeld is an exterior 2-form on the manifold V . The
equivalence principle says that on the Lorentzian manifold (V, g),
the electromagnetic ﬁeld satisﬁes, in the absence of electric charges,
the vacuum Maxwell equations
dF = 0,∇ · F = 0.

(III.6.2)

Its stress energy tensor is the Maxwell tensor1ταβ := Fα λ Fβλ − gαβ F λμ Fλμ .
4(III.6.3)

As foreseen from the equivalence principle, if an electromagnetic ﬁeld
satisﬁes the sourceless Maxwell equations, then its Maxwell tensor isdivergence-free:∇α τ αβ = 0.

(III.6.4)

The covariant derivatives that appear in these equations of motion are
a symptom of the inﬂuence of gravitation on electromagnetism.

III.6.3

Charged ﬂuid

In an electrically charged ﬂuid, there is an electromagnetic ﬁeld generated by the ﬂuid motion, which in turn is inﬂuenced by the electromagnetic ﬁeld Fαβ . In the simplest classical model, the ﬂuid is a perfect ﬂuid
with unit velocity u, and energy and presure densities μ and p, and theelectric current is the sum of convection and conduction currents:
J α = quα + σ α
The stress–energy tensor is the sum of the ﬂuid stress–energy tensor andthe Maxwell tensor:
Tαβ = (μ + p)uα uβ + pgαβ + ταβ .
Exercise III.6.2 Given the metric g, write the coupled system of
equations satisﬁed by u, μ, p, q and F.
Hint: Use the equations∇α T αβ = 0and∇α τ αβ = uα J α .

61

62

General Relativity

III.7

Gravitation and curvature

There is no intrinsic splitting between gravity and inertial-type forces.
In a general spacetime (V, g), it is always possible at a given point(see Chapter I) to choose local coordinates such that the Christoffel symbols vanish at that point; gravity and relative acceleration arethen, at that point, exactly balanced. It is even possible to choose local coordinates such that the Christoﬀel symbols vanish along a given
geodesic—astronauts spacecrafts have made popular knowledge the factthat in free fall one feels neither acceleration nor gravity; in a smallenough neighbourhood of a geodesic, the relative accelerations of objects
in free fall are approximately zero.
Nevertheless, gravity is a physical reality that cannot be assimilatedwith the old notion of inertia. In a general Lorentzian manifold, there isno coordinate system (unless the metric is locally ﬂat) in which all theChristoﬀel symbols vanish in the domain of a chart; a family of geodesics cannot in general be represented by straight lines. In the presenceof a non-everywhere-vanishing mass or energy density, gravity manifests
itself by the non-vanishing of the Riemann curvature tensor Riem(g).
If Riem(g) does not vanish in an open set U of a spacetime (V, g),then the gravitational ﬁeld in U cannot be identiﬁed with an inertial
ﬁeld. The existence of a non-zero curvature of the metric is revealed,beyond the ﬁrst approximation, by the relative acceleration of test particles, as is predicted by the equation of geodesic deviation (Chapter I,
Section 7).

III.8
21 For details, see Will (2014).

Observations and experiments

The following observations have been made and experiments performed
to conﬁrm the predictions of General Relativity.21

III.8.1

The Einstein equivalence principle

The fact that the acceleration at a point in a gravitational ﬁeld of bodies
small at the considered scale is independent of their nature is common
to both Newton’s and Einstein’s theories of gravitation. It has been
veriﬁed with an accuracy of the order of 10−13 in several experiments—
in particular in the laboratory with atoms of titanium and beryllium
and a very sophisticated apparatus based on a torsion balance, and also
at an astronomical scale by reﬁned analyses of results of lunar laser
ranging (LLR) to determine precisely the orbit of the Moon together
with sophisticated approximation methods to show the compatibilityof this orbit with the equivalence principle applied to the system Sun–
Earth–Moon.

III.8

III.8.2

63

Deviation of light rays

A consequence of Einstein’s postulate of the equivalence of mass andenergy is that particles of zero mass should also follow geodesics of thespacetime metric. Light rays in particular are null geodesics, and light isdeﬂected by a gravitational ﬁeld.22 This was conﬁrmed by observation
as early as 1919 by Eddington and Dyson during a total solar eclipse(though not with very high precision, because of large experimental errors). A solar eclipse makes it possible for a photograph of the sky to betaken in which stars are visible, so that their positions can be comparedwith those in a photograph taken in the absence of the Sun. Greater
precision is now obtained for the deﬂection of radio waves, which does
not need an eclipse to be observed.
A spectacular veriﬁcation of the deﬂection of light rays by a gravitational ﬁeld is provided by the observed phenomena called gravitational
lensing, in which the light rays from a star or galaxy, i.e. the nullgeodesics issuing from some point in curved spacetime, may intersect
again such that an observer can see several images simultaneously. Several gravitational lenses have now been observed,23 in particular the‘Einstein cross’, obtained by the Hubble telescope, consists of four images of the same quasar; in the centre appears a fainter image of thegalaxy whose gravitational ﬁeld acts as the lens.24

III.8.3

Observations and experiments

22 Newtonian gravitational theory also
predicts light deﬂection, with lighttreated as small but may be massive
particles. But the Newtonian light deﬂection is exactly half of that predicted
by Einstein and veriﬁed by Eddington.

23 See,for
et al. (1992).

instance,

Schneider

24 See for instance Figure 4.16 in Chapter 4 of Ohanian and Ruﬃni (2013).

Proper time, gravitational time delay

As in Special Relativity, the physically measurable quantity is propertime, which is the length in the spacetime metric of timelike curves. It
is again postulated that the period of speciﬁc spectral radiation fromatoms provides universal clocks for the measure of proper time.
A consequence of the reality of the dependence of this physically measurable time on the presence of a gravitational ﬁeld is the observation ofa redshift in the spectrum of a given atom in a gravitational ﬁeld, as
seen by a distant observer for whom this ﬁeld is weaker. Such shifts add
to the Doppler eﬀect for bodies in motion. The reality of the Einstein
eﬀect was veriﬁed in the years 1926–1928 with convincing accuracy bycareful analysis25 of the lines of various elements in the solar spectrumby St John, an astronomer working in the Mount Wilson observatory.
In 1960, the Mossbauer resonant eﬀect (emission of γ rays with anextremely narrow proﬁle, reabsorbed in crystals with a very sharp resonance) enabled Pound and Rebka to measure the shift of spectral lines
due to the variation of the gravitational ﬁeld with height above theEarth’s surface in their laboratory. We give below a brief account of thetheoretical prediction in this case.
Inspired by the Newtonian approximation (see Section IV.5 in Chapter IV), we assume the spacetime metric to be, in the laboratory,g = −N 2 (x1 )dt2 + g11 (x1 )(dx1 )2 + gab dxa dxb ,a, b = 2, 3,(III.8.1)

25 See details on pp. 126–130 of Bruhat(1931).

64

General Relativity

where x1 varies with the height of a point while xa , a = 2, 3, label the
position of this point in a horizontal plane. Consider two identical clocks,
one at rest on the ground, say at x1 = 0, the other at rest on the samevertical, with height labelled by x1 = h. The hypotheses are therefore
that, in the coordinates xα , the representatives of the world lines of theclocks lie in the plane (x1 , t); they are the lines where only t varies, given
respectively by x1 = 0 and x1 = h. Let T (0) and T (h) be the periods,
in proper time, of these clocks. The signals of the beginning and end ofa period of the clock on the ground are transmitted by light rays to an
observer handling the other clock. Such light rays obey the diﬀerential
equation√
g11
dt.
(III.8.2)=
1
dx
N
A signal emitted at parameter time t(0) reaches the clock at height h atparameter time
 h√
g11 1
t(h) =(x ) dx1 + t(0).
(III.8.3)
N0
The diﬀerence in the parameter times between two emitted and two
received signals is therefore the same. However, this statement is nottrue for the proper times, which are the physically measured quantities.
We denote by T (0) (respectively T (h)) the period in proper time of theclocks. Using the relation between the parameter time and the proper
time for each of the clocks, namely dT = N dt, we ﬁnd thatt2 (0) − t1 (0) = N −1 (0)T (0) = t2 (h) − t1 (h) = N −1 (h)T (h). (III.8.4)
The corresponding lapse of proper time marked by the clock of thisobserver is
T (h) =N (h)T (0).
N (0)(III.8.5)

A longer period is observed (redshift) if N (h) > N (0), which is the casein our laboratory example where the source of gravitation is the Earthwith mass m. Then (see the Newtonian approximation)
N=
˜ 1−

m
.
x1

(III.8.6)

Theory and experimental results have proven to be in excellent agreement.
A related eﬀect (which involves the geometry of null geodesics connecting the timelike world lines of various massive bodies) has been measured
directly for celestial objects, and called time delay by Shapiro, who
measured it ﬁrst, in 1966, by sending radar signals to planets (i.e. in thegravitational ﬁeld of the Sun) and measuring the time elapsed on Earthbetween their emission and their return after reﬂection. We will give thecalculations leading to the theoretical prediction in Chapter V, devoted
to the Schwarzschild spacetime.

III.9The reality of the proper time deﬁned on spacetime by a Lorentzian
metric that is not locally ﬂat in the presence of a gravitational ﬁeld has
been checked directly by carrying caesium clocks on satellites around theEarth and observing that they gain some hundred nanoseconds26 overidentical clocks that remain on the ground. The experiment has now
been done with clocks on satellites orbiting Mars and on its surface.
Finally, it has been found that the accuracy of the GPS positioning system depends upon General Relativistic correction of the physical(proper) time, both because the satellites that receive the signals from
Earth are far above the Earth’s surface and because, for high-precision
results, it is necessary to take into account that the Earth is not exactly
spherical, and has plains and mountains that induce variations in thegravitational ﬁeld.27

III.8.4

Conclusion

All experimental results to date28 are in agreement with the predictionsof Einstein’s General Relativity.

III.9

Problems

III.9.1

Newtonian gravitation theory in absolute space
and time E n × R

1. By analogy with the case n = 3, the Newtonian potential on the
Euclidean space E n of a gravitational mass density ρG is a scalarfunction U given by⎡
U (x) :=ρG (y)
dy,n−2
Rn |x − y|with|x − y| = ⎣⎤ 12(xi − y i )2 ⎦ .

i=1,...nShow that U satisﬁes the elliptic partial diﬀerential equation
ΔU :=



∂2
U = −(n − 2)(area S n−1 )ρGi )2(∂x
i=1,...,n2. Show that the Newtonian potential of a pointlike particle with gravitational mass density mG δ, where δ is the Dirac measure at the origin
O, is
U=

mG
.
rn−2

Problems

65

26 After correction for the kinematic
eﬀect of Special Relativity. An approximately drag-free satellite describes ageodesic of the Lorentzian (Schwarzschild) metric, but the observer on
Earth, not in free fall because held by
the ground, does not.
27 We compute in Problem IV.11.5 in
Chapter IV the ﬁrst correction (quadrupole moment) to the Newtonian potential due to non-sphericity of thesource. In Newton’s theory of gravity, the potential has no inﬂuence ontime. We will study in Chapter IV the
Newtonian approximation of the Einstein theory of gravity, in which time isdependent of the gravitational ﬁeld.
28 For a description of basic experiments, their results and discussions,see, for instance, Chapter 4 of Ohanianand Ruﬃni (2013). For up to date results, see papers by Cliﬀord Will, forinstance his recent review (Will, 2014).

66

General Relativity

Hint: The Dirac measure at O is by deﬁnition such that if f is acontinuous function at O, thenf (y)δ dn y = f (0)(area S n−1 ).
Rn

3. Assuming the laws of Newtonian dynamics and gravitation,
F = mI γ,with mI the inertial mass and γ = grad U,

show by using Newton’s principle of action and reaction that mG is
proportional to mI by a ﬁxed constant.

III.9.2

Mass in length units (case n = 3)

The value of the Newton gravitational constant GN is, in cgs units witha 10−13 accuracy,
GN,cgs = 6.67259 × 10−8 =˜ 7 × 10−8 .
Compute the unit of mass in the centimetre–gram system (in which thesecond is expressed in terms of the centimetre by taking c = 1) under
the condition that the Newton gravitational constant is equal to 1.
Compute in this unit the mass of the Earth, whose mass is about
6 × 1024 cgs grams in the cgs system.

Solution
This uses only the dimensions L3 M −1 T −2 of GN . Recall that1s=
˜ 3 × 1010 cm-s.

(III.9.1)

Set 1 g = μ cm-g. The trivial fact that the quotient of the values ofa quantity in diﬀerent units is the inverse of the quotient of the units
gives
GN,geom,cm = GN,cgs μ−1

1
× 10−20 = 1,9

with
GN,cgs =˜ 7 × 10−8 .
29 See more comments in Chapter V.

Hence, GN,geom = 1 if μ = 79 ×10−28 . Then mEarth = 6×1027 g implies29
mEarth,geom =
˜

42
cm.
90

III.9

III.9.3

Planck units

The dimensional constant that has been measured with the greatest
accuracy is the Planck constant h. Planck remarked that it is possible
to deﬁne from GN , h and c units for time, length and mass, called Planck
units or geometric units, such these constants all take the value 1.
1. Show that the dimensions of h are M L2 /T −1 .
2. Assume that the three constants c, GN and h are known in cgs units.
Denote by LP = x cm, Mp = y grams and TP = z seconds new units
such that these constants are equal to 1. Such units are called Planck
units. Write the three equations satisﬁed by x, y and z.
3. Compute the unique solution giving the Planck units.

Solution
1. h = E/ν. The dimensions of energy E are M L2 /T −2 and those of
frequency ν are T −1 . The result follows immediately.
2. The relation between cgs and Planck units and the dimensions of thegiven constants implyzyz 2
cP = ccgs , GP = Gcgs 3 ,xx

hP = hcgsz,yx2

and setting these constants equal to 1 givesx= ccgs ,z

x3= Gcgs ,yz 2which imply triviallyxx= ccgs ,= c−2
cgs Gcgs ,zyyx2= hcgs ,zz= c−3
cgs Gcgs ,y2
c3cgs G−1
cgs x = hcgs .

Therefore, the unique values of the unknowns x, y and z arehcgs Gcgsx=, y = xc2cgs G−1
z = xc−1
cgs ,
cgs ,
c3cgs
which give
x=




hcgs
,
c3cgs Gcgs

y=hcgs ccgs
,
Gcgs


z=

hcgs Gcgs.
c5cgs

Of course, the use of Planck units in formulas does not eliminate
the need for at least one of them to be deﬁned through a physicalphenomenon, to link equations with reality.

Problems

67

IV

The Einstein equations

IV.1
IV.2

Introduction
68
The Einstein equations 69

IV.3

The cosmologicalconstant
General Einsteinian
spacetimes

IV.4

72
72

IV.5

Newtonianapproximation75
IV.6 Minkowskian
approximation
79
IV.7 Strong high-frequency
waves
82
IV.8 Stationary spacetimes
IV.9 Lagrangians
IV.10 Observations andexperiments
IV.11 Problems

87
91
94
95

1 Except for the advance of the perihelion of the orbit of Mercury, which
behaves approximately like a slowly
rotating ellipse whose perihelion advances 42 per century more than thatcomputed using Newton’s law when
taking into account the gravitationalﬁelds due to other planets.
2 Einstein was helped in obtaining theﬁnal result by his friend, the mathematician Marcel Grossmann.

IV.1

Introduction

The previous chapter replaced the Minkowski spacetime (R4 , η) of Special Relativity by a general Lorentzian manifold, (V 4 , g)—hence thename ‘General Relativity’. The connection coeﬃcients of the metric g
replaced both the gravitational and the inertial forces. There are spacetime frames where at a given point, or even along a given spacetime
curve, they cancel each other. This statement is in agreement with the
weak Einstein equivalence principle, which concerns the motion of one
test particle. However, in contrast to inertial forces in Newtonian theory,
which disappear in inertial frames (without canceling gravitation), there
are no reference frames deﬁned on an open set of a spacetime of General Relativity where all the connection coeﬃcients vanish, except if themetric is ﬂat, i.e. free of gravitation. In Einstein gravity, the existenceof a non-zero gravitational ﬁeld manifests itself by the curvature of thespacetime metric.
The non-Minkowskian metric g replaces in a sense Newton’s gravitational potential, which satisﬁes the Poisson equation, a linear elliptic
second-order partial diﬀerential equation with mass density as a source.
The metric g should satisfy equations that give in ﬁrst approximation
the same results as Newton’s law for the motions of bodies in a weakgravitational ﬁeld and with velocities small with respect to the speedof light. Indeed, in these circumstances, Newton’s law had been veriﬁed with excellent agreement.1 Einstein’s geometric foresight looked forgeometric relations between the metric g of General Relativity and possible sources of gravitation. The equivalence of mass and energy that hehad discovered before, the symmetric 2-tensor character of the metricg whose curvature he wanted to link with the gravitation sources, andhis genius all led Einstein, after various unsuccessful attempts,2 to thefamous Einstein equations that are at present more fundamental thanever for theoretical physics at all scales.
We will write the equations in spacetimes of arbitrary dimension n+1,
since they are now used by physicists in a wider context than the original
dimension n = 3. We will specify that n = 3 only when this leads tospecial properties.

IV.2

IV.2

The Einstein equations

IV.2.1The Einstein equations in vacuum

The vanishing in a domain of spacetime of the Riemann curvature tensor of the metric g implies that this metric is locally ﬂat in this domain(see Chapter I), and hence without gravitational eﬀects. It is therefore
too strong a condition to impose in domains empty of matter or ﬁeld
energy sources in a spacetime that is not globally empty. A natural candidate for an equation to impose in vacuo3 on the Lorentzian metric g, asymmetric 2-tensor, is the vanishing of the symmetric 2-tensor Ricci(g),which is related to the Riemann curvature tensor by contraction. The
tensor Ricci(g) is a second-order partial diﬀerential operator on g, likethe Laplace equation for the Newtonian potential, though it has quite
diﬀerent properties, as we will see in the following sections: ﬁrst, it isnonlinear;4 second, it presents both elliptic properties like the Newtonian potential and hyperbolic ones like the wave equation, corresponding
to propagation of the gravitational ﬁeld with the speed of light. We
formulate the basic proposition5 of Einsteinian gravitation.
Proposition IV.2.1 In an open set U devoid of energies other thangravitation, the Lorentzian metric g of an Einsteinian spacetime (V, g)satisﬁes the following tensorial Einstein equations in vacuum:
Ricci(g) = 0.

∂ λ
∂ λ
Γ −
Γ + Γλαβ Γμλμ − Γλαμ Γμβλ = 0,∂xλ αβ ∂xα βλ

3 The notion of vacuum seems clear tocommon sense: it is a region of spacetime where there is neither matter nor
electromagnetic ﬁeld. Modern physics
has shown that what we called the vacuum is indeed full of strange things—
pairs of particles and antiparticles constantly annihilating each other and liberating virtual photons or neutrinos,
which in turn give birth to pairs of particles. But these stange phenomena arein the realm of quantum ﬁeld theory
and do not concern us in this book.
4 Gravitation is also its own source.

5 It

is known now that the Einsteinequations are the only tensorial ones
(up to addition of a ‘cosmologicalterm’, see Section IV.3) that are of second order and quasilinear for the metricg in four dimensions.

(IV.2.2)

6 A partial diﬀerential equation of orderm is called quasilinear if it is linear in
the derivatives of order m. It is called
hyperquasilinear if, in addition, the coeﬃcients of these derivatives do notcontain derivatives of order m − 1.

where the Γλαβ are the Christoﬀel symbols of the metric g, given in local
coordinates (see I.7.13) by
1 ∂gαμ
∂gβμ
∂gαβ
+
−
Γλαβ := g λμ [αβ, μ], with [αβ, μ] :=
.
2 ∂xβ
∂xα
∂xμ
These equations are not independent; they satisfy the identities deducedin Chapter I from the Bianchi identities:∇α S αβ ≡ 0,1
Sαβ := Rαβ − gαβ R,
2

69(IV.2.1)

The equations (2.1) are represented in each chart (see Chapter I)with domain included in U by the following system of second-order hyperquasilinear6 partial diﬀerential equations for the components gαβ ofthe metric g:Rαβ ≡The Einstein equations

R := g λμ Rλμ .

(IV.2.3)

The tensor with components7 Sαβ is called the Einstein tensor, it is
the symmetric 2-tensor with geometric expression
1
Einstein(g) := Ricci(g) − gR(g).
2

7 Denoted by G
αβ by many authors.

70

The Einstein equations

IV.2.2Equations with sources

It took a considerable time for Einstein to settle on the choice of equations connecting the metric g with sources. From a physical point ofview, he was inspired on the one hand by the relativity to observersof the splitting beween energy and momentum (see Chapter II) and by
the conservation laws for the various stress–energy–momentum tensors
T found in Special Relativity and on the other hand by the equivalence principle. From a mathematical point of view Einstein’s choice ofequations was motivated by the contracted Bianchi identities.
The geometric equations found by Einstein in 1915, with the help of
the mathematician Marcel Grossmann, are the very simple ones
1
Einstein(g) := Ricci(g) − gR(g) = GE T ;2i.e. in local coordinates and arbitrary dimension n + 1 ≥ 3,1
Sαβ := Rαβ − gαβ R = GE Tαβ .
28 It is only in the classical dimension n+

1 = 4 that d−2 = 2—a fact that should
not be forgotten when working in otherdimensions.

9 In particular when there is a densityof linear momentum, which is naturally represented by a non-symmetric
2-tensor. This is the case for electromagnetic ﬁelds with induction.
10 Denoted by κ by some authors.

(IV.2.4)

Exercise IV.2.1 Show that in a spacetime of dimension n + 1 > 2,
the Einstein equations with sources are equivalent to the following:8Rαβ = GE ραβ ,with

ραβ ≡ Tαβ −

Tλλ
gαβ .
n−1

(IV.2.5)

The source T is the stress–energy symmetric 2-tensor, supposed torepresent the pointwise value of all the energies, momenta and stresses
present in the spacetime. It is a phenomenological tensor whose choice
is not always easy,9 even in the classical dimension n + 1 = 4 and even
without taking quantum mechanical considerations into account.
The factor10 GE is a phenomenological dimensional constant. We will
explain later, by using the Newtonian approximation, why in spacetime
dimension n + 1 = 4 physicists takeGE = 8πGN ,with GN the Newtonian gravitational constant.
Note that the gravitational constant does not appear in Special
Relativity—hence, the Einstein equivalence principle does not imply
that it is a constant. In fact, several physicists (Jordan, Thiry, Dicke,
Dirac, and others) have conjectured that it is a spacetime scalar function.
However, no experiment has conﬁrmed this conjecture.
The geometric units of time and length have been deﬁned in Chapter III. In these units, the speed of light is equal to 1.
In this book, as in most mathematical studies, except when making
numerical estimates, we assume units of mass–energy chosen such that
GE takes its geometric value: GE = 1.

IV.2

The contracted Bianchi identities (2.3) show that the Einsteinequations are compatible only if the tensor T satisﬁes the equations,called conservation laws,∇α T αβ = 0.

(IV.2.6)

We have written in Chapter III the equations of motion of the simplest,
most common energy sources on a general Lorentzian metric of arbitrary dimension by using their expression in Special Relativity and theEinstein equivalence principle.

IV.2.3Matter sources

In the case of dust (incoherent matter, also called pure matter),the stress–energy tensor isTαβ = ruα uβ ,(IV.2.7)

with u the unit ﬂow velocity and r the rest-mass density.
In the case of a perfect ﬂuid, the stress–energy tensor isTαβ = μuα uβ + p(gαβ + uα uβ ),(IV.2.8)

with u again the unit ﬂow velocity and with μ and p respectively theenergy and the pressure densities, both of which are non-negative inclassical situations. The equations of motion of the phenomenologicalquantities μ, p and u (see Chapter III) imply that T αβ satisﬁes theconservation law∇α T αβ = 0.

(IV.2.9)

Note that we could have made the converse argument and deduced
the equations of motion of the sources from the conservation laws implied by the Einstein equations satisﬁed by the spacetime metric. In
Einstein’s theory, in contrast to Newton’s, the equations of motion are
a consequence of the equations satisﬁed by the gravitational potential.

IV.2.4

Field sources

For an electromagnetic ﬁeld, the stress–energy tensor is the Maxwell
tensor1(IV.2.10)ταβ := Fα λ Fβλ − ηαβ F λμ Fλμ .
4
The Maxwell tensor satisﬁes the conservation laws
∇α τ αβ = 0(IV.2.11)

if the 2-form F that represents the electromagnetic ﬁeld satisﬁes theMaxwell equations of General Relativity
dF = 0 and ∇ · F = 0.

(IV.2.12)

The Einstein equations

71

72

The Einstein equations

IV.3
11 Thereexist other possibilities(Gauss–Bonnet equations) in higher
dimensions, which seem to play a rolein string theory.

The cosmological constant

It is known that the only tensorial operator for a metric g on a fourdimensional11 spacetime that is second-order and quasilinear is theEinstein tensor with the possible addition of a linear term Λg, with Λ
an arbitrary constant, called the cosmological constant, which Einstein
did not include, because it has no a priori physically ﬁxed value. The
Einstein equations with cosmological constant Λ read1
Einstein(g) + Λg ≡ Ricci(g) − gR(g) + Λg = T.
2
In local coordinates, these take the form

12 Calling it ‘the greatest blunder of mylife’, because its value was neither logically nor experimentally determined.

1Sαβ + Λgαβ ≡ Rαβ − gαβ R + Λgαβ = Tαβ ,(IV.3.2)
2
where T is the stress–energy tensor considered before, supposed to represent the density of all the energies, momenta and stresses of thematter and classical ﬁeld sources. Einstein introduced the cosmologicalconstant when looking for a stationary model for the cosmos (see Problem IV.11.1) and removed it12 after the interpretation of astronomical
observations as showing the universe to be expanding. It is again included by cosmologists, although its value is controversial and may havebeen diﬀerent at diﬀerent epochs. It is generally considered to be verysmall now but to have been large in the early universe. In fact, thetendency now is to put Λ among the sources, interpreting −Λgαβ as anenergy–momentum tensor of the vacuum generated by quantum-particle
processes; the constant Λ is then replaced by a function of a new ﬁeld φsatisfying a wave equation with a non-linear potential. A ‘cosmological’term also appears in supergravity and string theories.
The equations (3.2) on an n + 1 = d-dimensional spacetime are
equivalent to the following:


d
Tλλ
Λ−
gαβ . (IV.3.3)Rαβ = ραβ , with ραβ ≡ Tαβ +
d−2
d−2

IV.4
13 Notto be confused with the Einstein spaces of geometers, which are
Riemannian spaces of constant curvature.

(IV.3.1)

General Einsteinian spacetimes

A spacetime of General Relativity is a pair (V, g), with V a diﬀerentiable
manifold and g a Lorentzian metric on V, both a priori arbitrary. It iscalled an Einsteinian spacetime13 if the metric g satisﬁes on V the
Einstein equations with source a phenomenological 2-tensor T assumed
to represent all the non-gravitational energies, momenta and stresses:
Einstein(g) = T,
∇ · T = 0,

i.e

i.e.

Sαβ = Tαβ ,∇α T αβ = 0.

(IV.4.1)(IV.4.2)

The deﬁnition of T is clear in the cases mentioned in Section IV.2.3 andIV.2.4, but is not so clear and is still controversial for more complicatedIV.4

sources, even in the classical cases of dissipative ﬂuids with or without
heat current and of electromagnetic ﬁelds with inductions. Coupling withquantum ﬁelds raises the deep problem of quantum gravity. New andmysterious ﬁelds are now being discovered in cosmological studies (seeChapter VII).
Two isometric 14 spacetimes (V, g) and (V  , g  ) are considered as
identical.
The spacetime is called a vacuum spacetime if T ≡ 0.

IV.4.173

14 That is, such that there exists a diffeomorphism f of V onto V  such thatf ∗ g  = g.

Regularity

In an Einsteinian spacetime, the manifold V can be assumed to be C ∞without restricting its generality.15 We have already said that we assume
in this book that the metric is as smooth as necessary for any given
statement to be true. It is clear that if the source T is discontinuous,
then the metric g cannot be C 2 . Using the deﬁnition of generalized
derivatives, it may be possible to deﬁne the Einstein tensor16 for metrics
that do not admit second derivatives in the usual sense.
The largest simple Sobolev Hilbert space for the components of themetric g for which the components of the Riemann tensor in four dimensions are almost everywhere deﬁned functions on a coordinate patch
diﬀeomorphic to a ball Ω of R4 is H3 (Ω), the space of functions that,together with their generalized partial derivatives of order up to 3, aresquare-integrable on Ω.
For physical interpretation, one is led to split, at least locally, space
and time and consider coordinate patches Ω = ω × I, with ω a ball in R3
and I an interval of R. The restriction to ω of gαβ is then assumed to bein H2 (ω), and hence uniformly continuous on ω, while the restrictionsof the ﬁrst derivatives ∂gαβ /∂xλ and second derivatives ∂ 2 gαβ /∂xλ ∂xμbelong respectively to H1 (ω) and L2 (ω). The restriction of the Riemanntensor is then deﬁned as a function in L2 (ω). For more comments, see
Chapter VIII on the Cauchy problem.

IV.4.2

General Einsteinian spacetimes

Boundary conditions

The Einstein equations are partial diﬀerential equations, and so theirsolutions are linked with the data of boundary conditions. We will see
in Chapter VIII on the Cauchy problem that, even in a vacuum, theEinstein equations present both elliptic aspects like the Poisson equationof Newtonian gravity and hyperbolic aspects like the wave equation oflight propagation. We just mention here that in the primary cases the
manifold V of an Einsteinian spacetime (V, g) is a product M × R, with
M spacelike and R timelike.
For cosmological studies, the case of a compact manifold M withoutboundary has been used, although this seems to be in conﬂict with recent
observations hinting at a ﬂat universe—but cosmology is a very debatedsubject.

15 It is known that any C 1 manifold canbe given a C ∞ structure that induces
this C 1 structure.

16 This possibility is limited, however,by the nonlinearity of the Riemanncurvature tensor.

74

The Einstein equations

17 This model is only an approximation,

like all mathematical models of physical situations. The system considered
as isolated is in fact embedded in a
curved universe.
18 Higher-dimensional

asymptoticallyEuclidean manifold can also be deﬁned.

19 AsymptoticallyEuclidean manifolds with several ends can also be
considered, with just some additional
complications in writing down theappropriate equations.

In the astrononphysical context, particularly relevant situations thatcan be confronted with observation are isolated systems of a few bodies
far from any other source of gravitation, for example the Solar Systemand binary star systems. In these cases, it is legitimate to assume that
(M , ḡ), with ḡ the Riemannian metric induced on M by the spacetime
metric, tends to be ﬂat far away from the studied system. It is thenassumed to be an asymptotically Euclidean manifold, which we deﬁnebelow.17
A three-dimensional18 asymptotically Euclidean Riemannian
manifold (M, ḡ) is deﬁned as a smooth manifold M union of a compact set and an19 end M end diﬀeomorphic to the exterior of a ball of
R3 . This end is given local coordinates x1 , x2 , x3 by this diﬀeomorphism.
We denote by r, θ, φ the corresponding polar (pseudo) coordinates,
1

i 2 2
. The Riemannian manifold (M, g) is called asymptotr :=i (x )
ically Euclidean if in M end it holds thatgij = δij + O(r−1 ), ∂ k gij = O(r−|k|+1 ),with the notation∂ k :=

20 Formetrics approaching the Euclidean metric in weighted Hölder
spaces, see Choquet-Bruhat (1974) and
Chaljub-Simon and Choquet-Bruhat(1979), and for those in weighted
Sobolev spaces, see Cantor (1979),
Cantor and Brill (1981), and ChoquetBruhat and Christodoulou (1981).

∂ k2
∂ k3
∂ k1
,
(∂x1 )k1 (∂x2 )k2 (∂x3 )k3|k| = k1 + k2 + k3 ,and where a function f is said to be O(r−k ) if rk f is uniformly boundedon M end . The useful maximum order of the derivatives appearing in thedeﬁnition depends on the circumstances. Variants of hypotheses on the
behaviour of metrics at inﬁnity appear in the solution of problems on
asymptotically Euclidean manifolds.20

IV.4.3

Physical comment

Even in the vacuum case, there exist many Einsteinian spacetimes. They
have underlying manifolds with diﬀerent topologies, as well as diﬀerent
metrics on each of these manifolds. A great number of exact solutionsof Einstein equations possessing isometry groups have been constructed,
in vacuum or with various sources. Some of these spacetimes have (atleast at present) a purely mathematical interest, but a few of them aremodels of known physical situations, at diﬀerent time or space scales.
There is no universal Einsteinian spacetime as a model for reality—
this is in disagreement with Newton’s concepts, and also with Special
Relativity.
Before giving the main lines of construction and properties of general Einsteinian spacetimes, we justify their validity by their possible
approximation by the Newton spacetime for sources with low velocity,
and we give for weak gravitation the approximation of Einsteinian spacetimes by Minkowski spacetime. The Minkowskian approximation reveals
a property of gravitational ﬁelds that has no analogue in Newton’s theory, namely the existence of gravitational waves and their propagation

IV.5

with the speed of light—this property eliminates the need for ‘action at
a distance’ that was so puzzling for Newton.

IV.5

Newtonian approximation

We have said that Einstein’s theory of gravitation must nearly coincidewith Newton’s in the physical situations where the latter proved to beaccurate. These situations are slow (compared to the speed of light)
velocity of the gravitating bodies with respect to Newton spacetime andweak gravitational ﬁelds. In Einstein’s theory, the spacetime coincides
with Minkowski spacetime in the absence of gravitation, that is withthe manifold R3 × R endowed with the ﬂat metric, which, in inertial
coordinates and geometrical units, reads as follows:−(dx0 )2 +3(dxi )2 .

i=1

An Einsteinian spacetime with a weak gravitational ﬁeld will be R3 × R
with a metric such thatg00 = −1 + h00 ,g0i = h0i ,gij = δij + hij ,with hαβ of order  << 1.
In a spacetime with masses moving slowly (compared with 1, the speedof light), the components hiα are small with respect to h00 and the time
derivatives ∂0 hαβ are small with respect to the space derivatives ∂i hαβ ;these are considered to be of the same order as hαβ .
Exercise IV.5.1 Justify these statements by considering units wherec=˜ 3 × 105 km/s.
Hint: Change the timescale by seting dx0 = c dt.

IV.5.1

Determination of GE

The component R00 in the considered approximation is equivalent to1
R00 =˜ ∂i Γi00 =˜ − Δh00 ,2with Δ the Laplace operator of Euclidean space. In Newtonian theory, the gravitational ﬁeld is the gradient of the Newtonian potential, afunction U that satisﬁes the Poisson equation:
ΔU = −ρG ≡ −4πGN ρ(IV.5.1)

where ρ is a positive scalar function on R3 equal to the mass density of the sources and GN is the Newtonian gravitational constant (seeChapter III).
In Einstein’s theory, the energy sources are represented by the stress–
energy tensor. Under usual circumstances, the most important energy

Newtonian approximation

75

76

The Einstein equations

source is pure matter, i.e. the stress–energy tensor is of the following
form, with μ some positive scalar function and u the unit ﬂow vector:
Tαβ =˜ μuα uβ .
In the case where the matter has a small velocity (with respect to the
speed of light, taken to be c = 1) for observers following the time linesof the coordinate system, it holds that u0 is equivalent to −1 and ui tozero; therefore, Tiα is negligible and11ρ00 ≡ GE (T00 − g00 T ) =˜ GE μ.
(IV.5.2)
2
2
We see that in the approximation that we are making, called the
Newtonian approximation, the equation R00 = GE ρ00 gives
Δh00 =
˜ − GE μ,that is, the Poisson equation of Newton’s theory for the gravitational potential if h00 is identiﬁed with 2U and one sets GE := 8πGN . This is thereason why many physicists write the Einstein equations in dimension
3 + 1 asSαβ = 8πGN Tαβ .

(IV.5.3)

The value of the Newtonian gravitational constant depends on the
choice of units for time, length, and mass. As we said before, GN isapproximately, in CGS units,
GN,cm3 g−1 s−2 = 6, 67259 × 10−8 .
Of course, making GE = 1 instead of GN = 1 implies a change by afactor 8π in the unit of mass–energy.
In this book, we write the Einstein equations in geometric units, i.e.
Sαβ = Tαβ , introducing the relevant scale factor to the usual physicalunits only when this is useful for interpretation of observations.

IV.5.2Equations of motion

In Newton’s theory, the equation of motion of a test particle is
d2 xi
∂U
=
.
2
dt
∂xi
In General Relativity, a test particle follows a geodesic of the spacetime
metric, i.e.
d2 xα
dxλ dxμ
= 0.
+ Γα
μλ
2
ds
ds ds
For small velocities, we have already seen that dx0 /ds ∼ 1 anddxi /ds ∼ 0; therefore,d2 xi
1∼ −Γi00 ∼ ∂i h00 .
(dx0 )2
2(IV.5.4)

IV.5

Newtonian approximation

We see that the Einstein law coincides with the Newton law in the
Newtonian approximation where h00 = 2U .

IV.5.3Post-Newtonian approximation

One can improve the Newtonian approximation by retaining further
terms in the expansions. This leads to the so-called post-Newtonian(PN) approximation. To make these approximations, it is more convenient not to replace the gravitational constant and speed of light by theirgeometric value 1.
Let us brieﬂy explain the structure of the ﬁrst post-Newtonian (1PN)
approximation, where one keeps all the corrections to the Newtonian
approximation containing one power of 1/c2 . There are three sourcesof such 1/c2 corrections: (i) in algebraic (v/c)2 terms, where v is a velocity variable; (ii) in U/c2 terms, where U is a Newtonian-potential-likevariable; or (iii) in time-derivative terms ∂02 = c−2 ∂t2 . At the 1PN approximation level, one must go beyond the linearized approximation to
the Einstein equations, and keep some of the quadratically nonlinear
terms in the time–time Einstein equation
R

00

−4

= 8π GN c


T

00


1
00
Tg
−
.
n−1

A convenient way of doing so is to parametrize the metric in the following
exponential form:21
2
g00 = − exp − 2 V ,c4g0i = − 3 Vi ,c2gij = exp + 2 V γij .
c(IV.5.5)

Inserting this form in Einstein’s equations, and using for simplicity wave
coordinates, one ﬁnds (in dimension n = 3) that the auxiliary metric γij
must be ﬂat modulo corrections of the next PN order O(1/c4 ), i.e.

γij = δij + O

1
c4


.

(IV.5.6)

In addition, one ﬁnds that the gravitational ‘scalar potential’ V andthe gravitational ‘vector potential’ Vi satisfy the following simple, linear
equations:
ΔV −

1 2
∂ V = −4π GN σ,
c2 t
Δ Vi = −4π GN σ i ,(IV.5.7)

21 Blanchet and Damour (1989).

77

78

The Einstein equations

where the source terms σ and σ i are simply given in terms of thecontravariant components of the stress–energy tensor by
T 00 + T ss
,
c2
0i
T
.
σ i :=cσ :=(IV.5.8)

The linear form of the equations (5.7) for V and Vi allows one to solve
them explicitly in terms of σ and σi . At the 1PN approximation, one
can write
 



1
1 2
3  σ(t, x )V (t, x) = GN d x+∂ σ(t, x ) |x − x | + O 4,|x − x | 2c2 t
c



1
σ i (t, x )+O 2 .
Vi (t, x) = GN d3 x
|x − x |c(IV.5.9)
The approximate form of the metric obtained by inserting
Eqs. (IV.5.9) in to the expressions (5.5), the ﬁrst of which explicitly
reads (upon expanding the exponential to show the level of nonlinearity
actually contained in the 1PN approximation)

 
1
2
2 2
,
(IV.5.10)
g00 = − 1 − 2 V + 4 V + O 6ccc

22 See, for instance, Will (2014).

23 See Brumberg and Kopejkin (1989)

and Damour, Soﬀel, and Xu (1991,
1992).

is suﬃciently accurate for describing all current experiments and observations in the Solar System, including perihelion advances of planetary
orbits, the bending and delay of electromagnetic signals exchanged
between the Earth and planets or satellites, and the very accurate
laser ranging data to the Moon. Let us, however, note several diﬀerent
extensions of the above 1PN formalism.
First, to compare not only General Relativity but also diﬀerent (nonEinsteinian) gravitational theories with experiments and observations,an extension of the above general relativistic 1PN metric, containing awhole collection of adjustable parameters, has been introduced, and iscalled the parametrized post-Newtonian (PPN) formalism.22
Second, when developing a general relativistic theory of the motion of
an N -body system (such as the Solar System), it has been found usefulto generalize the PN formalism into a multichart approach to generalrelativistic celestial mechanics.23 In contrast to the traditional one-chart
approach to the general relativistic N -body problem (in which a single
coordinate system is used to describe both the gravitational ﬁeld andthe motion of N bodies), the multichart approach uses N + 1 coordinate
systems: a global coordinate system xμ and N local coordinate systemsα(with A = 1, 2, . . . , N ), each of which is attached to one of the
XA
N bodies. This multichart approach has been found useful not only in
the Solar System, but also in binary systems containing strongly selfgravitating objects such as neutron stars or black holes (see Section VI.6
in Chapter VI).

IV.6

IV.6

Minkowskian approximation

To compute the Minkowskian approximation of solutions of the Einsteinequations, one considers Lorentzian metrics on Rn+1 with components
in some coordinate frames:gαβ = ηαβ + hαβ ,with ηαβ = diag(−1, 1, . . . ., 1). One assumes hαβ and its derivatives tobe small with respect to 1; their products are then even smaller.

IV.6.1

Linearized equations at η

A ﬁrst approximation is obtained by neglecting in the Einstein equations
all terms containing products of the perturbation h, that is, by considering a solution of the Einstein equations linearized at η. The following
linearizations are particular cases of those computed in Chapter I and
result easily from the expressions for the Christoﬀel symbols and the
Ricci tensor when products of h’s and of their derivatives are neglected:(δΓλαβ )η (h) =1 λμ
1η [μ, αβ] = η λμ (∂α hμβ − ∂μ hαβ )221(δRαβ )η (h) = − hαβ + ∂α fβ + ∂β fα ,2

(IV.6.1)

2
where  := η λμ ∂λμ, with ∂α := ∂/∂xα , is the Minkowskiand’Alembertian operator and fα is the set of functions (indices are raisedwith the Minkowski metric η)


1
1
fα :=
(IV.6.2)∂λ hλα − ∂α hλλ .
22The decomposition (6.1) is non-tensorial; the conditions f α = 0 can
be satisﬁed by choice of coordinates.
Exercise IV.6.1 Show that the equations f α = 0 are equivalent to thelinearized wave equations for the coordinates xα , i.e.
f λ ≡ (δF λ )η (h),withF λ := g xλ ≡ g αβ Γλαβ .

Hint:δ(g αβ Γλαβ )|η = η αβ (δΓλαβ )|η .

IV.6.2

Plane gravitational waves

The formula (6.1) shows that if f α = 0 the ﬁrst approximation is solutionof the linearized Einstein equations at η in vacuum, (δRαβ )η (h(1) ) = 0
which reduce to2hαβ := η λμ ∂λμhαβ ≡ −
(1)

(1)

 ∂2
∂2
(1)
hαβ = 0.
+
i )2
(∂t)2
(∂x
i

(IV.6.3)

Minkowskian approximation

79

80

The Einstein equations

These equations are a system of ordinary wave equations for the perturbation h(1) ; the solutions propagate on Minkowski spacetime withvelocity 1, i.e. the speed of light.
(1)
In the case of a Minkowski background, the equations hαβ = 0 imply
trivially fα = 0. However, fα = 0 implies fα = 0 only if fα satisﬁes
appropriate initial data. We will study such problems in Section IV.7
for the full Einstein equations. Here, for weak ﬁelds, we will proceed
directly, considering the general plane waves
hαβ = cαβ ϕ(aλ xλ ),(1)

with cαβ

and aλ constants.

Such a perturbation of the Minkowski metric is a solution of thelinearized Einstein equations at η if(ai )2 ϕ (aλ xλ ) = 0(IV.6.3a)1with χαλ := cαλ − ηαλ c.
2(IV.6.3b)(1)hαβ ≡ cαβ ϕ ≡ cαβ (−a0 )2 +iand (linearized gauge condition)
2fα ≡ aλ χαλ ϕ (aλ xλ ) = 0,

Equation (6.3a) implies that aα is a null vector,
(ai )2 = 0,(a0 )2 −(IV.6.4)iwhile (6.3b) implies that the constant tensor cαβ satisﬁes the polarization conditions
aλ χαλ = 0,
which are n+1 linear and homogeneous equations with (n + 1)(n + 2)/2
unknowns χαλ . The wave is said to be transverse-traceless, abbreviated to TT, because by a ﬁrst-order change of frame one can reduce thepolarization conditions to the equations
c00 = c0i = 0, cii = 0, and ai cij = 0.
Exercise IV.6.2 Prove this statement.
The polarized cαβ span a vector space of dimension (n + 1)(n − 2)/2:
in the classical dimension n = 3, weak gravitational waves have twodegrees of polarization. For example, choose a frame such that a2 =a3 = 0. The equations above then imply c1i = 0 and c22 + c33 = 0.
Examples of oscillating weak gravitational waves are(1)hαβ = 0,

(1)

(1)

(1)

except for h23 , h22 , and h33 ,

with
(1)

h23 = c23 sin(ωt − ωx1 ),

(1)

(1)

h22 = −h33 ≡ c22 sin(ωt − ωx1 ).

IV.6

IV.6.3

Minkowskian approximation

81

Further results on gravitational waves

We have seen that in situations where one can apply the linearized approximation to Einstein’s equations, a system emits gravitational waves
at inﬁnity, which are given, to the lowest approximation, in terms of thequadrupole moment24 of the system. More precisely, in a TT coordinate
system, the metric perturbation far from the system reads, to lowest
order,
 
 2


1
d
r
2 GN 11TThij (t, x)  4+O
Ikl t −,
Pik Pjl − Pij Pkl
c
r
2
dt2
c
r2
(IV.6.5)

24 See Problem IV.11.5.

where Pij ≡ δij −ni nj is a spatial projector orthogonal to the unit vectorni ≡ xi /r and where the quadrupole moment of the system, Iij (t), is
given (to lowest approximation) byT 00 (t, x )1 2 ij
i j
x
x
−
δ
x
.
(IV.6.6)
Iij (t) = d3 x
c2
3
The recent development of kilometre-size interferometric detectorsof gravitational waves (LIGO, VIRGO, etc.) has provided strong motivation for improving the theoretical treatment of the generation ofgravitational waves by astrophysically realistic systems. As many ofthese systems (such as binary neutron stars or binary black holes)
contain strongly self-gravitating systems, one needs to use the multichart formalism sketched in Section IV.5.3. In addition, it has beenfound necessary to (i) include higher multipoles than the quadrupole
Iij , (ii) include higher post-Newtonian corrections to the Newtonianlevel quadrupole formula, and (iii) push the post-Newtonian expansion
to the highest possible approximation. These developments, however, areoutside the scope of this book.25

IV.6.4

Tidal force

The inﬂuence of a gravitational wave on the trajectory of a single isolatedparticle, as noted before, cannot be observed. The tidal force due to a
gravitational wave acting on a pair of particles, i.e. the geodesic deviation
equation (see Chapter I), depends on the Riemann tensor. Considering
two nearby particles with initially parallel unit velocity uα = δ0α andspatial separation vector X i , the equation for the geodesic deviation
force readsuα uβ ∇α ∇β X i = X α uλ uμ Rλα β μ ≡ X j R0
(1)

(1)i

0j ,

(IV.6.7)

with, at ﬁrst order,
2
˜ ∂00
Xi
uα uβ ∇α ∇β X i =

and
(1)i

R0

(1)0
˜
0j ≡ −R0i j =∂∂δΓ0ij −
δΓ0 .
0∂x
∂xi 0i(IV.6.8)

25 For the generalization of (6.5) to theinﬁnite sequence of higher multipoles
see Damour and Iyer (1991). For areview of the current theory of gravitational radiation from post-Newtonian
sources, see Blanchet (2014). See also
Damour and Nagar (2011).

82

The Einstein equations(1)

In the TT gauge, where hα0 = 0,(δΓ0αβ )η (h(1) ) ≡1(1)∂0 hαβ ,2(IV.6.9)

and hence
1 2
0
δR0iJ
= − ∂00hij
2
We see that the equation of geodesic deviation implies at ﬁrst order ofthe Minkowskian approximation that2
(X i + X j hj
∂00

(1)i

IV.6.5) = 0.

Gravitational radiation

The word ‘radiation’ usually refers to energy transfer without material
support.
In the preceding subsections, the approximations used were mathematically well deﬁned and physically well understood. By analogy withother ﬁelds, in particular electromagnetism, one expects gravitational
waves to carry energy. However, as we have already mentioned, there
is no pointwise intrinsically deﬁned gravitational energy. At best, it ispossible to deﬁne some non-local quantities depending also in general onanother a priori given metric—quantities that possess some properties
analogous to those of the energies of other ﬁelds. Much important andvery complex work, analytical as well as numerical, has been (and iscontinuing to be) devoted to the problem of gravitational radiationenergy, but some fundamental questions are still open.

26 Wentzel–Kramers–Brillouin.

IV.7

Strong high-frequency waves

IV.7.1Introduction

It is of course possible to deﬁne weak gravitational waves as perturbations of a given non-ﬂat Lorentzian metric and to obtain for them linear
equations. It is more interesting to study nonlinear eﬀects by extending
to nonlinear equations the WKB26 anzatz, ﬁrst used to study approximate rapidly oscillating solutions of the Schrödinger equation in theformu(x) = a(x)eiωφ(x) ,(IV.7.1)

where x is a spacetime point and a and φ are functions on spacetime.
The function φ, called the phase of the wave, and ω, a number called thefrequency, are real. The number ω is assumed to be large compared withthe values of the functions a and φ and their derivatives with respect
to x. Such an approximation is called in mechanics a high-frequency
wave, or a progressive wave with wave fronts φ(x) = constant; thefunction u varies more rapidly in directions transverse to the wave fronts

IV.7

than on the wave fronts. The study of progressive waves, also called thetwo-timing method, has many applications in classical as well as inquantum physics.
The WKB ansatz, or its generalization by Lax to asymptotic series, is
not well adapted for the study of high-frequency waves associated withnonlinear equations, because the product of two functions like (7.1) is
not of the same type.27 J. Leray28 and his collaborators introduced forlinear equations a more general anzatz, replacing eiωφ(x) by a generalfunction of x and ωφ(x). For quasilinear equations, it is then possible29 toconstruct high-frequency waves that are approximate solutions in a welldeﬁned sense. The eﬀects of the nonlinearity on the waves are possible
distortion of signals and the appearance of singularities similar to shocks.
In the case of General Relativity, where the unknown is a Lorentzian
metric on a manifold V , a high-frequency gravitational wave in vacuum is a Lorentzian metric that is the sum of a non-oscillating part g
and a rapidly varying one depending on a large parameter ω called thefrequency:

Strong high-frequency waves

83

27 Isaacson applied the original WKB
method to the linearized Einstein equations and then looked for a solution ofequations with source a stress–energy
tensor obtained by averaging the obtained perturbation. Progressive wavesof the type (7.2) for the nonlinear
Einstein equations are constructed in
Choquet-Bruhat (1969a).
28 Gårding, Kotake, and Leray (1966).
29 Choquet-Bruhat (1969b). For coupling of high frequency gravitational andﬂuid waves see Choquet-Bruhat andGreco (1983).

g(x, ωφ(x)) = g(x) + h, h := {ω −1 v(x, ξ) + ω −2 w(x, ξ)}ξ=ωφ(x) .
(IV.7.2)
We say that (7.2) is an asymptotic solution of order p of the vacuum Einstein equations if it is such that ω p Ricci(g(x, ωφ(x)) remains uniformly
bounded30 as ω tends to inﬁnity.
To write the asymptotic expansion of the Ricci tensor of the metric(7.2), we set31
h := δg := g − g,hence

∂h = ∂(g − g), ∂ 2 h = ∂ 2 (g − g). (IV.7.3)

The Taylor formula gives the expansion
1
Ricci(g) − Ricci(g) = δRicci + δ 2 Ricci + . . . ,2(IV.7.4)

withδRicci ≡ Riccig · h + Ricci∂g · ∂h + Ricci∂ 2 g · ∂ 2 h.
The coeﬃcients Riccig , Riccig , Ricci∂g , and Ricci∂ 2 g depend only on
the background metric g. The second derivative δ 2 Ricci, a quadratic
form in (h, ∂h, ∂ 2 h), is computed analogously. We remark that δ 2 Ricci
does not contain the square of ∂ 2 h, because Ricci(g) is linear in ∂ 2 g.
To compute ∂h and ∂ 2 h, we use the deﬁnition of h. We denote by aprime a derivative with respect to ξ and underline partial derivatives
with respect to x. We set ϕλ := ∂λ ϕ ≡ ∂ λ ϕ. Elementary calculus implies(x, ξ) + ω −1 [∂ λ vαβ (x, ξ) + φλ wαβ(x, ξ)]∂λ hαβ (x, ωφ(x)) ≡ {φλ vαβ+ω −2 ∂ λ wαβ (x, ξ)}ξ=ωφ(x)30 The added term in ω −2 is introduced
to ensure rigorously the asymptotic
character of the constructed solution.
31 It is also possible to write acoordinate-independent formula byintroducing a given metric e. Notethat Ricci
and Ricci∂g∂g areg∂ 2 gindependent of the choice of the given
metric e.

84

The Einstein equations

and a corresponding formula for the second derivative,
+ ϕλ ∂ μ vαβ+ ϕμ ∂ λ vαβ+ vαβ∂ λ ϕλ∂λ ∂μ hαβ = ωϕλ ϕμ vαβ+ ϕλ ϕμ wαβ+ ω −1 Rλμ,αβwith

IV.7.2Rλμ,αβ := [∂ 2λμ vαβ + φλμ wαβ+ (ϕμ ∂ λ + ϕλ ∂ μ )wαβ+ω −1 ∂ λ ∂ μ wαβ ](x, ξ) ξ=ωφ(x)Phase and polarization

The asymptotic expansion in ω of the Ricci tensor is obtained by replacing the metric and its ﬁrst and second derivatives by the above
expressions. We see that this expansion starts with a term in ω 1 , whose
coeﬃcient must vanish for g to be an asymptotic solution of order zero.
We thus obtain the equations
Ricci∂ 2 g ϕλ ϕμ v  = 0.
λμ(IV.7.5)

This is a linear homogeneous system for the second derivative withrespect to the parameter ξ of the tensor v, which reads, in coordinates,
11− ϕλ ϕλ vαβ+ [ϕα Pβ (v  ) + ϕβ Pα (v  )] = 0,2
2(IV.7.6)

with
1
Pα (v  ) ≡ ϕλ v λ α − ϕα v λλ .
2
By analogy with what we saw in the Minkowskian approximation, wecall Pα the polarization operator. We state as a theorem the result
we have obtained.
32 That is, if its gradient is a null vector.

This condition is necessary for the wave
to be signiﬁcant.

Theorem IV.7.1 If the phase is isotropic,32 then the necessary andsuﬃcient condition for the progressive wave to satisfy the Einstein equations at order zero in ω is that the tensor v satisﬁes the four polarization
conditions
Pα (v) = 0.
Exercise IV.7.1 Prove that the polarization conditions express thevanishing at order zero of the perturbation of the harmonicity functions
g λμ Γαλμ .
Hint: For an arbitrary metric, the following identity can be proved bystraightforward computation (see Chapter VIII on the Cauchy problem):(h)Rαβ ≡ Rαβ + Lαβ ,(IV.7.7)1
gαλ ∂β F λ + gβλ ∂α F λ ,2(IV.7.8)

with
Lαβ ≡

IV.7

Strong high-frequency waves

λμααF α ≡ g λμ Γα
λμ . ≡ g ∇λ ∂μ x ≡ g x ,(h)while the Rαβ are a system of quasilinear, quasidiagonal (i.e. linear anddiagonal in the principal, second-order, terms) wave operators,
1(h)2gαβ + Pαβ (g)(∂g, ∂g),(IV.7.9)Rαβ ≡ − g λμ ∂λμ2where P is a quadratic form in the components of ∂g, with coeﬃcientspolynomial in the components of g and its contravariant associate.

IV.7.3

Propagation and backreaction

Using previous results, or by direct computation, we ﬁnd that thecoeﬃcients at zeroth order in ω in the expansion of Ricci(g) are
Rαβ = Rαβ + (Pv  )αβ + L(v  , w )αβ + Nαβ (v, v  , v  ),(0)with P a linear propagation operator along the rays of ∂ϕ, namely


1 

(Pv  )αβ ≡ − ϕλ ∂ λ vαβ+ vαβ∂ λ ϕλ ,2
while L reads
11Lαβ ≡ [ϕα Qβ (v  ) + ϕβ Qα (v  )] + [ϕα Pβ (w ) + ϕβ Pα (w )] ,
2
2
with
1
Qα (v  ) := ∂ λ v λα − ∂ α v λλ .
2The nonlinear term N (v, v  , v  ) comes from δ 2 Rαβ . We ﬁnd that forpolarized v it reduces toNαβ (v, v  , v  ) ≡




1
1
1
1


ϕα ϕβ v λμ vλμ− vλλ vμμ +− vλλ vμμ .
vλμ vλμ2222We can now prove the following theorem.
Theorem IV.7.2 The progressive wave
g αβ (x) + {ω −1 vαβ (x, ξ) + ω −2 wαβ (x, ξ)}ξ=ωϕ(x)is an asymptotic solution of order one of the vacuum Einstein equations
on a manifold V under the following hypotheses:(1) The phase ϕ is isotropic for the background g.
(2) v satisﬁes the linear, non-homogeneous, propagation systemP(v  ) = 0
along the rays33 of the phase ϕ and also satisﬁes the polarization
conditions on a hypersurface Σ transverse to these rays, assumed tospan V. The ﬁeld v is periodic in ξ on Σ.

33 Null curves tangent to ∂ϕ.

85

86

The Einstein equations(3) w is a solution of the linear system
11− vλλ vμμ ) + ϕα (E − E).
Pα (w ) = Qα (v  ) + ϕα (v λμ vλμ42(4) The background metric g satisﬁes the Einstein equations with source
a null ﬂuid:
Rαβ = Eφα φβ ,where, with T the period of v,
1 T
E(., ξ)dξ, with
E≡
T 0

1
E≡
4



1

− vλλ vμμ
v λμ vλμ2.

Proof.
(1) If v  satisﬁes the propagation equations P(v  ) = 0 on V × R and thepolarization conditions on Σ transverse to rays that span V , then it
satisﬁes the polarization conditions on V × R because the equation
P(v  ) = 0 implies the propagation both of ϕα vαβ = 0 and vαα = 0.
Indeed,1g αβ (Pv  )αβ ≡ −ϕλ ∂ λ vλλ + vλλ ∂ λ ϕλ2and also1ϕα (Pv  )αβ ≡ −ϕλ ∂ λ (ϕα vαβ) − ϕα vαβ∂ λ ϕλ ,2
because if ϕα is a gradient and isotropic, thenϕλ ∂ λ ϕα = ϕλ ∂ α ϕλ = 0.
The coeﬃcient of ω in the asymptotic expansion of Ricci(g) istherefore zero.
(2) The function x → v  (x, ξ), a solution of the linear diﬀerential equation P(v  ) = 0 with coeﬃcients independent of ξ, is determined on
V × R if it is known on a submanifold transverse to rays that span
V. It has period T in ξ if its data on the submanifold have period T .
(3) When v is known, the equations for w are non-homogeneous linear
equations, namely
Lαβ + Nαβ + Rαβ = 0.
(4) If v and w have period T in ξ, then the following relation holds,
because linear terms in v  or w integrate to zero: T
Nαβ (., ξ) dξ.
T Rαβ = −0

An elementary computation gives
Nαβ = −ϕα ϕβ

1
E−
2


v

λμ

1
vλμ − vλλ vμμ
2



IV.8

Stationary spacetimes

87

from which there follows the expression given for R and hence thelinear system for w given in the theorem. These linear equations
have solutions on V × R, periodic with period T in ξ, because thelinear transposed homogeneous system has an empty kernel and thecoeﬃcients have period T . The tensor w can be chosen also of period
T in ξ and the right-hand side has a zero integral on ξ on the interval
0 ≤ ξ ≤ T.
Note a most remarkable fact, which is not shared by other nonlinear
ﬁelds: the vector v obeys linear propagation equations; that is, there isno distortion under propagation of gravitational signals, in spite of thenonlinearity of the Einstein equations.

IV.7.4

Observable displacements

As in the case of weak gravitational waves, the observable displacements due to a strong gravitational wave are governed by the tidal force
determined by the highest-order terms in ω of the Riemann tensor.
Exercise IV.7.2 Consider two nearby particles with initially parallel
unit velocity uα = δ0α and spatial separation vector X i . Write the ﬁrst
approximation of the geodesic deviation due to a strong high-frequency
wave.

IV.8

Stationary spacetimes

IV.8.1Deﬁnition

We call stationary an (n + 1)-dimensional spacetime (V, g) that admits
a one-parameter group G1 ≡ R of isometries with timelike Killing vector
ξ whose orbits are diﬀeomorphic34 to R and span the manifold V. Moreprecisely,35 we assume that V is a product M × R, a point of V is a pair(x, t), x ∈ M, t ∈ R, the subspaces M × {t} of constant t are spacelike
submanifolds, and the Killing vector ξ is represented by ∂/∂t. In frames
adapted to the product structure, the spacetime metric g readsg ≡ −ψ 2 (dt + a)2 + ḡ,(IV.8.1)

with t ∈ R a time coordinate on the orbits of the vector ξ = ∂/∂t;
ψ, a, and ḡ are respectively a t-dependent scalar, a 1-form, and the
Riemannian metric on M. In local coordinates xi in the domain of achart of M, one hasa ≡ ai dxi ,ḡ ≡ gij dxi dxj ,with all coeﬃcients in g being independent of t.

(IV.8.2)

34 We exclude the case of orbits diﬀeomorphic to S 1 , i.e. the case of closed
timelike curves, because this is considered non-physical.
35 The assumptions that V is a product
M × R could be relaxed in the deﬁnition of stationarity, but the uniquenesstheorems given below would not necessarily hold.

88

The Einstein equations

A stationary spacetime is called static if the orbits are orthogonal to
the n-dimensional space manifolds. Its metric then readsg ≡ −N 2 dt2 + ḡ.

(IV.8.3)

Static spacetimes are considered to represent equilibrium situations,while stationary spacetimes model permanent motions. Both play animportant role in relativistic dynamics.
Lemma IV.8.1 A stationary spacetime is static if the 1-form a is anexact diﬀerential:
a = dφ,i.e.

ai = ∂i φ,with φ a t-dependent scalar function on M.
Proof. For a stationary spacetime, assume that a = dφ, and consider
the change of its time and space factorization under the change of timeparametert = t + φ.

(IV.8.4)

This puts its metric into the static formg ≡ −N 2 dt2 + gij dxi dxj ,with g and N ≡ ψ independent of t .

36 See, for instance, CB-DMI IV.

(IV.8.5)

Static spacetimes are invariant under time reversal, t → −t. This
property can also be taken as a deﬁnition to distinguish static spacetimes
among stationary ones.
One calls locally static those stationary spacetimes for which the
1-form a is closed, da = 0, but is not globally an exact form, i.e. there
does not exist on the whole of M a function such that a = dφ. This canhappen only36 if M is not diﬀeomorphic to Rn .

IV.8.2Equations¯ and R̄icci the covariant derivative and Ricci tensor in
We denote by ∇the Riemannian metric ḡ. One can show by straightforward computationthat the components of the Ricci tensor of the spacetime metricg := −ψ 2 (dt + ai dxi )2 + ḡ, with ḡ := gij dxi dxj ,reduce, with f = da, to
Rij ≡ R̄ij +ψ2 h¯ i ∂j ψ,fi fjh − ψ −1 ∇2

¯ k ai − ∇¯ i ak ),fij := g jk (∇(IV.8.6)

1 ¯∇j (fi j ψ 3 ),2ψ(IV.8.7)

1 4 j iψ fi fj + ψΔḡ ψ.
4(IV.8.8)

Ri0 ≡ −
R00 ≡

with

IV.8

Stationary spacetimes

89

Exercise IV.8.1 Prove these formulas using the general formulas forconnection and curvature given in Chapter I. More general formulas arederived in Chapter IX.
In the case of a locally static spacetime, the components of the Ricci
tensor reduce to¯ i ∂j ψ,
Rij ≡ R̄ij − ψ −1 ∇(IV.8.9)

Ri0 ≡ 0,(IV.8.10)

R00 ≡ ψΔḡ ψ.

(IV.8.11)

The second identity37 shows that the source must have zero momentum
for an Einsteinian spacetime to be locally static, justifying its name.
A reciprocal theorem is easy to prove under a physically meaningful
hypothesis, namely the following:
Theorem IV.8.1 A stationary Einsteinian spacetime with sources ofzero momentum, i.e. ρ0i ≡ 0, is locally static if either(1) M is compact.
(2) The spacetime is asymptotically Euclidean.
Proof. If ρ0i = 0, a previous identity gives
Ri0 ≡ −1 ¯
∇j (fi j ψ 3 ) = 0,2ψand hence¯ j (ai fi j ψ 3. ) = 1 ψ 3 fi j fj i .
∇2(IV.8.12)

If M is compact (and hence without boundary according to ourdeﬁnitions) integration of the above equality on M gives, by Stokes’s
formula,ψ 3 fi j fj i μM = 0,
M

i

j

i

and fj ≡ 0 since fi fj ≥ 0,
If M is asymptotically Euclidean, then we have, with Br a ball ofradius r, and using again Stokes’s formula,1
ψ 3 fi j fj i μM = lim
ψ 3 fi j fj i μBr = limai fi j ψ 3 nj μ∂Br .
r→∞
r→∞
2
M
Br
∂Br
(IV.8.13)
When the radius r of the ball tends to inﬁnity, the right-hand side tends
to zero if n > 2, because if the spacetime is asymptotically Euclidean, then the product af falls oﬀ like r−(2p+1) , 2p + 1 > n − 1, while
˜ rn−1 μS 2 , where μS 2 is the volume element of the sphere S n−1 .
μ∂Br =
We conclude again that fij ≡ 0 on M.
37 For a consequence of the last identity when M is compact, see Problem IV.11.9.

90

The Einstein equations

IV.8.338 Lichnerowicz (1939).

Non-existence of gravitational solitons

One calls a gravitational soliton a complete non-trivial (i.e. non-ﬂat)
stationary solution of the vacuum Einstein equations.
We extend to arbitrary dimensions the proof of a theorem given
in 3+1 dimensions by Einstein and Pauli in the static case and by
Lichnerowicz38 in the stationary case.
Theorem IV.8.2 In (n + 1)-dimensional spacetimes, the gravitational
solitons (M × R, g) with M compact or asymptotically Euclidean are
locally static with Ricci-ﬂat space metric and ψ a constant.
Proof. Since gravitational solitons are solutions of the vacuum Einsteinequations, Theorem IV.8.1 shows that they are locally static: fij ≡ 0.
The remaining vacuum Einstein equations readΔḡ ψ = 0,

(IV.8.14)

R̄ij = ψ −1 ∇i ∂j ψ.

(IV.8.15)

By well-known theorems resulting from the maximum principle applied
to solutions of elliptic partial diﬀerential equations, the equality (8.14)
implies that ψ = constant on a manifold M that is compact or asymptotically Euclidean (if ψ is smooth and uniformly bounded—assumptions
we implicitly include in the deﬁnition of gravitational solitons). The lastequation concludes the proof.
Corollary IV.8.1 Under the hypotheses of the theorem, the only
(3+1)-dimensional gravitational solitons are locally ﬂat.
Proof. In 3 dimensions, a Ricci-ﬂat manifold is ﬂat.
39 By

a theorem of Schoen and Yau,a locally ﬂat asymptotically Euclidean(complete—a property included in ourdeﬁnition) Riemannian 3-manifold isisometric to R3 .

40 See Chapter 3 of Wolf (2011).

In the classical case considered by Einstein where M is an asymptotically Euclidean39 manifold diﬀeomorphic to R3 , the only gravitational
soliton is Minkowski spacetime M 4 .
In the compact case, there are, besides the ﬂat torus T 3 , ﬁve nonisometric (i.e. non-diﬀeomorphic) Riemannian orientable compact ﬂat
manifolds, quotients of the Euclidean space E 3 by discrete isometrygroups.40
Remark IV.8.1 Corollary IV.8.1 is not true in higher dimensions.

IV.8.441 Fourès (Choquet)-Bruhat (1948).

Gauss’s law

A property of stationary spacetimes deduced easily from the previous
identities is the following, which we have called41 the relativistic Gauss
law.
Theorem IV.8.3 The time–time component Rtt of the Ricci tensor ofa stationary spacetime (M × R, ĝ) in a natural frame with time axis tangent to the timelines and space axis tangent to the manifold Mt satisﬁes
the divergence identity

IV.9



˜ i ∂i ψ + 1 (aj fji ψ 3 ) .
ψRtt ≡ ψRtt ≡ ∇2(IV.8.16)

Proof. The component Rtt is immediately deduced from the components
previously computed in the frame θ0 := dt + ai dxi . The change-of-frame
formula gives
Rtt ≡ R00 + ai R0i ,(IV.8.17)

with
R00 ≡ −ψ −2 R00 ,R0i ≡ g̃ ij R0j .

(IV.8.18)

Therefore,
Rtt ≡ −


1 2 j i1
ψ fi fj + ψ −1 Δg̃ ψ + ψ −1 ai ∇j (fi j ψ 3 ) ,4
2(IV.8.19)

which gives, using the deﬁnition of fij ,Rtt ≡ −ψ −1



1˜
i j 3
Δg̃ ψ + ∇j (a fi ψ ) ,2(IV.8.20)
that is, the given identity.

Corollary IV.8.2 If the space manifold of a stationary Einsteinian
spacetime with source ρ is asymptotically Euclidean, then it holds that
ﬂuxinﬁnity grad ψ := lim

ρ→∞

∂Bρ

ni ∂i ψμ∂Bρ =M

ψρtt μg̃ .

Exercise IV.8.2 Prove this corollary, which has an analogue in Newtonian mechanics, called Gauss’s law.

IV.9

Lagrangians

Lagrangians, arising from energies, play a fundamental role in physics.
The Lagragian formulation of the Einstein equations, found independently by Einstein and Hilbert, stands apart, being unrelated to apointwise intrinsically deﬁned gravitational energy. However, it plays animportant role in many modern developments, in particular for equations
with ﬁeld sources.

Lagrangians

91

92

The Einstein equations

IV.9.1

Einstein–Hilbert Lagrangian in vacuo

In this section we use the results of Chapter I, and its physicists’notation.
42 One equivalently says that a solutionof the Einstein equations in vacuo isa critical point of the Einstein–Hilbert
Lagrangian.

Theorem IV.9.1 The Einstein equations in vacuo are the Euler
equations42 of the Einstein–Hilbert Lagrangian, deﬁned by(IV.9.1)
Lgrav (g) := R(g)μg ,with R(g) the scalar curvature and μg the volume element of theLorentzian metric g.
Proof. We have
δLgrav (g) =[δR(g)μg + R(g)δμg ] ,(IV.9.2)withδR ≡ g αβ δRαβ + Rαβ δg αβand1
δμg ≡ − gαβ δg αβ μg .
2
We have found in Chapter I that g αβ δRαβ is the divergence of a vector,
g αβ δRαβ ≡ −∇λ v λ ,λαv λ := ∇λ hαα − ∇α h ,hαβ := δgαβ , (IV.9.3)

and hence the integral of this term vanishes if v λ vanishes on the boundary of the integration domain, as is assumed in computing the Euler
equations. We therefore have


 
1
Rαβ − gαβ R δg αβ μ(g).
δLgrav (g) = [δR(g)μg + R(g)δμg ] =2(IV.9.4)

Exercise IV.9.1 Prove the formula (IV.9.4).
Hint: The derivative of a determinant is ∂λ det g ≡ g αβ ∂λ gαβ det g.

IV.9.2

Lagrangians for Einstein equations with sources

A Lagrangian for the Einstein equations with sources (with the latterbeing generically denoted by φ) is the sum of the vacuum Lagrangian
and a Lagrangian for sources. By the equivalence principle,
the Lagran
gian in General Relativity of a source, L(g, φ) = L(g, φ)μg , should
be deduced from the Lagrangian in Special Relativity, invariant underdiﬀeomorphism, by replacing the Minkowski metric η by g.

IV.9

Lagrangians

93

General theorem
The vanishing of the variation with respect to g and φ of the sum of theEinstein–Hilbert Lagrangian and the Lagrangian of the sources gives theEinstein equations with source a tensor T, together with the equationsof motion for the sources.
The ﬁrst variation of the total Lagrangian
Ltot (g, φ) := [R(g) + L(g, φ)]μg(IV.9.5)
reads (the dot below denotes some algebraic linear form)δLtot (g, φ) ≡ {[S αβ (g) − M αβ (g, φ)]δgαβ − Φ(g, φ) · δφ}μg , (IV.9.6)
where Φ = 0 are the equations of motion of the ﬁelds φ on (V, g) andM αβ coincides with their stress–energy tensor.
Remark IV.9.1 The conservation laws ∇α M αβ = 0, necessary forthe solvability of the Einstein equations with source φ, are a consequenceof the invariance under diﬀeomorphisms of the source Lagrangian, which
implies that if Φ(g, φ) = 0, then it holds that for any vector X,
M αβ (∇α Xβ + ∇β Xα )μg = 0.
Matter and ﬁeld sources
In the case of matter sources, it is somewhat diﬃcult to ﬁnd a Lagrangian in Special Relativity, and even in Newtonian mechanics. A
simple-looking Lagrangian for barotropic perfect ﬂuids was proposed by
Taub.43 It reads44
Lﬂuid := p(μ)μg .
The Taub Lagrangian leads to the Euler equations through a somewhat involved process, introducing ﬁrst a Lagrangian with the constraintg(u, u) = −1 and a Lagrange multiplier λ, and using Lagrangian-type
coordinates where the timelines are the ﬂow lines.
Another proposal introducing velocity potentials is due to Schutz.45
None of these Lagrangians has been eﬀectively used in the study ofproperties of relativistic ﬂuids.
In the case of the usual ﬁeld sources, the Lagrangians in Special Relativity are well known and naturally transferred to General
Relativity.
The Maxwell equations in vacuo are the Euler equations with respect
to a closed 2-form F of the Lagrangian1
F αβ Fαβ μg .
(IV.9.7)
Le.m. (F, g) := −4
The Lagrangian of a scalar ﬁeld f with potential V (f ) is

 
1 αβ
g ∂α f ∂β f + V (f ) μg .
Lscal (f, g) := −2

(IV.9.8)

43 Taub (1954, ?).
44 Hereμ denotes the energy density
and μg the volume element.

45 Schutz (1971).

94

46 SeeThe Einstein equations

YCB-OUP 2009, Chapter III,
Section 6.5.

The Lagrangians for Yang–Mills ﬁelds and wave maps are obtained
analogously by replacing ordinary products by scalar products in the
Lie algebra in the case of Yang–Mills ﬁelds and in the metric of thetarget in the case of wave maps.46
In particle physics at a classical (i.e. non-quantum) level the dynamics
is ruled by the standard model Lagrangian with a structure of the
type
 
1 αβ
1
F .Fαβ + ψ̄γ μ Dμ ψ + g μν Dμ H · Dν H
Lstandard (φ) := −42
+V (H) + λψ̄Hψ μg ,with φ := (F, ψ, H, g), F a Yang–Mills ﬁeld with values in the Lie algebraof U (1) × SU (2) × SU (3), ψ various spinor ﬁelds, and γ μ Dirac gamma
matrices, while H is a complex-valued scalar doublet, the Higgs ﬁeld; V
is a potential for H and λ is some constant coeﬃcient.
Exercise IV.9.2 Check the general theorem in the Maxwell and scalarﬁeld cases.
Hint:





g αβ ∂α φ∂β φμg =






1
δg αβ ∂α φ∂β φ − gαβ g λμ ∂λ φ∂μ φ − 2 δφg αβ ∇α ∂β φμg .
2=

δg αβ ∂α φ∂β φμg + 2
+ g αβ ∂α φ∂β φδμgδg αβ ∂α δφ∂β φμg

The theorem is taken to deﬁne the equations for sources and thestress–energy tensor when only the Lagrangian is known.
Remark IV.9.2 A constant factor depending on the units in which
the sources are measured must be put in front of the Lagrangian for thesources when it is added to the vacuum Einstein Lagrangian in the caseof interpretation of observations in non-geometric units.

IV.10

Observations and experiments

In Chapter V, we will study the spherically symmetric vacuum Eisteinian
spacetime, the ﬁrst exact solution constructed by Schwarzschild in 1916.
We will explain most of its observable consequences, and brieﬂy quote
the results of observations, which are all in very good agreement withthe predictions.
We will discuss the reality of the revolutionary prediction of Einstein’s
equations—black holes—in Chapter VI.
The recently operational gravitational wave detectors VIRGO and
LIGO have not so far been able to detect any gravitational wave signal.

IV.11This is not so surprising, given the predicted weakness of such signals.
However, the reality of energy loss by gravitational radiation was
conﬁrmed as early as 1979 by the observation by Taylor and collaborators of the slowing down of the period of a binary pulsar,47 which
was interpreted as a shrinkage of its orbit due to gravitational radiation. Many other binary pulsars have now been discovered. More than
30 years of observations and recording of data from the original Hulse–
Taylor pulsar have led (not without hard work from theoreticians andphysicists) to information being obtained on several physical properties,
including the masses, orbits, rotation, advance of the periastron, andother parameters, of the pulsar and its invisible companion (both are
now thought to be neutron stars). All the results obtained agree withthose predicted by Damour and Deruelle48 from the Einstein equations,rather than those predicted by alternative gravitation theories.49
General Relativity and the Einstein equations are universally adopted
as the best model for gravitation at planetary and astronomical scales.
Their use at the atomic, quantum, scale requires new tools, still
the subject of much research and discussion. The sub-Planckian andcosmological scales pose serious problems that probably require newideas.
If gravity is to become a quantum ﬁeld theory like others, there must
exist an elementary particle called the graviton. This particle should
be massless like the photon, since gravitation propagates with the speedof light. The putative graviton50 is thought to be spin51 2 for the same
reasons why the photon is called spin 1 and the electron and the neutrino
spin 12 .

IV.11

Problems

IV.11.1

The Einstein cylinder

The Einstein cylinder52 is the manifold S 3 × R endowed with thestatic metric− dt2 + a20 γ+ ,(IV.11.1)

with a0 a constant and γ+ the metric of the unit 3-sphere S 3 , which is,in the usual angular coordinates,γ+ := dθ2 + sin2 θ (dφ2 + sin2 φ dψ 2 ).
1. Show that setting sin θ = r gives the form familiar to geometers:
γ+ =

dr2
+ r2 (dφ2 + sin2 φdψ 2 ).
1 − r2(IV.11.2)

Problems

95

47 A binary pulsar is a system of twostars gravitating near enough to eachother to be considered as an isolated
system, one of them being a pulsar. A
pulsar is a star from which we receive
strong electromagnetic signals at regular intervals. It is now thought that thesignals are emitted in the direction of
the axis of a rapidly rotating neutronstar.
48 Damour and Deruelle (1981, 1986)
and Damour (1982).
49 For observational and experimental
results up to the end of 2013, see the review by Damour for the Particle Data
Group (Damour, 2013b).

50 The observation of a graviton is very

unlikely in the near future.
51 Spin is a quantum notion whose precise deﬁnition is outside of the scopeof this book. It takes only integer orhalf-integer values. A supersymmetricpartner of the graviton, the gravitino,would have spin 32 .

52 The Einstein cylinder, also called theEinstein static universe, was the ﬁrst
Einsteinian cosmological model to beconstructed.

96

The Einstein equationsShow that the components of the Ricci tensor are (whatever coordinates are used)+
R0α ≡ 0, Rij ≡ 2γij.

(IV.11.3)

2. Show that the Einstein cylinder is a solution of the Einstein equations
with perfect ﬂuid source of constant positive energy and negative
pressure:−2
μ0 = 3a−2
0 , p0 = −a0 < 0.

(IV.11.4)

3. A negative pressure is unacceptable on classical physical grounds.
Show that the Einstein cylinder is a solution of the Einstein equations
with a positive cosmological constant and source a perfect ﬂuid withpositive energy and pressure.
4. Show that the Einstein cylinder is locally conformally ﬂat. Determine
the image of Minkowski spacetime under a diﬀeomorphism into the
Einstein cylinder.

Solution
1. This is a straightforward computation.
2. The Einstein equations with cosmological constant Λ are
Sαβ = Tαβ − Λgαβ .
We deduce from (11.3) and (11.4) thatR ≡ 6a−20 ,+and S00 ≡ 3a−2
0 , Sij ≡ −γij .

The stress–energy tensor Tαβ ≡ (μ + p)uα uβ + pgαβ of a perfect ﬂuid
in the Einstein static universe with static ﬂow vector u0 = 1, ui = 0 is+
T00 ≡ μ0 , T0i ≡ 0, Tij ≡ p0 a20 γij,+)
and hence (recall that on the Einstein cylinder, g00 ≡ −1, gij ≡ a20 γij+
Λ
S00 = T00 + Λ, Sij = Tij − a20 γij

if−2
μ0 = 3a−2
0 − Λ, p0 = Λ − a0 .

IV.11.2

de Sitter spacetime

The de Sitter spacetime has for supporting manifold S 3 × R, like theEinstein cylinder. Its metric, analogous to the Einstein cylinder metric, is− dt2 + a2 γ+ ,but the coeﬃcient a is now time-dependent.

(IV.11.5)

IV.11

Problems

97

1. Show that the de Sitter metric is a solution of the vacuum Einsteinequations with positive cosmological constant Λ if and only if thecoeﬃcient a is such thatä − k 2 a = 0, with ȧ :=∂2a∂a, ä := 2 ,∂t∂tk 2 :=

Λ,3(IV.11.6)andȧ2 − k 2 a2 = −1.

(IV.11.7)

2. Show that the general solution for (11.5) is, with A and B a pair ofconstants,
a = Aekt + Be−kt .

(IV.11.8)

Show that such a function a satisﬁes (11.6) if 4AB = k −2 . Show thatthe metric, called the de Sitter metric,53− dt2 + k −2 (cosh2 kt)[dα2 + sin2 α(dθ2 + sin2 θ dφ2 )](IV.11.9)

53 The de Sitter spacetime is timesymmetric. The radius of its sphericalspace sections expands to inﬁnity inboth time directions.

is a solution of the vacuum Einstein equations with positive cosmological constant.

IV.11.3

Anti-de Sitter spacetime

1. Show that the spacetime metric deﬁned on R3 × (−π/2, π/2) by− dt2 + (cos2 t) γ− ,(IV.11.10)

where γ− is the metric of the hyperbolic 3-space of constant negative
curvature,γ− := dχ2 + (sinh2 χ)(sin2 θ dφ2 + dθ2 ),satisﬁes the vacuum Einstein equations with negative cosmologicalconstant Λ = −3.
2. The spacetime classically called anti-de Sitter spacetime is the
manifold R3 × R with the static metricgAdS := −dt2 (cosh2 χ) + dχ2 + sinh2 χ(sin2 θ dφ2 + dθ2 ), 0 ≤ χ < ∞.
(IV.11.11)
Show that it is conformal54 to the half 0 < θ < π/2 of the Einstein
cylinder.
3. Show that no light ray emitted at time t0 will go beyond a space slice
at ﬁnite time t = t0 + T.
Remark IV.11.1 Five-dimensional anti-de Sitter spacetime plays
an important role in supersymmetric theories conformal ﬁeld, inparticular with regard to the so-called AdS–CFT correspondence.

54 This property has been used by
Choquet-Bruhat to prove the global existence of Yang–Mills ﬁelds in anti-deSitter spacetime. It plays an importantrole in recent work by H. Friedrich.

98

The Einstein equations

Solution
1. This is straightforward computation.
2. Setcosh χ =

1
,
cos α

hence sinh χ dχ = −

sin α
dα.
cos2 αComputation using the identities satisﬁed by sines and cosines gives
the announced identity:
gAdS ≡

1[−dt2 + dα2 + sin2 α(sin2 θ dφ2 + dθ2 )].
cos2 α3. We have on a radial light ray
dχ
dt =
,
cosh χ

 χ
hence

0

t=t +

dχ
,
cosh
χ
χ0

and hence, whatever χ0 is,
 ∞
 χ
dχ
dχ
<2
t − t0 ≤:= T < ∞.
eχ
0 cosh χ
0(IV.11.12)(IV.11.13)

Therefore, no light ray emitted at time t0 will go beyond the spaceslice t = t0 + T. The anti-de Sitter spacetime has no global causality
properties; it is not globally hyperbolic (see Chapter VIII).

IV.11.4

Taub–NUT spacetime

The underlying manifold of the Taub spacetime is S 3 × I with S 3 athree-dimensional topological sphere and I := (t− , t+ ) an interval of R.
This manifold is endowed with the Lorentzian metric−U −1 dt2 +(2 )2 U (dψ+cos θ dφ)2 +(t2 + 2 )(dθ2 +sin2 θdφ2 ), (IV.11.14)where
1t± = m ± (m2 + 2 ) 2(IV.11.15)

2(mt + 2 ).
t2 + 2(IV.11.16)

andU ≡ −1 +The numbers and m are positive constants. The coordinates ψ, θ, φ are
the Euler coordinates on the sphere S 3 .
1. Show that the Taub spacetime is a solution of the vacuum Einsteinequations.
2. Show that U tends to zero as t tends to t+ or t− . Show that the
metric is still Lorentzian when t is in the complement in R of theclosure of I, but ψ is now a time variable and t a space variable.

IV.11

Problems

99

3. Show that by settingψ = ψ +

1
2

 t

dτ
,
U
(τ )
0

(IV.11.17)

the Taub metric extends to a C ∞ metric on S 3 × R, called Taub–NUT:554 2 dψ 2 − 4 dψ  dt + 4 2 U cos θ dφ dψ  − 2 cos θ dt dφ+(t2 + 2 )(dθ2 + sin2 θ dφ2 ).

(IV.11.18)

4. Show that the two hypersurfaces, diﬀeomorphic to S 3 , t = m ± (m2 +2 12) , which bound the Taub spacetime in Taub–NUT are null manifolds, generated by closed null geodesics where ψ only varies. They
are the Cauchy horizons of the maximal development of any Cauchy
surface (see Chapter VIII) in the Taub spacetime.
5. Show that one family of null geodesics issuing from a point of the
Taub region crosses both horizons, and that another family spirals
near these surfaces and is incomplete. Show that in the NUT region
there are closed timelike curves (the surfaces t = const are timelike
there).
6. Construct another extension of the Taub spacetime with analogous
properties, though not isometric to it, by the change of coordinatesψ  = ψ −

1
2

 t

dτ
.
0 U (τ )(IV.11.19)

Remark IV.11.2 The Taub–NUT spaces give counterexamples to several conjectures (see Chapter VIII). The consolation is that they are notgeneric, because of their symmetries.

IV.11.5

The quadrupole formula

Many approximation methods in General Relativity use approximationsof a Newton-type potential U generated by a mass of density ρ given ingeometrical units (κ = 1) by the integralρ(y) 3d y, d3 y ≡ dy 1 dy 2 dy 3 .
U (x) =
R3 |x − y|
1. Show that 1/|x − y| admits the following Taylor expansion abouty = 0:

1
xi
xi xj
1 
1
= +
yi 3 +(3y i y j − r2 )δij 5 + . . . ,|x − y|r i=1,2,3 r
2 i,j=1,2,3rwith r2 :=



i 2
2
i=1,2,3 (x ) , and r :=i 2
i=1,2,3 (y ) .

55 For its discoverers, Newman, Unti,
and Tamburino.

100

The Einstein equations

2. Write, for a point x outside the support of the function ρ,
U (x) =

1  ij xi xj
M  i xi+
D 3+
Q+ ... .
r
r
2 i,j
r5
i

Give the expressions for M, Di , and Qij , which are called respectively
the mass, the dipole moment, and the quadrupole moment of thematter density ρ. Show that U reduces to −M/r if ρ is sphericallysymmetric. Show that Di can be made zero by choosing for origin
the Newtonian centre of gravity of the matter.
3. Show that for a non-spherical mass, the gradient of U will in generaldiﬀer from −M/r2 by a term in 1/r4 .

IV.11.6

Gravitational waves

Give details and draw images of the displacement of the plane-wave
perturbation of a Minkowski metric.

IV.11.7

Landau–Lifshitz pseudotensor

By analogy with other ﬁelds, it was thought long ago that a locallydeﬁned energy for the gravitational ﬁeld should be deﬁned in a postMinkowskian approximation by a quadratic form in the ﬁrst derivatives
˜ αβ − ηαβ . A simple-minded way to obtain such a∂λ hαβ , hαβ := δgαβ =gquadratic form is inspired on the one hand by the Einstein equations withright-hand side the stress–energy tensor of matter or other ﬁelds andon the other hand by the classical Picard iteration method for solving
quasilinear equations.
1. Show that in vacuum the ﬁrst approximation h := δg at η of a metricg solution of the Einstein equations satisﬁes the linear equations
11
δRαβ ≡ Lαβ (h) ≡ − ∂ λ ∂λ hαβ + ∂λ ∂α hλβ + ∂λ ∂β hλα − ∂α ∂β hλλ = 0.
2
2
(IV.11.20)
2. Show that the Picard method gives for the approximation h(2) oforder two the following linear equations with source quadratic in theﬁrst approximation h:Lαβ (h(2) ) = hλμ [∂λ (∂α hβμ + ∂β hαμ − ∂μ hαβ ) − ∂α ∂β hλμ ] + tαβ ,with tαβ quadratic in ﬁrst derivatives, namely1tαβ := −∂λ hλμ (∂α hβμ + ∂β hαμ − ∂μ hαβ ) + ∂β hλμ ∂α hλμ (IV.11.21)
2
+

1 λ ρ
∂ hρ (∂α hβλ + ∂β hαλ − ∂λ hαβ ) + ∂λ hμα ∂ λ hβμ − ∂λ hμα ∂μ hλβ .
2

IV.11

The tαβ are the components of the Landau–Lifshitz pseudotensor.
This was considered by Landau and Lifshitz as representing thestress–energy of the ﬁrst-order perturbation h.

IV.11.8

High-frequency waves from a sphericallysymmetric star

Take as background the spherically symmetric Vaidya metric, whichin the coordinates x0 = u, x1 = r, (xA ) = (θ, φ) reads


2m(u)
g := − 1 −(du)2 − 2 du dr + r2 (dθ2 + sin2 θ dφ2 ).
r
1. Check that the Vaidya metric coincides with the Schwarzschild metric(see Chapter V) when m is constant. Write down its contravariant
components.
2. Write the conditions that must be satisﬁed by the high-frequency
wave
g(x, ωu) := g + ω −1 v(x, ωu), +ω −2 w(x, ωu),with

x := (u, r, θ, φ)for it to be an asymptotic solution of the vacuum Einstein equations
with phase u(a) of order zero;(b) of order 1, in a gauge such that v01 = 0.
3. Show that the ﬁrst term v in the wave falls oﬀ at inﬁnity like r−1 ,as the background metric. Show that the second term w falls oﬀ liker−2 .

Solution
1. Elementary computation gives
2m(u) 4 2
Detg = − 1 −r sin θ,rg 00 = g 0A = g 1A = 0,

g 01 =

r4 sin2 θ 11
, g = 1.
Detg

2. The coordinate function u is isotropic for g because g uu = 0. The
gradient of the phase ϕ ≡ u ≡ x0 is ϕα = δα0 .
(a) For a solution of order zero, the polarization conditions
Pi (v) ≡ v 0i ≡ vi1 = 0and
1P0 (v) ≡ − v ii = 0
2

Problems

101

102

The Einstein equations

must be satisﬁed; that is, using previous relations,g iα viα = g 10 v10 + g AB vAB ,g AB vAB ≡

1
r2


v22 +

1
v33
sin2 θ

= −v10 .

(IV.11.22)(0)(b) To have an asymptotic wave of order one, we must have Rαβ = 0.
If we choose the gauge such that v10 = 0, then the propagation(0)
equations, Rij = 0 are found to reduce to
1∂1 vAB − vAB = 0.
r
These equations integrate tovAB (x, ξ) = rγAB (u, θ, φ, ξ).
We setγ22 =: α,γ23 =: β.

The polarization conditions are then equivalent to
γ33 = −α sin2 θ.
To have a progressive-wave solution of order one of the vacuum(0)Einstein equations, it remains to satisfy the equations R0α = 0.
(0)The equations R0i = 0 are found to be
1 (0)R01 ≡ − w11
= 0,
2


1 
1
1
(0)


 cos θR02 ≡ − w12 +∂3 β + 2α∂2 α +
= 0,
2
2r
sin θ
sin2 θ
1 
1(0)+R03 ≡ − w1322r

∂2 β  − ∂ 3 α  +cos θ 
β
sin θ


= 0.

We see that, given α and β uniformly bounded functions of ξ aswell as their primitives with respect to ξ, these linear equations
, also linear in α and β  , are satisﬁed by functions w1i
suchin wijthat there exist functions w1i that are uniformly bounded in ξ.
(0)
It remains to satisfy the equation R00 = 0, which reads





1 ij 
1
β2
1
β 2
2 dm
2
2
.
=− 2 α +− 2
g wij + 2 α +
2
2
2
r
2r
r
du
sin θ
sin θ
(IV.11.23)
This equation can be satisﬁed by a w uniformly bounded in ξ only
if the right-hand side is also the second derivative of a function
uniformly bounded in ξ. This will be the case if


1
β 2
dm
= 0,
(IV.11.24)
+
α2 +
4
du
sin2 θ

IV.11for example, by assuming dm/du < 0 and setting μ(u) :=12(−dm/du) 2 ,α = μ(u) sin ξ sin θ, β = μ(u) cos ξ sin θ cos θ.

(IV.11.25)

One ﬁnds bounded wij satisfying all the required equations:
w12 =

μ(u)
cos ξ (cos θ + 2),
r

w13 =

(IV.11.26)

μ(u)sin ξ sin2 θ,r(IV.11.27)

wAB = 0.

(IV.11.28)

andRemark IV.11.3 The asymptotic metric obtained is axially symmetric (the coeﬃcients do not depend on ϕ), but not sphericallysymmetric. It is conjectured and partially proved that, in agreement with physical intuition, no progressive gravitational wave withspherically symmetric background is spherically symmetric.
3. This is left to the reader.

IV.11.9

Static solutions with compact spacelike
sections

Show that in dimension 3+1, the only locally static solution of the Einstein equations with compact spacelike sections and source such that
ρ00 ≥ 0 is vacuum and locally ﬂat.

Solution
The Einstein equations for a locally static spacetime metric are¯ i ∂j ψ = ρ0i , ρi0 ≡ 0, ψΔḡ ψ = ρ00 .
R̄ij − ψ −1 ∇(IV.11.29)¯ i ∂ i ψ, then
If M is a compact manifold (without boundary) and Δḡ ψ ≡ ∇it holds thatΔḡ ψμḡ =ψ −1 ρ00 μḡ , hence ρ00 ≡ 0 if ψ > 0 and ρ00 ≥ 0.
0=
M

M

Therefore, Δḡ ψ = 0. On a compact manifold, this implies ψ = constant,and the equations imply Ricci(ḡ) = 0. We know that Ricci-ﬂat threedimensional manifolds are locally ﬂat.

Problems

103

104

The Einstein equations

IV.11.10
56 Arnowitt–Deser–Misner.

Mass of an asymptotically Euclidean
spacetime

The ADM56 mass of an asymptotically Euclidean manifold (M, g) is
deﬁned by a ﬂux integral through the sphere at inﬁnity of its end:mADM :=

1lim
16π r→∞

∂Br(∂i gij − ∂j gii )ni dμ.

Consider the asymptotically Euclidean metric on R3 ∩ r > 0,
r2 = x2 + y 2 + z 2 ,

m 4
1+
(dx2 + dy 2 + dz 2 ),2risometric with the space part of the Schwarzschild metric in standard
coordinates (see Chapter V).
Show that its ADM mass is equal to m.

Solution
For the given metric gij = (1 + m/2r)4 δij , δij = 0 for i = j,






m 3
m 3
m xj
m xj
∂i gij = 4 1 +− 2− 2, ∂j gii = 12 1 +
,
2r
2r r
2r
2r r



m 3 m xj
ni
mxi= 4m 2 ,(∂i gij − ∂j gii ) = 8 1 +
=
˜
4
2
3
2r
2r r
r
r
and hence, with ni ni = 1 and 4π the surface of the unit sphere S 2 ,1mADM =(∂i gij − ∂j gii )ni dμ = m.
16π S1

IV.11.11

Taub LagrangianConsider the Lagrangian with constraint g(u, u) = −1 and Lagrange
multiplier λ:L = {R(g) + [μ + p + λg(u, u)]}μg .
Prove by choosing coordinates adapted to the ﬂuid ﬂow that the Euler
equations for such a Lagrangian lead, with the choice λ = μ, to the usual
Euler equations for perfect ﬂuids and the Einstein equations with source
their stress–energy tensor.

V

The Schwarzschildspacetime
V.1

V.1 Introduction
105
V.2 Spherically symmetricspacetimes
105

Introduction

In 1916, soon after the publication of Einstein’s equations, an exact
solution was constructed by Schwarzschild that could model the gravitational ﬁeld outside a spherically symmetric isolated body such as theSun. It was shown that, in ﬁrst approximation, the relativistic planetary
trajectories, i.e. timelike geodesics of the Schwarzschild metric, coincidewith the Kepler orbits. However, in the case of Mercury, the planet nearest to the Sun, an additional advance of its perihelion of about 42 per
century beyond the Newtonian calculuation was found,1 in agreementwith astronomers’ observations. Other eﬀects predicted by the Einsteinian gravitation theory, namely deﬂection of light rays, redshift, and timedelay, have also been found to be in remarkable agreement with observations and experiments (see Sections V.8 and V.9). In Chapter VI, we
will treat a revolutionary property of Einsteinian spacetimes, the existence of black holes, which ﬁrst appeared through investigations of the
Schwarzschild metric.

V.2

Spherically symmetric spacetimes

We start with the following natural and elementary deﬁnitions.
Deﬁnition V.2.1 A three-dimensional Riemannian manifold (M, ḡ) is
said to be spherically symmetric if
1. The manifold M is represented by one chart (U, Φ) with Φ(U ) = R3 or
the exterior of a ball B of R3 centred at some point O. We denote byρ, θ, φ spherical (pseudo) coordinates in Φ(U ), linked to the canonicalcoordinates x, y, z of R3 by the usual relationsx = ρ sin θ sin φ, y = ρ sin θ cos φ, z = ρ cos θ.

(V.2.1)

2. In Φ(U ), given by ρ ≥ ρ0 ≥ 0, 0 ≤ θ < π, 0 ≤ ϕ < 2π, ḡ isrepresented by a metric of the formeh(ρ) dρ2 + f 2 (ρ)(dθ2 + sin2 θ dϕ2 ),(V.2.2)V.3 Schwarzschild metric
V.4 Other coordinates
V.5

V.6

V.7
V.8
V.9
V.10
V.11
V.12

107
108

Schwarzschildspacetime and event
horizon111
The motion of theplanets and perihelion
precession
112
Stability of circularorbits117
Deﬂection of light rays 118
Redshift and time
dilation
121
Spherically symmetricinterior solutions
122
Spherically symmetricgravitational collapse 127
Problems
132

1 Actually,

this was ﬁrst obtained by
Einstein, in November 1915, by a perturbative calculation where he solved
the Einstein equations by successive
approximations, going to second orderin GE .

106

The Schwarzschild spacetime

with f a monotonically increasing function of ρ. The interpretationis that Φ(U ) is foliated by metric 2-spheres ρ = constant, centred at
O; their areas in the metric (2.2) are 4πf 2 . The metric (2.2) is thegeneral form of a metric invariant under rotations in R3 , centred at
O. It is deﬁned on the whole of R3 if the ball B is empty.
Remark V.2.1 In the preceding deﬁnition, ρ = 0 represents, by definition, a single point O. The vanishing of f 2 (0) does not imply asingularity in the metric, but reﬂects the fact that spherical ‘coordinates’
are not admissible coordinates at ρ = 0.
The choice of the coordinate r given by r = f (ρ) is called thestandard choice.
Deﬁnition V.2.2 Consider a spacetime (V, g) with V contained in the
product R3 ×R, a point of V being labelled (x, t). Suppose the subsets Mtof constant t are spacelike submanifolds; then we denote by gt the Riemannian metric induced by g on Mt . The trajectories of the vectors ∂/∂tare supposed timelike. The spacetime is said to be spherically symmetric
if the following hold:
1. Each manifold Mt has a representation as the exterior R3 − Bt of
a ball Bt of R3 centred at the origin O. Each manifold (Mt , gt ) is
spherically symmetric. In R3 − Bt the metric gt reads, in standard
coordinates,
gt = eλ(r,t) dr2 + r2 (dθ2 + sin2 θ dϕ2 ).
2 Equivalently, the lapse and the shift ofthe slicing (see Chapter VIII).

(V.2.3)

2. For each t, the g length and the representative of the projection on Mtof the vector ∂/∂t tangent to the timeline2 are both invariant under
the rotation group deﬁned above.
Lemma V.2.1 A spherically symmetric spacetime (V, g) admits ametric of the formg = −eν dt2 + eλ dr2 + r2 (dθ2 + sin2 θ dϕ2 ),(V.2.4)

where λ and ν are functions of t and r only.
Proof. A scalar on R3 − B invariant under the rotation group is necessarily a function of r and t alone. A vector ﬁeld invariant under thisgroup is tangent to the radial lines (lines where only the r coordinate
varies) and its magnitude depends only on r for each t. Therefore, thegiven deﬁnition implies thatg = −a2 (r, t) dt2 +2b(r, t) dt dr+eλ(r,t) dr2 +r2 (dθ2 +sin2 θ dϕ2 ). (V.2.5)
We can eliminate the diagonal term in dt dr by computing an integrating factor for the 1-form in two variables ω := a dt − b dr, i.e. a functionof t and r, which we denote by e−2ν , such that its product by the 1-formis the diﬀerential of a function τ :e−2ν ω ≡ e−2ν (t, r)(a dt − b dr) ≡ dτ.

V.3

We take τ as a new time coordinate, keeping r, θ, ϕ as space pseudocoordinates. Writing the metric (2.4) with this new time coordinate,renamed as t, gives the formula (2.3). The coordinates t, r are called
standard coordinates.

Exercise V.2.1 Given a and b, determine ν.
Remark V.2.2 If the point with coordinate r = 0 belongs to the
manifolds Mt , it describes a timelike line called the central world line.

V.3

Schwarzschild metric

We will prove the following theorem:
Theorem V.3.1 A smooth spherically symmetric metric is a solutionof the vacuum Einstein equations if and only if it is the Schwarzschild
metric, which reads, in standard coordinates, with m a constant



−1
2m
2m
2
dr2 + r2 (dθ2 + sin2 θ dϕ2 ).
gSchw = − 1 −dt + 1 −rr(V.3.1)
Proof. We set t = x0 , r = x1 , θ = x2 , ϕ = x3 and denote by a prime thederivative with respect to r. We ﬁnd that the only non-zero Christoﬀelsymbols of the spherically symmetric metric (2.4) areΓ000 =11ν∂t λ λ−ν 1∂t λ∂t ν, Γ100 = eν−λ ν  , Γ001 = , Γ011 =e,, Γ01 =
2
2
2
2
2
(V.3.2)

Γ111 =λ, Γ122 = −re−λ , Γ212 = Γ313 = r−1 , Γ133 = −r sin2 θ e−λ ,2(V.3.3)
Γ233 = − sin θ cos θ, Γ323 = cot θ.

(V.3.4)Computing the components of the Ricci tensor, we ﬁndrR10 ≡ r−1 ∂t λ, R22 ≡ −e−λ 1 + (ν  − λ ) + 1, R33 ≡ sin2 θ R22 .
2(V.3.5)
We deduce from these identities that for a solution of the vacuum
Einstein equations, Rαβ = 0, λ is independent of t. Therefore, ν  is also
independent of t, ∂t ν  = 0, and ν is of the form
ν(t, r) = ν(r) + f (t).
(V.3.6)
 1 f (t)2
dt and we rename τ as t; we then have ∂t ν = 0. The
We set τ := e 2other non-identically zero components of the Ricci tensor then reduce to
 

ν
ν 2
ν  λ
ν
R00 ≡ eν−λ+−+,(V.3.7)2
4
4r

Schwarzschild metric

107

108

The Schwarzschild spacetime

R11 ≡ −ν 2λ ν 
λ
ν 
−
+
+ .
2
4
4
r

(V.3.8)The vacuum Einstein equations therefore implyr(eλ−ν R00 + R11 ) ≡ ν  + λ = 0.

(V.3.9)Modulo this relation, the equations R22 = 0 and R33 = 0 reduce to−e−λ (1 − rλ ) + 1 = 0;
that is,(e−λ ) +

1
e−λ
= .
r
r

(V.3.10)The general solution of this linear equation for e−λ ise−λ = 1 +

A
,
r



A
hence eν = B 1 +,r(V.3.11)

with A and B arbitrary constants. The constant B can be made equal
to 1 by a rescaling of t. The constant A is denoted by −2m; we will
see in Section V.6, Equation (6.8), that the Newtonian approximation
corresponding to the Schwarzschild metric then coincides with Newton’s
gravity, with the coeﬃcient m being the gravitational mass expressed inunits of length. We have thus obtained the metric (3.1).
To show that this metric satisﬁes the full vacuum Einstein equations,
we must check that it satisﬁes also the equations R00 = 0 and R11 = 0;
this can be done by using (3.9) and (3.7), (3.8), (3.11) to show thateλ−ν R00 − R11 = 0.
Exercise V.3.1 Prove this last statement by using the Bianchi identities.
Remark V.3.1 We have supposed in this theorem that the areas of theorbits of the symmetry group are monotonically increasing along theirorthogonal trajectories.
In the course of the proof, we have obtained Birkhoﬀ ’s theorem:
Theorem V.3.2 (Birkhoﬀ ) A smooth spherically symmetric metricsolution of the vacuum Einstein equations is necessarily static.

V.4

Other coordinates

In some problems, it is useful to use alternative, non-standard, coordinates for the Schwarzschild metric.

V.4

V.4.1

Other coordinates

109

Isotropic coordinates

One deﬁnes new coordinates X, Y, Z, called isotropic, on R3 × R thatare related to the standard r, θ, φ (i.e. x, y, z) by settingm 2,(V.4.1)r =R 1+2R
X = Rr−1 x, Y = Rr−1 y, Z = Rr−1 z.

(V.4.2)In terms of these coordinates, the metric reads, with R2 = X 2 +Y 2 +Z 2 ,2
2R − mm 4
gSchw = −dt2 + 1 +(dX 2 + dY 2 + dZ 2 ). (V.4.3)
2R + m2R
Exercise V.4.1 Prove this formula (see the solution of Problem
V.12.4).
It is clear from the above expression that the spaces t = constant of
the Schwarzschild metric are conformal to Euclidean space.
Remark V.4.1 r is a monotonically increasing function of R, from 2m
to inﬁnity, when R increases from m/2 to inﬁnity; r and R are equivalent
for large r.

V.4.2

Wave (also called harmonic) coordinates
3 Note that while the standard form ofTheorem V.4.1 The 3 + 1 metric,3 deﬁned for r̄ > m,r̄ − m 2 r̄ + m 2−
dt +dr̄ + (r̄ + m)2 (dθ2 + sin2 θ dφ2 )r̄ + mr̄ − m

(V.4.4)is isometric to the Schwarzschild metric by the mapping r = r̄ + m. The
corresponding Cartesian coordinates t and xi deﬁned by
x1 = r̄ sin θ sin φ, x2 = r̄ sin θ cos φ, x3 = r̄ cos θ
are wave coordinates.
Proof. We write an arbitrary spherically symmetric, static, metric on
R3 × R in the form
− A2 dt2 + B 2 dr̄2 + r2 (dθ2 + sin2 θ dφ2 ),(V.4.5)where θ, φ are spherical coordinates on S 2 . The coeﬃcients A, B, and r
are functions of r̄ only. The coordinate t is obviously a wave coordinate.
We look for a function r = f (r̄) such that the coordinates xi deﬁned by
the above relations are wave coordinates, i.e. such that the functions xi
satisfy the wave equation. For an arbitrary function ψ, the wave equationin the metric (4.4) reads




1 ∂
∂ψ
1
1 ∂2ψ
(V.4.6)− 2 2 + 2
AB −1 r2
+ Δ∗ ψ ,A ∂tr AB ∂ r̄
∂ r̄

the Schwarzschild metric generalizes tohigher values of n, its expression inwave coordinates does not (see Problem V.12.5).

110

The Schwarzschild spacetime

where Δ∗ is the Laplacian on the sphere S 2 . If xi is one of the functions
given above, thenxi ∂ 2 xi∂xi= ,= 0 and∂ r̄
r̄ ∂ r̄2

Δ∗ xi = −2xi .

(V.4.7)

Hence the condition that the xi be wave coordinates reduces to
1 d(AB −1 r2 ) − 2r̄ = 0.
AB dr̄
In the case of the Schwarzschild metric, we have for r > 2m

dr
2m
,
B ≡ A−1 ,
A≡ 1−
r
dr̄
and hence the harmonicity condition isd dr̄ 2 2d dr̄ 2A r − 2r̄ ≡(r − 2mr) − 2r̄ = 0.
dr dr
dr dr

(V.4.8)(V.4.9)(V.4.10)We setr = m(1 + z),
and (4.10) becomes


d
dr̄
(z 2 − 1)− 2z = 0,dz
dz

for z > 1.

(V.4.11)(V.4.12)This linear second-order diﬀerential equation is a Legendre equation,
whose general solution is, with C1 and C2 arbitrary constants


z z+1
ln
−1 .
(V.4.13)r̄ = C1 z + C2
2 z−1
To avoid introducing an additional logarithmic singularity on the horizon
r = 2m, we taker̄ = mz,i.e.

r = r̄ + m.

(V.4.14)

Using this value of r, we can check that the Schwarzschild metric in the
wave coordinates xi deﬁned above is the given metric.


V.4.3
4 Pretorius (2005a).

Painlevé–Gullstrand-like coordinates

The following stationary but non-static form of the Schwarzschild metrichas been used in numerical computations:4



2m
2 2m
2
2
2
2
(x dx + y dy + z dz) dt.
− 1−
dt + dx + dy + dz +rrr
(V.4.15)
It is called a boosted Schwarzschild metric.

V.5

V.4.4

Schwarzschild spacetime and event horizon

111

Regge–Wheeler coordinates

We deﬁne, in the region r > 2m, a tortoise radial coordinate
ρ = r + 2m log(r − 2m).

(V.4.16)

The metric then takes a form in which the timelike sections θ = constant,φ = constant are conformal to two-dimensional Minkowski space:
2mgSchw = 1 −(V.4.17)(−dt2 + dρ2 ) + r2 (dθ2 + sin2 θ dφ2 ),rwhere r is the function of ρ deﬁned by (4.16).

V.5

Schwarzschild spacetime and event
horizonIf m = 0, the Schwarzschild metric reduces to the Minkowski metric;
if m = 0, it is singular for r = 0 and has a coordinate singularity forr = 2m (see below).
The sign of the constant m = 0 is very important in determining theproperties of the Schwarzschild metric.
If m < 0, the Schwarzschild metric deﬁnes a spherically symmetricspacetime on the whole of R3 × R, except at {0} × R, where the metric is singular. At present, the Schwarzschild metric with m < 0 has no
physical interpretation. It can be shown that there exists no asymptotically Euclidean Einsteinian spacetime (R3 × R, g) with sources of positiveenergy that coincides for r > a ≥ 0 with a Schwarzschild metric withm < 0; this is a particular case of the positive-mass theorem.5
If m > 0, the Schwarzschild metric is a regular Lorentzian metric witht timelike and r spacelike and r > 2m; 2m is called the Schwarzschildradius. For r large with respect to m, we will see in the following sections
that the resulting physical properties coincide in ﬁrst approximation to
those of the Newtonian theory for a spherical body of gravitational6
mass m centred at r = 0.
It is possible to construct so-called interior Schwarzschild solutions
that are spherically symmetric spacetimes, not necessarily static, smoothon a manifold R × R3 ∩ {r ≤ a}, satisfy the Einstein equations withsources of positive energy, and lead to a complete admissible Einsteinian
spacetime on the whole manifold R×R3 , with, for r > a, a Schwarzschildspacetime of mass 2m < a (see Section V.10).
Since 1 − 2m/r vanishes for r = 2m, the Schwarzschild metric instandard coordinates appears to be singular there: g00 vanishes and grr
becomes inﬁnite. For r < 2m, r = 0, the Schwarzschild metric is again
a regular Lorentzian metric, but the timelike and spacelike character ofthe coordinates t and r are interchanged.
We will construct later, whatever the value of a > 0 , vacuum Einsteinian spacetimes that are regular for r ≥ a > 0 and isometric for

5 See, for instance, YCB-OUP2009 andreferences therein.

6 Recall that the gravitational and in-ertial masses diﬀer by a factor equal
to the gravitational constant. In thisbook, unless otherwise stated, we workin units such that this constant as wellas the speed of light are equal to 1.

112

The Schwarzschild spacetime

7 For the Sun, 2m
˜ 2.96 km and forSun =the Earth, 2mEarth =˜ 8.87 mm.

r > 2m to a Schwarzschild spacetime in canonical coordinates with radius 2m. They are interpreted as spacetimes modelling the exteriors ofbodies with Schwarzschild radius 2m. However, the apparent ‘Schwarzschild singularity’ r = 2m > a has a deep physical meaning that we will
discuss in Chapter VI: no light ray or other classical signal (i.e. one notdue to a quantum eﬀect) can escape from the regions r < 2m. For thisreason, the hypersurface R × {r = 2m} is called an event horizon anda spacetime with source of radius a < 2m is called a black hole (seeChapter VI).
If a is identiﬁed with the radius of the body in a classical CGS length
unit and the gravitational mass m determining the Schwarzschild radius
is expressed in the same unit of length, but with the units of time andmass chosen so that the speed of light and the gravitational constant areboth equal to 1, as we do in writing the equations (see Problem III.9.3
in Chapter III), then the Schwarzschild radii of the solar planets andnormal stars are very much inside these bodies:7 2m << a.
In the rest of this chapter, we will study properties of Schwarzschildspacetimes in the region r > 2m. We will compare their predictions with
the Newtonian ones and unravel new eﬀects of gravitation predicted by
Einstein’s General Relativity. We will also quote results of observations
and experiments, all in agreement with Einstein’s theory.

V.6

The motion of the planets andperihelion precession

V.6.1

Equations

The trajectories of bodies of small size and mass in a spherically symmetric gravitational ﬁeld, for instance the trajectories of the solar planets,
are timelike geodesics of this ﬁeld, i.e. of the Schwarzschild spacetime.
We denote by ds the element of proper time on a timelike curve; i.e.
with our signature convention,
ds2 ≡ −gαβ dxα dxβ .

(V.6.1)

As in the Newtonian case, we ﬁnd that the orbits remain in a ‘plane’of R3 by considering ﬁrst the equation (cf. the expressions for the
Christoﬀel symbols) 2dϕd2 θ 2 dr dθ
−
sin
θ
cos
θ
+
= 0.
(V.6.2)ds2
r ds ds
ds
We choose the coordinate θ such that at some initial instant s0 we havefor the considered motion of the planet θ(s0 ) = π/2 and (dθ/ds)(s0 ) = 0.
The equation satisﬁed by θ then implies that the orbit remains in the‘plane’ θ = π/2. With this choice of the coordinate θ, the equation forϕ reduces tod2 ϕ 2 dr dϕ
= 0,
+
ds2
r ds ds(V.6.3)

V.6

The motion of the planets and perihelion precession

which integrates to an analogue of the Newtonian area law, withconstant such that
r2

dϕ
= .
ds

some

(V.6.4)

Remark V.6.1 Equation (6.4) is a consequence of the invariance ofthe metric under rotations. The constant can be interpreted as theangular momentum per unit mass, as seen at large distance.
The equationd2 t
dν dr dt
= 0,+ds2
dr ds ds

with



2m
ν := log 1 −r(V.6.5)integrates to (‘energy’ integral due to t-translation invariance)


2m dt
= E, E a constant.
(V.6.6)1−r
ds
The remaining geodesic equation is
 2
 2
 2
d2 r 1 dλ drdϕeν−λ dν dt
−λ
+
−e
+
.
ds2
2 dr ds
ds
2 ds ds(V.6.7)

Using the expressions for λ and ν together with the previous integrals(6.4) and (6.6), we see that, when = 0, this equation reduces tod2 um1+ u = 2 + 3mu2 , with u ≡ .
dϕ2
r(V.6.8)This equation is formally the same as the linear equation in u of Newtonian mechanics, except for the addition of the nonlinear term 3mu2 .
Therefore, it holds thatu(ϕ) = uNewton (ϕ) + v(ϕ),(V.6.9)where v satisﬁes the equationd2 v+ v = 3mu2 .
dϕ2(V.6.10)

As is well known, the solutions of the Newtonian equation (forare the conics
1= m −2 (1 + e cos ϕ) =: uNewton ,r= 0)(V.6.11)

where e is the eccentricity and where the longitude of the perihelion, aconstant in this Newtonian case, has been taken to be equal to zero. The
constant e depends on the initial position and velocity of the planet. It
is not far from 0 for the solar planets—their orbits are almost circular.
For the solar planets, r is large with respect to , and hence 3mu2
can be considered as a small correction to m −2 ; the correction v will
therefore be small with respect to uNewton . An approximate solution of

113

114

The Schwarzschild spacetime(6.11), a small correction to the Newtonian expression, is obtained byreplacing (6.11) by the linear equationd2 v+ v = 3mu2Newton ≡ 3m3 −4 (1 + 2e cos ϕ + e2 cos2 ϕ). (V.6.12)
dϕ2This is the diﬀerential equation for a forced oscillation. The generalsolution is the sum of the general solution of the associated homogeneous
equation, i.e. an arbitrary periodic function of period 2π in ϕ, and aparticular solution, for instance



1 1
− cos 2ϕ .
(V.6.13)3m3 −4 1 + eϕ sinϕ + e2
2 6
The term that will give the observationally most signiﬁcant contribution
to the correction is the ‘secular’ (non-periodic) term in ϕ; that is, weconsider the Einsteinian approximation:uEinstein ∼ uNewton + 3m2 −4 eϕ sin ϕ= m −2 (1 + e cos ϕ + e3m2 −2 ϕ sin ϕ).

(V.6.14)

In geometric units, m2 −2 is small, and therefore 3m2 −2 ϕ is equivalent
to sin(3m2 −2 ϕ) and uEinstein is equivalent touEinstein ∼ m −2 {1 + e cos[(1 − 3m2 −2 )ϕ]}.

(V.6.15)

The orbit is no longer a closed curve (unless it is circular, i.e. if e = 0),
because uEinstein does not have period 2π in ϕ. It is only approximately
an ellipse; the closest point to the centre (perihelion), attained for ϕ = 0,
is attained successively after each increase of ϕ by 2π(1 − 3m2 −2 )−1 ∼2π + 6πm2 −2 . The additional 6πm2 −2 is the famous Einsteinianperihelion precession.
Remark V.6.2 Elementary calculus gives a perihelion precession per
orbit
6πm
,
p := a(1 − e2 ).
p
p is called the parameter of the Newtonian ellipse, a is its semi-majoraxis, and e its eccentricity. For equivalent a’s, the precession of theperihelion is greatest for eccentricities approaching 1.

V.6.2

Results of observations

It was observed long ago by astronomers that the orbits of the solar
planets are not exact ellipses, but slowly rotating ones. This phenomenon was interpreted in Newton’s theory as a result of the inﬂuence ofother planets. The perihelion precession of most planets could thus beaccounted for—except for that nearest the Sun, Mercury, for which a
precession of 42 per century over the 5600 observed (found by Leverrier in 1845) remained unexplained. The Einsteinian correction (1916)
just ﬁlled the gap, which was a remarkable success for the new theory.

V.6

The motion of the planets and perihelion precession

For a long time, it was thought that the not exactly symmetric shape of
the Sun could also play a role but was too diﬃcult to estimate,8 but recent data have estimated the quadrupole moment of the Sun and shown
its contribution to Mercury’s perihelion advance to be quite small.
In 1974, a pulsar (a rotating neutron star, PSR 1913+16) was observed
by Taylor and collaborators, orbiting around a companion that is mostprobably also a neutron star. Its orbit shows a precession of about 4.2 deg
per year, which is about 27 100 times the precession rate of Mercury.
This is believed to be an Einsteinian eﬀect, but since the masses of theorbiting objects are not known, the precession cannot be used directly
to test General Relativity; it is used instead to estimate these masses.
To do this, one must generalize the above treatment of the dynamics ofa small mass around a large one to the study of the relativistic dynamics
of two comparable masses.
In order to compare the predictions of General Relativity with theobservations of binary pulsars, it is necessary to go beyond the ﬁrst postNewtonian approximation presented in Section IV.5.3 in Chapter IV.
First, as a neutron star is an object that generates at its surface a very
surface −1 + hsurface,
strong deformation of the Minkowski metric (g0000
surface2with h00
 2 GM/c R  0.4 for a neutron star, to be compared withhsurface(Sun) ∼ 10−6 and hsurface(Earth) ∼ 10−8 ), one needs to develop
00
00
approximation methods that go beyond the usual weak-ﬁeld expansion
and are able to deal with the motion of strongly self-gravitating bodies.
Such methods, based on the multichart approach mentioned in Section IV.5.3, were developed by several authors in the 1970s and 1980s.9
These approaches use the method of matched asymptotic expansions
to transfer information between the tidal-like expansions of the metricγαGAαβ (XA ) considered in the local coordinate system XA attached to each
body (labelled by A) and the post-Minkowskian (or post-Newtonian)
expansion of the metric gμν (xλ ) considered in the global coordinate
system xμ . In addition, it was necessary to push the accuracy of the
post-Newtonian expansion used in the global chart xμ much beyond the
1PN level. More precisely, it was necessary to go to the 2.5PN level,
corresponding to keeping terms (v/c)5 smaller than the Newtonian-levelterms. The ﬁrst complete results in the 2.5PN approximation were obtained in the early 1980s.10 At this level of approximation, the fact thatthe gravitational interaction between the two bodies propagates (via a
retarded Green function of the wave equation) at the velocity of lightentails an observable eﬀect: namely a slow decrease of the orbital period
Pb given by11
37 4
2
1 + 73192 π
dPb
24 e + 96 e=−
ν(GM Ω)5/3
,
5
2
7/2
dt
5c
(1 − e )(V.6.16)

where ν ≡ mA mB /(mA + mB )2 is the symmetric mass ratio, M ≡mA + mB the total mass, Ω = 2π/Pb the orbital frequency, and e
the eccentricity of the orbit. This eﬀect has been observed in several

115

8 In the case of satellites orbiting the

Earth, the irregularities of their motions are used to determine the shapeof the Earth.

9 See,

notably, D’Eath (1975) andDamour (1983a).

10 See Damour and Deruelle (1981) andDamour (1982).

11 See Damour (1983b).

116

12 SeeThe Schwarzschild spacetime

Damour and Deruelle (1985,

1986).

13 Fora review, see, for instance,
Damour (2013b).

binary pulsars, with a magnitude in precise agreement with the General Relativity prediction (6.16). As this eﬀect is a direct consequenceof the propagation of the gravitational interaction at the velocity oflight, it constitutes an observational proof of the reality of gravitationalradiation.
In addition, the development of the relativistic theory of the ‘timing’ of binary pulsars12 has allowed several other comparisons between
the predictions of Einstein’s theory and binary pulsar data. More precisely, the measurement of n ‘post-Keplerian’ parameters appearing in
the Damour–Deruelle timing formula allows one to extract n − 2 testsof General Relativity (or alternative theories of gravity). For instance,in the case of PSR 1913+16, one could use pulsar timing data to measure the following three ‘relativistic’ or ‘post-Keplerian’ observables: (i)advance of periastron, (ii) apparent modiﬁcation of the proper time ofthe pulsar by combined Doppler and gravitational eﬀects, and (iii) rateof change of the orbital period of the pulsar linked to gravitational radiation damping. Comparison of these three ‘post Keplerian’ observables
with their General Relativity prediction leads to an accurate (10−3 level)
conﬁrmation of Einistein’s theory in a regime that involves both strongﬁeld eﬀects (in the neutron star) and radiative eﬀects. Other binary
pulsars13 have recently allowed further accurate conﬁrmation of GR.

V.6.3

Escape velocity

As an exercise in the physical interpretation of General Relativity, we
compute the radial velocity with respect to an observer at rest in the
Schwarzschild metric that must be applied to a test object for it toescape the gravitational attraction.
Let r0 be the r coordinate of the static observer in a Schwarzschildspacetime with mass m < r0 /2. Denote by ṙ0 = dr/ds(0) the propertime initial velocity, supposed to be radial, of the launched rocket atparameter time t = 0. This rocket, supposed to be in free motion after
its launch, follows a radial geodesic curve and hence satisﬁes, by (6.6),
the equation




2m
2m
ṫ0 .
ṫ = E = 1 −1−
rr0(V.6.17)

We also have, as a result of the deﬁnition of ds,

1=

1−

2m
r




−1
2m
ṙ2 .
ṫ2 − 1 −r

(V.6.18)2m.
r(V.6.19)Henceṙ2 = E 2 − 1 +V.7

Stability of circular orbits

117

The rocket can attain a maximum of the parameter r (and then turnback) when ṙ = 0, that is, when r takes the value
rM =

2m
.
1 − E2(V.6.20)

The number rM is an attained maximum of the parameter r if it ispositive and ﬁnite, that is, if E < 1. The escape velocity for which rM
is inﬁnite corresponds to E = 1, and hence, by (6.16), toṙ02 =

2m.
r0(V.6.21)

The relativistic escape velocity V for the observer at rest is given by
the ratio of the space (radial) and time components V 1 and V 0 in theproper frame of this observer of the velocity vector, which has components (ṙ0 , ṫ0 ) in the natural frame of the coordinates t, r. The proper
frame of the static observer is


1
− 12
2m 2
2m
θ0 = 1 −dt, θ1 = 1 −dr,(V.6.22)
r0
r0
and therefore

1

V =

2m
1−
r0



− 12
ṙ0 ,

0

V =

2m
1−
r0

 12
ṫ0 .

Hence, using (6.20) and (6.18) when E = 1, we obtain

2m
V1
.
V =: 0 =V
r0(V.6.23)(V.6.24)

This escape velocity coincides with its Newtonian value. It tends to 1, thevelocity of light, when r0 tends to 2m, in agreement with interpretations
that we will give in a forthcoming section.

V.7

Stability of circular orbits

To study the stability of circular orbits, we use the identity
gαβ ẋα ẋβ = −1and (6.4) and (6.6) to obtain


 2
1 2 1
1
2m
ṙ +
+ 1 = E2.
1−
2
2
r
r2
2

(V.7.1)(V.7.2)This type of diﬀerential equation occurs in classical mechanics andgoverns the motion of a particle of unit mass and energy 12 E 2 in thepotential14


 2
1
2m
V (r) ≡+
1.
(V.7.3)1−
2
rr2

14 Our consideration of the dynamics ofa test particle by means of an eﬀective
potential can actually be generalized
to the case of the motion of a binary
system of comparable masses by using the ‘eﬀective one-body’ approachto two-body systems (see Buonanno
and Damour, 1999; Damour and Nagar,
2011).

118

The Schwarzschild spacetime

By diﬀerentiating (7.3) with respect to s, we obtain the following secondorder diﬀerential equation for r
..

r+

dV
= 0.
dr

(V.7.4)..

A circular orbit r = r0 is one for which ṙ = 0; hence r = 0 and(dV /dr)(r0 ) = 0. That is, for such an orbit, the potential V has a criticalpoint. Computation givesdV≡ r−4 mr2 − 2 r + 3m 2 .
dr
The critical points are therefore given by√2± 4 − 12 2 m2
R± =
.
2mr2

(V.7.5)(V.7.6)

For 2 < 12m2 , there is no circular orbit. For 2 > 12m2 , there are two
possible circular orbits: r0 = R+ and r0 = R− . The circular orbit r = r0
is stable if the critical value r0 is a minimum of V ; it is unstable if r0
is a maximum. Indeed, linearization around r0 of dV /dr shows that theequation (7.4) leads to oscillations of r around r0 if (d2 V /dr2 )(r0 ) > 0
and exponential growth of r if (d2 V /dr2 )(r0 ) < 0. Elementary calculus
shows that R− is a maximum of V and hence the orbit r0 = R− isunstable, while R+ is a minimum and hence r0 = R+ is stable.
We see from (7.6) that, given m, the smallest possible value of R+ is
6m, while the smallest possible value of R− (obtained when tends toinﬁnity) is 3m. We have proved the following theorem:
Theorem V.7.1 In a Schwarzschild spacetime of mass
√ m, there is no
circular orbit with angular momentum less than m 12. The Schwarzschild coordinates r of circular orbits all satisfy r > 3m. The last stable
circular orbit has r ≥ 6m.

V.8

Deﬂection of light rays

V.8.1

Theoretical prediction

Light rays are null geodesics, so they are bent by the curvature of thespacetime. The diﬀerential equations they satisfy are the same as theequations for timelike geodesics, except that the derivative denoted by a
dot is now a derivative with respect to the aﬃne parameter on the nullgeodesic and (7.1) is replaced by
gαβ ẋα ẋβ = 0.
That is, using (6.4) and (6.6), we have

 2
1 2 1
1
2m
ṙ +
= E2.
1−
2
2
r
r2
2

(V.8.1)(V.8.2)

V.8

We deduce from this equation and from (6.4) that on a null geodesic,
 2
1 dr
1
2m
+ 2 − 3 = k2 ,(V.8.3)r4 dϕrrwith k = E 2 −2 a constant. Setting u ≡ r−1 gives
 2
du+ u2 − 2mu3 = k 2 .
dϕ(V.8.4)Hence, after diﬀerentiation, we obtaind2 u+ u = 3mu2 .
dϕ2(V.8.5)The term 3mu2 in (8.5) is the Einsteinian correction to the following
equation:d2 u
1+ u = 0, with u ≡ .
dϕ2
r(V.8.6)This gives, in the absence of gravitation, straight lines as light rays, withequations in spherical coordinates taking the form
1
1
= usphr (ϕ) ≡cos(ϕ − ϕ0 ),rr0(V.8.7)

where r0 is the displacement from the centre. If we consider the Einsteincorrection 3mu2 as small, we obtain an approximation to Einstein’s lightrays by solving the diﬀerential equationd2 u+ u = 3mu2sphr .
dϕ2(V.8.8)

Setting ϕ0 = 0 for simplicity, we ﬁnd that the general solution to (8.8)
is
1
1
m
=
cos ϕ + 2 (1 + sin2 ϕ).
(V.8.9)r
r0
r0
On the straight line given by (8.7), the coordinate r tends to inﬁnity
when φ tends to −π/2 or π/2. On the curve given by (8.9), r tends
to inﬁnity when φ tends to ±(π/2 + α) with, by (8.9) and elementary
trigonometry, α satisfyingm(V.8.10)− sin α + (1 + cos2 α) = 0.
r0
That is, if α is small,
α=
˜

2
.
r0

(V.8.11)

The total deﬂection of a light ray is therefore estimated to be
δ=
˜

4m
.
r0

(V.8.12)For light rays grazing the solar surface, this angle is about 1.75 .

Deﬂection of light rays

119

120

The Schwarzschild spacetime

V.8.2

Fermat’s principle and light travel parametertime

We have said that light rays follow null geodesics of spacetime. The
following property is a generalization to static spacetimes of a classicaltheorem of Fermat in Newton’s space E 3 .
Theorem V.8.1 In a static spacetime with metricds2 = −g00 dt2 + gij dxi dxj ,the projections on space of light rays are geodesics of the Riemannian
metricdσ 2 =gij dxi dxj;
g00

i.e. they are relative minima of the integral
 t2 
=t1

On a null curve,

gij dxi dxj.
g00

is the parameter-time duration.

Proof. This is a straightforward computation, using the geodesic equations in ds and dσ. We leave it as an exercise.

15 Ferrarese (2004).

In addition, we note that Fermat’s principle can be generalized toarbitrary spacetimes as follows.15
Theorem V.8.2 The ‘proper time of arrival’ of null curves to a given
timelike curve admits a critical point (is a relative minimum) at alight ray.

V.8.3

Results of observation

It was ﬁrst necessary to wait for an eclipse to be able to observe theapparent displacement of the stars due to the deﬂection by the Sun oflight coming from them. As early as 1919, an expedition was organized
by Eddington to measure the bending of light by the Sun. The observeddeﬂection was in reasonable agreement with the prediction but was notvery precise, until data from the Hipparcos satellite veriﬁed the deﬂection of light to a 10−3 precision. More precise results have long since beenobtained with the use of radio waves: it was not necessary in this case to
wait for an eclipse. Measurements using several radio telescopes at intercontinental separations (very long baseline interferometry, VLBI) now
give a precision of about 10−4 . They strongly support General Relativity, in particular in constrast to the Jordan–Brans–Dicke scalar–tensor
theory.

V.9

V.9

Redshift and time dilation

The time dilation and redshift eﬀects are not due to a change of thevelocity of light in a gravitational ﬁeld—in General as well as in Special
Relativity, the speed of light is a universal constant. Rather, the eﬀects
are caused by dependence on the observer of the measure of the proper
time (see Chapter III).

V.9.1

Redshift

We have already considered the gravitational redshift in Chapter III, in
any static spacetime. Its measurement in a Schwarzschild spacetime isone of the classical tests of Einstein’s equations.
Suppose for simplicity that the emitting atom and the observer areboth at rest with the same angular coordinates, and with radial Schwarzschild coordinates rA and rO , respectively. The emitted period TA andthe observed period TO are then linked by the relation (see (III.8.5))

TO =

2m
1−
rO

 
1−

2m
rA

−1
TA .

(V.9.1)Hence TO > TA if rO > rA , and so a redshift (smaller frequency ν) of
spectral lines is then observed. If m/r0 is small, then
TA
νO
≡
=
˜ 1−m
TO
νA



1
1
−
rA
rO


.

(V.9.2)This formula shows that the time between signals emitted regularly in
its own time by a source located at a point with parameter rO tending
to the horizon 2m seem progressively longer to a stationary faraway
observer; it tends to inﬁnity as the source tends to the horizon rO = 2m.
This eﬀect, already noticed by Oppenheimer and Snyder, is called theinﬁnite-redshift eﬀect.

V.9.2

Time dilation

To state the elements of the theoretical prediction of time dilation, we
treat the case of a rocket sent from the Earth in a radial direction witha velocity less than the escape velocity and moving then in free fall, i.e.
following a timelike geodesic of the Schwarzschild metric representing the
Earth’s gravitational ﬁeld. The parameter time tM it takes the rocket
to fall back to Earth is twice the parameter time it takes to attain its
maximum rM value (see Section V.6.3). We calculate the proper timeusing the formula

Redshift and time dilation

121

122

The Schwarzschild spacetime



 tM
ds
2m
−1
dt =
sA =
E
1−
dt
dt
r
0
0
 tM 1 − 2mr ds (0) dt,
=
2m dt
0
1−
r0
 tM

with



ds
dt



2
 
−1 
2 
dr
2m
2m
(0) .
− 1−
(0) = 1 −r0
r0
dt

(V.9.3)(V.9.4)We set (dr/dt)(0) = v and we obtain for sA , when m/r0 is small, theapproximate expression
 tM 
2m m mv 2
+
˜
+
sA =
1−
dt.
(V.9.5)rr0
r00
The proper time observed on Earth at the ﬁxed r0 between the origin
and the impact point is tM 
 tM 
2m
m
sO =
1−
dt =
˜
1−
dt.
(V.9.6)r0
r0
0
0
Therefore, a standard clock carried by the rocket shows a delay over thesame standard clock of the observer, given approximately by tM 
m mv 22m+2 +sA − sO =˜−
dt > 0,(V.9.7)rr0
r00
since r > r0 on the trajectory. Experiments made with caesium clocks
have conﬁrmed this time delay and its estimates with great precision.
Remark V.9.1 On a Lorentzian manifold, timelike geodesics are, asin Minkowski spacetime, local maxima of the length of timelike curves
joining two points. In the given example, it is the rocket that follows ageodesic, while in the twin paradox (see Chapter II), it is the travelling
twin who is not in free fall—he has to use an engine to come back.

V.10

Spherically symmetric interior
solutions

A relativistic model for a spherically symmetric isolated star is a spherically symmetric spacetime (V ≡ R3 × R, g) such that g satisﬁes theEinstein equations on V ,1Sαβ ≡ Rαβ − gαβ R = Tαβ ,2where Tαβ , the source stress–energy tensor, is zero outside the star anddepends inside it on the star’s physical constitution—this is not well

V.10

Spherically symmetric interior solutions

known and is diﬃcult to model by a single formula. For this reason,
there is no physically reliable exact solution for the considered problem.
However, important qualitative features can be obtained from generalconsiderations.
Remark V.10.1 As in Newtonian theory, a spherically symmetric
Einsteinian gravitational ﬁeld vanishes in the interior of a hollow
sphere—that is to say, namely a spherically symmetric vacuum solution
deﬁned for r < a is necessarily ﬂat in this domain. Indeed, the construction of Section V.3 shows that such a solution is a Schwarzschild metric(3.1), with m some arbitrary number. The only solution of this form thatis continuous at r = 0 is the ﬂat solution with m = 0.

V.10.1

Static solutions. Upper limit on mass

We look for equilibrium conﬁgurations, i.e. static spacetimes. We use
Schwarzschild coordinates. For r > rstar , with rstar the radial Schwarzschild coordinate of the star’s boundary, we have vacuum; hence thesolution is Schwarzschild. For r < rstar , we again look for a static
spherically symmetric metric, of the formg = −eν dt2 + eλ dr2 + r2 (dθ2 + sin2 θ dϕ2 ),(V.10.1)

but now we solve the equations with a non-vanishing stress–energy tensor, which we take, in the absence of better modelling, to represent a
perfect ﬂuid; that is, the Einstein equations are
Sαβ = Tαβ ,with Tαβ ≡ (μ + p)uα uβ + pgαβ ;(V.10.2)

equivalently,
1Rαβ = ραβ ≡ (μ + p)uα uβ + (μ − p)gαβ .
2(V.10.3)

For a static solution, the ﬂuid unit velocity is ui = 0, u0 = e−ν/2 ,tangent to the timelines. Then T0i = R0i = 0,1
T00 ≡ μeν , R00 = eν (μ + 3p),2

1(μ − p)gij .
2(V.10.4)
The scalar functions μ and p, like λ and ν, depend only on r.
The identities found in Section V.3 give (a prime denotes diﬀerentiation with respect to r)


r 
1
−λ

1
+
R
(ν
R22 ≡
≡
−e
−
λ
)
+ 1,
33
2
sin2 θ
Tij ≡ pgij , Rij =r(eλ−ν R00 + R11 ) ≡ ν  + λ ,and for the spacetime scalar curvatureR ≡ −e−ν R00 + e−λ R11 + r−2 (R22 + sin−2 θ R33 ).

(V.10.5)

123

124

The Schwarzschild spacetimeν  λ
ν  − λ
2
ν 2
2
−
+
+ 2 + 2.
≡ −e−λ ν  +
2
2
r
r
r

(V.10.6)

Elementary computations give1
1
S00 ≡ R00 + eν R ≡ [R00 + eν−λ R11 + eν r−2 [R22 + (sinθ)−2 R33 ]}.
2
2
16 We will see in Section VIII.7.2 thatthe fact that the component S00 does
not contain the lapse (here eν ) is a
general property of the Einstein tensor.

The equation S00 = T00 ≡ μeν is then found to reduce16 to a diﬀerential
equation for λ that can be written as
 r


1
d −λ
2
−λ
1 − ρ2 μ(ρ) dρ + constant .
1− (e r) = r μ, hence e =drr0(V.10.7)
Smoothness of the metric in a neighbourhood of the origin r = 0 of polar
coordinates imposes the vanishing of the integration constant; hence
1 r 2
2M (r), with M (r) ≡e−λ = 1 −
ρ μ(ρ) dρ.
(V.10.8)r
2 0This solution deﬁnes a Lorentz metric in a domain r ≤ a only if a
2M (a) ≡ρ2 μ(ρ) dρ ≤ a.
(V.10.9)
0

Having computed λ, we use the equationν  + λ ≡ r(eλ−ν R00 + R11 ) = r(eλ−ν ρ00 + ρ11 ) ≡ reλ (μ + p) (V.10.10)
to ﬁnd thatν  = (eλ − 1)r−1 + reλ (p − μ) =2M (r) + r3 (p − μ).
r[r − 2M (r)](V.10.11)

On the other hand, the ﬂuid equation (cf. Chapter IV) gives(μ + p)uα ∇α u1 + ∂1 p ≡ (μ + p)u0 u0 Γ101 + ∂1 p = 0;
that is,1p = − (p + μ)ν  ,2and we ﬁnd that
2M (r) + r3 (p − μ)1.
p = − (p + μ)2
r[r − 2M (r)](V.10.12)

This is known as the Tolman–Oppenheimer–Volkov equation ofrelativistic hydrostatic equilibrium. It can be integrated if one assumes
some equation of state inside the star. In most situations, it is quite
diﬃcult to know with any reliability an equation of state, because of thecomplexity of the phenomena that occur. However, one can deduce someconclusions from this formula.
When the density μ is a constant, μ0 , the integral (10.8) gives
 r
1
ρ2 dρ = μ0 r3 .
(V.10.13)
2M (r) = μ0
3
0

V.10

Spherically symmetric interior solutions

125

Equation (10.12) then reduces to a diﬀerential equation for p that canbe integrated exactly. Assuming that p vanishes at the boundary r = aof the star, one ﬁnds for the pressure at its centre
p(0) = μ0

1
1 − 1 − 2a−1 M (a) 2
1.

(V.10.14)

4a.
9(V.10.15)

3[1 − 2a−1 M (a)] 2 − 1This pressure becomes inﬁnite if
1

3[1 − 2a−1 M (a)] 2 = 1, i.e.

M (a) =The pressure p(0) becomes negative if M (a) > 4a/9.
Exercise V.10.1 Prove these results.
The classical conclusion is that stars with M (a) ≥ 4a/9 cannot exist.
The same result holds17 if one supposes only that μ is a non-increasing
function of r.
Theorem V.10.1 There exist no equilibrium conﬁguration of spherically symmetric stars ﬁlled with a perfect ﬂuid with μ a non-increasing
function of r and such that
M (a) ≥

4a,9

17 The integration was ﬁrst performedby Schwarzschild in 1916. For details,
and a discussion of the case of anon-constant density, see Wald (1984),
p. 129.

(V.10.16)

where a is the standard radius of the star and M (a) is given by
1 a
M (a) =
μ(ρ)ρ2 dρ.
2 0

V.10.2

Matching with an exterior solution

An Einsteinian model for the exterior and the interior of a sphericallysymmetric star is a manifold R3 × R with a Lorentzian metric g thatsatisﬁes the Einstein equations on the whole manifold, induces an interior
Schwarzschild metric in B × R, with B a ball of R3 ﬁlled with matter,and an exterior, vacuum, Schwarzschild metric in the complementarydomain. The metric, spherically symmetric, reads on R3 × R
g = −eν dt2 + eλ dr2 + r2 (dθ2 + sin2 θ dϕ2 ,with eλ and eν equal to the coeﬃcients of the interior metric gstar forr < rstar , denoted by a above, and to those of a vacuum Schwarzschild
metric in the exterior. For such a metric to be a solution on R3 × R,
some continuity18 properties are required from the functions λ and ν atthe boundary r = a between the interior and exterior regions. Equation(10.8) in particular imposes thate−λext = 1 −2m
2M (r)= e−λint = 1 −rr

for r = a;18 If these quantities were not continuous, their diﬀerentiation would introduce a measure with support r = rstar
in the components of the Ricci tensor
that are not present in the source.

126

The Schwarzschild spacetimethat is,m = M (a) ≡

1
2

 a
r2 μ(r) dr.

(V.10.17)

0

The number M (a), sometimes called the mass function, can be computed in terms of the density in classical units by reinserting in the above
formula the Einstein gravitational constant GE = 8πGN . This leads tothe integral of the density on the volume of a sphere Ba of radius a,namely acgsM cgs (a) = 4πGcgs
r2 μcgs (r) dr.
N
0Note, however that M (a) is not the proper mass of a static spherical starof standard radius a, which should be computed with the space proper
volume element
1

r2 e 2 λ dr sin θ dθ dφ,resulting in a smaller quantity; the diﬀerence represents the gravitationalbinding energy of the star in equilibrium.

V.10.3

Non-static interior solutions

Birkhoﬀ’s theorem does not apply to interior solutions. There exist timedependent, spherically symmetric solutions of the Einstein equations
with sources in a domain r < rstar ; the radius rstar of the star may be afunction of t. The full solution is the considered interior solution gint forr ≤ rstar and a Schwarzschild exterior solution with mass m for r ≥ rstar ,provided that rstar > 2m, the Schwarzschild radius of the star, which isconstant for an exterior solution and is linked with the energy content
of the star, which is considered as an isolated object. When the star
contracts so much that rstar becomes smaller than 2m, it is no longer
observable from the region r > 2m of the spacetime (see Chapter VI).
It is physically clear that there can be no static interior solution withzero pressure, because there is nothing in that case to resist the gravitational attraction. This can be checked mathematically by solving (10.12)
with p = p = 0, which gives rμ(r) = 2r−3 M (r) ≡ 2r−3
ρ2 μ(ρ) dρ,(V.10.18)
that is, setting y(r) =

r
0

0

ρ2 μ(ρ) dρ, the diﬀerential equation

y =

2
2
y, hence y = Cer ,rwith C a constant. Then2

μ = Cr−2 er ,which is always inﬁnite for r = 0.

(V.10.19)

V.11

V.11

Spherically symmetric gravitational collapse

Spherically symmetric gravitational
collapse

It is believed that when a star has exhausted all thermonuclear sourcesof energy, it will collapse under its own gravitational ﬁeld. Oppenheimer
and Snyder19 made a rigorous study of this process for a sphericallysymmetric dust cloud by using a special solution given by Tolman20 in
the restricted case of a spatially constant matter density starting fromrest. This study was revisited by Gu21 with the restriction to spatiallyconstant matter lifted. Gu analysed shell-crossing singularities as wellas the essential singularity at the centre of symmetry. He gave a detailed mathematical discussion of the formation of horizons and of what
is now called the nakedness of the singularities. Hu22 constructed thefully general spherically symmetric solutions for dust, not restricted to
the Tolman class, and discussed their properties. The work done by Gu
and Hu for non-homogeneous dust clouds was repeated, in ignoranceof these papers published in China, by Müller zum Hagen, Seifert, andYodzis23 , who proved the existence of shell-crossing singularities in thecase of the Tolman class; later, Christodoulou proved the existence ofcentral singularities,24 starting from rest. Finally, Newman25 obtained
the general solution. These western papers were motivated by the cosmic censorship conjecture, which is violated both at shell-crossing
singularities and at the centre, as was already shown in Gu’s paper.

V.11.1127

Tolman, Gu, Hu, and Claudel–Newman metrics

In the case of dust, the ﬂow lines are the timelike geodesic trajectories of the particles; we take these geodesics as timelines (comoving
coordinates). A spherically symmetric metric can then be written as− dt2 + e2ω dr2 + R2 (dθ2 + sin2 θ dφ2 ),(V.11.1)

where ω and R are functions of r and t. The ﬂow lines of the dust are
the curves on which the coordinates r, θ, φ are constant; a so-called dust
shell is labelled by its coordinate r. The metric is regular as long as e2ω
and R are smooth positive functions. R2 is (up to multiplication by 4π)
the area of the spherical dust shell of parameter r, at time parameter t.
One ﬁxes the parameter r by choosing it to be such that r = R(r, 0).
The stress–energy tensor T of the dust reduces toTαβ = μuα uβ ,(V.11.2)

with uα = δ0α , since the timelines are the matter ﬂow lines. The only
non-zero component of the stress energy tensor is
T00 = μ,(V.11.3)withμ = μ(r, t) for r ≤ a and μ(r, t) = 0 for r > a,

(V.11.4)

19 Oppenheimer and Snyder (1939).
20 Tolman (1934) and, Bondi (1947).
21 Gu (1973).

22 Hu (1974).

23 Müller

zum Hagen, Seifert, andYodzis (1973).

24 Christodoulou (1984).
25 Newman (1986).

128

The Schwarzschild spacetime

26 Which may not remain the outermost

shell if there is shell crossing as time
evolves.

where a is the r parameter of the outermost dust shell26 at time t = 0.
We denote by primes and dots respectively diﬀerentiation with respect
to r and t. We compute the general solution in the chosen notation. The
equation R10 = 0 is equivalent to the second-order equationṘ − ω̇R = 0,

(V.11.5)

which integrates to the ﬁrst-order equationR e−ω = f (r),(V.11.6)

with f (r) an arbitrary function of r.
The conservation equations reduce to−1∇α T α0 ≡ μ̇ + ΓαṘ) = 0;α0 μ ≡ μ̇ + μ(ω̇ + 2R

(V.11.7)

hence, with φ an arbitrary function of r, we obtain
μ(t, r) =

e−ω
φ(r).
R2

(V.11.8)

Using (12.6), this equation becomesr2 μ0.
(V.11.9)
R2 R
Here μ0 is an arbitrary function of r that we identify with the initial
density because we have chosen r such that R(0, r) = r and henceR (0, r) = 1. It has been shown by R. Newman that the Einsteinequations then admit the ﬁrst integral
μ(t, r) =1 2 M (r)1Ṙ −= [f 2 (r) − 1],
2
R
2
with

 r
M (r) =f (ρ)μ(t, ρ)R2 (t, ρ)eω dρ.

(V.11.10)(V.11.11)

027 Note as before that a factor 4π has

been included in our deﬁnition of densities; the integral giving M (r) is takenover a 3-sphere.

The function M (r) is the integral over the volume occupied by the dust
shells 0 ≤ ρ ≤ r of the density weighted by the factor f (ρ).27 It isindependent of t, as can be veriﬁed by using (12.6) and (12.9), whichshow that r
M (r) =
μ0 (ρ)ρ2 dρ.
(V.11.12)
028 Hu (1974) and Newman (1986).

The explicit formulas for R(r, t) and ω(r, t) depend on the sign off 2 (r) − 1; they have been computed and studied in the general case by
Hu and Newman, to whom we refer the reader.28 The case where thedust cloud starts from rest, i.e.
M (r)+ f 2 (r) − 1 = 0,
(V.11.13)rhas been studied by Christodoulou, using parametric equations for R
and t.
Ṙ(0, r) = 2

V.11

Spherically symmetric gravitational collapse

Here, following Gu29 we study the case f 2 (r) = 1, in which exact
integration permits a clear and comparatively short discussion. In thecase f 2 (r) = 1, (11.10) reads1(V.11.14)
R 2 Ṙ = ± 2M (r).
We suppose that the star starts contracting; then the minus sign must
be chosen when t increases, and the equation integrates to33
R(r, t) 2 = Φ(r) −2M (r)t,2
with Φ an arbitrary function. Since we have normalized the radial par3
ameter r by the condition R(r, 0) = r, it holds that Φ(r) = r 2 . We use
the notation
13
2M (r) =: h 2 (r),(V.11.15)
2
and then3

1

2

R(r, t) = [r 2 − h 2 (r)t] 3 .

(V.11.16)

Using (11.6), we then ﬁnd
 1

3
1
1
1 1
eω = R = [r 2 − h 2 (r)t]− 3 r 2 − h− 2 h (r)t .
3Here, as a consequence of the deﬁnition (11.15), we have
9 2
r μ0 (r).
2
On the other hand, the equation for μ givesh (r) =μ(t, r) =

r2 μ0
3
2

1
2

1

1

[r − h (r)t][r 2 − 32 h− 2 h (r)t].

(V.11.17)

We make a change of coordinates: instead of choosing as radial parameterthe number characterizing a dust shell, we take the area R of a dust shellat time t. Since we have chosen f (r) = 1, i.e. R = eω , it holds that1
(V.11.18)
dR = eω dr + Ṙ dt = eω dr − R− 2 2M (r) dt.
The change of coordinates from r, t to R, t is admissible if R does
not vanish. In the new coordinates, the spacetime metric becomes, for
0 ≤ r ≤ a,

2M (r)
1
dt2 +2R− 2 2M (r) dR dt+dR2 +R2 (dθ2 +sin2 θ dφ2 ).
− 1−
R(V.11.19)
We see that the metric of the space sections t = constant reduces to the
Euclidean metric in polar coordinates. If the space manifold is, for everyt, homeomorphic to a ball of three-dimensional Euclidean space and R isan admissible polar coordinate, then R(t, r) is also the distance at time
t between the centre and the dust shell with parameter r.

29 Gu (1973).

129

130

The Schwarzschild spacetime

V.11.2

Monotonically decreasing density

Collapse of dust shells
In this subsection, we suppose that μ0 is a monotonically decreasing
function of the parameter r, as r increases from 0, the centre of the star,
to a, the value of r at the surface boundary of the star at time t = 0.
To study the possible collapse, we study the evolution in proper time
t of the function R(t, r) for a given dust shell, i.e. for a given value of r.
We see from (11.16) that the shell collapses at the centre of symmetryat the time t1 (r) where R(t1 , r) = 0; that is,
13

t1 (r) = h− 2 (r)r 2 .
It follows that
1 1 3
9 1 3
dt1 (r)= r 2 h− 2 [3h(r) − rh (r)] = r 2 h− 2 [3M (r) − rM  (r)] .
dr
2
4(V.11.20)
If μ0 is monotonically decreasing, then, using the deﬁnition (11.11) of
M (r), we ﬁnd that r
μ0 (ρ)ρ2 dρ − μ0 (r)r3 ≥ 0.
(V.11.21)
3M (r) − rM  (r) ≡ 3
030 Shell crossing, leading to non-central

singularities, exists when the density
is not monotonically decreasing. See
Gu (1973), Hu (1974), Müller zum Hagen et al. (1984), and Newman (1986).

31 Vanishingcrossing.

of R

signals a shell

Hence the shells with increasing parameter r arrive successively at thecentre, and there is no shell crossings.30 If the density is uniform (theOppenheimer–Snyder case), then the dust shells all arrive at the sametime at the centre.
The metric (11.19) is a regular Lorentzian metric if the linear formdR − Ṙ dt does not vanish, i.e. if R = eω > 0. We deduce from theexpression (11.16) for R that


3 1
3 3 
1 −1

2
2
2
R R =
r − h (r)h (r)t = 0.
(V.11.22)2
2
2
Hence R (t, r) vanishes31 at a time t2 given by1
1
1
2M (r)
2h 2 (r)3h 2 (r)r 2
= 3
t2 =
≡ 3
.
h (r)
3r 2 μ0 (r)r 2 μ0 (r)(V.11.23)

We note thatt1h (r)r
.
=
t2
3h(r)

(V.11.24)

The monotonicity of μ0 and the mean value theorem led Gu to theconclusion thatt1 ≤ t2 ;(V.11.25)

that is, the dust evolution does not induce a singularity in the metric
before a dust shell arrives at the centre, in agreement with the previous

V.11

Spherically symmetric gravitational collapse

conclusion of the absence of shell crossing. The ﬁrst occurence of thesingularity is at a time t0 , with
 3  12r
1;
(V.11.26)=!
t0 = lim
r=0 h(r)
3
μ
(0)
0
2
the greater μ0 (0) is, the sooner the singularity appears.
Matching with an exterior metric
The interior metric is given by (11.19) for r ≤ a, i.e.
 3
 23
1
0 ≤ R ≤ a 2 − h 2 (a)t .

(V.11.27)

The computations that we have made are still valid outside the star,
where r > a and μ0 (r) = 0, but for r > a it holds that a
M (r) ≡ Ma :=μ0 (ρ)ρ2 dρ, r > a,
(V.11.28)
0

and the exterior metric reads



2Ma
2Ma
2
− 1−
dR dt + dR2 + R2 (dθ2 + sin2 θ dφ2 ).
dt + 2
R
R(V.11.29)
It is a boosted Schwarzschild metric with horizon
R = 2Ma .

(V.11.30)

This solution takes the usual Schwarzschild form



−1
2Ma
2Ma
− 1−
dR2 + R2 (dθ2 + sin2 θ dφ2 )dt2 + 1 −R
R
(V.11.31)
if we make the change of time coordinate

−1 
2Ma
2Ma
dR.
(V.11.32)dτ = dt − 1 −R
R
The full spacetime metric g is deﬁned on R3 × R+ , with R, θ, φ polar
coordinates on R3 and t ∈ R+ (i.e. t ≥ 0) by
 3
 23
1
g = gdust , 0 ≤ R ≤ a 2 − h 2 (a)t ,(V.11.33)
g = gext ,

 3
 23
1
R ≥ a 2 − h 2 (a)t .

(V.11.34)

The interior dust solution is hidden behind the horizon when
 3
 23
1
a 2 − h 2 (a)t < 2Ma .
(V.11.35)
This begins at a positive time t3 given by1

3

3

t3 = h− 2 (a)[a 2 − (2Ma ) 2 ](V.11.36)

131

132

The Schwarzschild spacetime

if a > 2Ma , that is, if the star is initially visible. We have seen thatthe outer shell of the star collapses into a singularity when t1 (a) =13h− 2 (a)a 2 , hence after the star has ceased to be visible. This fact was aninspiration for the formulation of the cosmic censorship conjecture
by Penrose (see Chapter VIII).

V.12

Problems

V.12.1

Relativistic and Newtonian gravitational
massesShow that the relativistic gravitational proper mass of a spherical starwith radius a and density μ(r) is larger that its Newtonian gravitational
mass.

Solution
The Newtonian gravitational mass is, with GN the Newtonian gravitational constant,
  a
 a
2
M (a) = GNr μ(r) dr sin θ dr dθ dφ ≡ 4πGN
r2 μ(r) dr.
S2

0

0

The relativistic gravitational mass, computed with the proper volume
element, is

− 1
 a
2M (r) 2
2
2Ma := GEr μ(r) 1 −dr.
(V.12.1)r
0
1

The proper mass Ma is greater than M (a) because [1 − 2M (r)/r]− 2 > 1.
The diﬀerence represents the gravitational binding energy of the star inequilibrium.

V.12.2The Reissner–Nordström solution

Show that the Reissner–Nordström metric (found in 1916 by
Reissner) and given in standard coordinates by


−1

2m Q2
2m Q2
2
+ 2 dt + 1 −+ 2
dr2 + r2 (dθ2 + sin2 θ dϕ2 )
− 1−
r
r
r
r
(V.12.2)
is a spherically symmetric solution of the Einstein–Maxwell equations
with electromagnetic potential
Ai = 0,A0 = −

Q,r(V.12.3)

V.12

identical with the classical electrostatic potential of a spherical bodywith charge Q.
Show that the metric (12.1) is smooth and Lorentzian with t a time
variable, provided that
2m Q2− 2 < 1.
rr

(V.12.4)

Remark V.12.1 For large r, the term Q/r2 is small in comparison
with m/r. Since the total charge of celestial bodies is observed to be negligible in comparison with their mass, the Reissner–Nordström solution
has little application in astrophysics. It was also abandoned as a possiblemodel for the electron, for which m−1 Q2 = 2.8 × 10−13 cm, and so theinﬂuence of the term r−2 Q2 would be important only at distances where
quantum eﬀects cannot be neglected.
The Reissner–Nordström solution has attracted interest in connection with modern aspects of mathematical General Relativity. Indeed, if
m2 > Q2 , it possesses two event horizons (see Chapter VI) given instandard coordinates by
r± = m ± m2 − Q2 .
(V.12.5)
It would invalidate a censorship conjecture if the word ‘generic’ was notincluded in the hypotheses (see Chapter VIII).
Reissner–Nordström-type solutions have recently become important instring theories.

V.12.3

Schwarzschild spacetime in dimension n + 1

Construct by reasoning and computations analogous to those forn + 1 = 4 the (n + 1)-dimensional Schwarzschild metric in sphericalstandard coordinates, r ∈ R+ ,



−1
2m
2m
gSchw = − 1 − n−2 dt2 + 1 − n−2
dr2 + r2 dω 2 , (V.12.6)
rrwhere dω 2 is the metric of the sphere S n−1 . The Schwarzschild spacetime
is deﬁned as for n = 3 by this metric supported by the manifold M × R,
with M the exterior of the ball rn−2 = 2m.

V.12.4

Schwarzschild metric in isotropic
coordinates, n = 3

Show that in isotropic coordinates the four-dimensional Schwarzschild
metric reads, with R2 = X 2 + Y 2 + Z 2 ,2
2R − mm 4
gSchw = −dt2 + 1 +(dX 2 + dY 2 + dZ 2 ). (V.12.7)
2R + m2R

Problems

133

134

The Schwarzschild spacetime

Solution
Recall that, in standard coordinates,



−1
2m
2m
2
gSchw = − 1 −dr2 + r2 (dθ2 + sin2 θ dφ2 ).
dt + 1 −rr(V.12.8)
The relation between r and R ism 2
4R2 + 4Rm + m2,
(V.12.9)
r =R 1+=
2R
4R
and hence the coeﬃcient of dt2 becomes
8Rm
2m
=1−
1−
=
r
(2R + m)2

2R − m2R + m2.

To compute the space metric in isotropic coordinates, we ﬁrst remark
that


−1
2
2R + m
r
2m
=
=
1−
r
r − 2m2R − m
and

m2
m 2
,
=R+m+
r =R 1+2R
4R
dr = dR(V.12.10)


m2
1−
.
4R2

We then proceed as follows. From the deﬁnitions
R−1 X = r−1 x, R−1 Y = r−1 y, Z = Rr−1 z, andR−2 (X 2 + Y 2 + Z 2 ) = r−2 (x2 + y 2 + z),(V.12.11)
it results that,d(r−1 x) = r−1 dx − r−2 x dr = R−1 dX − R−2 X dR
with analogous relations for y and z, and hencer−2 (dx)2 − 2r−3 x dx dr + r−4 x2 (dr)2 = R−2 (dX)2− 2R−3 X dX dR + R−4 X 2 (dR)2 ,and, by summation and simpliﬁcation,r−2 (dx2 + dy 2 + dz 2 − dr2 ) = R−2 (dX 2 + dY 2 + dZ 2 − dR2 ).
Using
r2 (dθ2 + sin2 θ dφ2 ) = dx2 + dy 2 + dz 2 − dr2= r2 R−2 (dX 2 + dY 2 + dZ 2 − dR2 )V.12shows that the space part of gSchw can be written as

2
2 

2R + m
m 4
m2
2
dR
+
1
+
(dX 2 + dY 2 + dZ 2 − dR2 ),1−
2R − m4R22R
and, after simpliﬁcation, the Schwarzschild metric is obtained in isotropic coordinates.

V.12.5

Wave coordinates for the Schwarzschild metricin dimension n + 1

Show that the requirement that xμ = (t, xi ) be wave coordinates,
2g xμ = 0, with xi = r̄(r)ni , ni ∈ S n , reduces to the equationd dr̄ n−1r
−∗ r̄−(n−1)r̄ =(1 − 2mr2−n ) −(n−1)r̄ = 0, (V.12.12)
dr drwith Δ∗ the Laplacian on the sphere S n−1 .
Show that setting s = 1/r gives as equation singular at s = 0:


d
dr̄
s3−n (1 − 2msn−2 )= (n − 1)s1−n r̄ .
ds
ds
Show the asymptotic expansions
 m2
−3
m
ln r + O(r−5 ln r), n = 4,4 r+r̄ = r +
5−2n
n−3
),
n≥5
O(r
(n − 2)rSolution
See Choquet-Bruhat, Chruściel, and Loiselet (2006).

.

Problems

135

VI
VI.1 Introduction
VI.2 The Schwarzschild
black hole
VI.3
VI.4

Black holes
136
136

Eddington–Finkelstein
extensions
137
Stationary black holes 141

VI.5

The Kerr spacetimeand black hole142
VI.6 Uniqueness theorems
for stationary black
holes
146
VI.7 General black holes
147
VI.8 Conclusions
VI.9 Solution of
Exercise VI.3.1

150

VI.10

151

Problems

149

VI.1

Laplace and Michell had already foreseen that light can become trappedby a massive body, so that the latter becomes black to observers and isperceptible only through its gravitational ﬁeld. However, the apparent‘Schwarzschild singularity’ r = 2m is a phenomenon that has no analogue in classical mechanics, and the completed spacetimes are indeed
very strange. It was long believed that these extensions have no physicalreality, i.e. that matter cannot be compressed so that it is included in
the region r < 2m. It was mainly through the vision of Robert Oppenheimer and John Archibald Wheeler that the reality of black holes wasseriously considered. They cannot be seen directly, but astronomical observations, in particular perturbations of motions of various stars, reveal
gravitational ﬁelds too strong to be explained by an invisible massive
companion other than a black hole. Some X-ray sources and active galactic nuclei are interpreted in terms of black holes. Also, some γ-ray
bursts are thought to be due ultimately to the explosion of matter accreted near the boundary of a black hole by the gravitational ﬁeld it
generates. There is now a large consensus among astrophysicists on theexistence of many black holes in the universe, even at the centre of ourown galaxy.

VI.2

HorizonSingularity
Fig. VI.1 Schwarzschild black hole.

Introduction

The Schwarzschild black holeThe ﬁrst model of a black hole appears in the ﬁrst exact solution of theEinstein equations constructed by Schwarzschild in 1916, the sphericallysymmetric one that we have studied in Chapter V (Fig. VI.1). We havedeﬁned the Schwarzschild spacetime as the manifold r > 2m in R3 × R,
with r a polar coordinate in R3 , with Lorentzian metric given in standard
coordinates by




2m
2m
− 1 dt2 +− 1 dr2 + r2 (dθ2 + sin2 θ dφ2 ), (VI.2.1)−tt
and we have called the submanifold r = 2m of R3 × R, diﬀeomorphic to
the product S 2 × R, the event horizon.
The Schwarzschild metric in standard coordinates with m > 0 ceases
to be a smooth Lorentzian metric for r = 2m; at this value of r, thecoeﬃcient g00 vanishes and g11 becomes inﬁnite.

VI.3

Eddington–Finkelstein extensions

137

For 0 < r < 2m, the Schwarzschild metric in standard coordinates isagain a smooth Lorentzian metric, but t is a space coordinate while r isa time coordinate.
Exercise VI.2.1 Show that the Schwarzschild metric with 0 < r < 2m
is neither spherically symmetric nor static.
The volume element of gSchw in standard coordinates,
r2 sin θ dr dθ dϕ dt, is smooth1 and non-vanishing. More importantly,
a straightforward computation shows that the coordinate-independentscalar Rαβγδ Rαβγδ , called the Kretschman scalar, is ﬁnite at r = 2m,
in fact for all r > 0, being equal to 48m2 /r6 . This property led tothe belief that the Schwarzschild spacetime (S 2 × {r > 2m}) × R is
extendible, in the sense that it can be immersed in a larger Einsteinian
spacetime, whose manifold is not covered by the Schwarzschild standard
coordinates with r > 2m.

VI.3

1 If r, θ, φ are interpreted as polar co-ordinates on R3 , then θ, φ are only
locally coordinates in the usual sense;
the sphere S 2 is not diﬀeomorphic to
R2 and hence cannot be covered by asingle coordinate patch.

Eddington–Finkelstein extensions

Let us consider (following Eddington in 1924, Lemaı̂tre in 1926, andFinkelstein in 1958) the change of coordinates, deﬁned for r > 2m,
obtained by replacing the canonical Schwarzschild time t by the ‘retarded
time’ v given by,2

 r
−1 .
(VI.3.1)v = t + r + 2m log
2mUsing this new coordinate v together with r, θ, ϕ, the Schwarzschild metric can be expressed as the so-called Eddington–Finkelstein (EF)metric2m(VI.3.2)− 1−
dv 2 + 2 dr dv + r2 (sin2 θ dϕ2 + dθ2 ).
r
The EF metric is a smooth metric of Lorentzian signature3 on the
manifold S 2 × R+ × R, (θ, ϕ) ∈ S 2 , r ∈ R+ , v ∈ R deﬁning a vacuum Einsteinian spacetime called the Eddington–Finkelstein black hole.
By its construction, the Schwarzschild spacetime is isometric with thedomain r > 2m of the EF black hole.
Exercise VI.3.1 Prove these statements.
The submanifold r = 2m is null (isotropic), since g rr = 0 for r = 2m.
One family of radial (i.e. θ = constant, ϕ = constant) light rays isrepresented by straight lines v = constant; the other family is given by


2m
− 1−
dv + 2 dr = 0;rthat is,dv = −2r
dr =2m − r−
4m+ 2 dr,2m − r2 Thischange of coordinates fromt, r, θ, φ to v, r, θ, φ is singular for
r = 2m.

3 The

vanishing of the coeﬃcient gvv
at r = 2m does not correspond to a
singularity of the metric, because its
determinant, det g ≡ 1, does not vanish
there.

138

Black holes

which integrates for r < 2m tov = 2r + 4m log(2m − r) + constant.
Under the change of coordinatest = v − r − 2m log

(VI.3.3)


2m
−1 ,
r

the EF metric takes, in the domain r < 2m, the Schwarzschild form

−1


2m
2m
− 1 dt2 −−1
dr2 + r2 (dθ2 + sin2 θ dφ2 ),rr
but t is now spacelike, and r timelike. The metric is no longer static.
It is singular for r = 0—in general interpreted as a limiting spacelike
3-surface. The following theorem justiﬁes the name of black hole given to
the EF spacetime: no future light ray issuing from a point where r < 2mcrosses the event horizon r = 2m, as can be seen from the light cones
and null geodesics in Fig. VI.2. More generally, the theorem is as follows:
Theorem VI.3.1 On a timelike line issuing from a point with r < 2m,in standard coordinates, the variable r is always decreasing and tends tozero in a ﬁnite proper time. Hence any observer crossing the Schwarzschild radius r = 2m attains the singularity r = 0 in a ﬁnite proper
time.
Proof. This is conveniently done in terms of the t, r coordinates, notwithstanding the fact that t is now a space coordinate and r a timeone.
tr = 2m(a)rt´r = 2m(b)

Fig. VI.2 (a) Orientation of the light
cones in the standard Schwarzschild coordinates. (b) Null geodesics and light
cones in the extension. Here t = v − r,where v is the ‘retarded’ time deﬁned
in (3.1).

r

VI.3

Eddington–Finkelstein extensions(1) Radial geodesics: we have found in Chapter V that on such geodesics,
with s the proper time,4ṙ2 = E 2 − (1 − 2mr−1 ),

ṙ :=

dr
.
ds

(VI.3.4)139

4 We are in the region r < 2m, andhence

E 2 − (1 − 2mr −1 )≡ r −1 2m − r(1 − E 2 ) > 0.

Hence the point r = 0 is attained from r = r0 < 2m after the ﬁnite
proper time
 r0
1
r 2 drds
dr ≡.
(VI.3.5)dr
2m − (1 − E 2 )r0(2) Non-radial geodesics and non-geodesic5 motions lead to a smaller
lapse of proper time between points with coordinates r0 < 2m and 0.


5 Recall that in a Lorentzian manifold

causal geodesics realize a local maximum of length between two causally
related points.

The amount of proper time taken for an astronaut entering the blackhole to reach the singularity is indeed very short—it is of the order of
the Schwarzschild radius estimated in geometric units. For a black holewith mass of the order of the solar mass, this gives a time of the orderof 10−5 s.
Remark VI.3.1 Because of the singularity for r = 0, the Finkelsteinblack hole cannot be considered as a generalized solution of the Einsteinequations, in contrast to the 1/r singularity in the potential of Newtonian
gravitation, which is solution of the Poisson equation with right-handside a delta measure mass located at the origin.6 The idea now is rather
to consider that, near this singularity, the gravitational ﬁeld is so strongthat it becomes a quantum ﬁeld or even a string ﬁeld, for which the
Einstein equations are no longer applicable.

VI.3.1

Eddington–Finkelstein white holeBy time reversal, one obtains manifestly another extension of the
Schwarzschild spacetime. The manifold is again S 2 × (r > 0) × R andthe metric is


2m
− 1−
dv 2 − 2 dr dv + r2 (sin2 θ dϕ2 + dθ2 ).
r
The extension to r < 2m now appears to observers in the Schwarzschildspacetime as a white hole: nothing can penetrate into it, but every past
inextendible light ray or timeline in the Schwarzschild spacetime emanates from this white hole, as can be seen from the radial null geodesics
and light cones in Fig. VI.3.

VI.3.2

Kruskal spacetime

It is possible to embed the Schwarzschild spacetime and both of its
extensions in a larger spacetime containing an additional copy of the

6 Remember also that for r < 2m,
the coordinate r is timelike. In a twodimensional, necessarily somewhat misleading, conformal diagram, r = 0 isrepresented by a spacelike line.

140Black holest´rr = 2mFig. VI.3 Radialnull
geodesics
and light cones of the Eddington–
Finkelstein white hole. Here t = v − r,where v is the ‘retarded’ time deﬁned
in (3.1).

Schwarzschild spacetime for which the previous black hole extension now
plays the role of a white hole and vice versa. The support of the obtained
spacetime is the manifold S 2 × R2 .
The metric of the Kruskal spacetime reads, in Kruskal coordinates,
32m3 −r/2m 2e(dz − dw2 ) + r2 (dθ2 + sin2 θ dφ2 ),rwhere θ, φ are coordinates on S 2 while z, w are coordinates on the open
set diﬀeomorphic to R2 deﬁned byz 2 − w2 > −1,
and r is the function of z and w deﬁned byz 2 − w2 =1(r − 2m)er/2m .
2m

In Kruskal coordinates, the radial light rays are represented by straightlines.
Exercise VI.3.2 Obtain the portion of the Kruskal spacetime isometric to the Schwarzschild spacetime by the change of coordinatesz = ev/4m + e−u/4m ,w = ev/4m − e−u/4m ,with
u = t − r − 2m log

 r
2m


−1 .

(VI.3.6)(VI.3.7)

Obtain other portions by analogous changes of coordinates.

7 Tachyon is the name given to a (so-far
unobserved) particle travelling fasterthan light.

The Kruskal spacetime has two asymptotically ﬂat regions (I and I in
Fig. VI.4(a)), each of which is isometric to the Schwarzschild spacetime.
A section through the Kruskal spacetime connecting these two regions,for instance w = 0, is called the Schwarzschild throat (Fig. VI.4(b)).
No non-tachyonic7 signal can travel from one of the asymptotically ﬂatregions to the other.
The Kruskal spacetime cannot be embedded in a larger spacetime.

VI.4
(a)

ω

141

r = constant < 2m

t=∞

r= 2m
(Singularity)
r= 0

r = constant > 2m

Stationary black holesI´

r = constant > 2m

I

t= 0

z

r = constant

t=∞

t= 0

r= 0
(Singularity)

r=2m

(b)

Fig. VI.4 (a) Kruskal(b) Schwarzschild throat.

VI.4

spacetime.

Stationary black holes

Most stars are not static8 with respect to their local inertial reference
frame, nor are they exactly spherically symmetric, so they cannot bemodelled by a Schwarzschild spacetime or, after their collapse, by a
Schwarzschild black hole. The simplest physical situations after static
are the stationary ones (see Chapter IV), where the spacetime is of thetype (M × R, g) with g invariant under translations along R.

8 In particular, their motion is not in-variant under time reversal.

9A

non-axisymmetric rotating body
would lose energy by gravitational radiation and hence cannot be stationary.

VI.4.1

Axisymmetric and stationary spacetimes

In General Relativity, physical considerations9 led to the conjecture10
that a stationary black hole is necessarily axisymmetric, a spacetime(V, g) being said to be axisymmetric if the underlying manifold is of
the type M × R, with Mt ≡ M × {t} spacelike and {x} × R timelike, andthe spacetime metric is such that Mt is diﬀeomorphic to the exterior ofa subset of R3 , gt admits a group S 1 of isometries that acts on R3 like10 The conjecture was formulated by
Hawking in the 1970s and proved byhim under an analyticity hypothesis
and some restrictive geometric assumptions (see Hawking and Ellis, 1973).
The analyticity hypothesis and someof the geometric assumptions haverecently been removed by Alexakis,
Ionescu, and Klainerman (2013).

142Black holesa rotation group with given origin Ot , and the lapse and shift of g arealso invariant under this group.
A spacetime is called stationary axisymmetric if, the origins Ot
being derived from each other by time translations, the two Killing ﬁelds∂/∂φ and ∂/∂t commute. Moreover, for such a spacetime, it is requiredthat the metric be invariant under simultaneous reversals of time t → −t
and angle of rotation x1 ≡ φ → −φ, as is consistent with physicalinterpretation. It holds then thatg02 = g03 = g12 = g13 = 0,
since the signs of these coeﬃcients change under such reversals. Using thefact that a two-dimensional metric can always be put in diagonal form,
we write the metric of an arbitrary stationary axisymmetric spacetime
as follows:g = g00 dt2 + 2g01 dt dφ + g11 dφ2 + g22 (dx2 )2 + g33 (dx3 )2 .

VI.5

The Kerr spacetime and black hole

In 1963, Roy Kerr found a stationary axisymmetric metric, an exact
solution of the Einstein vacuum equations, which was later interpreted
as a black hole and became very important in General Relativity (seethe following sections).

VI.5.1

Boyer–Lindquist coordinates

The Kerr metric, written in Boyer–Lindquist coordinates (found byBoyer and Lindquist in 1967), readsA
sin2 θ
2mr
dt dφ + dr2 + A dθ2− 1−
dt2 − 4mra
A
A
B

2 
2
2mra sin θ+ r2 + a2 +(VI.5.1)
sin2 θ dφ2 ,Awith a and m constants andA ≡ r2 + a2 cos2 θ,B ≡ r2 − 2mr + a2 .

(VI.5.2)

The coordinate r appearing in this metric can be considered as deﬁning
the manifold structure of the support of this metric by interpreting it
together with θ and φ as polar coordinates on the outside of the subset
of R3 such that A > 2mr and B > 0; t ∈ R is a time coordinate.
The Kerr metric (written here in Boyer–Lindquist coordinates) is axisymmetric (it does not depend on φ) and stationary but not static (itdoes not depend on t, but it is not invariant under time reversal). It
is invariant under simultaneous reversals of time t → −t and angle ofrotation φ → −φ, as required. The Kerr metric is interpreted as the

VI.5

The Kerr spacetime and black holegravitational ﬁeld outside a rotating black hole. It has not been foundto be the gravitational ﬁeld outside any realistic rotating star.
To interpret the constants a and m, we go to Cartesian-type coordinates
x2 + y 222
2
2.
(VI.5.3)r = x + y + z , θ = Arctanz
We then see that for large r, the Kerr metric tends to the Minkowski
metric and we have the following equivalence for its deviation from
Minkowski:




2m
2m
2
∼
gKerr = − 1 −dt + 1 +dr2 + r2 (dθ2 + sin2 θ dφ2 )rr(VI.5.4)
4ma(VI.5.5)− 3 (x dy − y dx) dt.
r
The ﬁrst line is equivalent for large r to the Schwarzschild metric: we
identify m with the mass of the system. The last term leads to theidentiﬁcation of a as the angular momentum per unit mass of the system.
The Kerr metric is invariant under the change (a, t) → (−a, −t), inagreement with this interpretation of a.

VI.5.2The Kerr–Schild spacetime

The Kerr metric in Boyer–Lindquist coordinates becomes singular on
various surfaces. As with the Schwarzschild metric, it is possible to transform it to another metric admissible in a larger domain. The following
coordinates adapted to light propagation and redeﬁnition of angularvariable adapted to the rotation were introduced by Kerr and Schild in
1965:
r2 + a2drB
adΦ = dφ + dr.
B
The Boyer–Lindquist metric then takes the Kerr–Schild form


2mr
Σ2
sin2 θ dΦ2
gKS ≡ − 1 −dv 2 + 2 dr dv + A dθ2 +AA
dv = dt ++ 4amrsin2 θ
dv dΦ + 2a sin2 θ dr dΦ,AwithΣ2 ≡ (r2 + a2 )2 − Ba2 sin2 θ.
This metric extends to a regular Lorentzian metric for all A > 0. It
reduces to the Minkowski metric in retarded time and polar coordinatesof R3 if m = a = 0, and it reduces to the Eddington–Finkelstein metricwhen a = 0.

143

144

Black holes

VI.5.3

Essential singularity

The Kerr metric becomes singular for A ≡ r2 + a2 cos2 θ = 0. This is agenuine singularity: the Kretschman scalar tends to inﬁnity when A
tends to zero.
In order to have the Kerr metric reducing to a form of the Minkowski
metric when m = 0 and a = 0, the coordinates are reinterpreted asfollows: θ and φ are coordinates on S 2 , but r = 0 is not a single pointof R3 ; it is assumed instead that the manifold R3 deﬁned by the Cartesian coordinates x, y, z is represented in the variables r, θ, φ through themapping (oblate polar ‘coordinates’)11

x = (r2 + a2 ) 2 sin Φ sin θ, y = (r2 + a2 ) 2 cos Φ sin θ,z = r cos θ.
The singularity A = 0 is then interpreted as the circle r = 0, θ = π/2,i.e.
z = 0,

VI.5.4x2 + y 2 = a2

Horizons

The case |a| > m
This would apply to very rapidly rotating bodies. We then have B > 0
for all r, no surface r = constant is a null surface, and the essential singularity is naked. This case is presently considered as physicallyunrealistic.
The case |a| < m
The Boyer–Lindquist metric then appears as singular for r = r+ or
r = r− , solutions of B = 0 given byr± = m ± m2 − a2 .
When a tends to zero, the Kerr metric tends to the Schwarzschild metric, r+ tends to 2m, and r− tends to zero. The surfaces r = r± are
not singular in the Kerr–Schild spacetime, but they are null surfaces—
the contravariant component g rr of the Kerr–Schild metric vanishes for
B = 0, as can be foreseen and checked by direct calculus.
The surface r = r+ is the event horizon: no particle entering the regionr < r+ can escape from it: the future light cone at points where r = r+points entirely towards the interior. The surface r = r− has no obvious
physical meaning.

VI.5.5

Limit of stationarity. The ergosphere

The variable r becomes a time variable when
2mr
> 1.
A

VI.5

The Kerr spacetime and black hole145

The metric is then no longer stationary. The surface r = rstat , with rstat
the largest root of A − 2mr = 0, is called the limit of stationarity. It
does not coincide with the horizon when a = 0 : it holds that
rstat = m + m2 − a2 cos2 θ ≥ r+ .
The domain between the limit of stationarity and the horizon r+ is called
the ergosphere.
The Penrose process is the extraction of energy from a Kerr blackhole by dropping a particle into the ergosphere, which will emit a particle
with a greater energy. The process can be described as follows. Consider
the Killing vector X ≡ ∂/∂v, with components (X α ) = (1, 0, 0, 0) in
Kerr–Schild coordinates. This vector is timelike outside the ergospherer > rstat ,2mrgαβ X α X β ≡ − 1 −< 0 when r > rStat ,A
and spacelike when r+ < r < rstat . A particle with timelike 4-momentum
p has an energy E = −pα Xα with respect to the vector ﬁeld X; thisscalar is constant along the trajectory (a geodesic) because of the lawof dynamics and the fact that X is a Killing vector. It holds that E > 0
outside the ergosphere. Suppose that the particle enters the ergosphereand splits there into two particles with momenta p1 and p2 . By theconservation of 4-momentum, it holds thatp = p1 + p 2 ,E = E1 + E 2 ,with Ei = −pαi Xα . Since X is spacelike inside the ergosphere, it
is possible to have the timelike vector p1 such that E1 < 0; hence
E2 = E − E1 > E. Both scalars E1 and E2 are conserved along the trajectories of the respective fragments. If the second fragment returns to
the outside of the ergosphere, it will be with a greater energy (with respect to the X observer) than the whole piece which left it. It can beconsidered that it has extracted energy from the rotating black hole.

VI.5.6

Extended Kerr spacetime

The Kerr–Schild spacetime that we have deﬁned is not complete. It is
possible to extend it (even to negative r), but the results11 are much
more complicated than in the Schwarzschild case and their physicalinterpretation very unclear.

VI.5.7

Absence of realistic interior solutions or modelsof collapse

Despite considerable eﬀort, no realistic source has been found to makean exterior domain of a Kerr metric a complete admissible Einsteinian
spacetime.

11 Cf. for instance, Wald (1984) andChandrasekhar (1983).

146Black holes

No construction of gravitational collapse analogous to those due to
Oppenheimer–Snyder, Gu–Hu, and Claudel–Newman in the sphericallysymmetric case has been made for axisymmetric gravitational collapse.

VI.6

Uniqueness theorems for stationary
black holes

We have seen in Chapter IV that there are no gravitational solitons
with support compact in space and sources with non-negative energy.
There is no theorem on the non-existence of spacetimes with compact
space outside the black hole region—but nor are there any examples ofsuch spacetimes. Unless otherwise speciﬁed, a black hole is considered
as an isolated object in spacetime; the black hole region in space is an
asymptotically Euclidean manifold with boundary its event horizon.

VI.6.1The Israel uniqueness theorem

The uniqueness of a static black hole in vacuum, the Schwarzschild blackhole, had been conjectured in the early 1950s by G. Darmois. It wasproved with simple and precise assumptions by W. Israel.

VI.6.2

Uniqueness of the Kerr black hole

The following uniqueness theorem was proved in 1975 by Robinson,
improving previous results of Carter from 1972.

12 Fordetails on the state ofthesehypotheses—inparticularon the horizon—see Chapter XIV,
Section 11 of YCB-OUP 2009 (contributed by P.Chruściel), Bray andChruściel(2004),
andAlexakis
et al. (2013).

13 This uniqueness was predicted in theearly 1960s by J. A. Wheeler and described by the picturesque statement‘Black holes have no hair’.
14 See again Chapter XIV, Section 11

by Chruściel in YCB-OUP 2009.

Theorem VI.6.1 Stationary, axisymmetric spacetimes that are solutions of the vacuum Einstein equations are uniquely speciﬁed by
two parameters—the mass m and the angular momentum a—if they
have a regular event horizon, are smooth outside the horizon, and are
asymptotically ﬂat.
Since the Kerr metric provides a solution of the vacuum Einstein equations satisfying the hypothesis of the theorem for any given m and a, theKerr metric represents the only such axisymmetric stationary black hole.
The Hawking–Klainerman theorem on the axisymmetric property ofa stationary black hole completes, under its hypotheses,12 the ‘nohair’ theorem13 for a stationary black hole in a vacuum spacetime ofdimension 4.
In higher dimensions, other stationary black holes have been constructed by Emparan and Reall.14

VI.6.3

Stability of the Kerr black hole

The intuitive deﬁnition of the stability of a dynamical system is thatits evolution does not change much under a change in its state at an
initial time. A stronger requirement if the system admits an asymptotic

VI.7

General black holes

147

state for given initial data is that the system admits the same asymptotic
state for data that are perturbations of these data. If the dynamics obeys
nonlinear diﬀerential equations
N (u) ≡ f (Dk u, . . . , Du, u) = 0,

(VI.6.1)

one is tempted to study the stability of a solution u, at least in a ﬁrst
step, to look for the evolution of a perturbation δu that satisﬁes thesystem linearized at u:∂N∂N(u)Dk δu + . . . +(u)δu = 0.
k∂(D u)∂u(VI.6.2)

If δu remains small for all times for small initial data, the solution u
is called linearization-stable. If δu tends to zero when the time tends
to inﬁnity, u is called asymptotically linearization-stable. Linearization
stability is often treated by physicists by considering perturbations given
by Fourier series. The linearization stability of the Schwarzschild spacetime was thus found by Zerilli and Wheeler in the early 1960s. Resultson the linearized stability of the Kerr spacetime have been obtained.15
Nonlinear stability is a much more diﬃcult problem, requiring global
existence of solutions of nonlinear diﬀerential equations. The nonlinear
stability of the Kerr spacetime is a subject of active research, inparticular by S. Klainerman and collaborators, with conjectures andpartial results.

VI.7

General black holes

An intuitive physical deﬁnition of a black hole is easy to give: a blackhole is a region of spacetime in which the gravitational ﬁeld is so strongthat no signal, in particular light, can get out. A black hole may reveal its
presence by the motion of surrounding stars that indicates the presenceof an enormous mass conﬁned in a region too small for any known matterto be contained in it. The presence of a black hole can also sometimes
be deduced from the very short and bright emission16 of γ rays (a γ-ray
burst) due to the explosion of an accretion ring formed around the blackhole by its tremendous gravitational pull. Black holes are the subjectof active observational research. The black holes interior is physicallyterra incognita, and very strange indeed. It is the subject of fascinating
advanced and varied mathematical studies and conjectures.

VI.7.1

15 See,inparticular,
Dafermos,Holzegel, and Rodnianski (2013).

Deﬁnitions

A mathematical deﬁnition of an isolated black hole given by Penrose
through a conformal diagram of an asymptotically Euclidean spacetime
is often used. A deﬁnition that does not appeal to conformal compactiﬁcation can be given for isolated black holes, or for a group of black
holes and material bodies that is isolated in the sense that it is so far
from other bodies that it may be considered to be embedded in an

16 Although it is rather thought now
that this emission is in general quasistationary.

148Black holes

asymptotically Euclidean spacetime (V, g). The following is a possible
deﬁnition:
Deﬁnition VI.7.1 The black hole region B of the spacetime (V, g)
is the complement of the past of the set covered by the null geodesics thathave an inﬁnite future canonical parameter.
The event horizon H is the boundary ∂B of the black hole region B
in the spacetime V. Under appropriate reasonable assumptions, the event
horizon H is a null hypersurface (C 0 but not necessarily C 1 ), generated
by null geodesics.

VI.7.217 See, for instance, Penrose (1979).

Weak cosmic censorship conjecture

The original idea of Penrose,17 coming from the study of spherical gravitational collapse where a black hole forms and hides the singularity fromtimelike observers, is the conjecture that generic Einsteinian spacetimes
with physically reasonable sources do not admit any naked singularity, that is, a singularity visible by an observer. With the deﬁnition of
singularity by incompleteness, trivial counterexamples can be obtainedby cutting out regions of any spacetime, for example Minkowski spacetime. Therefore, the mathematics must be more precise to grasp thephysical content. We give the following deﬁnition of a nakedly singularspacetime.
Deﬁnition VI.7.2 An inextendible spacetime (V, g) is said to be future
nakedly singular if it admits a future inextendible causal curve that lies
entirely in the past of some point x ∈ V .
We formulate the conjecture as follows:

18 Genericis usually taken to mean
without exceptional properties or stable under small perturbations.

Deﬁnition (No naked singularity conjecture). An inextendible,
generic,18 Einsteinian spacetime with physically reasonable sources admits no naked singularity.
Note that the big bang is not a counterexample to this conjecture—it
has no past, and hence does not correspond to any future inextendiblecausal curve.
A generic spacetime can be understood as a spacetime that is stable, insome sense to be deﬁned, under small perturbations. Reasonable sources
are physical sources that have a hyperbolic, causal evolution and do nothave their own singularities (shocks, shell crossings, etc.).
We will brieﬂy return to cosmic censorship in Chapter VIII.

VI.7.3

Thermodynamics of black holes

There are many theorems and conjectures about the thermodynamics
and quantum properties of black holes—diﬃcult to prove and even often to state in a precise way. Several properties have analogies with

VI.8

the fundamental laws of thermodynamics. They have led physicists toattribute to general black holes both entropy and temperature.
The ﬁrst black hole ‘thermodynamic’ property is the Hawking area
theorem, which says that the surface area A of the horizon H at agiven time cannot decrease with time: it led to the identiﬁcation of A,
up to a constant factor, with the entropy of the black hole.
The temperature of a black hole has been deﬁned through its surface
gravity,19 often denoted by κ (not to be mistaken with the gravitational
constant GN , which is also often denoted by κ). The usual deﬁnition of
the surface gravity does not apply to the horizon of a black hole. It
is deﬁned for stationary black holes by the formula, with k the Killing
vector tangent to the horizon,

Conclusions

149

19 For material bodies, the surface gravity in Newtonian theory is the gravitational acceleration at points of its
surface, about 9.81 m s−2 in the case of
the Earth.

k α ∇α k β = κk β .
This deﬁnition has been challenged by many authors and is still the subject of active discussion and possible generalizations to non-stationary
black holes.
Hawking radiation from black holes is a quantum eﬀect. It is wellknown that there are no absolute barriers in quantum theory. Hawking
introduced quantum waves to study radiation from black holes and provide a consistent deﬁnition of their entropy and its variation (i.e. thedecrease of their horizon area) with ‘black hole evaporation’. Such radiation could be due to splitting of virtual pairs of particles emerging
from the quantum vacuum outside the black hole, with one particle falling into the black hole and the other escaping from it. Hawking radiationis very weak and has not been observed.
These properties of black holes are fascinating and the subject of active
study and discussion, involving also quantum gravity, a largely open subject. They are at present beyond any possible physical conﬁrmation andmostly also any rigorous mathematical proof, although progress in theconstruction of a quantum theory of gravitation, in particular throughstring theory, opens new roads to the modelling of black hole interiors.
An exception, as far as mathematical proofs are concerned, is the proof
through sophisticated mathematics by Huisken and Ilmanen and independently by Bray20 of the Riemannian Penrose inequality, which
admits a purely geometric formulation. It is linked with several theorems
or conjectures due to Penrose (see Problem VI.10.5).

VI.8

Conclusions

VI.8.1

Observations

The conjectured existence of the ﬁrst black hole, Cygnus X1 in 1972,
was deduced from the study of a star’s orbital parameters, the values ofwhich implied the existence of an invisible companion too massive to bethought to be even a neutron star. An enormous number of candidate

20 See, for instance, Bray and Chruściel(2004).

150

Black holes

black holes have now been detected. Astronomers usually classify black
holes as follows.
Stellar-Mass black holes with masses between 10 and 24 solar masses
reveal their presence by the motion of companion stars due to theexistence of a strong gravitational ﬁeld that cannot be explained otherwise. Another indication of the presence of a black hole is radiation
emitted from matter falling into the black hole under its tremendous
gravitational pull.
Supermassive black holes, millions (or billions) times more massive
than the Sun, are thought to be in the centres of many galaxies, including
our own Milky Way. They are detected by the motion of nearby stars
and gas.
Recently, evidence for intermediate-mass black holes has been obtained by powerful new telescopes, such as Chandra, XMM-Newton,
and Hubble.
There is an abundant literature on the subject in relevant scientiﬁcjournals and on the Internet.

VI.8.221 Although

not so short for a verylarge mass M , since it is estimatedas 10−5 s × M/MSun ∼= 103 s for
M = 108 MSun .

The interiors of black holes

We have seen that the representation of the interior of a Schwarzschild
black hole by a Lorentzian manifold ceases to be meaningful at the singularity r = 0, attained in a very short time21 by any infalling object.
The representation of the singularity in the Kruskal diagram by a spacelike submanifold is misleading. The interior of the Kerr black hole andits singularity are very strange. They cannot be considered as Einsteinian spacetimes in any classical sense. The general belief is that a newtheory of gravitation must be constructed to represent them. This is asubject of very active research, linked with the old problem of the quantization of the gravitational ﬁeld. Previous research in this diﬃcult ﬁeld
has known dramatic transformations through the introduction of new
and fascinating ideas, in particular string theory. However, further discussions of black holes and quantum gravity are outside the scope of this
book.

VI.9

Solution of Exercise VI.3.1(1) With t now denoted by ρ and r by τ , the Schwarzschild metric reads

−

−1


2m
2m
−1
− 1 dρ2 + τ 2 (dθ2 + sin2 θ dφ2 ),dτ 2 +
ττIt has Lorentzian signature for 0 < τ < 2m, with τ a timelike coordinate. For any such given τ , the space metric is deﬁned on the
product R × S 2 ; it is degenerate at τ = 2m and singular for τ = 0.

VI.10(2) The deﬁnitionv = t + r + 2m log

 r
2m


−1 ,

(VI.9.1)i.e.
dv = dt + dr +

2m dr
r dr
= dt +,
r − 2mr − 2mimplies by a simple computation the announced result:




−1

2m
2m
2m
2
2
− 1−
dr = 1 −dt + 1 −(dv 2 + 2 dv dr).
rrr(3) The metric is Lorentzian because
2 dv dr =
1
2
2(dv + dr) − (dv − dr) .
2

VI.10

Problems

VI.10.1

Lemaı̂tre coordinates

As was done by Lemaı̂tre in 1933, consider coordinates τ, ρ, θ, φ given interms of the standard Schwarzschild coordinates t, r, θ, φ on R4 by thefunctions""√" 2m − √r "√""
τ := t + 2 2mr + 2m log " √√ "," 2m̄ + r "2
ρ − τ := r3/2 √ .
3 2m
1. Show that the coordinates change is singular for r = 2m.
2. Show that in the Lemaı̂tre coordinates, the Schwarzschild metric takes
a form that is singular only for r = 0, but non-static:−dτ 2 +

2m 2
dρ + r2 (dθ2 + sin2 θ dφ2 ).
r

3. Determine the radial geodesics.

Solution
1. The logarithm tends to −∞ when r tends to 2m.
2. This is a straightforward computation.
3. The radial geodesics are lines where only τ varies.

Problems

151

152

Black holes

VI.10.2

Reissner–Nordström black hole

Discuss for which values of the constants m and Q the Reissner–
Nordström metric (see Chapter V)



−1
2m Q2
2m Q2
2
+ 2 dt + 1 −+ 2
dr2
gRN := − 1 −rrrr(VI.10.1)+ r2 (dθ2 + sin2 θ dϕ2 ),with electromagnetic potential Ai = 0, A0 = −Q/r describes a blackhole and study its event horizon.

Solution
The metric gRN is a smooth Lorentzian metric for all r = 0 if
r2 − 2mr + Q2 > 0,that is, m2 < Q2 , since the roots of this polynomial in r,
r± = m ± m2 − Q2 ,(VI.10.2)

are then imaginary. For Q2 < m2 , the metric becomes singular in thestandard coordinates on the two manifolds r = r+ and r = r− . The
manifold r = r+ is the event horizon.

VI.10.3

Kerr–Newman metric

The Kerr–Newman metric reads


A
2mr
sin2 θ
dt dφ +
dr2
gKN = − 1 −dt2 − 4mraA
A
B + Q2
2mra2 sin2 θ222+ A dθ + r + a +(VI.1.2)
sin2 θ dφ2 ,Awith a and m constants andA ≡ r2 + a2 cos2 θ,B ≡ r2 − 2mr + a2 .

(VI.1.3)

Check that gKN coincides with the Kerr metric in Boyer–Lindquist
coordinates if Q = 0, and with the Reissner–Nordström metric if a = 0.
Show that gKN satisﬁes the Einstein equations with electromagnetic
source.
22 Christodoulou and Ruﬃni had shownin 1971 (see Ohanian and Ruﬃni, 2013)that the irreducible mass increases in
any Penrose problem, proving thus in aparticular case the Hawking area theorem. For the general case, see Hawking
and Ellis (1973).

VI.10.4

Irreducible mass (Christodoulou–Ruﬃni)

Assume the Hawking area theorem,22 which says that no physical process
can make the area of the event horizon of a black hole decrease. Show
that the energy loss in a Penrose process in Kerr spacetime is at best50%.

VI.10

Solution
The radius of the event horizon of a Kerr spacetime with parameters mand a isr+ = m + m 1 − |J|2 ,where J denotes the spin of the black hole.
The event horizon of the static spacetime resulting from an optimal(i.e. absorbing all the rotational energy) Christodoulou–Ruﬃni–Penrose
process with mass m is r = 2m . The Hawking area theorem implies
m ≥


1
m(1 + 1 − |J|2 ).
2

The gain in mass is⎛⎞1
1⎠ < 1 m.
m − m ≤ m ⎝ 1 − ! 2
2
1 − |J|2
It tends to m/2 when |J| tends to its maximum value 1.

VI.10.5

The Riemannian Penrose inequality

We have given in Chapter V the Schwarzschild metric in isotropic
coordinates,gSchw = −2R − m2R + m2

m 4
dt2 + 1 +(dX 2 + dY 2 + dZ 2 ) (VI.10.3)
2R

and have shown in Chapter IV that it has ADM mass mADM = m.
Prove that

A
,
m=
16π
where A is the area of the horizon.

Solution
The horizon, deduced from the expression for the spacetime metric, is
R=

m
.
2

Straightforward classical calculus gives the area of this 2-sphere:

m2 4
A=
2 4π = 16πm2 .
24 dX dY dZ =4
Bm/2

Problems

153

154

Black holes

23 For details, see Bray and Chruściel(2004) and references therein.

Comment. The above equalities and the Beckenstein–Hawking area
theorem led Penrose to conjecture that general black holes should satisfy
an inequality of the form

A
.
m≤
16π
This inequality considering only the space manifold, then called the ‘Riemannian Penrose inequality’, has been proved by Huisken and Ilmanen
under the hypotheses that the space manifold is a Riemannian asymptotically Euclidean manifold with non-negative scalar curvature that admits
a compact boundary composed of minimal surfaces but no other minimal surface.23 The considered area A is the area of any disconnected
such surface. The relevance to physics is that such boundary surfaces are‘trapped surfaces’ conjectured to be linked with black holes containing
a singularity of the corresponding spacetime.

Introduction to cosmology
VII.1

Introduction

A cosmological model is a spacetime that is supposed to represent thewhole past, present, and future of our universe. It has long been argued
that in the framework of General Relativity, gravitation should model
the geometry of the cosmos because its forces are long range (in contrast
to nuclear forces) and non-compensated (there are no negative masses—
in contrast to the existence of positive and negative electric charges).
Since Einstein’s formulation of his equations, it was assumed by most
cosmologists that the cosmos in which we live is a Lorentzian 4-manifold(V, g) that satisﬁes the Einstein equations with the non-gravitationalenergies as sources. In the beginning, cosmological models relied essentially on a priori conjectures based on philosophical prejudices, the aimof simplicity, and the desire for uniﬁcation.
Because of considerable progress in astronomical observations by both
Earth-based telescopes and satellites,1 cosmology has become a fully
ﬂedged part of physics. There is a wealth of information that has been
accumulated in recent years from observations of the cosmos and thatneeds to be analysed. There have been remarkable advances in our knowledge, but also new and puzzling questions. We cannot in this book enter
into a detailed exposition of the amazing number of observations madesince the beginning of this century, and we can only give some indicationof how remarkably precise conclusions, often convincing but sometimes
speculative, have been deduced from these observations.
We will in this chapter give the main facts that led a majority of cosmologists (although there are still dissidents) to believe that our cosmos
is represented by a four-dimensional Robertson–Walker spacetime (seeSection VII.4) solution of the Einstein equations with source all the
non-gravitational energies. Radiation energy is today negligible with respect to matter energy, but is thought to have been dominant at earlystages. The matter source was considered until fairly recently as a perfectﬂuid, whose particles would be galaxies or clusters of galaxies. Observations made at various scales have now shown that the usual mattersources represent only a quite small fraction, an estimated 4.5%, of theenergy content of the universe.2 Modern cosmologists have introduced
new types of sources: dark matter (about 25%) and dark energy (about70%). There are numerous physical conjectures regarding the nature ofthese sources. A possible interpretation of dark matter is the existenceof WIMPS (weakly interacting massive particles), particles that interact

VII
VII.1 Introduction
155
VII.2 The ﬁrst cosmologicalmodels
156
VII.3 Cosmologicalprinciple
157
VII.4 Robertson–Walker
spacetimes
158
VII.5 General propertiesof Robertson–Walker
spacetimes
161
VII.6 Friedmann–Lemaı̂tre
universes
164
VII.7 Primordial
cosmology
VII.8 Solution of
Exercises VII.6.1
and VII.6.2
VII.9 Problems

167

168
169

1 Such as COBE (Cosmic Background

Explorer), and WMAP (Wilkinson
Microwave Anisotropy Probe), and theinformation gathered by the Plancksatellite (analysed and published in
March 2013).

2 Dynamical properties of celestial bod-ies, for instance the ﬂat aspect of somegalaxies, cannot be explained by their‘normal’ mass content.

156

Introduction to cosmology

3 Though

not quite by all of them—
there are still adherents of the steadystate and continuous-creation theory ofGold, Hoyle, and Narlikar and also of
I. E. Segal’s theory of chronogeometry(Segal, 1976).
4 See, for instance, for results and references prior to 2002, Cotsakis andPapantonopoulos (2002) and for references prior to 2010, Cheng (2010). Most
recent observations can be found in
papers on the arXiv (gr-qc or astro-ph).

only through the weak ﬁeld and gravitation. Dark energy is often interpreted as an energy of the vacuum, represented by a cosmologicalconstant, or possibly a scalar ﬁeld, called quintessence.
In Section VII.7, we give a brief description of what is currently
thought by most cosmologists3 to be the birth and infancy of the cosmos
in which we live.4

VII.2

The ﬁrst cosmological models

Soon after the Einstein equations were ﬁrst framed, models were
proposed for the whole cosmos.

VII.2.1

5 Olbers pointed out in 1823 that if
space was inﬁnite and ﬁlled with ahomogeneous distribution of stars, thesky would appear uniformly bright—
in clear contradiction with the darknight sky. Historians of astronomy havefound that Olbers was not the ﬁrst to
make this remark, but it is his namethat is still attached to it.

Einstein static universe

Historically, the ﬁrst cosmological model was found in 1917 by Einstein himself, looking for a solution that was homogeneous and eternal,
would solve Olbers’ paradox,5 and would be in agreement with Mach’s
principle. He was thus led to the Einstein static universe (see Problem IV.11.1) with closed space manifold S 3 , and spacetime manifold
S 3 × R endowed with the static metric− dt2 + a20 γ+ ,(VII.2.1)

with a0 a constant and γ+ the metric of the unit 3-sphere S 3 , that is, inthe usual angular coordinates,γ+ := dθ2 + sin2 θ (dφ2 + sin2 φ (dψ 2 )Setting sin θ = r gives the form familiar to geometers:
dr2+ r2 (dφ2 + sin2 φ dψ 2 ).
(VII.2.2)
1 − r2
We have seen that the Einstein static universe is a solution of the Einstein equations with static perfect ﬂuid source of constant density andpressure given by
γ+ =

μ0 =
6 Another obstacle to the acceptance of

the Einstein static universe as a modelfor our cosmos is that it is unstable.
7 An idea Einstein did not like, because
it introduces a new parameter, a priori
not geometrically deﬁned.

8 Theseproperties have been used by
Penrose to give a geometric deﬁnitionof asymptotic ﬂatness; see, for instance,
Hawking and Ellis (1973) or Section 6
in Appendix VI of YCB-OUP2009.

3 −2
a , p0 = −a−2
0 < 0.
2 0(VII.2.3)

A negative pressure is unacceptable on classical physical grounds.6
This diﬃculty was remedied by Einstein by introducing a positive
cosmological constant.7 The formulas (2.3) are then replaced by
μ0 =

3 −2
a − Λ,
2 0

p0 = −a−2
0 + Λ;(VII.2.4)

Einstein choose Λ = a−20 so that p0 = 0.
The Einstein static spacetime, also called the Einstein cylinder, is(but with a diﬀerent interpretation) the arena of the Segal cosmos.
The manifold S 3 × R is the universal cover of compactiﬁed Minkowski
spacetime; it is conformally locally ﬂat,8 and the conformal group acts

VII.3on this geometric, non-metric, structure. The fact that representationsof the conformal group govern the world of elementary particles seemedto I. E. Segal a justiﬁcation of his cosmological model. He explains thecosmological redshift by the existence of a local time diﬀerent from thecosmic time.
A few years after introducing the cosmological constant (in order toobtain a physically realistic time-independent cosmos), Einstein abandoned it (calling it the greatest blunder of his life) and accepted
time-dependent models, motivated by the observation by astronomersof the redshift of light coming from distant stars, interpreted as being
due to the expansion of the universe.9

Cosmological principle

157

9 Recent observations indicate that thecosmological expansion is accelerating.
This has led cosmologists to reintroduce a cosmological term.

VII.2.2

de Sitter spacetime

The de Sitter spacetime (see Problem IV.11.2) was constructed in 1917
by Willem de Sitter. It is a solution of the four-dimensional Einsteinequations with positive cosmological constant. It is not realistic, because
it is a vacuum solution, but was the ﬁrst example of a time-dependent,
expanding, Einsteinian spacetime.
Einstein and de Sitter had long discussions about their respectivemodels. These discussions, and the discovery of the expansion of theuniverse, led Einstein to abandon the cosmological constant, but accept
time-dependent spacetimes.

VII.2.3

General models

In the classical models, still basic for cosmological studies, the cosmos
is modelled by a four-dimensional Einsteinian spacetime. Nowadays, allcosmologists consider, as it is experienced by everybody in daily life,that there is a ﬂow of time—that is, a past and a future. They speak of
the universe as the cosmos at a given time. Note that we do not observe
the present state of the cosmos, since information comes to us at mostwith the speed of light. Most cosmologists now accept that the universeis expanding—indeed that the expansion is accelerating, according torecent observational data.

VII.3

Cosmological principle

VII.3.1

Assumptions

The standard cosmological models are based on the so called
cosmological principle, which is composed of two assumptions.
The ﬁrst assumption is that the cosmos is a manifold M × R endowedwith a Lorentzian metric, four-dimensional in the usual case, such thatthe lines x × R, x ∈ M, called trajectories of the ‘fundamental observers’, are timelike geodesics orthogonal to the manifold M × {t} at each

158

Introduction to cosmology

point (x, t) ∈ M × R. The Lorentzian metric of the cosmos can therefore
be written as(4)

g := −dt2 + (3) g.

The Riemannian manifold (M,(3) g) is the universe at time t. The proper
time of fundamental observers is called cosmic time.
The second assumption of the cosmological principle is that the universe at each time should look the same in all directions and also to
any fundamental observer. This assumption is called the Copernican
principle—in the name of Copernicus, who deprived us of a central position in the Solar System. We also know that the Sun does not occupy aremarkable place in our galaxy. The mathematical content of the second
assumption, modulo the ﬁrst one, is as follows:

10 That is, for each pair of points x andy in Mt , there exists an isometry of(M, g) that brings x to y.

(1) Isotropy: the Riemann tensor of the space metric (3) g is at eachpoint x invariant under rotation in the tangent space to M centred
at x.
(2) The space metric is homogeneous, i.e. it admits a transitive10
group of (global) isometries.

VII.3.2

11 First found by Penzias and Wilson in

1964, but predicted earlier by Gamovfrom the big bang theory.
12 A black body is one that absorbs allradiation. Contrary to its name, a black
body emits radiation when heated, but
the graph of the emitted energy interms of wavelength (the black body
spectrum) is a roughly Gaussian-typecurve with a peak depending on thetemperature.
13 Planck satellite in 2013.

14 Thereis no real diﬃculty in extending the deﬁnition to n + 1 dimensions, although the space Riemanntensor is then no longer equivalent
to the Ricci tensor. Note also thatthe topological classiﬁcation of threedimensional manifolds does not extend
to higher dimensions.

Observational support

We see the stars very unevenly distributed in the night sky. Galaxies,
and even clusters of galaxies, are also observed in our telescopes as being
very anisotropically and inhomogeneously distributed. An argument infavour of adopting the cosmological principle is that, at a still larger
scale, isotropy and homogeneity seem to be attained.
For most astrophysicists, the strongest evidence for the validity of thecosmological principle is the isotropy of the CMB (cosmic microwave
background) radiation. This is the faint background glow11 that sensitive radio telescopes detect in the sky, which otherwise appears dark
between stars and galaxies to traditional optical telescopes. The CMB
radiation is measured to be very nearly isotropic, with a temperature ofabout 2.725 K and a black body12 spectrum. An observed discrepancy
with isotropy of order 10−3 is interpreted as being due to our own motion relative to fundamental observers. There are further anisotropies ofthe order of 10−5 : the most recently observed13 anisotropic structures atthis scale seem to be best explained by random inhomogeneities in the
big bang.
An argument in favour of homogeneity is that the fundamental physical constants seem to be and have been the same throughout the cosmos
with a remarkable accuracy.

VII.4

Robertson–Walker spacetimes

The Robertson–Walker spacetimes are (3 + 1)-dimensional14 models
satisfying the assumptions of the cosmological principle; that is, theyVII.4are represented by metrics that read, on a product M × R with M athree-dimensional manifold,(4)g ≡ −dt2 + (3) g,(VII.4.1)

where (3) g is a t-dependent Riemannian metric on M whose Riemanniancurvature is isotropic at each point of M. We will see that this implies
that the metric (3) g is homogeneous.

VII.4.1

Robertson–Walker universes, metric at given t

The curvature tensor of a Riemannian metric g at a point x ∈ M isisotropic at x, that is, invariant under rotations in the tangent space
to M at x, if and only if it is of the form
Rij,hk (x) = K(x)(gih gjk − gjh gik )(x).

(VII.4.2)

Exercise VII.4.1 Prove the ‘if ’ part of this property.
Since in dimension 3 the Riemann tensor is equivalent to the Ricci
tensor (4.2) can be replaced in that case by (recall that Sij ≡ Rij − 12 gij R
denotes the Einstein tensor)(3)

Rij ≡ 2K (3) gij ,

hence

(3)

Sij ≡ −K (3) gij .

(VII.4.3)

The contracted Bianchi identity implies
(3)

(3)

∇j

S ij ≡ 0,i.e. ∂i K = 0.

Hence, K is a constant; the isotropic metric (3) g is also homogeneous.
Riemannian spaces with curvature of the form (4.2) with K a constant
are called spaces of constant curvature.
To determine the general Riemannian spacetimes (M,(3) g) of constantcurvature, it is convenient to use, in a neighbourhood of an arbitrarily chosen point, polar pseudo-coordinates centred at that point, inwhich the spherical symmetry resulting from the cosmological principle
is manifest. In these coordinates, the metric (see Chapter V) takes the
form
(3)

g ≡ eμ dr2 + r2 (dθ2 + sin2 θ dφ2 ),withμ = μ(r).

(VII.4.4)

Equations (4.2) then read (the prime denotes the derivative with respect
to r)
(3)

Rik ≡ 0, i = k,(3)

R11 ≡ r−1 μ = 2Keμ ,1 (3)r
R33 ≡ (3) R22 ≡ −e−μ + 1 + e−μ μ = 2Kr2 .
2
2
sin θ

(VII.4.4)(VII.4.5)

The general solution of (4.4) is trivially found to be e−μ−Kr2 +constant, and (4.5) then gives
e−μ = 1 − Kr2 .

=(VII.4.6)

Robertson–Walker spacetimes

159

160

Introduction to cosmology

We already know that a metric of constant curvature K = 0 is locallyﬂat.
A metric with constant curvature K > 0 [respectively K < 0] is lo1
cally a 3-sphere of radius K 2 [respectively locally a hyperbolic 3-space
1of radius |K|− 2 ]. In these cases, one classically scales r by setting
1
r = |K|− 2 r̄ and relabels r̄ as r. In the new coordinate r, the metrictakes one or other of the standard forms according to the sign of K:(3)g ≡ |K|−1 γε , γε ≡dr2
+ r2 (sin2 θ dφ2 + dθ2 ),1 − εr2

ε = sign K.

(VII.4.7)
The metric γε can be transformed into the familiar forms of the unit
sphere or pseudosphere metrics as follows:when ε = 0, r ≡ χ, we have
γ0 = dχ2 + χ2 (sin2 θ dφ2 + dθ2 );(VII.4.8)

when ε = 1, r ≡ sin χ, we haveγ+ = dχ2 + sin2 χ (sin2 θ dφ2 + dθ2 );(VII.4.9)

when ε = −1, r = sinh χ, we haveγ− = dχ2 + sinh2 χ (sin2 θ dφ2 + dθ2 ).

15 In an old terminology, universes sup-ported by compact manifolds werecalled ‘closed’ (recall that, unless otherwise speciﬁed, manifolds are withoutboundary). Those with non-compact
support were called ‘open’.

(VII.4.10)

The isotropic homogeneous simply connected Riemannian manifold
corresponding to ε = 1 with 0 ≤ χ < π and θ, φ angular coordinates on
S 2 , is the sphere S 3 , a compact Riemannian manifold.15
The metrics with ε = 0 or ε = −1 can both be supported by the noncompact manifold R3 , with coordinate 0 ≤ χ < ∞. The corresponding
isotropic, homogeneous, simply connected Riemannian manifolds are theEuclidean space E 3 or the hyperbolic 3-space H3 .
We have computed Robertson–Walker spaces supported by simply
connected manifolds by looking for globally spherically symmetric metrics around one point. There are other metrics that are isotropic andhomogeneous but not globally spherically symmetric, which can be obtained as quotients of the previous metrics by an isometry group. They
are supported by manifolds with diﬀerent topologies. An example in thecase ε = 0 is the ﬂat 3-torus.

VII.4.2

Robertson–Walker cosmologies

From Section VII.4.1, there are three types of Robertson–Walker spacetime metrics:− dt2 + a2 (t)γε ,1

ε = 0, a(t) is equal to |K|− 2 ,K being the constant-in-space, timedependent, scalar Riemann curvatureof (3) g.

16 Whenε = 1, −1, or 0,(VII.4.11)

where a is an arbitrary function16 of t. We see that a(t) is a scaling factor of local spatial distances of fundamental observers. In an expanding
universe, (i.e. if ȧ := da/dt > 0), this distance increases proportionallyto a and to the original distance, as in an inﬂated balloon. Indeed, on

VII.5General properties of Robertson–Walker spacetimes

the trajectory of a fundamental observer, only t varies; the space distance at time t of two fundamental observers moving respectively on thetimelines x = x0 and x = x1 is the product by a(t) of the distance inthe metric γε between these two points of M.

VII.5

General propertiesof Robertson–Walker spacetimes

VII.5.1

Cosmological redshift

The ﬁrst essential cosmological data came from the observation of theredshifts of stars and galaxies. The redshift parameter is deﬁned to be
z≡

νS
− 1,
ν0

(VII.5.1)

where ν0 is the observed frequency and νS the emitted frequency. Thereis a shift towards the red [respectively towards the blue] if z > 0[respectively z < 0]. We now prove the following proposition.
Proposition VII.5.1 In an expanding Robertson–Walker spacetime,
there is a cosmological redshift with parameter given approximately by
1+z ≡a(t0 )νS,=˜
ν0 a(tS )(VII.5.2)

if we assume that the variation of a is negligible during a period of theemitted and a period of the received light signal.
Proof. Recall that a Robertson–Walker metric is of the form
− dt2 + a2 (t) dχ2 + f 2 (χ)(dθ2 + sin2 θ dφ2 ) .
(VII.5.3)
Let O0 and OS be two fundamental observers. Take coordinates suchthat they have the same θ and φ and have χ coordinates 0 and χS ,respectively. An observer O with space coordinate χO = 0 (this is norestriction) and t = tO receives at time tO light emitted at time tS < tO
by a source S with space coordinate χ(tS ) = χS > 0 if it is on the light
ray joining these two spacetime points. This light ray is the solution of
the diﬀerential equationdt= a(t)dχ(VII.5.4)

given by
 tO
χS =

tS

dt
.
a(t)

(VII.5.5a)

Photons emitted by OS situated at χs at times tS and tS + TS follow
null rays and arrive at O0 situated at χO = 0 at times t0 and t0 + T0

161

162

Introduction to cosmologythat satisfy
 tO
tS

dt
=
a(t)

 tO +TO
tS +TSdt.
a(t)(VII.5.6)

Elementary calculus shows that the times TS and T0 are therefore relatedby tO +TO
 tS +TSdt
dt=.
(VII.5.7)
a(t)
a(t)
tS
tO
Hence, if a is considered as constant during the small amounts of time
TS (period of the emission) and TO (period of the reception), we obtainTSa(tS ).
=TOa(tO )(VII.5.8)

Since the frequency is the inverse of the period, this result leads toνSa(t0 ),=
ν0a(tS )



which is the desired relation (5.2).

17 A statisticaltreatment is necessary
because an individual star, even an individual galaxy or cluster of galaxies,
cannot be considered as a fundamental
observer.

(VII.5.9)

A positive cosmological redshift, ν0 < νS signals an expansion of theuniverse, a(t0 ) > a(tS ).
Statistical observations17 of distant galaxies conﬁrm, if we analysethem in the framework of a Robertson–Walker spacetime, that our universe is at present expanding, a (t0 ) > 0. Over the last few years, it has
been observed that the expansion is accelerating: a (t0 ) > 0.

VII.5.2

The Hubble law

In a Robertson–Walker spacetime, the distance between two given fundamental observers at some cosmic time t is proportional to a(t). The
Hubble parameter is deﬁned to be the constant-in-space, t-dependentscalarH ≡ a−1 a .

18 This again is a statistical statementwith regard to distances of fundamental observers emitting and receiving
signals.

(VII.5.10)

It measures the rate of expansion (or possibly contraction) of theuniverse. It has dimension (time)−1 . An expanding Robertson–Walker
universe has a positive Hubble parameter.
The Hubble law says that the observed redshift is proportional to the
distance of the source.18 It is true as consequence of the theory only inﬁrst approximation for not too distant sources, as we now show.
Approximate computation of the redshift gives
1 + z :=z :=a(t0 )νS,=˜
ν0 a(tS )νSa (tO )(tO − tS )a(tO )
− 1=
˜
.
−1=
ν0
a(tS )a(tS )(VII.5.11)

VII.5

Recall that

 tO
χS =

tS

dt
=
˜ (t0 − tS )a−1 (tS ),a(t)General properties of Robertson–Walker spacetimes

163(VII.5.12a)and hence, with dS = a(t0 )χS ,z=
˜ H(tO )a(t0 )(t0 − tS ) =˜ H(t0 )dS .
a(tS )For a given observer at a given time, the redshift is, in rough approximation, proportional to the distance of the source.
The Hubble law has long been used by astronomers to estimate distances from the Earth of various astronomical objects that are too far
away for these distances to be measured by triangulation. It is then notlogical to use these redshifts to estimate distances. Fortunately, distances
can also be estimated by comparing the luminosity of a star as observed
from Earth (the ‘apparent luminosity’) with the emitted luminosity (the‘absolute luminosity’) as calculated from the physical nature of the star.
In the past, cepheid variables (whose absolute luminosity is related totheir period of variation) were used as such ‘standard candles’, but it was
discovered that there are two kinds of cepheids. More reliable ‘standard
candles’ have been found by astronomers, particular so-called type Ia
supernovae.19

VII.5.3

Deceleration parameterA second approximation of the relation between redshift and distanceis obtained by introducing the deceleration parameter linked withthe second derivative of the scale factor a. This is a dimensionless scalar that measures the rate of variation with time of the expansion (orcontraction), and is deﬁned by
q=−

aa
.
a2

(VII.5.13)

An expanding accelerated universe has a negative deceleration parameter.
Taylor formula gives


1
2
2
a(tS ) = a(t0 ) 1 + (tS − t0 )H(t0 ) − qH (tS − t0 ) + . . . (VII.5.14)
2
and
−1

χS = aH2(t0 ) t0 − tS + (tO − tS ) + . . . ,2(VII.5.15)

which gives the next approximation,1z = H(t0 )dS + (q + 1)H(t0 )d2S + . . . .
2(VII.5.16)

19 Supernovae of type Ia have weakhydrogen lines and a strong Si line at
6150 Å.

164

Introduction to cosmology

VII.5.4
20 Although

there were, and still are,some cosmologists who believe in an
eternal universe.

Age and future of the universe

It is generally believed that the universe started with a big bang,20
the structure of which is a subject of active debate (see Section VII.7).
We denote by t = 0 the time at which it is believed to have started
expanding as a Robertson–Walker universe. We set a(0) = a0 . We denote
by t0 the present time, that is, the age of the universe. By deﬁnition,a(t0 ) > 0; also, a (t0 ) > 0, since we observe (statistically) redshifts, notblueshifts—the universe is presently expanding.
Until near the end of the twentieth centuries, it was thought for classical physical reasons that the expansion of the universe has been, is,and will be always slowing down—that is, a (t) < 0 for all t > 0, andthe curve is concave-downwards. The universe would end in a big crunch
at a time t = T < ∞. In fact, more recent observations have led to thebelief that the expansion of the universe is now accelerating a (t0 ) > 0.
Estimates of t0 and the future of the universe depend on the properties ofthe function a(t), that is, of the qualitative and quantitative properties
of sources.

VII.6

Friedmann–Lemaı̂tre universes

VII.6.1

Equations

A Friedmann–Lemaı̂tre cosmos is a Robertson–Walker spacetime whose
metric satisﬁes the equations
Sαβ = −Λgαβ + Tαβ ,1with Sαβ ≡ Rαβ − gαβ R,
2(VII.6.1)

with Tαβ such that
T00 := μ,Tij = pa2 γij ,Ti0 = 0.

The non-zero Christoﬀel symbols of the spacetime metric (4.11),−dt2 + a2 γε , are computed to be
Γ0ij ≡ aa γij ,Γj0i ≡ a−1 a δij ,i
Γijh = γjh, witha :=da, (VII.6.2)
dt

iwhere γjh
are the Christoﬀel symbols of the relevant metric γε .
The non-zero components of the Ricci tensor of the spacetime metricare, using the value rij = 2εγij for the Ricci tensor of the spatial metricγε ,R00 ≡ −3a−1 a ,
Rij ≡ 2ε + aa + 2a2 γija :=

d2 a(VII.6.3)
dt2

The scalar curvature isR ≡ −R00 + a−2 γ ij Rij ≡ 3a−2 2ε + 2aa + 2a2 .

VII.6Friedmann–Lemaı̂tre universes

165

Hence the component S00 of the Einstein tensor reads as a ﬁrst-order
diﬀerential operator on a:1
S00 ≡ R00 − g00 R ≡ 3a−2 ε + a2
2
The corresponding Einstein equation21 gives μ in terms of a, a , and Λ;
it is called the Friedmann equation:
(VII.6.4)
μ = S00 − Λ ≡ 3a−2 ε + a2 − Λ.
The equation
γ ij Sij ≡ γ ij Tij − 3Λ ≡ 3pa2
gives p in terms of a, a , and a , namelyp = −a−2 ε + a2 + 2aa + Λ.

(VII.6.5)

Exercise VII.6.1 Prove (6.4) and (6.5).
Exercise VII.6.2 Using the formulas obtained for μ and p, prove thatthe stress–energy tensor satisﬁes the conservation laws, which read, sinceΛ is a constant,∇α T αβ = 0.

VII.6.2

Density parameter

In discussing cosmological observations, it is useful to replace the density function μ by a dimensionless quantity Ω, called the ‘density
parameter’, which is deﬁned by
Ω≡

μ+Λ
,
3H 2

(VII.6.6)

where H := a−1 a is the Hubble parameter. Note that Ω depends only
on cosmic time. The Friedmann equation (6.4) reads(VII.6.7)
3 a−2 ε + H 2 = μ + Λ,that is,Ω≡1+ε.
a2 H 2(VII.6.8)

If our universe is modelled by a Robertson–Walker cosmology, thenthe value of Ω determines the type of Robertson–Walker spacetime inwhich we live: Ω > 1 implies that ε = 1, with closed space sections; Ω < 1
implies ε = −1, with the spatial sections being open if they are simply
connected. The critical case Ω = 1 implies ε = 0, and hence locally ﬂatspace (Euclidean space if simply connected). Physicists argue that in themodels we are studying, if Ω > 1 by an appreciable amount at an earlytime, the universe will subsequently collapse in an extremely short time.
If Ω < 1, it will expand so fast that no stars could form. Thus one must

21 It is a constraint on a and a , as pre-dicted from a general analysis of theEinstein equations.

166

Introduction to cosmology

suppose that Ω was initially very close to 1; this is called the ﬂatness
problem.

VII.6.3

Einstein–de Sitter universe

At the present cosmic time, p is negligible with respect to μ.
The Einstein–de Sitter universe (found by Einstein and de Sitterin 1932) is a Friedmann–Lemaı̂tre universe such that p = 0 (dust model),Λ = 0, and ε = 0 (ﬂat case). Then a satisﬁes the second-order diﬀerential
equationa2 + 2aa = 0,(VII.6.9)

which can be integrated easily. Indeed, it can be written as
 

a
1 a

2

2aa + a ≡ 2aa
+
= 0,
a
2a
which is equivalent to1dln(a a 2 ) = 0.
dt(VII.6.10)

Hence, if a(0) = 0, it holds that3a 2 = kt, k a constant.
By setting χ := kr, the metric reads4 − dt2 + k 2 t 3 dr2 + r2 (dθ2 + sin2 θ dφ2 ) .

(VII.6.11)

Exercise VII.6.3 Find this metric by using (6.5) and (6.6).

VII.6.4

General models with p = 0

In the general case with p = 0, one can use the Friedmann equation
μ = 3a−2 ε + a2 − Λ(VII.6.12)
and the conservation lawd(μa3 ) = 0
dt
to obtain
a−1 C + a2 Λ− ε, C = μa3 a constant.
3
Hence one has the diﬀerential equation

1
dt
=
.
da
f (a)
a2 = f (a) :=(VII.6.13)

for the function a → t(a), whose inverse gives the variation of a infunction of t.

VII.7

Primordial cosmology

167

In fact, it is now possible with a good approximation to deduce density μ from observations by a dynamical analysis, and the distance aχ byusing standard candles. Analysis of the redshift then gives the Hubble
parameter H and the deceleration parameter q, and hence a and a .
It is found that today a is positive (expansion) and also a > 0 (acceleration). The belief among most cosmologists is that at present theuniverse is in accelerated expansion.

VII.6.5

ΛCDM cosmological model

The standard model of the universe adopted at present by most cosmologists is, up to small corrections, a Friedmann–Lemaı̂tre universe;that is, a Robertson–Walker spacetime solution of the Einstein equations with cosmological constant and source a stress–energy tensor ofmatter, with negligible contribution from radiation and neutrinos (evenif the latter have a tiny non-zero mass). Observations of the motions ofstars and galaxies using powerful Earth-based and satellite telescopes
seem to imply that at a cosmological scale classical baryonic matter22
is a very small part of the energy content of the cosmos, around 4.5%.
The observations indicate the existence of cold dark matter, estimatedat about 25% of the universe energy content and probably composed of
WIMPS (weakly interacting massive patricles) with only weak and gravitational interactions. The acceleration of the universe is explained in theΛCDM model by the existence of dark energy represented by the term
Λg, usually considered as an energy of the vacuum, of quantum origin
and constant in spacetime, in agreement with the observational result of
the constancy throughout spacetime of the fundamental dimensionlessparameters, in particular the ﬁne-structure constant (see Chapter IV).
Remark VII.6.1 The expansion history for the ΛCDM cosmologicalmodel is presently obtained from the supernova distance–redshift relation
deduced from observation of type Ia supernovae. It assumes the validity of
the Friedmann equation. However, as pointed out by some cosmologists,
the Friedmann equation itself has not been independently tested.

VII.7

Primordial cosmology

The general belief is that the cosmos started with a singularity23 about
14×109 years ago.24 There was a primeval phase25 for which the physics
was very diﬀerent from anything we have ever investigated experimentally. This physics, linked with the search for quantum gravity, is the
subject of intensive investigation, mainly in the context of string theories, with amazing results. In the most popular string theory, the universe
is an (n + 1)-dimensional manifold that has n − 3 of its spacelike dimensions compactiﬁed or a very small size (possibly of the order of the
Planck length) and that is ﬁlled with one-dimensional strings. In sometheories, space and time began to appear only after the Planck time.26

22 Protons, neutrons, and electrons.

23 The structure of the conjectured initial singularity (big bang or oscillatory)
has led to interesting mathematicalwork but remains a mystery.
24 This estimated age was augmented
by a few million years after analysis
in 2013 of data from the Planck satellite, which led to a reduction of thevalue of the Hubble parameter (seeSection VII.5) by about 10%.
25 The duration of this primeval phaseis estimated as a Planck time, that is,about 10−44 s (see Chapter III). Notethat giving an estimate for the durationof this preliminary stage is somewhat
contradictory to the fact that time andspace perhaps did not exist at thisstage.

168

Introduction to cosmology

26 Or

rather no time less than the
Planck time 10−44 s exists.

27 Thetransition phase of inﬂation
could have been a weakening of theinﬂaton called the slow roll. It is believed that damping oscilations were
probably present (see Damour and
Mukhanov, 1998).

28 Protons and neutrons.
29 By

a process that is not yet fully
understood, as a result of the existence at some time of a slightly greaternumber of baryons than antibaryons,
most of the latter were annihilated,leaving behind an overwhelming predominance of matter over antimatter inthe universe.

30 In the radiation epoch, the photons

lost energy by interations with electrons (Thomson scattering) and theuniverse was opaque. After recombination and the capture of electrons byatoms, the universe became transparent.

For the next earliest phase, the favoured paradigm is that the universe(that is, a section t = constant of the Friedmann–Lemaı̂tre model), entered a phase of quasi-exponential growth, called inﬂation, driven by anenergy content approximately equivalent to a large cosmological term,a constant or a function of a scalar ﬁeld, called the inﬂaton. Inﬂationis generally considered by cosmologists as solving the ﬂatness problem,
every region of space being stretched almost ﬂat by the expansion. Inﬂation is also considered as solving the ‘horizon problem’, that is, theobserved similarity in properties (cosmological principle) in regions ofspace so far apart now that no signal emitted near the big bang could
have reached them both without inﬂation.
The inﬂationary stage (during which the universe is similar to a de Sitter spacetime) was followed by a transition27 to a radiation-dominatedera, during which the universe was ﬁlled by an extremely hot plasmaof quarks, gluons, leptons, photons, and neutrinos. The evolution of theuniverse was governed until a time of the order of 10−6 seconds by quantum mechanics, and the physics was more like that which is or soon will
be accessible to laboratory experiments, albeit under extreme conditions.
The universe was expanding and cooling; there came periods ofso-called recombinations and the formation of baryons28 and antibaryons.29 A later recombination between ions and electrons led to
the formation of atoms and molecules. The universe became transparent.30 The cosmic microwave background (CMB) radiation observed
now is thought to be made of the free photons surviving from this epoch.
Finally, stars and galaxies appeared, and the universe as we know it wasborn at a time estimated at about 108 years.
The modelling of these fascinating early phases is, however, outside
the scope of this book.

VII.8

Solution of Exercises VII.6.1
and VII.6.2

We need to show that in a Friedmann–Lemaı̂tre universeμ = 3a−2 ε + a2 − Λ,
p = −a−2 ε + a2 + 2aa + Λ
∇α T

αβ

=0

Solution(VII.8.1):
S00 = T00 − g00 Λgives
μ = S00 − Λ ≡ 3a−2 ε + a2 − Λ.

(VII.8.1)(VII.8.2)(VII.8.3)

VII.9(VII.8.2):
Sij ≡ Rij −a2
γij R ≡ 2ε + aa + 2a2 − 3(ε + aa + a2 ) γij
2When n = 3, γ ij γij = 3, and hence
γ ij Sij ≡ 3 −ε − 2aa − a2 = γ ij Tij − 3Λ = 3(pa2 − Λ)
impliesp = −a−2 ε + a2 + 2aa + Λ.
(VII.8.3): The stress–energy tensor is such that T i0 = 0, and hence∇α T ai ≡ ∂j T ji + Γhhj T ji + Γihj T hj .
The non-zero Christoﬀel symbols of the spacetime metric areiΓ0ij ≡ aa γij , Γj0i ≡ a−1 a δij , and Γijh = γjh. Denoting by(3)∇ the covariant derivative in the metric γ, we have
∇α T ai ≡ ∂j T ji +Γhhj T ji +Γihj T hj ≡ (γ) ∇j T ji ≡ γ ij ∂j (pa2 ) = 0,
because a and p depend only on t. We also have
∇α T a0 ≡ ∂0 T 00 +3a−1 a T 00 +aa γij T ij ≡ μ +3a−1 a (μ+p).
Using the previous expressions for μ and p, we ﬁndμ = −6a−3 a ε + a2 + 6a−2 a a ,
μ + p = 3a−2 ε + a2 − a−2 ε + a2 + 2aa≡ 2a−2 ε + a 2 − aa ,and henceμ + 3a−1 a (μ + p) = 0.
Thus
∇α T a0 = 0.

VII.9

Problems

VII.9.1

Isotropic and homogeneous Riemannian
manifolds

Isotropy of the space section Mt at a point x means that there is no
privileged direction in the tangent space Tx to Mt at x.

Problems

169

170

Introduction to cosmology

1. Show that if the Riemann tensor of the metric g takes at x the form,
with K(x) some number,
Rij,hk (x) = K(x)(gih gjk − gjh gik )(x),(VII.9.1)

it is invariant on the tangent plane at x under rotations centred at x.
2. Show that the tensor (9.1) is locally homogeneous.

Solution
1. The tangent space to an n-dimensional Riemannian manifold at x is
an n-dimensional Euclidean vector space. A rotation around its origin
x is represented by an n × n matrix A (called an orthogonal matrix)mwhose elements Am
p acting on tangent vectors X at x, Yp = Ap Xm ,are such thatq
Yp2 :=AmXq2 ;p Xm Ap Xq ≡pthat is,

p



q

q
Am
p Ap = 0, m = q,

p



q
Am
p Ap = 1, m = q.

(VII.9.2)

p

In an orthonormal frame at x, the components of the Riemann tensor(9.1) readRij,hk (x) = K(x)(δih δjk − δjh δik )(x).
Under a rotation A, the transformed curvature tensor R has components
Ri  j  ,h k (x) = K(x)Aii Ajj  Ahh Akk (δih δjk − δjh δik )(x);that is, using the formulas (9.2),
Ri  j  ,h k (x) = K(x)(δi h δj  k − δj  h δi k )(x).

VII.9.2
31 See, for instance, see CB-DMI, p. 81

Age of the universe

1. Use the Taylor formula31

or Bony (2001), p. 61.

a(t) = a(t0 ) + a (t0 )(t − t0 ) + (t − t0 )2

 1(1 − λ)a (t0 + λ(t − t0 )) dλ0

to ﬁnd estimates of the age t0 and the time T of the end of an expanding Robertson–Walker universe when we assume that a (t) < 0
for 0 ≤ t ≤ T .
2. What can we say if a (t0 ) > 0.

VII.9

Solution
1. The Taylor formula implies
a(0) = a(t0 ) − t0 a (t0 ) + t20a (t0 (1 − λ)) dλ,

0 ≤ λ ≤ 1.

Hence, if a(0) = 0, a (t0 ) > 0, and a (t) ≤ 0 for t ≤ t0 , thent0 ≥a(t0 )= H(t0 ).
a (t0 )Suppose that a (t) ≤ −C, some positive constant for t ≥ t0 ; then1a(t) ≤ a(t0 ) + a (t0 )(t − t0 ) − C(t − t0 )2 ,2and hence a(T ) = 0 (big crunch) as soon as1
a(t0 ) + a (t0 )(t − t0 ) − C(t − t0 )2 = 0;2that is,T − t0 =−a (t0 ) +a2 (t0 ) + 2Ca(t0 ).
C

2. The age t0 depends on the behaviour of a (t), and hence of a (t) for
0 < t < t0 . Assume a (t) ≥ 0 for 0 ≤ t ≤ t1 ; hence a(t1 ) ≥ 0 and(inﬂation) a (t) ≥ C > 0 for t1 ≤ t ≤ t0 . Then the Taylor formula
reads
 1

2
a(t0 ) = a(t1 )+a (t1 )(t0 −t1 )+(t0 −t1 )(1−λ)a (t1 +λ(t0 −t1 )) dλ,0

and impliesa(t0 ) ≥VII.9.3

C(t0 − t1 )2 .
2Classical Friedmann–Lemaı̂tre universes

Consider a Friedmann–Lemaı̂tre universe with Λ = 0 ﬁlled with a perfect
ﬂuid with equation of state p = (γ − 1)μ, with γ a constant. Note thatγ = 1 corresponds to dust and γ = 43 to pure radiation.
1. Show that the velocity of acoustic waves is less than the speed of light
if γ ≤ 2 (see Chapter IX).
2. Show that in such a Friedmann–Lemaı̂tre universe, it holds that, with
C a constant,μa3γ = C.
Therefore, the density μ must decrease with increasing a.

Problems

171

172

Introduction to cosmology

Solution
If p = (γ − 1)μ, the equation (6.5) readsd 3(a μ) + 3a−1 a (γ − 1)μ = 0dt
and simpliﬁes toμ + 3a−1 a γμ = 0.
Hence, by integration, with C a constant,μa3γ = C.

VII.9.4

Milne universe

The Milne universe is a Robertson–Walker spacetime with ε = −1,
Λ = 0, space support R3 , and is a vacuum (μ = p = 0). The Milne
universe, or analogous spacetimes with compact (non-simply connected)
space sections, play an important role as asymptotic states in cosmology.
Write the spacetime metric of the Milne universe. Show that it islocally ﬂat, isometric to a wedge of Minkowski spacetime.

Solution
The Friedmann equation impliesa2 = 1.
Hence a = 1, and, up to the choice of label of the time origin, a(t) = t.
The spacetime metric on the manifold R3 × (0, ∞) then takes the form
−dt2 + t2 [dr2 + sinh2 r(dθ2 + sin2 θ (dφ2 )],where r, θ, φ are polar (pseudo-)coordinates on R3 . The space metriccollapses for t = 0 and expands indeﬁnitely when t tends to inﬁnity.
The spacetime metric is locally ﬂat, as can be seen by computing its
Riemann tensor, for instance by using the 3 + 1 decomposition given in
Chapter VIII.

Part B
Advanced topics
This part provides a deeper study of the properties of general solutionsof the Einstein equations either in vacuum or with stress–energy–
momentum sources. Particular attention is paid to relativistic ﬂuids andkinetic models, both of which have become important in astrophysics
and cosmology.

General Einsteinian
spacetimes. The Cauchy
problem
VIII.1The Cauchy problem for a partial diﬀerential equation of order m on
Rn+1 with unknown a function f is the search for a solution such thatf and its derivatives of order less than m take given values on an ndimensional submanifold M. In coordinates x0 , xi , i = 1, 2, . . . , n, with
M represented by x0 = 0, the independent data are∂pf
f (0, x ) and(0, xi ), p = 1, . . . , m − 1.
(∂x0 )pi

In the regular1 analytic case, this problem has one and only one analytic
solution in a neighbourhood of M.
The stability2 of the solution of the Cauchy problem, and its existencein the non-analytic case, depend crucially on its characteristic determinant, the polynomial obtained in replacing in the terms of order m thederivative ∂f by a vector X. For a linear second-order equation∂2f∂f+ bα α + cf = h,∂xα ∂xβ
∂x

175
176

VIII.5 Local existence anduniqueness181
VIII.6 Solution of the wave
gauge constraints
181
VIII.7 Geometric n + 1
splitting
184
VIII.8 Solution of theconstraints
by the conformal
method190

VIII.9 Motion of a systemof compact bodies
193
VIII.10 Global properties
195
VIII.11 Singularities andcosmic censorshipconjectures

196

VIII.12 Problems

198

1 The regularity condition for a quasi-(VIII.1.1)

the characteristic polynomial is the scalar function
P (X) := aαβ Xα Xβ .

VIII.1 Introduction
VIII.2 Wave coordinates

VIII.3 Evolution in wave
gauge
178
VIII.4 Preservation of the
wave gauges
179

Introduction

aαβ

VIII

(VIII.1.2)

The properties of solutions depend essentially on the signature of thisquadratic form. In a domain U of Rn+1 , the equation is elliptic if P (X)is positive- (or negative-) deﬁnite; it is hyperbolic if it is of Lorentziansignature.
For elliptic equations, the solution of the Cauchy problem exists inthe analytic case, but is not stable: one says that the Cauchy problem is
not well posed.3 Possibly well-posed problems for elliptic equations areglobal problems, for instance the so-called Dirichlet problem, whichis the data of the unknown on the boundary of the domain.4
For hyperbolic equations, the Cauchy problem is well posed in theanalytic case, but also for more general functional spaces,5 in particular smooth functions. For linear equations, the solution is global, i.e. it

linear equation with analytic coeﬃcients and analytic data on M, represented by x0 = 0, is that the coeﬃcientof the derivative ∂ m f /(∂x0 )m does notvanish for x0 = 0.
2 Thatis, contiguous dependence on
initial data.
3 A problem is said to be well posed if
it has one and only one solution andthis solution depends continuously on
the data.
4 In

this case, the well posedness depends also on the lower-order terms.
5 An

important property for General
Relativity, since the relativistic causality imposes that a solution of a relativistic equation depends only on the pastof this point, while analytic functions
are entirely determined by their values
in any open set.

176

General Einsteinian spacetimes. The Cauchy problem

6 Diagonal in the highest-order terms.

exists on the whole of Rn+1 , although it may grow, even exponentially,
with t ≡ x0 . Results extend to some (see J. Leray (1953)) quasidiagonal6
systems and to systems on manifolds. That is, for linear second-ordersystems on a Lorentzian manifold (V n+1 , g) that readJ
I J
Ig αβ ∇2αβ f I + bα,IJ ∇α f + cJ f = h ,(VIII.1.3)

the Cauchy problem with data on a spacelike section M n is globallywell posed in relevant functional spaces, with relevant hypotheses on thecoeﬃcients and Cauchy data. In particular, the solution f I is smooth if(V n+1 , g) is a smooth and regularly sliced Lorentzian manifold and the
Cauchy data are smooth. An important property for Relativity is thatthe solution at a point depends only on the values of the initial data inthe past of this point.
A quasidiagonal, quasilinear system of second-order diﬀerential equations on a manifold V n+1 reads, in a domain U of local coordinates,J
I J
Ig αβ ∇2αβ f I + bα,IJ ∇α f + cJ f = h ,7 The introduction of ∇ permits a glo-

bal geometric formulation, impossible
with ∇ for the Einstein equations,where g itself is the unknown and satisﬁes ∇g ≡ 0.

8 In the classical physical case n+1 = 4,
the number of independent coeﬃcientsgαβ of a general metric g is 10.

where g, b, and c depend on the unknown f and its ﬁrst derivative. The
covariant derivative ∇ is taken in an a priori given7 metric g in U. It isalways possible to take for g in U the ﬂat metric, i.e. ∇α := ∂/∂xα . The
Cauchy problem for the system (1.4) is well posed for some initial data
if the linear system obtained by replacing in the coeﬃcients g, b, and cthe quantities f and ∂f by their initial data is a well-posed system forthese initial data. Note that the existence of the solution (which can beobtained by iteration), is in general for nonlinear equations only local intime.
The Einstein equations in vacuum are a geometric system for the pair(V, g), invariant under diﬀeomorphisms of the manifold V and the associated isometries of the Lorentzian metric g. These equations constitute,from the analyst’s point of view, a system of (n + 1)(n + 2)/2 secondorder quasilinear partial diﬀerential equations for the (n + 1)(n + 2)/2
coeﬃcients8 of the metric in local coordinates. However, the equations
are not independent, because of the contracted Bianchi identities, a consequence of the invariance under diﬀeomorphisms. We will see that the
Einstein equations have both hyperbolic and elliptic aspects.

VIII.29 Einsteinused them in the Minkowskian approximation. It seems thatde Donder was the ﬁrst to introduce
them for the full Einstein equations,shortly before G. Darmois.
10 In

computations done in local coordinates (xα ), it holds that ∂α ≡∂/∂xα .

(VIII.1.4)

Wave coordinates

The Einstein equations are invariant under diﬀeomorphisms; to construct generic Einsteinian spacetimes, one ﬁxes the local coordinates by
a general condition, called a gauge choice. One then has to prove thatthe constructed spacetime satisﬁes the gauge condition.
The results of the Minkowskian approximation have led various authors, over many years,9 to select in various problems what are now
called harmonic or wave coordinates, that is, to impose that the scalarfunctions x → xα deﬁning local coordinates satisfy the wave equations10

VIII.2

F α := g xα ≡ g λμ ∇λ ∂μ xα = 0.

Wave coordinates(VIII.2.1)

In terms of the Christoﬀel symbols associated with the coordinates xα ,it holds thatF α ≡ g λμ Γα
λμ .

(VIII.2.2)

The following identity results from a straightforward computation:(h)Rαβ ≡ Rαβ + Lαβ ,(VIII.2.3)1
gαλ ∂β F λ + gβλ ∂α F λ2(VIII.2.4)

where
Lαβ ≡

(h)

vanish in wave coordinates while the Rαβ are a system of quasilinear,quasidiagonal (i.e. linear and diagonal in the principal, second-order,
terms) wave operators1(h)2Rαβ ≡ − g λμ ∂λμgαβ + Pαβ (g, ∂g),
2

(VIII.2.5)

where P is a quadratic form in the components of ∂g, with coeﬃcients that are polynomials in the components of g and its contravariant
associate.
The system of partial diﬀerential equations(h)Rαβ = ραβ

(VIII.2.6)

are called the (harmonically) reduced Einstein equations (thevacuum reduced Einstein equations if ραβ ≡ 0).

VIII.2.1

Generalized wave coordinates

It is clear that one can also deduce from the Ricci tensor a quasidiagonal
operator on the components of the metric, namely
(h,H)

Rαβ

(h)

≡ Rαβ +1gαλ ∂β H λ + gβλ ∂α H λ ,2(VIII.2.7)

if the wave coordinate conditions are replaced by the more general ones11
α
FH:= F α − H α = 0,with

F α ≡ g λμ Γα
λμ ,(VIII.2.8)

where the H α are known scalar functions. Their advantage is that they
contain a new set of functions H α , which can be freely speciﬁed, or
eventually chosen to satisfy ad hoc equations.

11 Friedrich (1986).

177

178

General Einsteinian spacetimes. The Cauchy problem

VIII.2.2
12 Gundlach,Calabrese, Hinder, andMartin-Garcia(2005),
Pretorius(2005b),and
Lindblom,
Scheel,Kidder, Owen, and Rinne (2006).

Damped wave coordinates

Recently, numerical analysts12 have been led to introduce reduced
Einstein equations in a wave-related gauge, called damped wave(h,H)
coordinates. The operator Rαβ is replaced by(h,H,γ0 )Rαβ13 For the Einstein equations with non-zero sources, one must solve the coupled system of these equations andthe equations satisﬁed by the sources,
which depend on their nature.

14 A

constructive method inspired byrepresentation formulas for solutions oflinear equations given in dimension 4
by Sobolev and in even dimension n +1 > 4 by de Rham (restricted to constant coeﬃcients) was used for the ﬁrst
proof of existence of solutions of the
Cauchy problem for quasilinear, quasidiagonal systems of wave equations
by Fourès (Choquet)-Bruhat (1952) forthe case n + 1 = 4 and by Fourès(Choquet)-Bruhat (1953) for the casen + 1 > 4. The Leray theory of general hyperbolic systems with its energy
method can also be used. The multiplicity of roots of the characteristic
determinant is irrelevant for the application of Leray theory because thecharacteristic matrix is diagonal. This
multiplicity destroys the Leray hyperbolicity when the characteristic matrix cannot be diagonalized into blocks
with determinants with distinct roots.
K. O. Friedrichs’s theory of ﬁrst-ordersymmetric hyperbolic systems has nodiﬃculty with multiple characteristics,
but such ﬁrst-order systems destroy the
spacetime appearance of the problem
and hide the propagation properties ofthe solution. Friedrichs hyperbolic systems have no analogue for higher-order
Leray hyperbolic systems. Moreover
the weak hyperbolicity of Leray–Ohya(see Chapter IX), important for relativistic causality, has no analogue inFriedrichs’s theory of symmetric hyperbolic ﬁrst-order systems.

15 For results in classical Sobolev spa-ces, see YCB-OUP2009, Appendix III.

11λgαλ ∂β H λ + gβλ ∂α H λ + γ0 FH(gβλ nα22+ gαλ nβ − gαβ nλ ),(h)≡ Rαβ +

with n the unit vector normal to spacelike slices and γ0 a constant.
The presence of this non-zero constant seems to stabilize the results ofnumerical calculations of the evolution system; it determines the time
λ= 0 are damped under evolution.
rate at which the gauge conditions FH
The use of a wave gauge gives the basic elements for the proof of fundamental properties of Einstein gravity using known results for partial
diﬀerential equations.

VIII.3Evolution in wave gauge

VIII.3.1

Solution of the reduced equations in vacuum

We consider the vacuum case,13 that is, the search for solutions of asystem of quasilinear wave equations. The natural well-posed problem,
called the Cauchy problem, is the construction of a solution taking together with its ﬁrst derivatives given values on a submanifold M of V
assumed to be spacelike for these initial values. If M has equation x0 = 0
in a local coordinate system and the analytic data are gαβ (0, xi ) (a quadratic form of Lorentzian signature) and (∂gαβ /∂x0 )(0, xi ), then the
manifold M is spacelike for these data if the initial data ḡ (the induced
metric on M with components gij (0, xh )), is positive-deﬁnite. Known
theorems for quasidiagonal quasilinear systems of wave equations14
lead to the following theorem, valid in relevant functional spaces,15in particular spaces of smooth functions with some ﬁnite number ofderivatives.
Theorem VIII.3.1 The Cauchy problem for the vacuum Einsteinequations in wave gauge is well posed; that is, it has one and only one
local solution depending continuously on the initial data. This solution
has the physically important property of exhibiting propagation of thegravitational ﬁeld with the speed of light: its value at a spacetime point
depends only on the data in the past of this point.
The equations being nonlinear, the solution in general is deﬁned only
in a neighbourhood of the initial space manifold. The appearance of singularities and their nature is a fundamental ﬁeld of research in General
Relativity.

VIII.4

VIII.3.2

Preservation of the wave gauges

179

Equations with sources

In the presence of sources, the reduced Einstein equations contain otherunknowns in addition to the metric g, namely ﬁelds or matter. The
necessary conservation laws satisﬁed by the sources, ∇α T αβ = 0, areequations that contain both g and these other unknowns; they must be
coupled with the Einstein equations. The hyperbolicity of this coupledsystem with a dependance domain of a solution of the Cauchy problem
determined by the null cones of the metric g is important for the satisfaction of the Einstein principle of causality (see Problems VIII.12.6
and VIII.12.7 and Chapters IX and X).

VIII.4Preservation of the wave gauges

A solution of the reduced Einstein equations that is a metric in wave
gauge satisﬁes the full Einstein equations, and, indeed, for any solutionof the reduced equations to be a solution of the full equations, it has to
be in wave gauge, that is satisfy the equations
F α ≡ g λμ Γαλμ = 0.

(VIII.4.1)

We use the Bianchi identities to show that if g is a solution of the
reduced Einstein equations16 in a wave (harmonic) gauge, then thegauge conditions satisfy a second-order linear quasidiagonal homogeneous diﬀerential system, because of the Bianchi identities, which imply
the following identities for the Einstein tensor S:∇α S αβ ≡ 0.

16 This preservation of the wave gauge
property holds for Einstein equations
with sources satisfying the conservation
laws.

(VIII.4.2)

Indeed, if g is a solution of the equations in wave gaugeαβ= ραβ ,
R(h)(VIII.4.3)

then it holds that, with T αβ := ραβ − 12 g αβ ρ the stress–energy tensor ofsources,171
S αβ − T αβ = − (g αλ ∂λ F β + g βλ ∂λ F α − g αβ ∂λ F λ ).
2

(VIII.4.4)

An elementary computation using the Bianchi identities and the conservation laws for the source T shows that, the functions F α then satisfy alinear homogeneous system of wave equations
2αF β + Aβλg αλ ∂αλ
α ∂λ F = 0,

(VIII.4.5)

where the A’s are linear functions in the Christoﬀel symbols of g.
Exercise VIII.4.1 Compute the coeﬃcients A. Write similar equations in the cases of generalized and damped generalized wave
coordinates.

17 Recall that we use geometric units
where GE = 1; equivalently, in the casen + 1 = 4, 8πGN = 1.

180

General Einsteinian spacetimes. The Cauchy problem

The homogeneous wave equations (4.5). imply F α = 0 if the initial
data for g are such that the F α and their ﬁrst derivatives vanish on theinitial manifold M; that is,
F α |M = 0 and ∂0 F α |M = 0.

(VIII.4.6)

Corresponding properties hold for the generalized and damped wave
gauges.

VIII.4.1

Wave gauge constraints

We will deduce from (4.6) relations that must hold between initial dataon the submanifold x0 = 0 for the solution of the reduced Einsteinequations to be a solution of the full Einstein equations, that is, to be a
metric in wave gauge. The simplest way to ﬁnd these relations is to use
the contravariant tensor densities
1

G αβ := |Det g| 2 g αβ .

(VIII.4.7)

It holds that
1

−2
F α := g λμ Γα∂β G αβ .
λμ ≡ −|Det g|(VIII.4.8)

Exercise VIII.4.2 Prove this identity.
Hint: The derivatives of the determinant of g are ∂α (Det g) ≡(Det g)g λμ ∂α gλμ .
Straightforward computation using the identity (2.3) satisﬁed by the
Ricci tensor shows that the Einstein tensor S satisﬁes the identity
1
1  αλαβg ∂λ F β + g βλ ∂λ F α − g αβ ∂λ F λ ,+S αβ := Rαβ − Rg αβ ≡ S(h)22(VIII.4.9)
αβwhere the S(h)are a quasidiagonal system of wave operators in the metric
αβg for the G .
Suppose that F α |t≡x0 =t0 = 0. Then also ∂i F α |x0 =t0 = 0, and theidentity (4.9) implies the equality1α0|M + (g 00 ∂0 F α )|M .
S α0 |M = S(h)2(VIII.4.10)

α0The expressions for S(h)and F α show that S α0 |M does not contain any
second time derivative of g; that is, the equations

C α := S α0 |M − T α0 |M = 0(VIII.4.11)

are constraints on the initial data. If they are satisﬁed, then a solution of the reduced Einstein equations with these initial data, i.e. suchα0
|M − T α0 |M = 0, satisﬁes also ∂0 F α |M . We have proved thethat S(h)following theorem:
Theorem VIII.4.1 A solution of the reduced Einstein equations withinitial data such that F α |M = 0 satisﬁes the full Einstein equations if
and only if the initial data satisfy the constraints C α = 0.
An analogous theorem holds for the other types of wave gauges.

VIII.6

VIII.5

Solution of the wave gauge constraints

181

Local existence and uniqueness

An Einsteinian spacetime is constructed in local coordinates,18 in aneighbourhood of a hypersurface M with equation x0 ≡ t = t0 , frominitial data gαβ |M , a Lorentzian metric for which M is spacelike, and(∂0 gαβ )|M .

18 See the geometric intrinsic formulations in Section VIII.7.

Theorem VIII.5.1 (Fourès (Choquet)-Bruhat, 1952)
1. The Cauchy problem for the vacuum Einstein equations with initial
data on a spacelike hypersurface gαβ |M and (∂t gαβ )|M satisfying thewave gauge constraints has a local-in-time solution, a Lorentzian metric in wave gauge depending continuously on the initial data. Its value
at a given point depends only on the past of this point.
2. The solution is locally geometrically unique.
Part 1 of this theorem is a consequence of Theorems VIII.3.1 andVIII.4.1.
For Part 2, we ﬁrst recall that the solution constructed in wave gauge
from given initial data gαβ |M , (∂0 gαβ )|M satisfying the constraints is
unique. To show local geometric uniqueness, we show that any given
vacuum Einsteinian spacetime deﬁned in a neighbourhood of M can be
put in wave gauge in a possibly smaller neighbourhood of M, taking
initial data that depend only on the original data. We will state a moregeneral, global, uniqueness property after deﬁnining geometric initial
data.

VIII.6

Solution of the wave gauge
constraints

The expressions for S α0 and F α show that the wave gauge constraints
are of the form, for the Einstein equations with source the stress–energy
tensor T,
Cα ≡1 ij 2 α0 1 00 2 αi
2 α0
G ∂ij G − G ∂it G +G i0 ∂it
G +Kα0 −T α0 = 0, (VIII.6.1)
2
2

where Kα0 depends only on G and its ﬁrst derivatives.
Remark VIII.6.1 Giving G αβ , ∂t G αβ for t = t0 is equivalent to giving
gαβ , ∂t gαβ , for t = t0 .
The constraints appear as a system of n+1 equations for (n+2)(n+1)
unknowns, G αβ , ∂t G αβ , on the manifold x0 = 0. This system, with moreunknowns than equations, is undetermined. It is natural to try to split
the initial data into speciﬁed quantities and n+1 unknowns, for which wewish to ﬁnd a well-posed system of partial diﬀerential equations, which
we expect to be elliptic because this is a pure, non-evolutionary, space
problem. This was done for the ﬁrst time in the general case19 by using
the equations (6.1). Two choices were suggested for splitting the initial
Cauchy data between unknowns and given data:

19 Bruhat (Choquet-Bruhat) (1962).

182

General Einsteinian spacetimes. The Cauchy problem(1) Give arbitrarily on the initial manifold M the quantities G αi . The∂j G αi are then known and the ∂t G α0 are determined by the harmonicity conditions. The remaining unknowns, in this ﬁrst step, areu := G 00 and v ij := ∂t G ij . Equations (6.1) can be written in thefollowing form:• For α = 0,C0 ≡1 ij 2 00
G ∂ij G − A = 0,
2

(VIII.6.2)

where
1 00 2 0i
2 00
G ∂it G + G i0 ∂it
G + K00 − T 00
(VIII.6.3)
2depends only on the speciﬁed quantities and on the space derivatives of the unknowns of order at most 1• For α = h,A :=1
C h ≡ − G 00 ∂i (∂t G hi ) + B h = 0,
2

(VIII.6.4)

1 ij 2 h0
2 h0
G ∂ij G + g i0 ∂it
G + Kα0 − G α0 .
2

(VIII.6.5)

where
20 The principal symbol of a scalar par-tial diﬀerential operator is the polynomial obtained by replacing in thehighest-order terms the partial derivative ∂i by the component ξi of a vector.
In the case of a system, the principal
symbol is a matrix (see the example
below).
21 A system of partial diﬀerential oper-

ators is elliptic if its principal symbol
is an isomorphism (i.e. an invertible
square matrix) for each non-zero vector ξ. For one scalar equation, thismeans that the principal symbol does
not vanish for any non-zero set of real
numbers ξi .

B h :=

Equation (6.2) is a semilinear second-order equation for G 00 withprincipal symbol20 the positive-deﬁnite quadratic form G ij ξi ξj ; itis an elliptic21 equation.
To write the system (6.4) of n equations with the n(n + 1)/2
unknowns X hi := ∂t G hi as an elliptic system for n unknowns
and n(n − 1)/2 speciﬁed functions, one can use the Berger–Ebindecomposition theorem for a functional space E as
E = ker D + range D∗ ,where D and D∗ are a diﬀerential operator and its adjoint, one ofwhich has injective symbol; the product DD∗ is then elliptic. Since
in this section we work in local coordinates, we simply recall here
the classical procedure for decomposing a symmetric unknown X ij
on Rn . We set
X ij ≡ Y ij + Z ij ,with ∂i Y ij = 0

and Z ij = ∂i U j + ∂j U i .

Equations (6.4) take the form

2 j
∂ii
U + ∂i ∂j U i = f j .
iThey imply

i

2
∂ii
(∂j U j ) =1∂j f j ,2which is an elliptic equation for ∂j U j and an elliptic system for U jwhen ∂i U i is known.

VIII.6

Solution of the wave gauge constraints

183(2) In the spirit of Wheeler’s ‘thin-sandwich conjecture’, we give arbitrarily on M the quantities G ij and ∂t G ij . The harmonicity
conditions then determine ∂t G 0i = −∂j G ji and imply the relation∂t G 00 = −∂i G 0i .

(VIII.6.6)

The constraints now read, without needing the introduction of afurther splitting, as a system of n+1 equations for the n+1 unknowns
G 00 , G 0i :1 ij 2 0h
G ∂ij G − F h = 0,
2

(VIII.6.7)

1 ij 2 00
2 j0
G ∂ij G − G 0i ∂ij
G − F 0 = 0,
2

(VIII.6.8)

Ch ≡

C0 ≡

where F 0 and F h depend only on the given quantities and on theﬁrst space derivatives of the unknowns.
The system is elliptic because of the expression of its characteristic
determinant, the determinant of its principal symbol.
Exercise VIII.6.1 Show that this determinant is proportional to a
power of the determinant of a Laplace operator.
Hint: The principal part in the equation C h = 0 contains only G 0h .
The solution of an elliptic system on a manifold is a global problem.
Two cases are of particular interest in General Relativity: the case of
asymptotically Euclidean manifolds and the case of compact manifolds.

VIII.6.1

Asymptotically Euclidean manifolds

A particularly physically relevant case is the study in Einsteinian gravitation of isolated systems, composed of a few bodies far from any othersource of gravitation, so that the metric tends to be ﬂat far away fromthe studied system. Such systems are modelled in general by asymptotically Euclidean manifolds, deﬁned in Chapter IV. For mathematical
study of solutions of elliptic diﬀerential equations on asymptotically Euclidean manifolds, one uses either weighted Hölder spaces22 or weighted
Sobolev spaces.23
The linearization of the nonlinear constraints (6.7) and (6.8) for given
Euclidean data reduces to the Euclidean Laplace equations, for which
solutions are known in the mentioned functional spaces. The nonlinear
constraints have been shown to have a solution24 for given data near
Euclidean data.
Various numerical methods are available for solution of the relevant
equations. It would be interesting to solve them numerically, their solution giving initial data directly for the harmonically reduced evolution
equations, which form a well-posed system.

22 Choquet-Bruhat (1974), ChaljubSimon and Choquet-Bruhat (1979).
23 Cantor (1979), Choquet-Bruhat andChristodoulou (1981).

24 Vaillant-Simon (1969).

184

General Einsteinian spacetimes. The Cauchy problem

VIII.6.2

Compact manifolds

Mathematicians are especially interested in the case of compact space
manifolds without boundary, where topology plays an important role.
Solutions of the constraints on compact manifolds have been extensively studied, but after writing them as elliptic equations for geometricunknowns—the induced metric ḡ and extrinsic curvature K.
Compact space manifolds are of interest as cosmological models.
However, the choice of a model for our whole cosmos is very speculative.

VIII.725 Fourès (Choquet)-Bruhat (1956).
26 Arnowitt et al. (1962).

Geometric n + 1 splitting

We will in the following sections formulate geometric, i.e. coordinate
independent, theorems about the existence and uniqueness of solutionsof the Cauchy problem. We ﬁrst give the following decomposition,25
often called the ADM decomposition, although Arnowitt, Deser, andMisner26 are only accountable for its Hamiltonian interpretation. This
decomposition has been for many years intensively used for numericalcomputation, in spite of the fact that the evolution equations that itgives are not a hyperbolic system.

VIII.7.1

Adapted frame and coframe

We consider a spacetime with manifold V = M ×R and hyperbolic metricg such that the submanifolds Mt ≡ M × {t} are spacelike. We take aframe with space axes ei tangent to the space slice Mt and time axis
e0 orthogonal to it. Such a frame, particularly adapted to the solutionof the Cauchy problem, is called a Cauchy adapted frame. We take
local coordinates adapted to the product structure, (xα ) = (xi , x0 = t),
and we choose for ei the vectors ∂/∂xi of a natural frame on Mt , i.e.
∂i = ∂/∂xi .

(VIII.7.1)

The dual coframe is found to be such that, with β a time-dependentvector tangent to Mt called the shift,θi = dxi + β i dt,(VIII.7.2)

while the 1-form θ0 does not contain dxi . We chooseθ0 = dt.

(VIII.7.3)

The vector e0 , i.e. the Pfaﬃan derivative ∂0 , is then∂0 ≡ ∂ t − β j ∂ j ,with∂t := ∂/∂t.

(VIII.7.4)

The vector e0 is timelike since it is orthogonal to spacelike surfaces. We
suppose that it deﬁnes the positive time orientation. In the coframe θα ,one has gi0 = 0, and the metric reads
ds2 = −N 2 (θ0 )2 + gij θi θj ,

θ0 = dt, θi = dxi + β i dt.

(VIII.7.5)

VIII.7

The function N is called the lapse. We shall assume throughout thatN > 0. The time-dependent, properly Riemannian, space metric induced
by g on Mt is denoted by either gt or ḡ. An overbar denotes a spatial
tensor or operator, i.e. a t-dependent tensor or operator on M . Note thatin our frame, ḡij = gij and ḡ ij = g ij .
A spacetime (V, g) with V = M × R and metric (7.5) is called a sliced
spacetime.
The non-zero structure coeﬃcients of a Cauchy adapted frame arefound to beii= −Cj0= ∂j β i .
C0j(VIII.7.6)¯ the covariant derivative corresponding to the space
We denote by ∇metric ḡ. Using the general formulas from Chapter I, we ﬁnd thati≡ Γijk ≡ Γ̄ijk ,ωjki≡ N g ij ∂j N,
ω00

0
0
ω0i
≡ ωi0
≡ N −1 ∂i N,(VIII.7.7)
0ω00≡ N −1 ∂0 N,

and
0
≡
ωij

1 −2 hhN∂ 0 gij + ghj Ci0,+ gih Cj0
2

(VIII.7.8)

from which we obtain0
ωij≡

1 −2 ¯N ∂0 gij ,2(VIII.7.9)

where the operator ∂¯0 is deﬁned on any t-dependent space tensor T by
the formula∂− L̄β ,∂¯0 :=∂t(VIII.7.10)

where L̄β is the Lie derivative on Mt with respect to the spatial vector β.
Note that ∂¯0 T is a t-dependent space tensor of the same type as T .
The extrinsic curvature (also called the second fundamental
form) of Mt is the t-dependent symmetric space tensor K given by1
0
0
Kij ≡ − N −1 ∂¯0 gij ≡ −ωij
n0 ≡ −N ωij.
2(VIII.7.11)

The remaining connection coeﬃcients are found to be (indices are raisedwith ḡ)iω0j≡ −N Kji + ∂j β i ,iωj0≡ −N Kji .

(VIII.7.12)

The trace in the metric ḡ of the extrinsic curvature, often denotedby τ, is called the mean curvature of Mt :τ ≡ trḡ K ≡ ḡ ij Kij .

(VIII.7.13)

Geometric n + 1 splitting

185

186

General Einsteinian spacetimes. The Cauchy problem

K and τ play an important role in the initial-value formulation and othergeometric problems of General Relativity.
Exercise VIII.7.1 Show that the extrinsic curvature K of Mt is the
projection on Mt of the spacetime gradient of the past-oriented unit normal n to Mt (some authors take the opposite orientation of n to deﬁneK). Show that τ := Trḡ K of K is equal to the spacetime divergence of n.
Remark VIII.7.1 The tensor K is symmetric. It can equivalently bedeﬁned, up to a factor 1/2, as the Lie derivative of the spacetime metricin the direction of n. It does not depend on the value of n outside Mt .
Remark VIII.7.2 A positive value of τ signals a positive divergenceof the past-directed normals. A negative value of τ corresponds to convergence of the past-directed normals, and hence future expansion of thesubmanifolds Mt .
We deduce from the general formula giving the Riemann tensor andthe splitting of the connection the following identities:
Rij,kl ≡ R̄ij,kl + Kik Klj − Kil Kkj ,(VIII.7.14)¯ Kki − ∇¯ k Kji ),R0i,jk ≡ N (∇j

(VIII.7.15)¯ i ∂j N )R0i,0j ≡ N (∂¯0 Kij + N Kik K k j + ∇(VIII.7.16)

From these formulas, we obtain the following expressions for the Ricci
curvature:¯ i ∂j N, (VIII.7.17)
N Rij ≡ N R̄ij − ∂¯0 Kij + N Kij Khh − 2N Kik K k j − ∇¯ hK hj ,N −1 R0j ≡ ∂j Khh − ∇(VIII.7.18)

R00 ≡ N (∂¯0 Khh − N Kij K ij + Δ̄N ).

(VIII.7.19)

Also, with R̄ := g ij R̄ij ,g ij Rij = R̄ − N −1 ∂¯0 Khh + (Khh )2 − N −1 Δ̄N,

(VIII.7.20)

R ≡ −N −2 R00 +g ij Rij = R̄+Kij K ij + (Khh )2 −2N −1 ∂¯0 Khh −2N −1 Δ̄N,
(VIII.7.21)
and1
1
S00 ≡ R00 − g00 R ≡ (R00 + g ij Rij ).
2
2
It follows that
2N −2 S00 ≡ −2S00 ≡ R̄ − Kij K ij + (Khh )2 .

(VIII.7.22)

VIII.7

VIII.7.2

Geometric n + 1 splitting

187

Dynamical system with constraints
for ḡ and K

We see in the above decomposition of the Ricci tensor that none of thecomponents of the Einstein tensor contains the time derivatives of thelapse N or the shift β. One is thus led to consider the Einstein equations
as a dynamical system with constraints for the two fundamental tensorsḡ and K of the space slices Mt , while N and β are gauge variables.27
Constraints
The expressions for the Ricci tensor lead to the following constraints
for ḡ and K on space slices Mt for solutions of the Einstein equations:28

28 These constraints correspond to theGauss and Codazzi equations wellknown to geometers.

Hamiltonian constraint
2N −2 S00 ≡ −2S00 ≡ R̄ − Kij K ij + (Khh )2 = ρ,

27 Darmois (1927) was the ﬁrst to viewthe Einstein equations (in a Gaussiangauge N = 1, β = 0) as an evolution
system for ḡ, K supplemented by initialconstraints.

with ρ := −2T00 ;(VIII.7.23)

for the usual classical sources, ρ ≥ 0.
Momentum constraint¯ h K h j = Jj ,N −1 S0j ≡ N −1 R0j ≡ ∂j Khh − ∇with

Jj := −N −1 T0j .
(VIII.7.24)

These constraints coincide with those found previously by using thewave gauge and shown to be preserved under evolution for stress–energy
tensor sources satisfying the conservation laws.
Evolution
The equations¯ j ∂i N∇∂ 0 Kij− 2Kjh Kih + Kij Khh −= ρij , (VIII.7.25)
NN
together with the deﬁnition
Rij ≡ R̄ij −∂¯0 gij = −2N Kij ,(VIII.7.26)

determine the derivatives transverse to Mt of ḡ and K when these tensors
are known on Mt , as well as the lapse N and shift β and the source ρij .
It seemed natural to look at these equations as evolution equations determining ḡ and K, while N and β, which are projections of the tangent
vector on the timeline respectively on e0 and on the tangent space to M,
are considered as gauge variables. These evolution equations have beenused extensively in the past for numerical computations. The Cauchy
problem for these equations is clearly well posed for analytic data—aphysically unsatisfactory condition because analytic functions have nonlocalizable support. The data of lapse and shift does not seem to be agood gauge choice. However, it can be proved29 that, given N and β, theoperator Rij given by (7.25), with Kij given by (7.26), is a second-order
diﬀerential system on the gij that is Leray–Ohya30 hyperbolic and

29 See YCB-OUP2009, Chapter VIII,
Section 3.

30 See J. Leray and Y. Ohya (1968).

188

General Einsteinian spacetimes. The Cauchy problem

for zero shift; Choquet-Bruhat andYork (1996) for arbitrary shift.

causal. The Cauchy problem for such systems has solutions in Gevrey
classes,31 causally determined from initial data. However, the inﬁnite
number of derivatives required from a Gevrey class foreshadow the poor
stability of results of numerical computation, as has been conﬁrmed bynumerical analysis.
More promising, appropriate combination of the evolution equationand the constraints have been shown to lead,32 modulo a gauge choice33
for N, to a quasidiagonal system of wave equations for K that together,with (7.1), give an hyperbolic causal system for ḡ and K.
Another possibility, leading to a mixed elliptic–hyperbolic system, is
to impose on N an elliptic equation that implies that the trace of K
takes a given value.34

33 Called an algebraic gauge in the ﬁrst

Preservation of constraints

31 Gevrey

classes are C ∞ functions
whose derivatives satisfy inequalities
weaker than those satisﬁed by analytic
functions and assure in this case theconvergence of their expansion in Taylor series. In contrast to an analytic
function, a Gevrey class is not determined by the values of all its derivatives
at a given point, nor by its value in an
open set; it can have compact support.

32 Choquet-Bruhat and Ruggeri (1983)

formulation of Choquet-Bruhat andRuggeri, but now called densitizing thelapse; it is equivalent to a wave gauge
choice for the time variable.
34 Christodoulou and Klainerman(1989).
35 Choquet-Bruhat and Noutchegueme(1986) and, independently, Anderson
and York (1999).
36 A

system of N ﬁrst-order partial
diﬀerential equations in N unknown
scalar functions of n + 1 variables withprincipal part
AI,α
J

∂uJ
,
∂xα

I, J = 1, . . . , N

is symmetric hyperbolic if the matrices
are symAα with (real) elements AI,αJ
metric and the matrix A0 is positivedeﬁnite. The Cauchy problem for such
systems has one and only one solution, in relevant functional spaces(Friedrichs, 1954). See, for instance,
YCB-OUP2009, Appendix IV.

As shown before for the wave gauge constraints, the geometrically formulated constraints (7.23) and (7.24) are preserved under evolution when
the stress–energy tensor source satisﬁes the conservation laws. A simple
proof follows.35
Theorem VIII.7.1 If Rij − ρij = 0 holds, then the constraints satisfy
a linear homogeneous ﬁrst-order symmetric hyperbolic system.36 If they
are satisﬁed initially, then they are satisﬁed for all time.
Proof. If Rij − ρij = 0, then we have, in the Cauchy adapted frame,with ρ := g αβ ραβ ,R − ρ = −N 2 (R00 − ρ00 ).
Hence
S 00 − T 00 =1 00
(R − ρ00 ) and R − ρ = −2N 2 (S 00 − T 00 ) = 2(S00 − T00 )2

and1
S ij − T ij = − ḡ ij (R − ρ) = −ḡ ij (S00 − T00 ).
2
With these identities, we may derive from the Bianchi identities and theconservation laws a linear homogeneous system for Σi0 ≡ S0i − T0i andfor Σ00 ≡ S00 − T00 with principal parts
N −2 ∂0 Σi0 + ḡ ij ∂j Σ00 , and∂0 Σ00 + ∂i Σi0 .

Since this system can be made symmetric hyperbolic, it has a uniquesolution, which is zero if the initial values are zero.
Exercise VIII.7.2 Show that Σi0 and Σ00 satisfy a quasidiagonal
homogeneous second-order system with principal part the wave operatorof the spacetime metric g.

VIII.7

VIII.7.3

Geometric n + 1 splitting

189

Geometric Cauchy problem. Regularity
assumptions

To give a geometric formulation of the solution of the Cauchy problem
for the Einstein equations with data on a manifold M , we introduce thefollowing deﬁnitions (in the usual case, n = 3).
Deﬁnition VIII.7.1
1. An initial data set is a triple (M, ḡ, K) with (M, ḡ) a Riemanniann-dimensional manifold and K a symmetric 2-tensor on M.
2. An Einsteinian vacuum development of (M, ḡ, K) is a Lorentzian(n+1)-manifold (V, g) solution of the vacuum Einstein equations suchthat M is an embedded submanifold of V and ḡ is induced by g on
M , while K is the extrinsic curvature of M in (V, g).
The following existence theorem holds as a direct consequence of previously stated and known results on hyperbolic diﬀerential equations. The
weakest hypotheses on initial data using only classical Sobolev spaces
and energy estimates are when n = 3, ḡ ∈ H3loc (M ), K ∈ H2loc (M ).
Recall that Hsloc (M ) is a space of tensor ﬁelds with components that,along with their derivatives of order less than or equal to s, are squareintegrable in relatively compact domains of local coordinates covering
M.37 The local uniqueness up to isometries is a consequence of thegeometric uniqueness given in Theorem VIII.5.1.
Theorem VIII.7.2 (existence and local uniqueness up to isometries)
An initial data set (M, ḡ, K) for the Einstein vacuum equations satisfying the constraints admits a vacuum Einsteinian development (V, g).
This development is locally unique, up to isometries.

37 Various, more intrinsic, equivalentdeﬁnitions can be found, for instance,
in YCB-OUP2009, Appendix III, Section 3.7.

Remark VIII.7.3 The hypothesis g ∈ H3loc when n = 3 is not suﬃcient to ensure the uniqueness of geodesics issuing from one point witha given tangent. This uniqueness would be ensured if the Christoﬀelsymbols were Lipschitzian, which is not the case for H3loc metrics on
three-dimensional space.
Lowering the regularity required of data is conceptually important for
the understanding of the mathematics and the physics of the theory,and essential in the study of global problems. Geometric hyperbolic evolution systems involving the Riemann curvature tensor instead of wave
gauges have been considered and have allowed for the broadening of admissible functional spaces for the initial data. Klainerman, Rodnianski,
and Szeftel38 have conjectured that suﬃcient conditions for the existenceof a Lorentzian metric solution of the vacuum Einstein equations in afour-dimensional neighbourhood of a 3-manifold M supporting initial¯ ∈ L2 locally on M , anddata (ḡ, K) are that Ricci(ḡ) ∈ L2 , that ∇K
39
that the volume radius of (M, ḡ) is strictly positive. In several longand diﬃcult papers using reﬁned functional analysis, they have proved
their conjecture interpreted as a continuation argument for the Einsteinequations; that is, the spacetime constructed by evolution from smooth

38 Klainerman, Rodnianski, and Szeftel (2012).

39 The volume radius is the lower boundof the quotient by r 3 of the volume ofgeodesic balls of radius r in (M, ḡ).

190

General Einsteinian spacetimes. The Cauchy problem

data can be smoothly continued, together with a time foliation, as longas the curvature of the foliation and the ﬁrst covariant derivatives of
its second fundamental form remain L2 -bounded on the leaves of thefoliation.

VIII.840 Lichnerowicz (1944) for τ = 0; York(1972) for τ = constant.

41 Choquet-Bruhat(1971) for τ = 0;
York (1972) for τ = constant.

The conformal method, initiated by Lichnerowicz and developed by
Choquet-Bruhat and York, has been intensively used to prove existenceof solutions of the constraints. Variants of it are now used in numerical computations of initial data for the two-body problem. It has theadvantage of turning the momentum constraint into a linear system independent of the conformal factor for a weighted extrinsic curvaturewhen the mean extrinsic curvature τ is zero or constant.40 The conformal factor then satisﬁes a semilinear elliptic operator with principal
part the Laplace operator of the conformal metric. Given a TT (transverse traceless) tensor, the momentum constraint can be written as alinear elliptic system for a vector ﬁeld.41 The method was used in earlynumerical computations, essentially by taking conformally ﬂat metrics.

VIII.8.1

42 The case of arbitrary n is treated indetail in YCB-OUP2009, Chapter VII,
Section 3.

Solution of the constraints
by the conformal method

Conformally formulated (CF) constraints

When the sources ρ and J are known, the unknowns in the constraints(7.24) and (7.23) are the metric ḡ and the tensor K. It is mathematically clear that these scalar and vector equations do not have a uniquesolution, even geometrically. Physically, this fact corresponds to theproperty of Einsteinian gravity that it is its own source. Roughly speaking, ‘radiation data’ should also be given, the constraints then becoming
an elliptic system for a scalar function and a vector ﬁeld. The geometricmeaning of the lower-order terms permits discussion of the existence anduniqueness of solutions on general manifolds. We treat for simplicity ofnotation the usual physical case n = 3.42
The Hamiltonian constraint (7.23) reads, for general n,R̄ − (|K|2ḡ + ρ − τ 2 ) = 0, R̄ := R(ḡ), τ := trḡ K.

(VIII.8.1)

This equation is turned into an elliptic equation for a scalar function ϕby considering the metric ḡ as given up to a conformal factor. In thecase n = 3, one setsḡij = ϕ4 γij ,with γ a given metric on M and ϕ a function to be determined.
The scalar curvatures of the conformal metrics ḡ and γ are found bystraightforward computation to be linked when n = 3 by the formulaR(ḡ) ≡ ϕ−5 [ϕR(γ) − 8Δγ ϕ] .

(VIII.8.2)

VIII.8

Solution of the constraints by the conformal method

Exercise VIII.8.1 Prove an analogous formula for general n ≥ 3 with
ϕ4/(n−2) .
Hint: The choice ϕ4 when n = 3 is made to eliminate quadratic terms
in ﬁrst derivatives of ϕ.
The identity (8.2) implies the following theorem:
Theorem VIII.8.1 When γ, K, and the source ρ are known, the Hamiltonian constraint is a semilinear second-order elliptic equation for ϕ,linear also in ﬁrst derivatives, which, in the case n = 3, reads
8Δγ ϕ − R(γ)ϕ + (|K|2ḡ − τ 2 + ρ)ϕ5 = 0.

(VIII.8.3)

To solve the momentum constraint, we split the unknown K intoa weighted traceless part K̃ and its trace τ. In the case n = 3, we set
1
Kij = ϕ−2 K̃ij + ḡij τ.
3(VIII.8.4)

Equivalently, with indices in K and K̃ being respectively lifted with ḡ ij
and γ ij ,1
K ij = ϕ−10 K̃ ij + ḡ ij τ,3ḡ ij = ϕ−4 γ ij .

(VIII.8.5)

The tensor K̃ is symmetric and traceless; indeed,1trγ K̃ ≡ γ ij K̃ij = ϕ−2 ḡ ij Kij − ḡij τ = 0.
3
Straightforward computation shows that the momentum constraint
becomes, with D the covariant derivative in the given metric γ,
Di K̃ ij =2 6 ijϕ γ ∂i τ + ϕ10 J.
3(VIII.8.6)

This equation has the interesting property that it does not contain ϕwhen J is zero and τ is constant on M, that is, when M is be a submanifold with constant mean extrinsic curvature in the ambient spacetime.
We have the following theorem:
Theorem VIII.8.2 In the case τ = constant and J = 0, the symmetric 2-tensor K̃ ij := ϕ−10 K ij is a TT tensor (transverse, traceless) andthe momentum constraint reduces to linear homogeneous sytem for K̃:
Dj K̃ ij = 0,

γij K̃ ij = 0.

(VIII.8.7)

Exercise VIII.8.2 Show that the space of TT tensors is the same fortwo conformal metrics.
Returning to the Hamiltonian constraint, we compute, using thedeﬁnition of K̃,1|K|2ḡ := ḡ ih ḡ jk Kij Khk = ϕ−12 |K̃|2γ + τ 2 .
3(VIII.8.8)

191

192

General Einsteinian spacetimes. The Cauchy problem

Exercise VIII.8.3 Prove this formula.
Hints: K̃ is traceless and γ ih γ jk γij γhk = δjh δhj = 3.
43 It

was obtained by Lichnerowicz(1944) for n = 3 and extended togeneral n by Choquet-Bruhat (1996).
Scaling of the sources was introduced
by York (1972). We will still refer to
as Lichnerowicz equations all the equations deduced from the Hamiltonian
constraint by the conformal method.
44 York (1972).

Exercise VIII.8.4 Write down the corresponding formula for arbitrary n.
The Hamiltonian constraint now reads
2
8Δγ ϕ − R(γ)ϕ + ϕ−7 |K̃|2γ + − τ 2 + ρ ϕ5 = 0.
3

(VIII.8.9)

It is a semilinear elliptic equation for ϕ when γ (a Riemannian metric),
τ, ρ, and K̃ are known. It is called the Lichnerowicz equation.43
A scaling of the momentum of the sources by setting, when n = 3,44
˜
J = ϕ−10 J,
with J˜ considered as a known quantity, permits the extension of the decoupling property to the non-vacuum case. The momentum constraints
now read
Dj K̃ ij = J˜i .
The scalar part ρ of the sources can also be scaled. The scaling proposedby York, justiﬁed by physical considerations (at least in the case n = 3
for electromagnetic ﬁeld sources), isρ = ϕ−8 ρ̃,with ρ̃ considered as a given function. The Hamiltonian constraint thenreads
2
8Δγ ϕ − R(γ)ϕ + |K̃|2γ ϕ−7 + ρ̃ϕ−3 − τ 2 ϕ5 = 0.
(VIII.8.10)
3

VIII.8.2

45 Choquet-Bruhat(1971a) and York(1972); Fischer and Marsden (1979),
Choquet-Bruhat and York (1980),
Choquet-Bruhat, Isenberg and York(2000) and see further references in
Choquet-Bruhat and York (2002). For
the coupling with a scalar ﬁeld see
Choquet-Bruhat, Isenberg and Pollack(2007).

Elliptic system

We complete the writing of the CF constraints as an elliptic partial
diﬀerential equation system on the initial manifold by a treatment of the
momentum constraint45 analogous to that indicated in Section VIII.6.
We can split the tensor K̃ into an element of the kernel of the homogeneous part of the momentum constraint operator, i.e. a TT tensor Y forthe given metric γ, and an element of the range of the dual operator,
the conformal Lie derivative Z of a vector ﬁeld X; that is, we set
K̃ij = Yij + Zij ,withDi Y ij = 0,

(VIII.8.11)

and, when n = 3,
2
(VIII.8.12)
Zij := (Lγ,conf X)ij ≡ Di Xj + Dj Xi − γij Dk X k .
3
A tensor of this form satisﬁes the momentum constraint if and only if
the vector X satisﬁes the second-order system˜Δγ,conf X := D · (Lγ,conf X) = D · J.

VIII.9

Motion of a system of compact bodies

193

Exercise VIII.8.5 Show that this system is a linear elliptic system
for X when γ and J˜ are known.
The search for the arbitrary TT tensor Yij can itself be done throughthe data of an arbitrary traceless tensor Uij by setting Y := Lγ,conf V +U
and imposing the requirement that the vector V satisfy the following
elliptic system:
Δγ,conf V ≡ D · (Lγ,conf V ) = −D · U.
The traceless tensor Y is then also transverse, i.e. satisﬁes D · Y = 0.
The arbitrary data in the extrinsic curvature K is in this scheme
the symmetric traceless tensor U ij . The physical extrinsic curvature isgiven by
 1
Kij = ϕ−2 Lγ,conf (X + V )ij + U ij + ḡ ij τ.
3The splitting of the solution K̃ into a given traceless tensor U and theconformal Lie derivative of an unknown vector X depends on the choiceof γ, not only on its conformal class.46
Exercise VIII.8.6 Prove this statement.

VIII.8.3

Physical comment

The conformal method is a mathematical, geometric convenience. The
tilde (∼) quantities are not directly observable. There is a large arbitrariness in the choice of their scaling, although some justiﬁcations are
given a posteriori, at least for electromagnetic ﬁeld sources. The tilde
quantities play the role of parameters to construct initial data solutionsof the constraints that can, hopefully, be used for evolution and lead toresults that can be confronted with observations.

VIII.9

Motion of a system of compact
bodiesThere is now a wealth of new results from observations of motions ofstars and galaxies obtained by powerful Earth-based and satellite telescopes. Also, a new generation of gravitational wave detectors (LIGO,
GEO, TAMA, and VIRGO) are now operational, and it is hoped thatwithin the next few years they will reach sensitivities that will allow forthe ﬁrst time observations of gravitational radiation. The prime targetsof these observations are the motions of compact binaries: black holes
and neutron stars. A fundamental problem for Einsteinian gravitation
theory is the modelling of the motion and the gravitational wave emission of systems of compact bodies. For many years, considerable eﬀort
has been spent towards this goal, with little success until fairly recently,
because of the nonlinearity of the equations, their hyperbolic–elliptic

46 More reﬁned splittings have been
given by York under the name of conformal thin sandwich.

194

General Einsteinian spacetimes. The Cauchy problem

47 The problem becomes arduous whenmore than two bodies are involved.

48 For recent comparisons, see, for in-character, and the insuﬃcient stability of the solution of the chosen evolution equations, in particular the so-called ADM ones for ḡ and K. In
Newtonian theory, two bodies represented by pointlike masses describe
conics with focus situated at the centre of gravity.47 In Einsteinian gravitation, the dynamics of two-body systems is already too complex to besolved exactly, and prediction of the gravitational radiation they will
emit in their inspiralling and merger can be obtained only by reﬁned
approximation methods (analytical or numerical) after a clever choiceof unknowns and gauges, and splitting of equations between constraints
and evolution. The mathematical well-posedness of the considered system is of course essential for the stability and reliability of the results.
Reﬁned analytical results have been obtained in the last decade bothby the high-order post-Newtonian approach and by using the eﬀective
one-body (EOB) method. In parallel, thanks to the advent of extremely
powerful computers, accurate numerical simulations of two-body systems
have been obtained using various numerical codes. The results of the two
types of methods (analytical and numerical) have been compared andfound to be in remarkable agreement.48

stance, Damour, Nagar, and Bernuzzi(2013) and Hinder et al. (2014).

VIII.9.1
49 See Damour and Nagar (2011).

The EOB formalism49 is an analytical approach that replaces the usual
post-Newtonian expanded approximations to the motion and radiationof binary systems by resummed expressions. This formulation allows one
to describe the dynamics of binary black hole systems up to the momentof the coalescence of the two black holes, and is further completed by
the description of the ﬁnal ringdown gravitational wave signal emitted
by the distorted black hole formed during the coalescence process. This
formalism thereby provides a quasi-analytical description of the entire
waveform emitted by a binary black hole system from early inspirallingup to the ﬁnal ringdown. It has also been extended to the description of
the coalescence of binary neutron star systems.

VIII.9.2

50 See Pretorius (2007) and references

therein.
51 Baumgarte–Shapiro–Shibata–

Nakamura.
52 See, for instance, Rezzolla and Zan-

otti (2013).

Eﬀective one-body (EOB) method

Numerical Relativity

Although its origins lie in the 1960s, it is in the years since 2005 that
Numerical Relativity has seen considerable expansion, owing to theenormous growth in the power of computers and to the developmentof stable codes describing the coalescence of compact binary systems.
This has permitted the modelling of more realistic situations than before, although diﬃculties remain. Two main codes are now in use: one
is based on damped generalized wave coordinates,50 while the other,called BSSN,51 is a modiﬁcation of the 3+1 decomposition using conformal weights. The code BSSNz4 improves the stability of results byintroducing four more unknowns linked with the harmonicity functions.
Numerical Relativity is a science in itself, outside of the scope of thisbook.52

VIII.10

VIII.10Global properties

VIII.10.1

Global hyperbolicity and global uniquenessA fundamental notion in the study of global properties of solutions ofhyperbolic partial diﬀerential equations is that of global hyperbolicity, deﬁned by Jean Leray53 for general hyperbolic diﬀerential equations
as compactness54 (in a functional space) of paths joining two points
whose tangent belong to the cone determining the domain of dependence of solutions of the Cauchy problem; that is, causal paths in the caseof relativistic causal systems. Global hyperbolicity forbids, in particular, the existence of closed causal curves.55 Later, Penrose56 introduced
what he called strong causality: a Lorentzian manifold (V, g) is said
to be strongly causal if any neighbourhood of any point x contains asmaller neighbourhood such that no causal curve penetrates in it morethan once. It can be proved57 that global hyperbolicity is equivalent tostrong causality together with compactness of the subsets of V that areintersections of the past of any point x with the future of any other pointy, traditionally denoted by I− (x) ∩ I+ (y).
A very useful deﬁnition due to R. Geroch is that of a Cauchy hypersurface in a Lorentzian manifold (V, g), that is, a submanifold S
of codimension 1 intersected once and only once by each inextendible
timelike curve. Geroch58 proved that the existence of a Cauchy surface is equivalent to global hyperbolicity. He also proved the important
property that the support of the spacetime is then a product S × R.
With these deﬁnitions, one can complete the existence theorem by aglobal uniqueness theorem proved by Choquet-Bruhat and Geroch;59 ituses in particular Zorn’s lemma. Proofs without the use of this lemma
have been published recently.60
Theorem VIII.10.1 (existence and geometric global uniqueness) A
vacuum Einsteinian development (V, g) of an initial data set (M, ḡ, K),
satisfying the vacuum Einstein constraints, exists and is unique (up to
an isometry) in the class of maximal 61 globally hyperbolic spacetimes.
The manifold M embedded in (V, g) is a Cauchy hypersurface.

VIII.10.2

Global properties

195

53 Leray (1953). The notion was appliedand described in detail for Lorentzianmanifolds by Choquet-Bruhat (1968).
54 The empty set is considered as compact.
55 A path is a mapping from an interval of R into the manifold; a curve isthe image of a path in the manifold.
A closed causal curve can be coveredby an inﬁnite sequence of causal paths
joining two points, a non-compact set.
56 See, for instance, Penrose (1968).
57 See Choquet-Bruhat (1968).

58 Geroch (1970). For a detailed proof,see, for instance, Ringström (2009).

59 Choquet-Bruhat and Geroch (1969).
60 See Ringström (2009), corrected by

Ringström in <http://www.math.kth.
se/∼hansr/mghd.pdf> and in Sbierski(2013).
61 Which cannot be embedded in a bigger one.

Global existence

A solution of the vacuum Einstein equations is called global if it is acomplete Lorentzian manifold. It is generally called singular if it is incomplete. Incompleteness can result from the appearance of a curvature
singularity (with the Kretschmann scalar62 tending to inﬁnity on a
causal curve for a ﬁnite value of the canonical parameter) or anotherphenomenon.63 Proving either global existence of solutions or, on thecontrary, their incompleteness and the formation of singularities, arediﬃcult problems and the subject of active research. Some remarkableachievements linked with the special properties of the Einstein equations, fundamental in physics but quite particular nonlinear geometric

62 The square of the Riemann tensor.
63 See Problem IV.11.4 on Taub spacetime.

196

General Einsteinian spacetimes. The Cauchy problem

64 Friedrich (1986).
65 Thisformalism uses on a fourdimensional manifolds Weyl spinors
represented by objects with 2-index
components.
For a tensorial formulation of
the Friedrich conformal system, see
Choquet-Bruhat and Novello (1987),
CB-DM II, Chapter V, Section 7.

66 SeeChristodoulou and Klainerman(1993). See also Klainerman and Nicolo(2003) and Bieri (2007).

67 This theorem showed that Einstein’s

intuition that the only global asymptotically Euclidean vacuum solution of
his equations is Minkowski spacetime
required one more hypothesis to betrue, namely a faster fall-oﬀ of the metric at spatial inﬁnity, which implies thevanishing of the ADM mass.
68 Their proof has been notably simpli-

ﬁed by Lydia Bieri. For the use of equations satisﬁed by the Riemann tensor,
see Problem I.14.5 in Chapter I.
69 Lindblad and Rodnianski (2005).

VIII.1170 Rodnianski and Speck (2009).
71 Seearticles and references
Chruściel and Friedrich (2004).

diﬀerential equations, have been obtained in the last twenty years withthe use of advanced and intricate mathematics. The details of theseachievements are outside of the scope of this book (in fact of any bookof reasonable size!). We will only give some deﬁnitions and quote someresults with references to original papers.
A future global existence theorem for data near Minkowskian dataon a manifold tending to a null manifold at inﬁnity was given by H.
Friedrich64 using an original conformal construction and the Newman–
Penrose formalism.65
A breakthrough in the problem of global existence for initial dataon a spacelike manifold was the proof in 1989 by Christodoulou andKlainerman66 of the global nonlinear stability of Minkowski spacetime—that is, the construction of complete globally hyperbolic solutions of thevacuum Einstein equations with asymptotically Euclidean initial datanear data for the Minkowski spacetime.67 They used in particular adouble null foliation and delicate estimates of the Riemann tensor.68
The proof of the global nonlinear stability of Minkowski spacetime wasobtained later by Lindblad and Rodnianski69 through wave coordinates
and the use of special properties (a kind of generalized null condition)of the Einstein equations. The global existence result has been extended
to cases of Einstein equations with ﬁeld sources, and even to the case or
irrotational ﬂuid sources.70
Proofs of global existence have been obtained for small initial data on
several categories of compact manifolds with symmetries by: Moncrief,
by Choquet-Bruhat and Moncrief, and by Choquet-Bruhat, and, in acase without symmetries, by Andersson and Moncrief.71

in

72 See Hawking and Ellis (1973).

The problem of the formation of singularities from generic initial data
was attacked by Penrose and Hawking in the 1970s.72 Inspired by the
Schwarzschild solution, they discussed what they called the strong andthe weak cosmic censorship conjectures. These conjectures concern generic Einsteinian spacetimes with physically reasonable sources.
A generic spacetime can be understood as a spacetime with no isometry group, or as a spacetime that is stable (in some sense to bedeﬁned), under small perturbations. Reasonable sources are physicalsources that have a hyperbolic, causal evolution and do not have theirown singularities (shocks, shell crossings, etc.).

VIII.11.1
73 Suggested

in private discussions by
Geroch and Penrose in 1969, andformalized by Eardley and Moncrief (1981).

Singularities and cosmic
censorship conjectures

Strong cosmic censorship conjecture

The strong cosmic censorship conjecture73 aims at proving the deterministic character of General Relativity at the classical (non-quantum)
level. It can be formulated as follows:

VIII.11

Singularities and cosmic censorship conjectures

Strong cosmic censorship conjecture The maximal globally hyperbolic vacuum Einsteinian development of generic74 initial data is
inextendible as a vacuum, even non-globally hyperbolic, Einsteinian
spacetime.75
The inextendibility is sometimes conjectured to hold for all developments, not necessarily solutions of the Einstein equations—this iscertainly the case when incompleteness is due to the appearance ofinﬁnite curvature.
Examples of non-globally hyperbolic extensions of a globally hyperbolic spacetime with initial data on S 3 are provided by the
Taub–NUT spacetimes.76 However, these extensions do not provide a
counterexample to the strong cosmic censorship conjecture, because the
Taub spacetime, because of its symmetries, is not generic. V. Moncrief
and J. Isenberg have shown that some qualitative features of Taub–NUT
spacetime imply in fact the existence of an isometry group.

VIII.11.2

197

74 Loosely speaking ‘generic’ means‘without special properties’. In a more
precise mathematical sense, it can beinterpreted that ‘generic initial data’form a dense subset of the set of possible initial data in some relevant topology, see for instance discussion in
Isenberg and Moncrief (2002).
75 The formulation leaves open thesmoothness required of extendibility
and also the hypotheses on initial data.
It was pointed out by P. Chruściel thatthe requirement that the initial manifold (M, ḡ) be complete is not suﬃcient
to make the conjecture plausible—
initial data for Minkowski spacetimeon a hyperboloid are then a trivial
counterexample.
76 See Problem IV.11.4 in Chapter IV.

Weak cosmic censorship conjecture

The original idea of Penrose came from the study of spherical gravitational collapse where a black hole forms, hiding the singularity totimelike observers. The weak cosmic censorship conjecture77 is thatin generic Einsteinian spacetimes with physically reasonable sources, it
is not possible for there to form any naked singularity, that is, a singularity visible by an observer; in other words, the past of no point x
contains a future causal curve that is inextendible.
Note that the big bang is not a counterexample to this conjecture—it
has no past, and hence does not correspond to any future inextendiblecausal curve. Nor is the Schwarzschild metric with m < 0 on the manifold (R3 − {0}) × R a counterexample to the ‘non-naked singularity’conjecture, because it is not considered to be a physically meaningful
metric.
A more elaborate version of the weak cosmic censorship conjectureuses the Penrose deﬁnition of conformal null inﬁnity—it raises technicaldiﬃculties regarding the existence and smoothness of the boundary of a
Penrose diagram for general spacetimes.
Christodoulou has obtained in a series of papers complete results supporting the weak cosmic censorship conjecture for the Einstein–scalar
equations with spherical symmetry.
An important theorem due to Penrose links the existence of singularities with trapped surfaces.78 In a four-dimensional spacetime, atrapped surface S is a two-dimensional compact spacelike surface
without boundary (typically a sphere) such that the two families of
future-directed null geodesics orthogonal to S have a positive convergence whether they are directed inwards or outwards. Penrose’s theorem
says that in a non-compact spacetime, a trapped surface always hides
an incompleteness of the spacetime.

77 The weak and strong cosmic conjectures are independent.

78 Penrose (1965).

198

General Einsteinian spacetimes. The Cauchy problem

79 Christodoulou (2009).
80 Klainerman,ski (2013).

Luk,

and

Rodnian-

In a 500-page book79 Christodoulou proves the formation oftrapped surfaces due to ‘short-pulse’ data on a characteristic cone.
Christodoulou’s80 construction has recently been extended by Klainerman, Luk, and Rodnianski to data on a null geodesic segment and his
proof has been simpliﬁed.

VIII.12

Problems

VIII.12.1

Symmetric hyperbolic systems

The system of N ﬁrst-order linear partial diﬀerential equations on
Rn × R,Mα∂u + Au + f = 0,∂xαi.e. Mα,IJ∂
uI + AIJ uI + f J = 0,∂xα(VIII.12.1)

with u = (uI , I = 1, . . . , N ) a set of N unknown functions, f = (f I , I =1, . . . , N ) a set of given functions, and Mα and A given N × N matrices,
is called symmetric hyperbolic (SH) if the matrices Mα are symmetric (Mα,IJ = Mα,JI ) and the quadratic form deﬁned by the matrix
M0 , the coeﬃcient of ∂/∂x0 , is positive-deﬁnite; that is,
M0 (u, u) ≡ M0,IJ uI uJ > 0 for all u = 0.
1. Deﬁne a function called the energy of u at time x0 = t by the
integral1
M0 (u, u) dn x, dn x := dx1 . . . dxn . (VIII.12.2)
Et (u) :=2 x0 =tShow that under appropriate smoothness and boundary conditions,
Et (u) satisﬁes the energy equality
 

1
∂
αIJ
M
uJ uI
∂xα
x0 =t 2IJ
J
+ A uJ uI + f uJ dn x dt.

 T
ET (u) = Et0 (u) +t0(VIII.12.3)
2. Show that the energy equality implies an integral energy inequality of the form
ET (u) ≤ Et0 (u) +

 T
t0


1
C1 (t)Et (u) + C2 (t)Et2 (u) dt. (VIII.12.4)

Deduce from it a bound of the energy.

VIII.12

Problems

199

Solution
1. It holds thatuJ Mα,IJ∂∂uI ≡ uI Mα,JI α uJ ,α∂x∂x

and hence, if the matrices Mα are symmetric (i.e. Mα,IJ = Mα,JI ),thenuJ Mα,IJ∂∂1
uI = Mα,IJ α (uI uJ )
∂xα
2
∂x
∂
1
1 ∂
(uI uJ Mα,IJ ) − uI uJ α Mα,IJ .
≡
2 ∂xα
2
∂x

Under appropriate smoothness and fall-oﬀ conditions at inﬁnity,
the integral on Rn of (∂/∂xi )(uI uJ Mi,IJ ) vanishes owing to Stokes’s
formula. There remains the equality
 

 T
1 ∂
∂
α,IJ
α,IJ
(uI uJ M) − uI uJ α M
2 ∂x0
∂x
t0
x0 =t
+ AIJ uI uJ + f J uJdxn dx0 = 0,

which gives the energy equality after performing in the ﬁrst term theintegrations ﬁrst with respect to xn and then over x0 .
2. Assume that M0 is uniformly positive-deﬁnite, i.e.
M0,IJ uI uJ ≥ C0 |u|2 ,C0 > 0, a constant,while (∂/∂xα )Mα,IJ and AIJ are uniformly bounded and f J issquare-integrable for each t. It is straightforward to deduce an integral energy inequality of the given form from the energy equality using
elementary properties of integrals and the Cauchy–Schwarz inequality" " 12 
 12
"
"
n "
2 n
2 n
"
uf d x" ≤|u| d x|f | d x
.
"
Rn

Rn

Rn

To prove the energy inequality, one uses a general theorem81 for differential equations that says that if f (t, y) is a function continuous in t
and Lipschitzian in y, then the diﬀerential equationz  = f (t, z)
has one and only one maximal solution taking a given initial value. Any
C 1 function y satisfying the inequalitiesy  ≤ f (t, y),y(t0 ) ≤ z(t0 )satisﬁes y(t0 ) ≤ z(t0 ).
This theorem implies that if y(t) satisﬁes the integral inequality T1C1 y(t) + C2 y(t) 2 dt,(VIII.12.5)
y(T ) ≤ y(t0 ) +t0

81 See the proof for instance in Choquet (2006).

200

General Einsteinian spacetimes. The Cauchy problem

82 A result sometimes called Grönwall’s

lemma.

with C1 and C2 non-negative, then y(t) is less than or equal to thefunction z satisfying the corresponding equality82 and such that z(t0 ) ≥y(t0 ).
In the considered case, y(t) := Et (u), one ﬁnds that z is the solutionof an equation of the form1z  = C1 z + C2 z 2 ,i.e. setting z = ζ 2 , the linear equation2ζ  = C1 ζ + C2 ,1which has a solution with initial data ζ0 := y02 continuous and bounded
for all ﬁnite t. Writing down this solution is left as an exercise for thereader.

VIII.12.2

The wave equation as a symmetric
hyperbolic system

1. The wave equation with mass a constant m for a scalar function u ona Lorentzian manifold (V, g) is
g αβ ∇α ∂β u − mu + f = 0.
For g = η, the Minkowski metric, write this as a symmetric hyperbolic
system.
For m a constant, compare the ‘mathematical energy’ of u deﬁnedby (12.2) with the physical energy as it would be deﬁned from thewave equation by multiplying it with ∂u/∂t.
2. Extend the study to the case when (V, g) is a sliced Lorentzianmanifold.

Solution
1. Recall our signature (−, + + . . . +). The equation reads2−η αβ ∇α ∂β u + mu ≡ ∂00
u−



2
∂ii
u + mu = f, ∂α :=

i

∂
.
∂xα

Set uα = ∂α u and consider the system Mα ∂α U with unknowns UI =(u, uα ):∂0 u = u0 , ∂0 ui − ∂i u0 = 0, ∂0 u0 −i∂i ui + mu = f

VIII.12

The principal operator Mα ∂α is the matrix
⎞
⎛
∂0
⎜
∂0
−∂1 ⎟
⎟
⎜
⎜
∂0
−∂1 ⎟
⎟.
⎜
⎝
∂0
−∂1 ⎠
−∂1 −∂2 −∂3∂0
The matrices Mα are obviously symmetric and M0 , the unit matrix,
is positive-deﬁnite.
Multiplication of the wave equation by ∂u/∂t gives by simple
computation the identity, if m is a constant:∂u ∂ 2 u  ∂ 2 u−+ mu − f∂t ∂t2(∂xi )2
i
1 ∂
≡
2 ∂t



∂u
∂t

2
+

  ∂u 2
i

∂xi

+ mu

2

 ∂  ∂u ∂u  ∂uf.
−=∂xi ∂t ∂xi∂ti

Hence, by integration when f ≡ 0, using Stokes’s formula, 2  
2

∂u
∂u
1
+
+ mu2 dx1 . . . dxn = constant if f ≡ 0.
i
2 Mt ∂t∂xi
The integrand is interpreted as the physical energy density at time t,
positive if m ≥ 0 and zero for m = 0 only if u = constant.
2. For a non-ﬂat Lorentzian metric on a sliced manifold, we take anorthonormal frame adapted to the slicing,g = ηαβ θα θβ ,with
θ0 = N dt, θi = aij dxj +β i dt, hence ∂0 =∂∂∂−β i i , ∂i = Aji j .
∂t
∂x
∂x

The wave operator reads
λg αβ ∇α ∂β u ≡ η αβ (∂α ∂β u + γαβ∂λ u).

Introducing the new unknowns uα = ∂α u and the identities
α∂0 ui − ∂i u0 = Ci0uαgives for the unknowns U = (u, uα ) a linear system with operator in
matrix formMα ∂α U + AU.
The matrices Mα are the same as in the Minkowski case, and hence
symmetric, and the matrix coeﬃcient of ∂/∂t in M0 is the Minkowski
M0 metric. The same reasoning applies as previously.

Problems

201

202

General Einsteinian spacetimes. The Cauchy problem

VIII.12.3

The evolution set of Maxwell equations
as a ﬁrst-order symmetric hyperbolic system

The evolution set of Maxwell equations on Minkowski spacetime M 4 ,with unknowns (E, H), comprises (see Chapter II)
∂E 1
∂H 3
∂H 2
=
−
− j,
2
∂t
∂x
∂x3

∂E 2
∂H 1
∂E 3
=
−
,
3
∂t
∂x
∂x2

(VIII.12.6)

together with the equations obtained by circular permutation of theindices 1, 2, 3. The characteristic matrix Mα ∂α is
⎛

∂0

⎜
⎜
⎜
⎜
⎜
⎜
⎝ −∂2
∂3

∂0
−∂2
∂1
0

∂0
−∂3
−∂1

−∂2
∂3
∂0

−∂2
∂1

∂0

⎞
∂3
0 ⎟
⎟
−∂1 ⎟
⎟.
⎟
⎟
⎠
∂0

It is symmetric, and M0 , the unit matrix, is positive-deﬁnite. Notethat in this case the energy densities corresponding to symmetric
hyperbolicity and to physics coincide, both being equal to 12 (|E|2 +|H|2 ).

VIII.12.4

Conformal transformation of the CF
constraints

Show that two diﬀerent choices of the given metrics γ and γ  leadto equivalent conformally formulated constraints if the sources areappropriately chosen.

VIII.12.5

83 SeeChoquet-Bruhat and Moncrief(2001) and Choquet-Bruhat (2004).

84 Moncrief (1986).

Einstein equations in dimension 2 + 1

The vacuum Einstein equations are trivial in the case n = 2 in thesense that Ricc(g) = 0 implies that the spacetime metric g is locally ﬂatwhen n + 1 = 3. However, the (2 + 1)-dimensional Einstein theory has
a topological content: (V, g) is not necessarily the Minkowski spacetime
M3 . In particular, V = S × R with S a two-dimensional compact surface
can be a Lorentzian ﬂat manifold with S the torus T 2 or a surface
of genus greater than 1. On the other hand, (2 + 1)-dimensional, nonﬂat, Einstein equations with sources appear for spacetimes that admit
a one-parameter spacelike isometry group.83
Write a conformal formulation of the constraints in the case n = 2 bysetting84
1ḡ = exp(2λ)γ and K ij = exp(4λ)K̃ ij + ḡ ij τ.
2

VIII.12

Solution
Elementary computation gives¯ i K ij − ḡ ij ∂i τ ≡ exp(4λ)Di K̃ ij − 1 γ̄ ij exp(−2λ)∂i τ.
∇
2
The constraints split again into a linear system for K̃ and a semilinear equation for λ when K̃ is known, if τ is a given constant and themomentum of the sources is zero, or properly weighted.

VIII.12.6

Electrovac Einsteinian spacetimes,
constraints

The Einstein equations with electromagnetic source are (see Chapter IV)
1(VIII.12.7)Sαβ = ταβ ≡ Fα λ Fβλ − gαβ F λμ Fλμ ,4where the electromagnetic 2-form satisﬁes the Maxwell equations
dF ≡ 0,i.e∇α Fβγ + ∇γ Fαβ + ∇β Fγα = 0,

(VIII.12.8)

δF = 0, i.e. ∇α F αβ = 0.

(VIII.12.9)

and, in vacuo,

The electromagnetic initial data on a hypersurface M are a 2-form F̄
and a vector ﬁeld Ē. The 2-form F̄ is the form induced on M by theelectromagnetic ﬁeld F, while Ē is the electric vector ﬁeld on M relative
to the unit normal n to M in the spacetime metric.
1. Show that in a Cauchy adapted frame, the components of Ē areĒ i := N̄ F̄ 0i ,with N the lapse of the metric g.

(VIII.12.10)

2. Show that a solution of the Maxwell equations must satisfy on M theconstraints
dF̄ = 0,i.e

¯ h F̄ij + ∇¯ j F̄hi + ∇¯ i F̄jh = 0,∇

(VIII.12.11)¯ i Ē i = 0.
∇

(VIII.12.12)

andSolution
1. E β = nα F αβ , nα = N δα0 , i.e E 0 = 0, E i = N F 0i .
2. dF = 0 and δF = 0 on M imply¯ h F̄ij + ∇¯ j F̄hi + ∇¯ i F̄jh = 0∂h F̄ij + ∂j F̄hi + ∂i F̄jh ≡ ∇
and
1 √
√
i0
0
αλ
g∂i F i0 + F i0 ∂i g = 0,
∇α F α0 ≡ ∂i F i0 +Γα≡√αi F +Γαλ F
Det g

Problems

203

204

General Einsteinian spacetimes. The Cauchy problem

that is, using Det g ≡ N 2 det ḡ,¯ Ē i = 0.
∂i (N̄ Det ḡ F̄ 0i ) ≡ ∇VIII.12.7(VIII.12.13)

Electrovac Einsteinian spacetimes,
Lorenz gauge

The vector potential A is a locally deﬁned 1-form such thatF = dA,
85 It

seems that this gauge, althoughused by Lorentz, was ﬁrst introduced
by another mathematician named Lorenz (without a ‘t’).

i.e.

Fαβ = ∂α Aβ − ∂β Aα .

(VIII.12.14)

The ﬁrst Maxwell equation dF = 0 is automatically satisﬁed. The
potential A is said to be in Lorenz85 gauge if it has zero divergence:∇α Aα ≡ 0.

(VIII.12.15)

Show that, in vacuo, the second Maxwell equation reduces in wave
coordinates and Lorentz gauge to a quasidiagonal semilinear system ofwave equations for the components of A of the formg αλ ∂λ ∂α Aβ = fβ (g, ∂g, ∂A).

(VIII.12.16)

State a local-in-time existence and uniqueness theorem for a solution of
the Cauchy problem for the Einstein–Maxwell system

Solution∇α Fαβ ≡ g αλ (∂λ Fαβ − Γμλβ Fαμ − Γμλα Fμβ ) = 0.
In wave coordinates, it holds that g αλ Γμλα = 0, and the above system
reads, in terms of A,g αλ ∂λ ∂α Aβ − ∂λ ∂β Aα − Γμλβ (∂α Aμ − ∂μ Aα ) = 0.
By elementary manipulations, these equations become
g αλ ∂λ ∂α Aβ − ∂β (g αλ ∂λ Aα ) + (∂β g αλ )∂λ Aα − Γμλβ (∂α Aμ − ∂μ Aα ) = 0.
(VIII.12.17)
If A satisﬁes the gauge condition g αλ ∂λ Aα = 0, they reduce to a quasidiagonal semilinear system of wave equations for the components of A,
of the formg αλ ∂λ ∂α Aβ = fβ (g, ∂g, ∂A).

(VIII.12.18)αλThe equation g ∂λ Aα = 0 is equivalent in wave coordinates, whereg αλ Γμαλ ≡ 0, to the Lorenz gauge condition, because∇λ Aλ ≡ g αλ ∇λ Aα ≡ g αλ (∂λ Aα − Γμλα Aμ ).
Equations (12.18) together with the Einstein equations in wave gauge,
with source the Maxwell tensor of F, constitute a quasidiagonal quasilinear system of wave equations for the pair g, A. Local existence andVIII.12

uniqueness for a solution of the Cauchy problem for the pair (g, A) inwave and Lorenz gauge results from the general theorem quoted before.
The proof that the constructed (g, A) satisﬁes the gauge conditions if
the initial data satisfy the constraints follows lines analogous to thosegiven in Section VIII.4, using, in addition, the identity ∇α ∇β F αβ ≡ 0.
Various other gauge conditions can be used on A to solve the Maxwell
equations: for example the temporal gauge A0 = 0 or the Coulomb gauge∇i Ai = 0. They have proved useful in diﬀerent domains.
All these gauges, like the Lorenz gauge, generalize to Yang–Mills ﬁelds.
They lead to equations with the same principal parts as in the case ofelectromagnetism, but in this case only semilinear.86

VIII.12.8

Problems

205

86 That is to say, only linear in the principal terms—in the case of the Yang–
Mills system, the ﬁrst-order derivatives.

Wave equation for F

1. Show that the Maxwell equations satisﬁed by the electromagnetic 2form F on a Lorentzian manifold (V, g) imply that F satisﬁes on V aquasidiagonal, quasilinear system of wave equations with coeﬃcientsdepending on the curvature of g.
2. Extend the result to the Yang–Mills case.

Solution
1. The Maxwell equations in vacuuo, dF = 0, δF = 0, imply(δd + dδ)F = 0;

(VIII.12.19)

that is,87 by straightforward computation using the Ricci identity,g αβ ∇α ∇β Fλμ + Rμα Fαλ − Rλα Fαμ + 2Rα λ ,β μ Fαβ = 0. (VIII.12.20)
2. The Yang–Mills gauge- and metric-covariant derivative of F is deﬁnedbyˆ := ∇F + [A, F ],∇F

(VIII.12.21)

with ∇ the Riemannian covariant derivative and [., .] the bracket in
the Lie algebra corresponding to the considered Yang–Mills model.
The Yang–Mills equations in vacuo areˆ α Fβγ + ∇ˆ γ Fαβ + ∇ˆ β Fγα = 0∇(VIII.12.22)

andˆ α F αβ = 0.
∇

(VIII.12.23)

These equations imply the following second-order semilinear equation
for F, depending on A:ˆ λ∇ˆ λ Fαβ − 2[F γ , Fγβ ] = 0,∇α

(VIII.12.24)

87 See, for instance, the expression forthis operator for an arbitrary p-form in
CB-DMI, V B 4.

206

General Einsteinian spacetimes. The Cauchy problem

whereˆ λ Fαβ ≡ ∇λ ∇λ Fαβ + 2∇γ [Aγ , Fαβ ] − [∇γ Aγ , Fαβ ]ˆ λ∇∇+ [Aγ , [Aγ , Fαβ ]].

88 Eardley

and Moncrief (1982). See
also the survey article by ChoquetBruhat (1983).

This equation for F plays an essential role in the proof of the global
existence of a solution of the Cauchy problem for the Yang–Mills
equations by Eardley and Moncrief.88

VIII.12.9

89 Bel (1958).

(VIII.12.25)

Wave equation for the Riemann tensor

Show that the Riemann tensor of an Einsteinian spacetime of arbitrary dimension satisﬁes a quasidiagonal semilinear system of wave
equations89 in the spacetime metric.

Solution
The Riemann tensor satisﬁes the Bianchi identities∇α Rβγ,λμ + ∇γ Rαβ,λμ + ∇β Rγα,λμ ≡ 0.

(VIII.12.26)

One deduces from this identity and the Ricci identity an identity of theform∇α ∇α Rβγ,λμ +∇γ ∇α Rαβ,λμ +∇β ∇α Rγα,λμ +Sβγ,λμ ≡ 0, (VIII.12.27)
where Sβγ,λμ is a homogeneous quadratic form in the Riemann tensor:
Sβγ,λμ ≡ {Rγ ρ Rρβ,λμ + Rα γ,β ρ Rαρ,λμ + [(Rα γ,λ ρ Rαβ,ρμ ) − (λ → μ)]}− {β → γ}.
(VIII.12.28)
Returning to the Bianchi identities gives, by contraction,∇α Rβγ ,α μ + ∇γ Rαβ,α μ + ∇β Rγα,αμ ≡ 0.

(VIII.12.29)

Therefore, using the symmetry Rαβ,λμ ≡ Rλμ,αβ ,∇α Rα β,λμ + ∇μ Rλβ − ∇λ Rμβ ≡ 0.

(VIII.12.30)

If the Ricci tensor Rαβ satisﬁes the Einstein equations Rαβ = ραβ , thenthe previous identities imply equations of the form∇α ∇α Rβγ,λμ + Sβγ;λμ = Jβγ,λμ ,(VIII.12.31)

where Jβγ,λμ depends on the sources ραβ and is zero in vacuum:Jβγ,λμ ≡ ∇γ (∇μ ρλβ − ∇λ ρμβ ) − (β → γ).

(VIII.12.32)

Note that (12.26) and (12.29) modulo the Einstein equations areanalogous to the Maxwell equations for the electromagnetic 2-form F :dF = 0,where J is the electric current.

δF = J,(VIII.12.33)

VIII.12

VIII.12.10

First-order symmetric hyperbolic system
for the Riemann tensor, Bel–Robinson
energyWrite in a Cauchy adapted frame a ﬁrst-order symmetric diﬀerential
system with constraints satisﬁed by the Riemann tensor of a (3 + 1)dimensional Einsteinian spacetime.

Solution
In a coframe θ0 , θi where g0i = 0 the Bianchi identities (12.26) with{αβγ} = {ijk} and the equations (12.30) with β = 0 do not containderivatives ∂0 of the Riemann tensor. We call them ‘Bianchi constraints’.
The remaining equations, called from here on ‘Bianchi equations’, readas follows:∇0 Rhk,λμ + ∇k R0h,λμ + ∇h Rk0,λμ = 0,

(VIII.12.34)∇0 R0 i,λμ + ∇h Rh i,λμ = ∇λ ρμi − ∇μ ρλi ≡ Jλμi ,(VIII.12.35)

where the pair (λμ) is either (0j) or (jl), with j < l. There are three of
one or the other of these pairs if the space dimension n = 3.
Equations (12.34) and (12.35) are, for each given pair (λμ, λ < μ), a
ﬁrst-order system for the components Rhk,λμ and R0h,λμ . If we choose
at a point of the spacetime an orthonormal frame, then the principal
operator is diagonal by blocks; each block corresponding to a choice ofa pair (λμ, λ < μ) is a symmetric 6 × 6 matrix that reads
⎞
⎛
∂0
0
0
∂2 -∂10⎜ 0
00∂3 -∂2 ⎟
∂0
⎟
⎜
⎜ 0
0
∂0 -∂3
0
∂1 ⎟
⎟.
⎜
M =⎜
0
-∂3 ∂00
0 ⎟⎟⎜ ∂2⎝ -∂1 ∂3
0
0
∂0
0 ⎠
0
0
∂0
0
-∂2 ∂1
The numerically valued matrix Mt of coeﬃcients of the operator ∂/∂tcorresponding to the Bianchi equations relative to the Cauchy adapted
frame is proportional to the unit matrix, with coeﬃcient N −2 , and hence
is positive-deﬁnite; thus the proof is complete. The energy corresponding
to the ﬁrst-order symmetric hyperbolic system (12.34), (12.35) is the
Bel–Robinson energy.

VIII.12.11

Schwarzschild trapped surface

Show that a 2-surface t =constant, r = r0 of the Schwarzschild spacetime
is not trapped if r0 > 2m. Show that it is trapped if r0 ≤ m.

Problems

207

IX
IX.1
IX.2

Relativistic fluids

Introduction
Case of dust

208
208

IX.3 Charged dust
IX.4 Perfect ﬂuid
IX.5 Thermodynamics

210
211
212

IX.6

Wave fronts andpropagation speeds
215
IX.7 Cauchy problem for
the Euler and entropy
system218
IX.8 Coupled Einstein–
Euler–entropy
system
221
IX.9 Dynamical velocity
222
IX.10 Irrotational ﬂows
225
IX.11

Equations in a
ﬂow-adapted frame
IX.12 Shocks
IX.13
IX.14
IX.15

Charged ﬂuids
Magnetohydrodynamics

226
226

229

231

IX.16 Viscous ﬂuids
IX.17 The heat equation
IX.18 Conclusion

231
234
235

IX.20

Solution of
Exercise IX.6.2
Problems

235
235

1 As said before (Chapter IV), thisproperty motivated Einstein in the
choice of his equations.

Introduction

A ﬂuid matter source in a domain of a spacetime (V, g) is such that
there exists in this domain a timelike vector ﬁeld u, called the unit
velocity, satisfying g(u, u) ≡ gαβ uα uβ = −1, whose trajectories are theﬂow lines of matter. A moving Lorentzian orthonormal frame is calleda proper frame if its timelike vector is u. In a proper frame, the unit
velocity has components u0 = 1, ui = 0.
One may also consider null ﬂuids, with ﬂow lines trajectories of a null
vector ﬁeld u, i.e. such that uα uα = 0.
The Einstein equations with ﬂuid source are1(IX.1.1)Sαβ ≡ Rαβ − gαβ R = Tαβ ,2where the stress–energy (momentum) tensor T is deduced from theequivalence principle and its expression in Special Relativity (see Chapters II and III) for the considered type of ﬂuid. The conservation
equations226

Yang–Mills ﬂuids(quark–gluon
plasmas)

IX.19

IX.1

∇α T αβ = 0,

(IX.1.2)

which in Special Relativity resulted from the physical laws of conservation of energy and momentum, are in General Relativity1 a consequenceof the Bianchi identities. The conservation equations (1.2) must sometimes be completed by equations satisﬁed by other physical quantities
appearing in T (e.g. the electromagnetic ﬁeld).
In this chapter, we will describe general properties of perfect ﬂuids.
When appropriate deﬁnitions are given, a number of these properties
generalize well-known properties of non-relativistic perfect ﬂuids. However, the equivalence of mass and energy in Relativity introduces a
number of fundamental diﬀerences. The relativistic causality principle,
limitation by the speed of light of the speed of any macroscopicallytransmitted signal, leads also to new considerations.
We will only brieﬂy touch on the case of dissipative ﬂuids, which arestill subject to controversies.

IX.2

Case of dust

A ﬂuid source is called pure matter or dust if in a proper frame, it
has neither momentum nor stresses; hence, in a proper frame, the only

IX.2

Case of dust

209

non-vanishing component of the stress–energy tensor is T00 = r, theproper mass density. In an arbitrary frame,
Tαβ = ruα uβ ,uα uα = −1,(IX.2.1)

where u is the unit ﬂow velocity.
At usual laboratory, and often at astronomical, scales, this dust stress–
energy tensor is a good approximation of a general ﬂuid stress–energy
tensor, since the mass energy is of order c2 (c being the velocity of light),
with respect to the other forms of energy.
The conservation laws for the dust stress–energy tensor read
∇α T αβ ≡ uβ ∇α (ruα ) + ruα ∇α uβ = 0.

(IX.2.2)

They give, using the property uβ uβ = −1, and hence uβ ∇α uβ = 0, thecontinuity equation (conservation of matter)∇α (ruα ) = 0(IX.2.3)

and the geodesic motion of the particles
uα ∇α uβ = 0.

(IX.2.4)

Similar equations are obtained for a null-dust model where uα uα = 0.
The geometric initial data for the spacetime metric g on an initial
manifold M are a Riemannian metric ḡ and a symmetric 2-tensor K.
The initial data for a dust source are a scalar function r̄ on M anda tangent vector ﬁeld v̄ to M. A solution (V, g, r, u) of the coupled
Einstein–dust equations is an Einsteinian development of the initial dataset (M, ḡ, K, r̄, v̄) if ḡ and K are respectively the induced metric and thesecond fundamental form of M as an embedded submanifold in (V, g),while r̄ is the function induced by r on M and v̄ is the value on M ofthe dust velocity with respect to the proper frame of an observer withtimelike vector orthogonal to M in (V, g). In local coordinates such thatthe values on M of the shift and the lapse of the development are respectively β̄ = 0 and N̄ = 1, it holds that v̄ i = (ū0 )−1 ūi , where ūα arethe components of u in the considered coordinate system at points of M.
It can be proved2 that an initial data set (M, ḡ, K, r̄, v̄) satisfying theconstraints for the Einstein equations with dust source admits a globally
hyperbolic and maximal Einsteinian development3 (V, g), unique up toisometries, with dust source (r, u).
The problem of how to use the general equations of ﬂuid evolution todetermine the motion of isolated bodies is a long standing one, which
has received only very partial answers. In the pure matter case, it canbe proved4 that a solution can be found for isolated bodies, i.e. when
the support ω of r̄ is the disconnected union of compact sets, by giving
to v̄ arbitrary values outside ω (see Problem IX.20.2).

2 Fourès (Choquet)-Bruhat (1958). For

more reﬁned statements on the required regularity, see YCB-OUP2009,
Chapter IX.
3 Non-complete in general.

4 Choquet-Bruhat and Friedrich (2006).

210

Relativistic ﬂuids

IX.3
5 This model is seldom valid in the realworld, where opposite charges interact.

Charged dust

The stress–energy tensor of charged pure matter5 (dust) is the sum of
the stress–energy tensor of the matter and the Maxwell tensor of theelectromagnetic ﬁeld F :Tαβ = ruα uβ + ταβ ,(IX.3.1)with1ταβ = Fα λ Fβλ − gαβ F λμ Fλμ .
4
The Maxwell equations are,∇α F αβ = J β := quβ ,dF = 0 and ∇ · F = J, i.e.

(IX.3.2)(IX.3.3)

where J is the convection electric current of the charge density q. They
imply the conservation of electric charge:∇α (quα ) = 0.

(IX.3.4)

We introduce the electromagnetic potential A, a 1-form such that (forsimplicity, we suppose that A exists globally on the considered domain)
F = dA.

(IX.3.5)

We take A in Lorenz gauge, i.e. such thatδA = 0.

(IX.3.6)

The Maxwell equations then read as a wave equation for A, namely∇α ∂ α Aβ − Rβ λ Aλ = J β = quβ .

(IX.3.7)

Modulo the Maxwell equations (and uα uα = −1), the stress–energy
conservation equations are equivalent to∇α (ruα ) = 0(IX.3.8)ruα ∇α uβ + quλ F βλ = 0.

(IX.3.9)andEquations (3.4) and (3.8) implyu α ∂α

q
r

= 0;

that is, the speciﬁc charge q/r is constant along the ﬂow lines; it isconstant throughout the spacetime that we construct if it is constant initially. We will make this simplifying (though not necessary) hypothesis,
and setq = kr,with k some given constant. Equation (3.9) can then be replaced byuα ∇α uβ + kuλ F βλ = 0.

(IX.3.10)

IX.4

Perfect ﬂuid

211

The proof of the existence of a solution (local in time) of the Cauchy
problem for the full Einstein–Maxwell–dust system when the initial datasatisfy the Einstein and Maxwell constraints with dust source follows thesame lines as in the vacuum Einstein–Maxwell case. Geometric global
uniqueness can also be proved.

IX.4

Perfect ﬂuid

IX.4.1Stress–energy tensor

We have written in Chapter IV the stress–energy tensor T of a perfect
ﬂuid in General Relativity:
Tαβ = μuα uβ + p(gαβ + uα uβ ),(IX.4.1)

where u is the unit ﬂow vector, while μ and p are respectively the energy
and pressure densities. In a proper frame, it holds that T00 = μ, T0i = 0,
and Tij = pδij , the isotropic-in-space pressure tensor. For classical ﬂuids,
μ and p are non-negative.6
Exercise IX.4.1 Show that in a spacetime of dimension n + 1, theEinstein equations with a perfect ﬂuid source can be written as
Rαβ = ραβ ,ραβ ≡ (μ + p)uα uβ +with

6 Quantum or cosmological phenomena

can lead to the appearance of negative
pressures.

1
gαβ (μ − p).
n−1

Hint: We have shown in Chapter IV (equation IV.2.5) that, with n thedimension of space (in the classical case, n = 3),ραβ ≡ Tαβ −1
gαβ Tλλ .
n−1

We have here
T ≡ g αβ Tαβ = np − μ.

(IX.4.2)

The energy–momentum vector relative to a timelike vector X isα
PX:= T αβ Xβ ≡ (μ + p)uα uβ Xβ + pX α .

(IX.4.3)

For X = −u, the components of P are P 0 = μ, the energy density, andP i = 0, zero momentum density.
We mention the following properties, which use a terminology frequently referred to, although misleading since weak and strong conditions are unrelated in the following propositions.
Proposition IX.4.1 If μ and p are non-negative, while X and Y are
causal with the same time orientation, then
1. The scalar T αβ Xα Yβ is non-negative; one says that a perfect ﬂuid
satisﬁes the weak energy condition.
7αβ2. The scalar ραβ X X is non-negative if X is causal; such a perfect
ﬂuid satisﬁes the strong energy condition.

7 Recall that the Einstein equations canbe written as Rαβ = ραβ ,1
T gαβ .
Tαβ − n−1ραβ :=

212

Relativistic ﬂuids

3. If μ > p and if X is past timelike, then the energy–momentum vector PX is future timelike like u; such a perfect ﬂuid satisﬁes thedominant energy condition.
Exercise IX.4.2 Prove these properties.
Hint: Compute these quantities in a proper frame and for Part 2 use theproperty (X 0 )2 ≥ (X i )2 .

IX.4.2Euler equations

The conservation laws of a perfect ﬂuid read
∇α T αβ ≡ (μ + p)uα ∇α uβ + g αβ ∂α p + uβ [∇α (μ + p)uα ] = 0. (IX.4.4)
By taking the contracted product with uβ , one deduces from theseequations the energy equation(μ + p)∇α uα + uα ∂α μ = 0,

(IX.4.5)

and, using this equation, one obtains the equations of motion(μ + p)uα ∇α uβ + (g αβ + uα uβ )∂α p = 0.

(IX.4.6)

The set of equations (4.4), (4.5) is called the Euler equations.
Exercise IX.4.3 Show that the condition uα uα = −1 is conserved
along the ﬂow lines.
Hint: The equations of motion imply uα ∂α (uβ uβ ) = 0.

8 See Taub (1959).

IX.5

Thermodynamics

IX.5.1Conservation of rest mass

In relativity, mass and energy are the same entity. However, there exists8
another scalar function of physical interest, namely the particle number
density r (better called the rest mass density if there are diﬀerent kinds ofparticles with non-zero rest mass). In the absence of chemical reactionsor quantum phenomena, conservation of particle number implies theequation∇α (ruα ) = 0.
The quantity r

IX.5.2−1(IX.5.1)

plays the role of a speciﬁc volume.

Deﬁnitions. Conservation of entropy

The diﬀerence between the total energy density μ and the rest massdensity r is called the internal energy density. One denotes by ε the
speciﬁc internal energy density; that is, one sets
μ = r(1 + ε).

(IX.5.2)

IX.5

Thermodynamics

In the case of local thermodynamic equilibrium (reversible thermodynamics), one deﬁnes a speciﬁc entropy density S and an absolutetemperature T by extending to relativistic perfect ﬂuids the identity of
the ﬁrst law of thermodynamics, namely
T dS := dε + pd(r−1 ).

(IX.5.3)

The thermodynamic quantities μ, p, S, and T are spacetime scalarfunctions.
Theorem IX.5.1 In a perfect ﬂuid, the thermodynamic identity (5.3)
and the matter conservation equation (5.1) imply the conservation of theentropy density along the ﬂow lines:9uα ∂α S = 0.

(IX.5.4)

Proof. The identity (5.3) and the deﬁnition (5.2) of ε give thatT uα ∂α S ≡ uα ∂α ε − r−2 puα ∂α r ≡ uα r−1 ∂α μ − (μ + p)r−2 uα ∂α r,
and hence, using the energy equation (5.1),
T uα ∂α S ≡ −r−2 (μ + p)∇α (ruα ).

IX.5.3Equations of state (n = 3)

For a perfect ﬂuid, only two thermodynamic scalars are independent;
the others are linked to them by relations that are assumed to depend
only on the nature of the given ﬂuid. Usually, the general formula called
the equation of state is the data of p as some function of μ and S:
p = p(S, μ),invertible as a function μ = μ(p, S).
Two circumstances are of particular physical interest for General
Relativity, astrophysics, and cosmology.
In astrophysics, one is inspired by what is known from classical ﬂuids, with additional relativistic considerations. Particularly interestingcases are those of barotropic and polytropic ﬂuids.
Barotropic ﬂuids
A ﬂuids is called barotropic when the equation of state reduces top = p(μ).
The ﬂuid dynamics is then governed by the energy and momentum equations. The particle number conservation equation decouples from theothers and can be solved after the ﬂuid motion has been determined.
Some physical situations that correspond to this model are the following:(1) Very cold matter, including models of nuclear matter, such as inneutron stars.

9 See Pichon (1965).

213

214

Relativistic ﬂuids(2) Ultrarelativistic ﬂuids, i.e. ﬂuids where the energy μ is largely dominated by the radiation energy. Then, by the Stefan–Boltzmann law,
it holds that μ = KT 4 and p = 13 KT 4 , and hence
p=

1
μ.
3

(IX.5.5)

The same equations for p and μ (with diﬀerent constants K) and theequation of state (5.5) are valid for a ﬂuid of neutrinos or electron–
positron pairs.
Remark IX.5.1 The stress–energy tensor of an ultrarelativistic ﬂuid
is traceless.
Polytropic ﬂuids
A ﬂuids is called polytropic if it obeys an equation of state of the form
p = f (S)rγ .
Several physical situations correspond to this case.
Frequently in stellar situations, only the internal energy ε and pressure
p are dominated by radiation; then ε = KrT 4 andp=1
KT 4 ,
3

hence

p=

1
rε.
3

(IX.5.6)

On the other hand, the thermodynamic identity together with theexpressions ε and p implydS =4K
d(r−1 T 3 ),
3

hence

S=

4KT 3
.
3r

(IX.5.7)

Eliminating T between (5.6) and (5.7) gives the polytropic equation ofstate of index γ = 43 :p=

10 In particular by analysing data fromthe Planck satellite (see Chapter VII).

K
3



3S
4K

 34

4

r3,

with

μ = 3p + r.

(IX.5.8)

More reﬁned equations of state adapted to various physical situations
have also been considered.
In cosmology, new information obtained from modern ground-based
and satellite telescopes has brought puzzling questions. It has been
known for a long time that radiation energy is presently only a verysmall fraction of the energy content of the cosmos, but it has recently
been found10 that baryonic matter itself represents at present only about
4% of this energy content. Another 24% is constituted by dark matter,
whose nature is conjectural, and the remaining 72% by what is called‘dark energy’, whose nature is a mystery.
In the early universe of the big-bang models, at very high temperature, the energy content of the sources can be roughly modelled as anultrarelativistic ﬂuid. For later times, after the formation of galaxies,

IX.6

Wave fronts and propagation speeds

215

cosmologists in general have assumed, for simplicity, a linear equationof state independent of entropy:p = (γ − 1)μ.

(IX.5.9)

In order that the speed of sound waves not be greater than the speed oflight (see Section IX.6), it has been supposed that 1 ≤ γ ≤ 2. The caseγ = 1 corresponds to dust. In the case γ = 2, the ﬂuid is called stiﬀ, orincompressible. In a stiﬀ ﬂuid, the speeds of sound and light are equal.

IX.6

Wave fronts and propagation speeds

In the spacetime of Galilean–Newtonian mechanics a wave front is a 2surface in space that propagates with time. Its propagation speed at apoint of space and an instant of time is the quotient by an inﬁnitesimalabsolute time interval δt of the inﬁnitesimal distance between two wave
fronts measured in the direction orthogonal to them, at times t andt + δt. In an Einsteinian spacetime, a wave front is a three-dimensional
timelike or null submanifold. The deﬁnition of its propagation speed
depends on the choice of an observer, and requires some thought.
Let us give ﬁrst some deﬁnitions.

IX.6.1

Characteristic determinant

The characteristic polynomial of a linear partial diﬀerential equation
at a point x ∈ Rn+1 is the polynomial obtained by replacing in its
higher-order terms the partial derivatives ∂/∂xα by the components ofa covariant vector.
In the case of a system of N ﬁrst-order partial diﬀerential equations11
with N unknowns U I and principal part
N

I=1

AJ,α
I

∂U I
,
∂xα

J = 1, . . . , N,

the characteristic polynomial Φ(x, p) is the determinant of the N ×N
matrix with elements AJ,α
J pα :
Φ(x, p) := Det AJ,α
(x)p
α .
I
At a given point x, the equation Φ(x, p) = 0 deﬁnes a cone in the
cotangent space called the characteristic cone.
The wave fronts associated to a system of ﬁrst-order partial diﬀerential equations on a spacetime are the submanifolds of this spacetime
whose normals are roots of the characteristic determinant; that is, wave
fronts satisfy in local coordinates an equation f = 0, where f is a solution of the ﬁrst-order nonlinear partial diﬀerential equation, called theeikonal equation,

11 For the deﬁnition for more generalsystems, see, for instance, YCBOUP2009, Appendix IV, Section 2.

216

Relativistic ﬂuids



∂f
Φ x,
= 0.
∂x

12 See for instance CB-DMI, IV Chap-ter 7.

Generically, discontinuities of the derivatives of the unknowns of theorder appearing in these principal parts can occur only across such
submanifolds.
By general results on ﬁrst-order partial diﬀerential equations,12 a wave
front is generated by the bicharacteristics of the eikonal equation, also
called rays, which are solutions of the ordinary diﬀerential system
dxα
dpα
=−
= dλ,
∂Φ/∂pα∂α Φ/∂xα(IX.6.1)

where λ is a parameter, called the canonical parameter. The tangentsto the rays issuing from a given point x generate a cone in the tangent
space to the spacetime at x, called the wave cone, which is dual to thecharacteristic cone. The wave cone is the envelope of the hyperplanes
whose normals (in the metric g) belong to the characteristic cone. A
wave front at x is tangent to the wave cone at x along the direction of
a ray.
Exercise IX.6.1 Show that Φ(x, p) is constant along a ray.
Exercise IX.6.2 The eikonal equation associated to the wave operator
g on a spacetime (V, g) is
1 αβ ∂f ∂fg= 0.
2∂xα ∂xβ(IX.6.2)

Show that the light cone and the characteristic cone of a spacetime (V, g)
can be identiﬁed by the usual identiﬁcation of the tangent and cotangent
spaces to V through the metric g.

IX.6.2

Wave front propagation speed

The propagation speed of a smooth wave front with respect to an observer is the propagation speed of the tangent plane to the wave front withrespect to the proper Lorentz frame of this observer, with the following
deﬁnition (one could give analogous deﬁnitions for higher dimensions,but to be clearer we prefer to stick to 3 + 1).

13 Two3-hyperplanesinfourdimensional
spacetimes
intersectgenerically along a 2-plane whose
normal in a 3-plane is a vector.

Deﬁnition IX.6.1 The propagation speed at a point x of a threedimensional hyperplane P with respect to an orthonormal Lorentzianframe at x is the velocity with respect to this frame of the vector ofgreatest slope in P, orthogonal to the 2-plane13 intersection of P withthe space hyperplane X 0 = 0.
Lemma IX.6.1 The propagation speed V of a hyperplane with normal
ν with respect to a Lorentzian frame at x with time vector u is given by
the formulaIX.6|V |2 =(uα να )2αβ(u u + g αβ )ν

α νβ

.

Wave fronts and propagation speeds

217(IX.6.3)

Proof. The vector of greatest slope in P with respect to the Lorentz
frame with time vector u is the vector in P orthogonal to the 2-plane I
that is the intersection of P and the hyperplane orthogonal to u. This
intersection I satisﬁes the equations
uα Xα = 0,

ν α Xα = 0.

(IX.6.4)

Choose for the ﬁrst space vector in the Lorentzian frame the projectionof ν on the hyperplane orthogonal to u; then (6.4) read
X0 = 0,ν 0 X0 + ν 1 X1 = 0.

(IX.6.5)

A vector in I therefore has components X0 = 0, X1 = 0, with X2 , X3
arbitrary. The vector of greatest slope Y in P orthogonal to I is uniquelydetermined by the conditions
Y ∈ P,i.e.

ν 0 Y0 + ν 1 Y1 = 0,

hence

ν0
Y1 = − 1 Y0 ,
ν

(IX.6.6)

and the orthogonality with I, that is, Y2 = Y3 = 0.
The propagation speed of P with respect to the considered Lorentz
frame is" " " 0"
" Y1 " " ν "(IX.6.7)|V | = "" "" = "" 1 "" .
Y0νThe given formula (6.3) takes the form (6.7) in a Lorentz frame where
the time axis is the unit vector u and the normal ν to P has components
ν2 = ν3 = 0.
IX.6.3

Case of perfect ﬂuids

The Euler and entropy conservation equations of a perfect ﬂuid are ofﬁrst order. Using the entropy equation (5.4) in the energy equation (4.5),
we see that the Euler–entropy equations written for the unknowns S, p, u
have a characteristic matrix composed of two blocks around the diagonal.
One of these blocks corresponds to the entropy S and its conservationlaw (5.4); it reduces to the linear form14a ≡ u α Xα .
The other block corresponds to the unknowns p and u and the equations (4.4) and (4.5). It is the following 5×5 matrix15 with ρ ≡ μ + p,μp = ∂μ/∂p:

14 To avoid confusion with pressure,
we denote by X the vector previously
denoted by p.

15 The four components of u are con-sidered as independent unknowns. The
identity g(u, u) = −1 is preserved by
the ﬂow.

218

Relativistic ﬂuids⎛a⎜⎜ X 0 + au0
⎜
1
1
M≡⎜
⎜ X + au⎜
⎝ X 2 + au2
X 3 + au3ρμp X0ρμp X1

ρμp X2

ρμp X3

ρa
0

0
ρa

0
0

0
0

0
0

0
0

ρa
0

0
ρa

⎞
⎟
⎟
⎟
⎟.
⎟
⎟
⎠

The determinant of this matrix is computed to be− ρ4 a3 D,(IX.6.8)

with D quadratic in X and given byD ≡ (μp − 1)(uα Xα )2 + μp X α Xα .

(IX.6.9)

We see that a perfect ﬂuid has two types of wave fronts:• Matter wave fronts, f = constant, such thatuα ∂α f = 0.

(IX.6.10)

They are submanifolds generated by the ﬂow lines. Their propagation
speed for a comoving observer (i.e. in a proper rest frame of the ﬂuid)
is zero.
• Sound wave fronts, whose normals satisfy D = 0. In a proper rest
frame gαβ = ηαβ (the Minkowski metric), u0 = 1, ui = 0. The
corresponding eikonal equation reads(∂i f )2 = 0.
(IX.6.11)− μp (∂0 f )2 +i

The propagation speed of these wave fronts is, assuming μp > 0,
1

|V | = (μp )− 2 ;16 Recall that we use geometric units;i.e. the speed of light is equal to 1.

it is less than the speed of light,16 as expected from a relativistic
theory, if and only ifμp ≥ 1.

(IX.6.12)

A ﬂuid such that μp = 1 is called an incompressible or stiﬀ

ﬂuid.

In such a ﬂuid, sound waves propagate with the speed of light.

IX.7

17 That is, linear in the highest-order,

m + 1, terms.

Cauchy problem for the Eulerand entropy system

The Cauchy problem is the determination of the solution of a partial
diﬀerential system from its initial data. Consider ﬁrst one linear17 partial
diﬀerential equation of order m + 1 with unknown a function U on Rn+1
where coordinates are denoted by x0 , xi , i = 1, . . . , n. The initial data
for U on the submanifold x0 = 0 are the values for x0 = 0 of U and
its partial derivatives with respect to x0 of order less than or equal tom. A Cauchy problem is said to be well posed, for initial data in some

IX.7

Cauchy problem for the Euler and entropy system

functional space, if it admits in a neighbourhood of x0 = 0 one and only
one solution depending continuously on the initial data. The equation iscalled hyperbolic if it admits a well-posed Cauchy problem with datain spaces where only a ﬁnite number of derivatives are involved18 andif the solution of this Cauchy problem manifests a ﬁnite propagation
speed. Hyperbolic well-posedness depends only on the coeﬃcients of theprincipal terms, i.e. terms of order m + 1. For a quasilinear diﬀerential
equation (i.e. one that is linear only in the highest (m + 1)-order terms),
the hyperbolicity depends on the values of the coeﬃcients of these terms
for the given initial data. By contrast, if the equation is semilinear(i.e. the coeﬃcients of the highest-order terms do not depend on theunknowns), hyperbolicity does not depend on the initial data.19
In General Relativity, causality is an essential feature, saying thatthe properties of physical phenomena at one point depend only on thepast of this point, determined by the causal paths20 of the Lorentzianspacetime metric g. The deﬁnitions of Cauchy problem and hyperbolicity
extend to systems of partial diﬀerential equations, in particular to quasidiagonal systems. The wave gauge reduced vacuum Einstein equations
are a quasidiagonal and quasilinear second-order partial diﬀerential systems as studied in Chapter VIII; they are a hyperbolic system for dataon a Lorentzian metric on a spacelike submanifold. They are a special
case of a quasidiagonal Leray hyperbolic systems.21

IX.7.1The Euler and entropy equations as a Leray
hyperbolic system

It can be shown22 that the Euler equations are equivalent to a quasidiagonal system with principal terms in the diagonal either uα ∂α or thethird-order operator
(IX.7.1)
uα (μp − 1)uλ uβ + μp g λβ ∂λ ∂β ∂α .
The characteristic cone at a point isuα Xα (μp − 1)uλ uβ + μp g λβ Xλ Xβ = 0.

(IX.7.2)

It is the union of a hyperplane uα Xα = 0, spacelike if u is timelike,reading (in a proper frame for u)(P ) := {X0 = 0},
and a second-order cone (C), reading (in a proper frame for u))*(C) := −(X0 )2 + μp(Xi )2 = 0 .
It is easy to see by drawing a ﬁgure that if μp > 0, then any straightline passing through a point y in the interior of (C), which is(yi )2 ≤ 0,−(y0 )2 + μp219

18 Mathematicians often use Sobolev
spaces Hs .

19 One calls hyperquasilinear equationsof order m + 1 in which the coeﬃcientsof the highest-order terms do not depend on derivatives of the unknowns oforder > m − 1. The Einstein equations
are hyperquasilinear.
20 In other words, physicists say thatthe speed of light is a maximum for the
propagation of all observables.

21 Leray (1953). See also Dionne (1962):
this article by a student of Leray reﬁnes his results in an important way.
It treats only the case of one equation, but the results extend trivially toquasidiagonal systems.
22 Choquet-Bruhat (1966).

220

Relativistic ﬂuids

23 Note that this geometric criterion re-fers to cones in the dual of the tangent
space to the spacetime manifold.

but not passing through the vertex, cuts the union (P ) ∪ (C) at three
distinct points.23 This is the criterion of Leray hyperbolicity for a thirdorder operator, or a quasidiagonal system of such operators.24

24 For details, see YCB-OUP2009, Ap-pendix IV, Section 2 or Leray (1953).

25 Friedrichs (1954). The Leray hyper-

bolic systems are much more general,containing equations of arbitrary orders, but some symmetric hyperbolic
systems do not satisfy the criteria for
Leray hyperbolicity.

26 Anile (1989).

27 A FOSH system, for barotropic ﬂu-

ids, was found in Special Relativity
by K. O. Friedrichs. Its constructionin General Relativity was sketched byRendall (1992).

IX.7.2First-order symmetric hyperbolic systems

An alternative to the Leray approach to proving the well-posednessof a Cauchy problem for evolution equations is the Friedrichs theory25 of ﬁrst-order symmetric hyperbolic (FOSH) systems. The
ﬁrst-order linear system of partial diﬀerential equations on Rn+1 ,
t
MIJ

J
∂U J
i ∂U
+ MIJ
= fI ,∂t∂xi

is called symmetric if the n+1, N ×N matrices Mt , Mi with elements
t,
i
α
α
, MIJ
are symmetric (i.e. MIJ= MJI). It is hyperbolic with respect
MIJ
0to the x ≡ t coordinate, and is then described as simply symmetric
hyperbolic, if, in addition, the matrix Mt is a positive-deﬁnite quadratic form. The Cauchy problem is then well posed if its initial data
are given on a submanifold S0 := {x0 = constant}. In a quasilinear
system, the matrices Mα are functions of the unknowns U, but not of
their derivatives. The symmetric hyperbolicity depends then on the values of these unknowns. The Cauchy problem with data on S0 is thenwell posed if the system is symmetric hyperbolic when in the coeﬃcientsthe unknowns are replaced by their initial data. Note that in general,
however, the solution exists only in a neighbourhood of S0 .
It is straightforward to write for wave equations equivalent FOSH systems (see Problem VIII.12.10 in Chapter VIII). In the case of the Euler
equations, the procedure is more subtle but may be preferred for somenumerical computations. To obtain FOSH systems, one can apply general methods inaugurated by Lax, developed by Boillat and Ruggeri, andexplained in the book by Anile.26 These authors use a convex functional
and auxiliary unknowns. In the case of relativistic perfect ﬂuids, one
can perform a direct computation to show the equivalence of the Euler–
entropy system to a FOSH system.27 The idea is to take as unknowns
the pressure p, denoted by U 0 , and the space components ui , denotedby U I , I = 1, 2, 3. The component u0 is determined through the identityuα uα = −1.

(IX.7.3)

From this identity, one deduces∇α u0 = −ui ∇α ui.
u0

(IX.7.4)

The energy–continuity equation yields the following ﬁrst-order evolution
equation for p and ui :∇i ui −∂α pui ∇0 ui
= 0.
+ μp uαu0μ+p(IX.7.5)

IX.8

Coupled Einstein–Euler–entropy system

221

The combination of the equations of motion of indices 0 and i,


ui
∂α p
∂α p
− 0 uα ∇α u0 + (g α0 + uα u0 )uα ∇α ui + (g αi + uα ui )
= 0,
μ+p u
μ+p
reduces to another ﬁrst-order evolution equation for p and ui :

 

∂α p
ui uj
ui
= 0.
(IX.7.6)uα ∇α ui + 0 ∇α uj + g αi − 0 g α0
u u0
u
μ+p
Setting A := (μ + p)−1 , the principal matrix of the system (7.5), (7.6) reads
M ≡ Mα ∂α ,





 ⎞
⎛
u1
u2
u3
2  α
A
μ
u
∂
A
∂
−
∂
−
∂
−
∂
A
∂
A
∂
α
1
0
2
0
3
0
p
⎜
⎟
u0 
u0
u0
⎜ 
⎟


1
1
2
1
⎜
⎟
u u1 αu u3 α⎜ A ∂ 1 − u ∂ 0 u α 1 + u u 1 ∂α
⎟
u
∂
u
∂
α
α
⎜
⎟
0
0
0
0
u
u u0
u u0
u u0
⎟.




:= ⎜
2
1
2
2
⎜
⎟
u u2 αu u3 αu u2α⎜ A ∂2 − u ∂0
⎟
u
∂
u
u
∂
1
+
∂
α
α
α
⎜
⎟
0
0
0
0
u
u u0
u u0
u u0
⎜ 
⎟



3
1
1
3
⎝
⎠
u u3 αu u3 αuu u3
αA ∂3 − 0 ∂0
u
∂
u
∂
u
1
+
∂
α
α
α
0
0
0
u
u u0
u u0
u u0
(IX.7.7)
It can be proved that the system is equivalent to a symmetric systemwith ∂0 = ∂/∂t and with Mt positive-deﬁnite28 if u is timelike andμp ≥ 1; that is, to a FOSH system.
Exercise IX.7.1 Show that (7.5), (7.6) are equivalent to the originalEuler–entropy equations.

IX.8

Coupled Einstein–Euler–entropy
system

IX.8.1

Initial data

An initial data set for the Einstein–Euler system with a given equationof state will be the usual data for the Einstein equations together withdata for the ﬂuid source.
We have seen that an initial data set for the Einstein equations is a
triple (M, ḡ, K) with M a 3-manifold, ḡ a properly Riemannian metric,
and K a symmetric 2-tensor on M . A spacetime (V, g) is said to admit
these initial data if M can be embedded in V as a submanifold M0 withinduced metric ḡ and extrinsic curvature K. If (V, g) is a solution of theEinstein equations with source the stress–energy tensor T, then (ḡ, K)
must satisfy the following constraints:R(ḡ) − K · K + (tr K)2 = 2ρ,(IX.8.1)¯ · K − ∇tr¯ K = J.
∇

(IX.8.2)

28 See YCB-OUP2009, Chapter IX,
Section 13.

222

Relativistic ﬂuids

In a Cauchy adapted frame, where the equation of M ≡ M0 in V isx0 ≡ t = 0 and the time axis is orthogonal to M0 , one has, in the caseof a perfect ﬂuid,
ρ ≡ N 2 T 00 ≡ N 2 (μ + p)(u0 )2 − p,

(IX.8.3)

J i = N T i0 ≡ N (μ + p)ui u0 ,1

N u0 = (1 + gij ui uj ) 2 .

(IX.8.4)

Using uα uα = −1, that is,1

N u0 = (1 + gij ui uj ) 2 ,(IX.8.5)

we see that on M0 the quantities ρ and J depend only on the values μ̄and p̄ of μ and p on M0 , and on the projection v of u, with components
v i = ui , on M0 ; they do not depend on the choice of lapse and shift. The
initial data on M0 for a perfect ﬂuid with equation of state μ = μ(p, S)
are two scalars p̄ and S̄ and a tangent vector v̄. Data for the third-ordersystem are obtained by using the restriction to M0 of the Euler equations
and its ﬁrst-order derivative in the direction of u. This computation now
requires a choice for initial lapse and shift, since this is also required forthe solution of the reduced Einstein equations.

IX.8.2Evolution

The Einstein equations reduced in harmonic gauge, with perfect ﬂuid
source, read as a second-order quasidiagonal system for the metric whose
principal part is the wave operator of the spacetime metric while theﬂuid variables appear at order zero. It is straightforward to show that,together with the Euler–entropy equations, they form a Leray hyperbolic
system provided that μp > 0. However, the system is causal (i.e. thedomain of dependence is determined by the light cone) only if μp ≥ 1.
It is also straightforward to write in that case the coupled system as
a FOSH system. We have already described in Chapter VIII the proof
that a solution of the reduced system is a solution of the full system if
the initial data satisfy the constraints.

29 SeeLichnerowicz (1955) and references therein. The dynamical velocity
was used by Fourès (Choquet)-Bruhat(1958) for the ﬁrst proof of the wellposedness of the Cauchy problem for
relativistic perfect ﬂuids, coupled or
not with the Einstein equations.

IX.9

Dynamical velocity

IX.9.1

Fluid index and Euler equations

Important properties of non-relativistic ﬂuids generalize to relativistic ﬂuids if one introduces a spacetime vector called the dynamical
velocity29 linked with both the kinematic unit vector u and thethermodynamic quantities.

IX.9

Dynamical velocity

223

In the case of barotropic ﬂuids, i.e. with equation of state μ = μ(p),the simplest way is to deﬁne a function f (p), called the index of theﬂuid, by p
dp
f (p) := exp(IX.9.1)
p0 μ(p) + p
and the dynamical 4-velocity by
Cα = f uα ,hence

C α Cα = −f 2 , C β ∇α Cβ = −f ∂α f.

(IX.9.2)

Theorem IX.9.1 For a barotropic ﬂuid, the Euler equations (4.4) and(4.5) are equivalent to the equation
C α (∇α Cβ − ∇β Cα ) = 0(IX.9.3)

and∇α C α + (μp − 1)

C αC β∇α Cβ = 0,C λ Cλwith μp ≡

∂μ
.
∂p

(IX.9.4)

Exercise IX.9.1 Prove this theorem.
Exercise IX.9.2 Show that the ﬂow lines of a perfect ﬂuids are
geodesics of a metric confomal to the spacetime metric.
Hint: Show that g̃ = F 2 g implies
˜ α Cβ = C α ∇α Cβ + C α Cα ∂β (log F ).
C α∇Corollary IX.9.1 A barotropic relativistic ﬂuid is incompressible(μp = 1) if and only if 30∇α C α = 0.

(IX.9.5)

Assuming an equation of state and expressing p as a function of f ,and hence of C, the equations (9.4), (9.5) read as a ﬁrst-order diﬀerential
system for C.
Exercise IX.9.3 Write these equations for the equation of statep = (γ − 1)μ.

IX.9.2

Vorticity tensor and Helmholtz equations

The vorticity tensor is deﬁned through the dynamical velocity as theantisymmetric 2-tensorΩαβ ≡ ∇α Cβ − ∇β Cα .
It results from the equations of motion (9.3) that this tensor is
orthogonal to the velocity; indeed, these equations readC α Ωαβ = 0.

(IX.9.6)

Theorem IX.9.2 (relativistic Helmholtz equations) The Lie derivativeof the vorticity tensor Ω with respect to the dynamic velocity C vanishes:
LC Ω = 0.

30 This property of incompressible ﬂuids generalizes the classical one for
Newtonian ﬂuids, ∂i v i = 0, implied inthis case by the constancy of density.

224

Relativistic ﬂuids

Proof. Since Ω is an exterior 2-form, the diﬀerential of the 1-form C, it
is a closed form, satisfying in local coordinates the identity∇α Ωβγ + ∇γ Ωαβ + ∇β Ωγα ≡ 0,
and henceC α ∇α Ωβγ + C α ∇γ Ωαβ + C α ∇β Ωγα ≡ 0.

(IX.9.7)

The equations of motion (9.3) imply, by the derivation ∇γ ,C α ∇γ Ωαβ + Ωαβ ∇γ C α = 0.
Using this equation, the identity (9.7) gives the equation
C α ∇α Ωβγ − Ωαβ ∇γ C α − Ωγα ∇β C α ≡ 0;that is, using the expression for the Lie derivative of a covariant tensor
and the antisymmetry of Ω,(LC Ω)βγ ≡ C α ∇α Ωβγ + ∇γ C α Ωβα + ∇β C α Ωαγ = 0.

(IX.9.8)

We have shown that the vorticity tensor Ω satisﬁes a linear diﬀerential homogeneous system along the ﬂow lines, and hence we have thefollowing corollary:
Corollary IX.9.2 If a smooth (barotropic) ﬂow has vanishing vorticity on a 3-submanifold transversal to the ﬂow lines, it has a vanishingvorticity on the domain of spacetime spanned by these ﬂow lines.

IX.9.3General perfect ﬂuid enthalpy h

For non-barotropic ﬂuids with equation of state depending on theentropy S, one introduces the enthalpy h, deﬁned through the thermodynamic identitydh ≡ r−1 dp + T dS,(IX.9.9)or, equivalently, because of the thermodynamic identity (5.3),
dh ≡ d[r−1 (μ + p)],we seth := r−1 (μ + p),(IX.9.10)

and now deﬁne the dynamical velocity by
C α := huα ,settingΩαβ := ∇α Cβ − ∇β Cα .

(IX.9.11)

Exercise IX.9.4 Show thatC α ∇β Cα = −h∂β h.

(IX.9.12)

Exercise IX.9.5 Show that the equations of motion are equivalent to(hT )−1 C α Ωαβ = ∂β S.

(IX.9.13)

IX.10

IX.10

Irrotational ﬂows

IX.10.1

Deﬁnition and properties

Irrotational ﬂows

225

A ﬂow with zero vorticity is called irrotational, its trajectories are
orthogonal hypersurfaces, since the equations ∇α Cβ − ∇β Cα = 0 imply
that there exists on spacetime (at least locally) a function Φ such thatCα = ∂α Φ.
Theorem IX.10.1 The irrotational ﬂow of a perfect ﬂuid is governedby a quasilinear wave-type equation.31
Proof. When the ﬂow is irrotational, (9.3) is satisﬁed identically while(9.4) reads αβ
g + (1 − μp )uα uβ ∇α ∂β Φ = 0.
(IX.10.1)
In this equation, the quantities u and p are given functions of Φ and ∂Φ.
Indeed, by deﬁnition,g αβ ∂α Φ∂β Φ ≡ −f 2

and

uα ≡

∂αΦ
,
f

while p can be expressed in terms of f, and hence of ∂Φ, by inverting
the relation (9.1).
The characteristic cone of this quasilinear second-order diﬀerential
equation for Φ is given by the quadratic form αβ
g + (1 − μp )uα uβ Xα Xβ = 0,
(IX.10.2)
whose dual is the sound cone. This quadratic form is of Lorentzian signature if and only if μp > 0; the equation satisﬁed by Φ is then hyperbolic.
It is causal (the sound cone is interior to the light cone) if μp ≥ 1. 
Remark IX.10.1 In the case of a stiﬀ ﬂuid, μp = 1, the equation forΦ reduces to the usual linear wave equation.
Exercise IX.10.1 Check the quadratic form (10.2) using (6.9).

IX.10.2

Coupling with the Einstein equations

The Einstein equations in harmonic gauge with source given by an irrotational ﬂow, together with the equation for Φ, form a second-order
quasidiagonal system whose principal parts are the wave operator of thespacetime metric or the sound wave operator. It is a causal hyperbolic
system if μp ≥ 1.
Remark IX.10.2 The solution can be interpreted as an Einsteinian
spacetime with source an irrotational ﬂow as long as ∂Φ is timelike, i.e.
g αβ ∂α Φ∂β Φ < 0.
Exercise IX.10.2 Formulate the Cauchy problem and a local existence theorem for the Einstein equations coupled with a perfect ﬂuid in
irrotational ﬂow.

31 Fourès (Choquet)-Bruhat (1958).

226

Relativistic ﬂuids

IX.11
32 Cattaneo(1959), Ferrarese (1963),
and Ferrarese and Bini (2007).

33 Friedrich (1998).

34 SeeYCB-OUP2009, Chapter IX,
Section 14, or Choquet-Bruhat andYork (1998), which uses the Riemanntensor, instead of the Weyl tensor used
by Friedrich.

Equations in a ﬂow-adapted frame

It was stressed long ago by Cattaneo and Ferrarese32 that the natural
physical quantities are timelines and not spacelike submanifolds. Using
the same kind of formalism as given in Chapter VII for Cauchy adapted
frames, but for Lorentzian frames with time axis tangent to the timelines(time-adapted frames) Friedrich33 has written the perfect ﬂuid equations
as a symmetric system for the pressure and the connection coeﬃcients.
This system is hyperbolic with respect to a time function, but only under
additional appropriate conditions.34

IX.12

Shocks

A shock is a discontinuity in the ﬂuid variables across a timelike nmanifold (n = 3 in the classical case). The stress–energy tensor is thendiscontinuous; its derivative is meaningful only in a generalized sense.
The relativistic Rankine–Hugoniot equations express the vanishing of the divergence of the stress–energy tensor in the space ofgeneralized functions (distributions); they arenα [T αβ ] = 0,35 Christodoulou (2007).

where n is the spacelike normal to the timelike shock front Σ and [T αβ ]is a measure with support Σ. A deep study of the formation of shocks
for ﬂuids in Special Relativity has been made by Christodoulou.35

IX.13

Charged ﬂuids

IX.13.1

Equations

The stress–energy tensor of a charged perfect ﬂuid is the sum of thestress–energy tensor of the ﬂuid variables and the Maxwell tensor of theelectromagnetic ﬁeld F :(ﬂuid)

Tαβ = Tαβ

+ ταβ ,

(IX.13.1)

with(ﬂuid)Tαβ:= (μ + p)uα uβ + pgαβ

(IX.13.2)and1ταβ = Fα λ Fβλ − gαβ F λμ Fλμ .
(IX.13.3)
4
The state of the charged ﬂuid is governed by the Einstein equations
Sαβ = Tαβ ,(IX.13.4)

the Maxwell equations, with J the electric current,
dF = 0and ∇ · F = J, i.e.

∇α F αβ = J B ,(IX.13.5)

IX.13and the conservation equations, which areαβ+ τ αβ ) = 0.
∇α T αβ ≡ ∇α (T(ﬂuid)(IX.13.6)

Using previous formulas, we see that these conservation equations read(μ + p)uα ∇α uβ + g αβ ∂α p + uβ [∇α (μ + p)uα ] + Jλ F βλ = 0. (IX.13.7)
Classically, the electric current J is the sum of a convection and a
conduction current:Jλ = quλ + σEλ ,Eλ = uμ Fμλ ,(IX.13.8)

where q is a scalar function (the electric charge density), σ is the electric
conductivity (in general assumed to be constant), and E is the electric
ﬁeld.
The current J satisﬁes the conservation equation∇α J α = 0.

(IX.13.9)

As for uncharged ﬂuids, taking the contracted product of (13.7) with ugives a continuity equation(μ + p)∇α uα + uα ∂α μ − σE α Eα = 0(IX.13.10)

and equations of motion(μ+p)uα ∇α uβ +g αβ ∂α p+uβ (uα ∂α p+σE α Eα )+Jλ F βλ = 0. (IX.13.11)
Modulo initial conditions, as in the vacuum case (see Chapter VIII), theMaxwell equations are equivalent to wave-type equations(dδ + δd)F = dJ.

(IX.13.12)

One can add to these equations the law of particle number conservation∇α (ruα ) = 0(IX.13.13)

and consider that μ is a function of p and r. If the equation of state isgiven by μ as a function of the pressure p and entropy S, then (13.13)
must be replaced by a thermodynamic law.

IX.13.2

Fluids with zero conductivity

When the conductivity is zero, the Lorentz force reduces to qE, which isorthogonal to the ﬂow vector u and hence does not furnish any work. Onesays that the ﬂuid is non-dissipative. The ﬁrst law of thermodynamics
and particle number conservation then imply, as in the uncharged case,the law of conservation of entropy:uα ∂α S = 0.

Charged ﬂuids

227

228

Relativistic ﬂuids

The Euler–entropy equations are then expected to be a hyperbolic
system. This is conﬁrmed by the following theorem:
Theorem IX.13.1 The Einstein equations in wave gauge with sources
an electromagnetic ﬁeld of potential in Lorentz gauge and a charged perfect ﬂuid with equation of state μ = μ(p, S) and with zero conductivity(σ ≡ 0) are, if μp > 0, a hyperbolic system for g, q, u, p, and theelectromagnetic potential A. This system is causal if μp ≥ 1.
A solution of the Cauchy problem for the full Einstein–Maxwell–charged ﬂuid system, for initial data satisfying the constraints, can be
deduced by methods analogous to those used for uncharged ﬂuids. Apartfrom the addition of electromagnetic wave fronts, the wave fronts are thesame as for uncharged ﬂuids.

IX.13.3

Fluids with ﬁnite conductivity

When the conductivity σ is ﬁnite, the Lorentz force is no longer orthogonal to u: it works under the ﬂuid ﬂow. The ﬂuid is called dissipative andits properties are expected to be diﬀerent from those of non-dissipative
ﬂuids. Indeed, experiments show, for example, that no shock wave propagates in charged mercury, a liquid with non-zero conductivity. This factsuggests that for such a ﬂuid, the Cauchy problem is not well posed inspaces of functions with a ﬁnite number of derivatives. Indeed the ﬁrst
law of thermodynamics now gives the entropy equation (S increases
along the ﬂow lines if σ > 0, as physically predicted)
T uα ∂α S = σE α Eα .
The equation expressing conservation of the electric current J,∇α (quα + σuλ F αλ ) = 0,36 Choquet-Bruhat (1965).

(IX.13.14)

contains derivatives of the electromagnetic ﬁeld F, and hence second
derivatives of the potential A. The characteristic determinant of theEuler–entropy–electromagnetic system is the same as in the case of zeroconductivity, but the characteristic matrix is no longer diagonalizable;
the system is not hyperbolic in the sense deﬁned before.
Fluids with ﬁnite, non-zero, conductivity are a physical example36
of Leray–Ohya hyperbolic systems (which the authors called ‘nonstrictly hyperbolic’). Such systems have solutions in Gevrey classes,
spaces of C ∞ functions whose derivatives satisfy inequalities weaker
than the inequalities satisﬁes by the derivatives of analytic functions
that ensure the convergence of their Taylor series. These Gevrey classes
enjoy the property fundamental in Relativity of being able to possess acompact support without being identically zero. This permits the limitation by the speed of light of signal solutions of Leray–Ohya hyperbolic
systems.
It can be proved that if μp ≥ 1, then the reduced Einstein–Maxwell–
Euler–entropy system of a charged perfect ﬂuid with ﬁnite, non-zero,IX.14

conductivity is Leray–Ohya hyperbolic (and causal) but not hyperbolic(in the strict sense).37

IX.14

Magnetohydrodynamics

In the case where the electric conductivity is so large that it can be considered as inﬁnite, the proper-frame electric ﬁeld Eα = uβ Fαβ becomes
negligible. The case of a zero electric ﬁeld is called magnetohydrodynamics.38 It plays a fundamental role in plasma physics.

IX.14.1

Equations

In three space dimensions, for a ﬂuid with inﬁnite conductivity
σ = ∞, the electromagnetic ﬁeld reduces, in the local frame deﬁned
by the ﬂuid velocity, to the magnetic vector H. The second Maxwell
equation δF = J is replaced by
E α := uβ F βα = 0.

(IX.14.1)

From the deﬁnition of the vector H (orthogonal to u),
H α = uβ (F ∗ )βα ,(IX.14.2)(F ∗ )βα = H α uβ − H β uα .

(IX.14.3)

one deduces

A straightforward calculation gives the following expression for the
Maxwell tensor:1(IX.14.4)ταβ ≡ uα uβ + gαβ |H|2 − Hα Hβ .
2
The ﬁrst Maxwell equation dF = 0 becomes∇α (F ∗ )βα ≡ ∇α (H α uβ − H β uα ) = 0.

(IX.14.5)

Modulo these equations, the divergence of τ is found to be


1 αβ
αβ
α β
∂α |H|2 + |H|2 ∇α (uα uβ ) − ∇α (H α H β ).
∇α τ = u u + g2(IX.14.6)
Straightforward computation shows that it is a vector orthogonal bothto u and to H.
The Lorentz force ∇α τ αβ being orthogonal to u, the continuity equation for a ﬂuid of inﬁnite conductivity is the same as for an uncharged
ﬂuid, namely∇α [(μ + p)uα ] − uα ∂α p = 0.

(IX.14.7)

The equations of motion read(μ + p)uα ∇α uβ + (g αβ + uβ uα )∂α p + ∇α τ αβ = 0.

(IX.14.8)

Magnetohydrodynamics

229

37 K. O. Friedrichs has obtained for acharged perfect ﬂuid with non-zero ﬁnite conductivity a symmetric hyperbolic system, but by modifying theequations in a way that is not physically justiﬁed.

38 Equations for magnetohydrodynamics in Special Relativity, keeping terms
neglected in classical magnetohydrodynamics, were obtained by Hoﬀman
and Teller (1950). In General Relativity, they were ﬁrst written down andstudied by Bruhat (Choquet-Bruhat)(1960).

230

Relativistic ﬂuids

IX.14.2

Wave fronts

A straightforward computation shows that the characteristic polynomialof the ﬁrst-order system (14.5), (14.7), (14.8) factorizes as[(μ + p)a2 − b2 ]2 D,

a := uα Xα , b := H α Xα ,with

(IX.14.9)

and with D a fourth-order polynomial in X:
D ≡ (μp − 1)(μ + p)a4 + [(μ + p + |H|2 μp )a2 − b2 ]X α Xα . (IX.14.10)
The fourth-order cone D = 0 is called the magneto-acoustic cone.
The second-order cone, which enters as a double factor in (14.9),(μ + p)a2 − b2 = 0,

(IX.14.11)

is called the Alfvén cone. It is composed of two hyperplanes1[(μ + p) 2 uα ± H α ]Xα = 0.

(IX.14.12)

The corresponding wave fronts, tangent to the dual of a Alfvén plane,
are called Alfvén waves.
In the proper frame of the ﬂuid, the normals ν to the Alfvén waves
satisfy the equations1(μ + p) 2 ν0 ± H i νi = 0,

(IX.14.13)

The propagation speed of the Alfvén waves is calculated using thegeneral formula|V |2 =(uα να )2αβ(u u + g αβ )ν

α νβ

.

(IX.14.14)

One ﬁnds that
1

|VAlf | = (μ + p)− 2 |Hν̄ |,(IX.14.15)

where Hν is the scalar product of H (a space vector) with the projection
on space (normed to 1) of the normal ν to the wave front.
The wave fronts whose normals lie on the magnetoacoustic cone D = 0
are called magnetoacoustic wave fronts. The equation of the magnetoacoustic cone is a polynomial of order 4 in ν, the normal to a wave
front, which reads, in a proper rest frame of the ﬂuid,μp (μ + p + H 2 )(ν̃0 )4 − ν̃02 (μ + p + |H|2 μp + Hν̃2 ) + Hν̃2 = 0, (IX.14.16)
where we have denotedνν̃ =  2 1/2
[ν
i i]

and

Hν̃ = Hi ν̃ i .

(IX.14.17)

We deduce from this formula, solved for the unknown ν̃02 , the two magnetoacoustic wave speeds. Both are well deﬁned, because |Hν̃ | ≤ |H|,
and less than 1, the speed of light, if μp ≥ 1. They are1|V |2MA =μ + p + |H|2 μp + Hν̃2 ± [(μ + p + |H|2 μp + Hν̃2 )2 − 4Hν̃2 μp (μ + p + H 2 )] 2.
2μp (μ + p + H 2 )(IX.14.18)

IX.16

The rapid wave speed corresponds to the plus sign and the slow wave
speed to the minus sign.
Study of the characteristic determinant shows39 that the system is a
Leray–Ohya-type hyperbolic system—the fact that Alfvén wave fronts
are tangent to the magnetosonic wave fronts does not permit one toconclude its (strict) Leray hyperbolicity. However, the system has been
proved to be symmetrizable hyperbolic40 , both by the general method of
Lax–Boillat and Ruggeri, and directly (at least in special Relativity) by
K.O. Friedrichs. This hyperbolicity could be foreseen physically by theexistence of shock waves41 and of high-frequency waves42 propagating
according to ﬁrst-order diﬀerential equations.

IX.15

Viscous ﬂuids

39 See Choquet-Bruhat (1966).

40 See Anile (1989).
41 General relativistic equations were
given by Bruhat (Choquet-Bruhat)(1960) and were studied in depth by
Lichnerowicz (1967).
42 Anile and Greco (1978).

Yang–Mills ﬂuids (quark–gluon
plasmas)Although Yang–Mills charges are not manifest at ordinary scales, plasmas of quarks and gluons exist under extreme circumstances.43 Theirproperties have analogies, but also diﬀerences, with those of electromagnetic plasmas. The equations look formally the same. The Maxwell
stress–energy tensor is replaced by the Yang–Mills stress–energy tensor
and the electric current by the Lie-algebra-valued Yang–Mills current

43 Extremely brieﬂy in Earth-based laboratories. Astrophysicists have conjectured that they are present in the coresof neutron stars.

J α = γuα + σF βα uβ ,where γ is a function on spacetime taking its values in the same
Lie algebra as the Yang–Mills ﬁeld F and σ is a number, the ﬂuid
conductivity.
The mathematical properties of Yang–Mills ﬂuids are44 quite similar to those of electrically charged ﬂuids, at least locally, in the case ofﬁnite conductivity.
In the case of inﬁnite conductivity, however, a remarkable property
occurs: the wave fronts do not split into analogies of Alfvén waves andacoustic waves, but are at each point tangent to an undecomposable
sixth-order cone.45

IX.16

231

Viscous ﬂuids

There have been many discussions of equations for relativistic viscous
ﬂuids and a relativistic heat equation. Most proposals for such equations
were motivated by the desire to obtain a causal theory, that is, one described by partial diﬀerential equations of hyperbolic type, with a causal
domain of dependence, whereas the usual Fourier law of heat transfer
as well as the viscous ﬂuid Navier–Stokes equations of Newtonian mechanics are of parabolic type, corresponding to an inﬁnite propagation
speed. We will see in Chapter X some results for the approximation
beyond perfect ﬂuids obtained in the framework of Relativity.

44 See Choquet-Bruhat (1992).

45 Choquet-Bruhat (1992b, 1993).

232

Relativistic ﬂuids

IX.16.1

46 Landau

and Lifshitz (1987) link uwith some density of energy, choosing usuch that the stress tensor is orthogonal
to u.

Generalized Navier–Stokes equations

The Einstein equivalence principle would give us an expression for the
stress–energy tensor of a viscous ﬂuid in General Relativity if we knew ofsuch as expression in Special Relativity. However, this is not the case—there is no general consensus about such a tensor in Special Relativity,
everyone being inﬂuenced in his choice by his own background. The difference between various authors begins already at the physical deﬁnitionof the ﬂow vector46 u. We will stick to the simplest deﬁnition, considering u as the ﬂux density vector for the particle number r. This quantity
satisﬁes the conservation law∇α (ruα ) = 0.

(IX.16.1)

It was remarked previously that the components T 00 , T 0i , T ij of thestress–energy tensor T in a proper frame represented respectively theenergy, momentum, and stresses of the ﬂuid with respect to this frame.
For perfect ﬂuids, the expressions for these quantities were taken as theirclassical analogues, which themselves result from ﬁrst principles.
In non-relativistic mechanics, the stress tensor of a viscous ﬂuid isobtained by considering it as a linearized perturbation σ of the perfect
ﬂuid stress tensor, a perturbation due to the deformation tensor of the
ﬂow lines. In inertial coordinates of absolute space, this deformation
tensor is
∂vi
∂vj
+
.
(IX.16.2)
Dij :=∂xj∂xi
One writes the classical Navier–Stokes equations for viscous ﬂuids byintroducing the expansion and the shear of the ﬂow lines (in Euclidean
space E 3 ). The contribution of rotational ﬂow is discarded, on the basisthat it corresponds to rigid motions, which do not generate stresses. The
Newtonian stress tensor of a viscous ﬂuid is then written as thesum of the stress tensor pδij of a perfect ﬂuid and a symmetric 2-tensor
given by∂v k
2 ∂v kσij := λδij k + ν Dij − δij k .
(IX.16.3)∂x
3 ∂x
The scalar coeﬃcients, in general considered as constants, are called thebulk viscosity and the shear viscosity.
The extension of the tensor σ to Relativity is ambiguous, owing on
the one hand to the absence of absolute space and on the other handto the equivalence of mass with energy, and hence with the work of thefriction forces due to viscosity. The following tensor has been proposed
as a relativistic generalization of the classical one:Tαβ := (μ + p)uα uβ + pgαβ + σαβ ,(IX.16.4)

with σ the viscosity stress tensor, linear in the ﬁrst derivatives of u
and orthogonal to u,σαβ = λπαβ ∇ρ uρ + νπαρ πβσ (∇ρ uσ + ∇ρ uσ ),(IX.16.5)

IX.16

where π is the projection tensor47
παβ := gαβ + uα uβ .

(IX.16.6)(IX.16.7)

supposed to represent a heat ﬂow resulting from friction. The problem isthen to write an equation to determine q and justify the symmetrizationin (16.7).
The equations for ∇α T αβ deduced from (16.4) and (16.5) are rather
complicated. They have been studied in detail by G. Pichon48 in the casewhen ∇α uα = 0, called an incompressible ﬂuid. He has shown that forlarge enough μ + p, they are of parabolic type, corresponding therefore
to inﬁnite propagation speed.

IX.16.2

A Leray–Ohya hyperbolic system for viscous
ﬂuids

We have seen that the vector that leads to the generalization of classicalproperties of irrotational motion, for perfect ﬂuids, is the dynamical
velocity C := h u, with h the enthalpy (see Section IX.9.3). We deﬁnefor this vector C the shear, the vorticity tensor, and the expansion of thecongruence of its trajectories (the ﬂow lines of the ﬂuid) by the usual
decomposition. In the case n + 1 = 4,∇α Cβ =11Ωαβ + Σαβ + Θgαβ ,24with Ω the vorticity tensor,
Ωαβ := ∇ρ Cσ − ∇ρ Cσ .
In the case of a perfect ﬂuid, Ω is orthogonal to C (i.e. to u). The scalarΘ is the expansion of the congruence deﬁned by the vector ﬁeld C:Θ := ∇α C α .
We have seen that Θ is zero if and only if the ﬂuid is incompressible (inthe relativistic sense). The symmetric tensor Σ is the shear, which has
zero trace:11Σαβ := (∇α Cβ + ∇β Cα ) − gαβ ∇λ C λ .
2
4
We propose to take as stress–energy tensor of a viscous ﬂuid the tensor
Tαβ = (μ + p)uα uβ + gαβ p + T̃αβ ,(IX.16.8)

where T̃αβ is the part of T due to viscosity, a perturbation of the perfect
ﬂuid stress–energy tensor, linear in ∇C, of the formT̃αβ := λΘgαβ + νΣαβ ,(IX.16.9)233

47 Note that the coeﬃcients λ and ν do
not have quite the same interpretation
as in Newtonian mechanics.

Some authors add to T a ‘momentum term’u α qβ + u β qα ,Viscous ﬂuids

48 Pichon (1965).

234

Relativistic ﬂuids

with λ and ν viscosity coeﬃcients depending on the particular ﬂuid
under consideration. The conservation laws ∇α T αβ = 0 are then secondorder equations for the dynamical velocity C with principal part11
λg αβ ∇α ∇λ C λ + ν∇α (∇α C β + ∇β C α ) − g αβ ∇λ C λ
2
4
=



1
ν
(∇α ∇α C β + ∇α ∇β C α ) + λ − ν ∇β ∇α C α .
2
4

The 4 × 4 characteristic matrix has elementsνν
Xα X α C β + aXα X β C α ,a := + λ.
2
4
To compute the characteristic determinant, we choose a frame whereX 2 = X 3 = 0. The matrix then reads⎛ν⎞
X α Xα + αX0 X 0
aX1 X 0
0
0
⎜2
⎟
ν α
⎜
⎟
X Xα + αX1 X 1
aX0 X 1
0
0
⎜
⎟
⎜
⎟.
2
ν α
⎜
⎟
X Xα
0
0
0
⎜
⎟
2
⎝
⎠
ν α
X Xα
0
0
0
2
(IX.16.10)
The characteristic determinant is thenν
2
Δ(X) ≡ P (X)+ a (X α Xα )2 ,
2
with
ν
 ν

X α Xα + αX0 X 0
X α Xα + αX1 X 1 − a2 X 1 X1 X 0 X0 ,P (X) :=
2
2
i.e.
2 ν

ν
ν ν
X α Xα + X α Xα (αXα X α ) ≡+ a (X α Xα )2 .
P (X) :=
22
2 2
The proposed system is Leray–Ohya hyperbolic (Gevrey class of index
2) and causal, a satisfactory property for a relativistic theory.
It can be conjectured that a solution of the proposed relativistic
Navier–Stokes equations converges to a solution of the perfect ﬂuid
equations when λ and ν tend to zero.

IX.17
49 A singular perturbation with higher

derivative has been proposed by Vernotte to give an account of the few
moments occurring before the steadystate is attained. Vernotte’s idea has
been extended to General Relativity by
Cattaneo.

The heat equation

We personally think that heat transfer is a collective eﬀect; the Fourier
equation corresponds in fact to some asymptotic steady state.49 It is
not surprising that it does not translate readily into relativistic causal
equations. The classical heat equation appears after the study of the
Brownian motion, unfortunately the extension to Relativity of the studyof Brownian motion is still in its infancy, owing to several diﬃculties, inparticular the lack of absolute time.

IX.20Problems

235

The Müller–Ruggeri extended thermodynamics theory oﬀers a way totreat dissipative phenomena in a relativistic context (see Chapter X).

IX.18

Conclusion

The relativistic theory of perfect ﬂuids is mathematically and physically50 satisfactory. With proper deﬁnitions, the theorems of classical(i.e. non-relativistic) perfect ﬂuid theory are valid. Global mathematical problems remain mostly open, but this is also the case for classicalﬂuids in spite of recent progress. A book by Christodoulou51 treats in
depth three-dimensional perfect ﬂuids in special Relativity up to shock
formation.

IX.19

Solution of Exercise IX.6.2

Light cone: The light rays issuing from a given point satisfy theequationsdxα= g αβ pβ ,g αβ pα pβ = 0.
(IX.19.1)dλHence, with the usual identiﬁcation of the tangent and cotangent space
to V through the metric g,
gαβ v α v β = 0,

IX.20

Problems

IX.20.1

Speciﬁc volume

v α := g αβ pβ .

Show by integrating the equation∇α (ruα ) = 0
and using the Stokes formula that the t-dependent space scalarr̄ ≡ ru0 N(IX.20.1)

represents the density of ﬂow lines crossing a spacelike submanifold,where N denotes the spacetime lapse.

IX.20.2

Motion of isolated bodies

Denote by (V, g, r, u) the manifold, spacetime metric, and perfect ﬂuid
ﬂow solution of the Einstein equation with perfect ﬂuid source withinitial data (ḡ, K, v̄, r̄) on a manifold M satisfying the hypothesis of

50 In spite of problems with the choiceof an equation of state for the ﬂuid,
particularly when the whole cosmos isconcerned, although it is unlikely thatits content can be modelled by a perfect
ﬂuid.
51 Christodoulou (2007).

236

Relativistic ﬂuids

the existence and uniqueness theorem. Assume that the support of theinitial density r̄ is the closure ω̄ of some open subset ω ⊂ M, which has
several disconnected components, spaces occupied by material bodies at
the initial time.
Show that the solution does not depend on the ‘unphysical’ data of v̄
where r̄ vanishes.
Extend the property to charged dust.

Solution
Equations (2.4) and (2.3) show that the support of r is contained in
the geodesic tube Ω generated by geodesics of g tangent to ū issuingfrom points in ω̄. Let v̄1 be another initial value for u, such that v̄1 = v̄on ω̄. The geodesic tube Ω1 of g corresponding to v̄1 coincides with Ω;
therefore, u1 = u in Ω. Since r = 0 outside Ω, the dust stress–energy
tensors ru ⊗ u and ru1 ⊗ u1 with u1 = u in Ω coincide on the whole ofV ; therefore, the triple (g, r, u1 ) satisﬁes the Einstein dust system on V.
By the uniqueness theorem, it coincides, up to isometry, with anothersolution (g1 , r1 , u1 ) taking the same initial values, i.e. g1 = g, r1 = r on
V and u1 = u in the domain Ω1 = Ω of the presence of a ﬂuid ﬂow.
The theorem on the motion of isolated bodies extends to the case of
charged dust if we reasonably make q̄/r̄ = 0 outside the support of r̄
because the ﬂow lines of u depend only on g and F, while the right-handsides of (3.10) vanish outside the support of r̄.

IX.20.3

Euler equations for the dynamic velocity

Recall thatC α Cα = −f 2 , hence C α ∇β Cα = −f ∂β f.
Write the equations of motion and the energy equation of a perfect ﬂuid
in terms of the dynamic velocity.

Solution
We deduce from the deﬁnition of the index f that1
∂ log f
=
,
∂p
μ+p

∂p
∂p
μ+p
= μ + p,=,2
∂ log f
∂f
2f 2

hence

∂p
∂p ∂ log f=∂f 2∂ log f ∂f



∂f 2
∂f

−1
= (μ + p)

1 1
.
f 2f

Hence
∂α p =∂p2f ∂α f, hence
∂f 2

∂α p
= f −1 ∂α f = −f −2 C β ∇α Cβ .
μ+p

IX.20Recall that the Euler equations of motion areuα ∇α uβ + (gβα + uα uβ )

∂α p
= 0.
μ+p

We haveuα ∇α uβ ≡ f −1 C α ∇α (f −1 Cβ ) ≡ f −2 (C α ∇α Cβ − f −2 C α Cβ f ∂α f )(IX.20.2)and∂α p= (gβα +uα uβ )f −2 f ∂α f ≡ f −2 (f ∂β f +f −2 C α Cβ f ∂α f ),(gβα +uα uβ )μ+p(IX.20.3)
which reads∂α p≡ f −2 (C α ∇α Cβ − C α ∇β Cα ),uα ∇α uβ + (gβα + uα uβ )μ+pand hence, for the equations of motion,
C α Ωαβ = 0.

(IX.20.4)

The energy (also called continuity) equation for a barotropic ﬂuid
reads
∂α p
dμ
= 0,
μp :=.
(IX.20.5)∇α uα + uα μp
μ+pdp
We have∇α uα + uα μp∂α p≡ ∇α (f −1 C α ) + f −1 C α μp f −2 f ∂α fμ+p≡ f −1 ∇α C α + C α f −2 (−1 + μp ) .

Hence, for the energy equation,∇α C α + f −2 (−1 + μp )C α ∂α f = 0.
This reduces to ∇α C α = 0 if μp = 1 (incompressible ﬂuids).

IX.20.4

Hyperbolic Leray system for the dynamical
velocity

1. Prove that the equations satisﬁed by the dynamical velocity C,
C α Ωαβ = 0 with

Ωαβ := ∇α Cβ − ∇β Cαand∇α C α = (μp − 1)CαC β ∇α Cβ ,C λ Cλimply, when δΩ is known, that C satisﬁes a quasidiagonal system of
second-order diﬀerential equations that is hyperbolic if dμ/dp > 0.
2. Show that C satisﬁes a quasilinear quasidiagonal third-order system,
which is hyperbolic and causal if dμ/dp > 1.

Problems

237

238

Relativistic ﬂuids

Solution
1. For C considered as a 1-form, we have the identity (see Chapter I){(dδ + δd)C}γ ≡ −g αβ ∇α ∇β Cγ + Rγα Cα .

(IX.20.6)

The deﬁnitions of the rotational and the energy equations givedC = Ω andδC :≡ −∇α C α = −(μp − 1)CαC β ∂α Cβ .
C λ Cλ
(IX.20.7)

It results from (20.6) and (20.7) that


Cα
αβ
α

β
g ∇α ∇β Cλ − Rλ Cα = −∇γ (μp − 1) λ C ∂α Cβ + ∇α Ωαγ .
C Cλ(IX.20.8)
We compute


Cα

β
∇γ (μp − 1) λ C ∇α Cβ ≡ A(μp − 1) + B∂γ (μp − 1),C Cλwhere we have setCαβA := ∇γC∇C
α β ,C λ Cλand B :=C αC β∇α Cβ .
C λ CλIt holds thatC αC βC αC βA ≡ λ ∇γ ∇α Cβ + ∇γ λ(∇α Cβ ).
C Cλ
C Cλ

By the antisymmetry of the Riemann tensor, we have
C α C β ∇γ ∇α Cβ ≡ C α C β (∇α ∇γ Cβ + Rγα ,βρ C ρ ) ≡ C α C β ∇α ∇γ Cβ .
By the deﬁnition of the vorticity Ω,
C α C β ∇α ∇γ Cβ ≡ C α C β ∇α (Ωγβ + ∇β Cγ )≡ C α C β ∇α ∇β Cγ + C β C α ∇α Ωγβ ,with, by the Helmholtz equation,
C α ∇α Ωγβ = Ωβα ∇γ C α + Ωαγ ∇β C α .

(IX.20.9)

Therefore,A≡
1  α βC C ∇α ∇β Cγ + C β (Ωβα ∇γ C α + Ωαγ ∇β C α )C λ CλC αC β+ ∇γ λ(∇α Cβ ).
C CλWe deduce from these computations, since Ω is of ﬁrst order inC, that when ∇α Ωαγ is known, the dynamical velocity C satisﬁes a system of quasidiagonal second-order equations with principal
operator α β C C
(IX.20.10)(g,C C)γ := g αβ + (μp − 1)∇α ∇β Cγ .
C λ Cλ

IX.20The ﬁrst-order terms in C areC αC βΩγβDγ := (μp − 1) − λ C α ∇α C β + ∇γ λ(∇α Cβ ) .
C Cλ
C Cλ
In a proper frame, the corresponding characteristic polynomial is− μp X02 +(Xi )2 .
(IX.20.11)
It is hyperbolic if μp > 0, causal if μp ≥ 1. It reduces to the Minkowski
polynomial for μp = 1.
2. Since ∇α Ωαβ is second-order in C, the principal part of the full operator does not reduce to the operator (20.10). To obtain a hyperbolic
operator for C, we take the derivative of (20.4) in the direction of C.
We haveC λ ∇λ ∇α Ωα β ≡ ∇α (C λ ∇λ Ωαβ ) − (∇α C λ )∇λ Ωαβ .
Hence, modulo the Helmholtz equations, C λ ∇λ ∇α Ωα β is a quadratic
polynomial in ∇C and ∇2 C, namely
Eβ := C λ ∇λ ∇α Ωα β
(IX.20.12)=∇α Ωβλ ∇α C λ + Ωλα ∇β C λ − ∇α C λ ∇λ Ωαβ .
We have proved that C α (∇α C,g C + δΩ) is a quasidiagonal thirdorder system with characteristic polynomial in a proper frame(Xi )2 .
f X0 −μp X02 +The lower second-order terms in C are
C α ∇α Dγ + Eγ .

IX.20.5

Geodesics of conformal metricShow that the trajectories of the dynamical velocity C := f u are
geodesics of the metric g̃ := f 2 g conformal to the original spacetime
metric g.

SolutionIf g̃ = f 2 g, it holds that˜ β Cα − ∇β Cα = (Γλ − Γ̃λ )Cλ
∇
βα
βα

−3
∂α f Cβ + ∂β f Cα − gαβ g λμ ∂μ f Cλ ,= −f(IX.20.13)
and hence, using (9.5) and (9.4),
˜ β Cα = C β ∇α Cβ − f −2 ∂α f C β Cβ = 0.
Cβ∇(IX.20.14)

Problems

239

240

Relativistic ﬂuids

IX.20.6

Cosmological equation of state p = (γ − 1)μTake the cosmological equation of state p = (γ − 1)μ.
1. Compute the index f.
2. Compute the energy equation.

Solution
1. If p = (γ − 1)μ, then the index f is given by(γ − 1) dp
f ≡ exp,
γpand hence, up to an irrelevant multiplicative constant,f ≡ p(γ−1)/γ ,p = f γ/(γ−1) .

2. If p = (γ − 1)μ, thenμp − 1 =

2−γ
,
γ−1

and the energy equation reads∇α C α +2 − γ C αC β
∇α Cβ = 0.
γ − 1 C λ Cλ(IX.20.15)

X

Relativistic kinetic theory
X.1

Introduction

In kinetic theory, the matter, usually called a plasma, is composed ofa collection of particles whose size is negligible at the considered scale:rareﬁed gases in the laboratory, stars in galaxies, or even clusters ofgalaxies at the cosmological scale. The number of particles is so great andtheir motion so chaotic that it is impossible to observe their individual
motion.
We have introduced in Chapter IX equations for perfect ﬂuids in General Relativity as a direct consequence of the perfect ﬂuid equations inSpecial Relativity and the equivalence principle. We have seen that they
are quite satisfactory mathematically as well as physically. We have also
seen that there are no compelling macroscopic considerations leading tosatisfactory relativistic equations for dissipative ﬂuids, even in Special
Relativity. The classical Navier–Stokes equations lead, like the Fourier
law of heat transfer, to equations of parabolic type, and hence to an inﬁnite propagation speed of signals, which is incompatible with relativistic
causality. The formal generalizations of these equations with the aim ofobtaining hyperbolic equations lack general justiﬁcation and there is noconsensus about them.
An a posteriori justiﬁcation of the non-relativistic Euler and Navier–Stokes macroscopic equations is to deduce them from the motion of theﬂuid particles at the microscopic scale and the statistical hypothesis ofclassical kinetic theory. It presents no conceptual diﬃculty to extend
the setting of kinetic theory to Special as well as to General Relativity.
It is straightforward to construct an energy–momentum vector and astress–energy tensor for kinetic matter and to couple the latter with
the Einstein equations. The collective motion of collisionless particles isnaturally modelled and leads to conservation equations for the stress–
energy tensor. A case of particular interest in General Relativity is whenthe ‘particles’ are stars in galaxies, or even galaxies in clusters of galaxies.
It is then appropriate to take the charges to be zero and the masses tohave values1 between two positive numbers.
When the particles undergo collisions, one can write a Boltzmann
equation, and a coherent Einstein–Boltzmann system, for an appropriate choice of the collision cross-section. The problem is the choice of aphysically reasonable cross-section; this problem arises already in the
non-relativistic case, but is especially delicate when it is stars or galaxies that collide. The Boltzmann equation arises in Special Relativity for

X.1 Introduction
X.2 Distribution function

241
242

X.3
X.4

Vlasov equations
Solution of a Vlasov
equation

246

X.5

The Einstein–Vlasov
systemThe Cauchy problem

X.6
X.7
X.8

249
251
252

The Maxwell–Einstein–
Vlasov system
254
Boltzmann equation.
Deﬁnitions
256

X.9

Moments andconservation laws
X.10 Einstein–Boltzmannsystem
X.11 Thermodynamics
X.12 Extended
thermodynamics
X.13 Solutions of selected
exercises

262

X.14

264

Problems

257
258
259
262

1 And not discrete values, while this is

the case for electromagnetic plasmas.

242

Relativistic kinetic theory

plasmas of elementary particles with a ﬁnite number of distinct propermasses and charges; the gravitational ﬁeld is then negligible under usual
circumstances. However, the general relativistic Boltzmann equation ispossibly important for obtaining equations for dissipative relativistic
ﬂuids as approximations.
Throughout this chapter, as in previous chapters, we assume that the
spacetime (V, g) is an oriented and time-oriented Lorentzian manifold,
the tangent bundle TV is oriented by the orientation of V, and all ﬁbres
are positively oriented. We treat the (n + 1)-dimensional case, specifyingn = 3 when it enjoys special properties.

2 Representation

by a many-particle
distribution function is possible, but
diﬃcult to handle. Representation by
a one-particle distribution function islinked with the onset of chaos. The
mathematical justiﬁcation of the onsetof chaos in relativistic dynamics is alargely open problem.
3 In

local
gαβ pα pβ .

coordinates,g(p, p)≡X.2

Distribution function

X.2.1

Deﬁnition

It is assumed that the state of the matter in a spacetime (V, g) isrepresented2 by a ‘one-particle distribution function’. This distributionfunction f is interpreted as the density of particles at a point x ∈ Vthat have a momentum p ∈ Tx V , the tangent space to V at x. In a
relativistic theory, the momentum p is a future timelike or null vector
whose components deﬁne the energy and momentum of the particle withrespect to a Lorentzian frame. With our signature conventions, it holds
that3 g(p, p) = −m2 , with m the rest mass of the particle. In view ofapplications to astrophysics, we do not assume a priori that it is thesame for all particles. We state a deﬁnition:
Deﬁnition X.2.1 A distribution function f is a non-negative scalar function on the so-called phase space PV , a subbundle of the tangent
bundle TV to the spacetime V :f : PV → R

by(x, p) → f (x, p),

with

x∈V,

p ∈ Px ⊂ Tx V.
(X.2.1)

If (V, g) is a Lorentzian manifold, then the ﬁbre Px at x is such thatgx (p, p) ≤ 0 and, in a time-oriented frame, p0 ≥ 0.
Remark X.2.1 If a particle has positive rest mass, then it holds thatp0 > 0 in a time-oriented frame.
If a particle has zero rest mass, then p0 can vanish only if p itself
vanishes. The value p = 0 is, from the mathematical point of view, asingular point of the vector ﬁeld X. On the other hand, relativistic physics says that particles with zero mass move with the speed of light, thecomponent p0 in an orthonormal frame is the energy with respect to an
observer at rest in this frame, and quantum theory tells us that it does
not vanish, i.e. p0 > 0.

X.2.2
4 In the sense of the Gibbs ensemble.

Interpretation

The physical meaning of the distribution function is that it gives a mean4‘presence number’ density of particles in phase space. More precisely,X.2

Distribution function

243

we denote by θ the 2(n + 1)-volume form on T V, i.e. with θx and θprespectively the volume forms on V and Tx V ,θ := θx ∧ θp .

(X.2.2)In local coordinates, θx and θp are given by1

θx = |Det g| 2 dx0 ∧ dx1 . . . ∧ dxn ,

X.2.3

1

θp := |Det g| 2 dp0 ∧ dp1 . . . ∧ dpn .
(X.2.3)

Moments of the distribution function

The moments of f are functions or tensors on V obtained by integrationon the ﬁbres of the phase space PV of products of f by tensor productsof p with itself.
Moment of order zeroThis is by deﬁnition the integral on the ﬁbre Px of the distributionfunction:5r0 (x) :=f θp .
(X.2.4)Px

It is a ‘density of presence’ in spacetime.
First and second moments

5 Recall that the integral on an oriented(n + 1)-manifold Px , with orientation
deﬁned by the order of coordinates
x0 , . . . , xn , of the exterior diﬀerential
form f (p)θp is equal to the calculus
integral
f (p)θp =f (p) dp0 . . . dpn .
Px

The ﬁrst moment of f is a vector ﬁeld on V deﬁned byα
P (x) :=
pα f (x, p)θp .

(X.2.5)Px

If the spacetime is time-oriented and the particles have non-negative rest
mass and travel towards the future, i.e. if pα is future causal (timelike
or null), then the ﬁbre Px is included in the subset p0 > 0. The vector
f pα is then also causal and future-directed, since f ≥ 0, and the sameis true of the ﬁrst moment P α .
Out of the ﬁrst moment, one extracts a scalar r ≥ 0 interpreted as
the square of a speciﬁc rest mass density given by
r2 := −P α Pα .

(X.2.6)

If P is timelike, then r > 0 and one deduces from the ﬁrst moment aunit vector U interpreted as the macroscopic ﬂow velocity and given by
U α := r−1 P α .

(X.2.7)A suﬃcient condition for P to be timelike is that the particles havepositive mass, since then all p, and hence also P, are timelike.
The second moment of the distribution function f is the symmetric
2-tensor on spacetime given by
T αβ (x) :=f (x, p)pα pβ θp .
(X.2.8)Px

It is interpreted as the stress–energy tensor of the distribution f.

Px

244

Relativistic kinetic theory

Theorem X.2.1 If a distribution function f depends on p only througha scalar product Uα pα , with U α a given timelike vector on spacetime,then the ﬁrst and second moments of f read respectively as the momentum vector and the stress–energy tensor of a perfect ﬂuid. The unit ﬂow
velocity is collinear with U.
Proof. Assume thatf (x, p) ≡ F (x, Uα (x)pα ).
The ﬁrst moment of f is thenP α (x) ≡Px

pα F (x, Uα (x)pα )θp .

(X.2.9)(X.2.10)

Take at x an orthonormal Lorentzian frame with time axis collinear with1
U. In this frame, Uα (x) = −λ(x)δα0 , with λ := (−Uα U α ) 2 , and hence wehave the following:
1. The ﬁrst moment of f has components at x such thatipi F (x, −λ(x)p0 )θp = 0,
P (x) =

(X.2.11)Px

because the integrand is antisymmetric in the pi while F is positive.
The vector P is therefore collinear with the time axis. The component
P 0 in the orthonormal frame,0
P (x) =
p0 F (x, −λ(x)p0 )θp ,(X.2.12)Px

is the macroscopic rest mass density (the presence number density if
all particles have the same mass normalized to 1).
2. In the same frame, the same antisymmetry considerations imply that
T 0i = 0 and T ij = 0 for i = j. The equality of the components
T ii , i = 1, . . . , n,
T ii =(pi )2 F (x, −λ(x)p0 )θp ,(X.2.13)Px

results from the invariance under rotations in momentum space ofthe function F and the volume element θp . The component T 00 is the
positive function00(p0 )2 F (x, −λ(x)p0 )θp .
(X.2.14)T =Px

The tensor T is therefore the stress–energy tensor of a perfect ﬂuid
with ﬂow vector u = λ−1 U, speciﬁc energy given by (2.17), andpressure computed from 2.13.

6 Marle (1969).
7 Eckart (1940).

Remark X.2.2 The physical interpretation of the ﬁrst moment that
we have given coincides with the interpretation chosen by Marle6 andwith Eckart’s original deﬁnition.7 For a discussion and interpretation

X.2

Distribution functionof the second moment in the general case (dissipative ﬂuids), see thediscussion in Marle’s paper on pp. 137–143.
Higher moments
Higher moments are deﬁned as totally symmetric tensors on V
given by
α1 ...αpM:=f (x, p)pα1 . . . pαp θp .
(X.2.15)
Px

They play an important role in the Müller–Ruggeri extended thermodynamics.

X.2.4

Particles of a given rest mass

Stars or galaxies do not have the same rest mass, but the original kinetic
theory was constructed for the case of gases in laboratories, composedof molecules with the same rest mass m. In that case, the phase spaceover (V, g), denoted by Pm,V , has for ﬁbre Pm,x the mass hyperboloid(also called the mass shell)Pm,x ≡ Px ∩ {g(p, p) = −m2 }.

(X.2.16)

In the case of particles of a given mass m, the volume form θm,p on themass shell Pm,x is, taking the pi as local coordinates on Pm,x (then p0
is a function of x and pi ),1

θm,p =|Det g| 2 1
dp ∧ . . . ∧ dpn .
p0(X.2.17)

Exercise X.2.1 Prove this formula using the Leray deﬁnitiond
1(g(p, p) − m2 ) ∧ θm,p = θp .
2

(X.2.18)

The moments can be deﬁned by the same formula as in Section X.2.3,
by integration on the mass shell and replacing θp by θm,p .
One can also consider the case of N diﬀerent particles with diﬀerent
masses mI , I = 1, 2, . . . , N ; there are then N diﬀerent phase spaces
PmI ,V , and diﬀerent distribution functions fI . The moments to consider
are the sums of the corresponding moments; for example, the second
moment isfI (x, p)pα pβ θmI ,p .
(X.2.19)T αβ (x) :=I

Pma ,xFor details see the article by Marle.8

8 Marle (1969).

245

246

Relativistic kinetic theory

X.3

Vlasov equations

In the Vlasov models, one assumes that the gas is so rareﬁed that theparticle trajectories do not scatter (i.e. one neglects collisions between
particles). Their motion is determined by the average ﬁelds they generateon spacetime.

X.3.1

General relativistic (GR)–Vlasov equation

In a curved spacetime of General Relativity, in the absence of nongravitational forces, each particle follows a geodesic of the spacetime
metric g. We have seen (Chapter I, Section I.8) that the diﬀerential
system satisﬁed by a geodesic in the tangent space T V of a pseudoRiemannian manifold (V, g) reads in local coordinates, with λ called acanonical aﬃne parameter,
pα :=

dxα
,
dλ

dpα
= Gα ,dλλ μwith Gα := −Γαλμ p p ,(X.3.1)

where Γαλμ are the Christoﬀel symbols of the metric g. In other words,
the trajectory of a particle in T V is an element of the geodesic ﬂow
generated by the vector ﬁeld X = (p, G) whose components pα , Gα ina local trivialization of T V over the domain of a chart of V are givenby (3.1).
Exercise X.3.1 Show by direct calculation that X = (p, G) is indeed
a vector ﬁeld on the tangent bundle T V.
In a collisionless model, the physical law of conservation of particles,
together with the invariance of the volume form in T V under the geodesic ﬂow (Liouville’s theorem; see Problem X.14.1) imposes that the
distribution function f be constant under this ﬂow, that is, that it satisfy the following ﬁrst-order linear diﬀerential equation, which we callthe GR–Vlasov equation:
LX f ≡ pα∂f
λ μ ∂f− Γα= 0.
λμ p p∂xα∂pα(X.3.2)

We have interpreted the scalar −g(p, p) as the square of the rest
mass of the particle. The following lemma, the formulation of a property already seen in other contexts, makes this interpretation physicallyconsistent:
Lemma X.3.1 In a GR–Vlasov plasma, the scalar g(p, p) is constantunder the geodesic ﬂow; that is,
LX {g(p, p)} := pα∂g(p, p)∂g(p, p)
+ Gα
≡ 0,
α
∂x
∂pα

Exercise X.3.2 Prove this identity.
Hint: Use the values of the Christoﬀel symbols.

λ μGα := −Γαλμ p p .

X.3

Vlasov equations

For particles with a given rest mass, it is convenient to take as coordinates on the phase space Pm the 2n+1 numbers xα , pi . The distributionfunction fm in such coordinates is
fm (xα , pi ) = f (xα , pi , p0 (pi )).

(X.3.3)

Exercise X.3.3 Prove that gαβ pα pβ = −m2 implies
∂p0
pi
=− ,
i
∂p
p0

∂p0
pλ pμ ∂gλμ
=
−
.
∂xα
2p0 ∂xα(X.3.4)

Deduce from this deﬁnition and identities that the Vlasov equation takes
the form, with bounded coeﬃcients,9∂fm
pi ∂fmGi ∂fm++= 0.
∂x0
p0 ∂xip0 ∂pi

X.3.2

EM–GR–Vlasov equation

When, in addition to gravitation, the particles are subjected to somegiven force, represented by a vector Φ tangent to the spacetime V, thevector Y tangent to the particle trajectories in the phase space PV overa spacetime (V, g) is
Y = (p, G + Φ),i.e.

(Y A ) = (pα , Gβ + Φβ ).

(X.3.5)

If the volume form θ is invariant under the ﬂow of Y, that is, if
LY θ = 0, then particle number conservation gives for the distributionfunction f a generalized GR–Vlasov equation
LY f ≡ pα

∂f
λ μ
α ∂f
+ (−Γα= 0.
λμ p p + Φ )∂xα∂pα(X.3.6)

Lemma X.3.2 LY θ = 0 if ∂Φα /∂pα ≡ 0.
Exercise X.3.4 Prove this result.
Hint: Use the identity LY θ ≡ d(iY θ) + iY (dθ).
If the particles have electric charge e and move in an electromagnetic
ﬁeld F , then their trajectories have tangent vectors in phase space
Y = (pα , Gα + Φα ),1with Φα ≡ e|g(p, p)|− 2 F αβ pβ .

(X.3.7)

We deduce from the expression for Φ and the antisymmetry of F that
∂Φα
≡ 0.
∂pα
Therefore, the volume form θ is invariant under the ﬂow of Y, and particle
number conservation gives the EM–GR–Vlasov equation
LY f ≡ pα

∂f
∂f
λ μ
− 12 αβ+ (−ΓαF pβ ) α = 0.
λμ p p + e|g(p, p)|∂xα∂p(X.3.8)9 Choquet-Bruhat (1971b).

247

248

Relativistic kinetic theory

The following lemma provides a coherent consideration of an EM–GR
plasma where particles have both the same mass and the same charge.
Lemma X.3.3 In an EM–GR–Vlasov plasma, the scalar g(p, p) isconstant along an orbit of Y in T V ; that is,pα∂g(p, p)∂g(p, p)+ (Gα + em−1 F αβ pβ )
≡ 0.
∂xα
∂pα

The proof is an easy consequence of the lemma proved in the purely
gravitational case, because∂g(p, p)∂(gλμ pλ pμ )
≡
= 2pα
∂pα
∂pα

and

F αβ pβ pα = 0,

by the antisymmetry of F.
Exercise X.3.5 Prove that if the particles all have the same mass m,then the distribution function fm satisﬁes on Pm the following reduced
EM–GR–Vlasov equation:∂fm
pi ∂fmGi + Φi ∂fm
+
+
= 0.
∂x0
p0 ∂xip0∂pi
The EM–GR–Vlasov equation can be generalized to particles withrandom charges (see Problem X.14.2).

X.3.3

10 Choquet-Bruhat and Noutchegueme(1986). See also YCB-OUP2009, Chapter X, Section 3.3.

Yang–Mills plasmas

Yang–Mills plasmas were not observed in laboratories until highenergy particle colliders were able to decompose baryons into theirconstitutive quarks and gluons. The Yang–Mills–Vlasov10 equation is
analogous to the EM–Vlasov equation, but the electromagnetic ﬁeld is
replaced by a Yang–Mills ﬁeld taking its values in a Lie algebra G and
the electric charge e, a constant, is replaced by a function q on the spacetime V with values in the Lie algebra G. The phase space for the kinetic
theory is now the product PV × G. Although the gravitational ﬁeld does
not seem to play a role at the scale of Yang–Mills plasmas, we write thegeneral relativistic equations here. The trajectory of a particle in thisphase space is a solution of the diﬀerential systemdxα= pα ,ds

dpα= Gα + q · F αβ pβ ,dsdq
= Q := −pα [Aα, , q], (X.3.9)
ds

where [.,.] is the bracket in the Lie algebra G, i.e. [Aα, , q]a :=cabc Abα q c . The distribution function f is now a function of x, p, q. In
the absence of collisions and other forces, f satisﬁes the Einstein–Yang–
Mills–Vlasov equationpα
11 A case treated in Marle (1969).

∂f
∂f
∂f
+ (Gα + q · F αβ pβ ) α + Qa α = 0.
α
∂x
∂p
∂q

(X.3.10)

A realistic plasma may contain N charged particles of diﬀerent
kinds,11 with rest masses mI and electric charges eI ; their momenta lie

X.4

Solution of a Vlasov equation249

in mass hyperboloids gαβ pα pβ = −m2I . If we take the pi as coordinates
on these mass hyperboloids and denote by fmI (x, p) their distributionfunctions, then each fmI satisﬁes a reduced EM–GR–Vlasov equation
on a phase space PmI .

X.4

Solution of a Vlasov equation

X.4.1

Construction

Let (V, g) be a Lorentzian manifold. Assume given on the tangent spaces
to V, TM0 V, at points of an initial hypersurface M0 ⊂ V , a function f¯that will be the initial value of a distribution function f solution of theEinstein–Vlasov equation, at least in a neighbourhood of TM0 V .
The GR–Vlasov equation is a linear ﬁrst-order partial diﬀerential
equation
LX f := pα∂f (x, p)∂f (x, p)+ Gα (x, p)
= 0.
∂xα
∂pα

As is classical for solving such an equation, we transform it into an
ordinary diﬀerential equation by solving the diﬀerential system called
the characteristic system:dxα= pα ,dλdpα
λ μ= Gα ≡ −Γαλμ p p .
dλ(X.4.1)

Let x0 = 0 be the equation of M0 . Denote by ξ i and π α coordinates ofa point of ξ ∈ M0 and a point π ∈ Tξ V, the tangent space to V at ξ.
Assume given a function f¯(ξ, π); then the quasilinear ﬁrst-order diﬀerential system (4.1) has, for λ small enough, one and only one solution,12a geodesic ﬂow that takes, for λ = 0, the given values (ξ, π):xα = φα (λ, ξ, π), pα = ψ α (λ, ξ, π),ξ := (ξ i ), π := (π α ),with
φ0 (0, ξ, π) = 0, φi (0, ξ, π) = ξ i , ψ α (0, ξ, π) = π α .

(X.4.2)Inserting these functions φα and ψ α into the Vlasov equation gives thefollowing ordinary diﬀerential equation along the trajectory issuing from(ξ, π):

12 A local C 1 solution exists, i.e. a
mapping (ξ, π) → (x(λ), p(λ)) for λsmall enough, if the coeﬃcients of thediﬀerential system are C 1 , i.e. if thespacetime metric is C 2 . The solution
is unique if the coeﬃcients are Lipschitzian in x and p, and hence if thespacetime metric is C 1,1 (i.e. C 1 with
Lipschitzian derivatives).

df {(φ, ψ)(λ, ξ, π)}= 0;dλthat is, f is constant along the trajectory, i.e.
f {(φ, ψ)(λ, ξ, π)} = f¯{(φ, ψ)(0, ξ, π)}.
The mapping Λ : (λ, ξ i , π α ) → (xα , pα ) reduces to the identity for
λ = 0; it is therefore invertible13 (if C 1 ) for small enough λ, i.e. in

13 In other words, the geodesic ﬂow has
no conjugate point.

250

Relativistic kinetic theory

a neighbourhood of M0 . The inverse mapping Λ−1 gives the following
solution of the GR–Vlasov equation:f (x, p) ≡ f¯(ξ(x, p), π(x, p)).

(X.4.3)The same procedure gives the general solution of an EM–GR–Vlasov
equation.
Exercise X.4.1 Write the characteristic system of an EM–GR–Vlasov
equation.

14 Thisglobal existence holds, in appropriate functional spaces, for linear hyperbolic equations, in particularwave equations in a Lorentzian metric
whose null rays issuing from a point do
not necessarily constitute (out of thispoint) a smooth hypersurface.

15 Choquet-Bruhat (1971b).

X.4.2

Global existence theorem

The construction given in Section X.4.1 breaks down when the geodesic
ﬂow ceases to be regular. However, this breakdown does not imply that
the Cauchy problem for the GR–Vlasov equation does not have a global
solution.14 This solution is not given by the formula (4.3), but its global
existence can be proven by using energy-type estimates for ﬁrst-orderlinear equations. We state a theorem,15 leaving to the interested reader
the formulation of relevant functional spaces.
Theorem X.4.1 The Cauchy problem for the GR–Vlasov equation on
the phase space PV of a regularly sliced (n + 1)-dimensional spacetime(V, g), with data f¯ on PM̄ , M̄ := M × {0}, admits one and only one
solution f on PV .
It is probably possible to prove global existence on the tangent bundleof a globally hyperbolic manifold with appropriate hypotheses on theinitial manifold and initial data.

X.4.3

Stress–energy tensor

The stress–energy tensor of a distribution f is its second moment:αβ
T (x) :=f (x, p)pα pβ θp .
(X.4.4)Px

α βIt is deﬁned at x ∈ V if f p p is integrable on the ﬁbre Px of the phasespace, which requires a proper fall oﬀ of f in p at inﬁnity in Px , satisﬁedin particular if f has compact support in p; this compactness results
in a neighbourhood of M̄ from the compactness of the support of theinitial f¯.
More generally, consider the case of interest for astrophysics:
0 < m2 ≤ −g(p, p) ≤ M 2 .
It holds that pα pβ /(p0 )2 is uniformly bounded on PV , and hence theintegrals (4.4) are deﬁned if
(p0 )2 f (x, p)θp
Px

is deﬁned.

X.5

X.5

The Einstein–Vlasov system

X.5.1

Equations

The Einstein–Vlasov system

251

The Einstein–Vlasov system comprises the Einstein equations for a
Lorentzian metric g with source the second moment of a distributionfunction f, coupled with the GR–Vlasov equation for f :Sαβ = Tαβ ,
Tαβ (x) :=Px

1Sαβ := Rαβ − gαβ R,
2with(X.5.1)1

f (x, p)pα pβ θp , θp := |Det g| 2 dp0 dp1 . . . dpn ,(X.5.2)∂f∂f
λ μ+ Gα α = 0, Gα := −Γαλμ p p .
∂xα∂p(X.5.3)

coupled with
LX f ≡ pα

X.5.2

Conservation law

The Einstein–Vlasov system is coherent if the stress–energy tensor (5.2)
satisﬁes the conservation law∇α T αβ = 0.
We prove the following more general theorem:16
Theorem X.5.1 If the distribution f satisﬁes the GR–Vlasov equation
LX f = 0, then its moments satisfy the conservation laws
∇α1 M α1 α2 ...αp = 0,

(X.5.4)

where ∇ is the covariant derivative in the spacetime metric g.
Proof. We choose at the point x coordinates such that the ﬁrst derivatives ∂α gλμ := (∂/∂xα )gλμ , and hence the Christoﬀel symbols, vanish at
x. At that point, in these coordinates, it holds that
∇α1 M α1 α2 ...αp =∂M α1 α2 ...αp .
∂xα1
1It also holds that (∂/∂xα )θp = θp , since (∂/∂xα )(Det g) 2 = 0 when
∂α gλμ = 0. Therefore,∂∂
α1 α2 ...αpM=f (x, p)pα1 . . . pαp θp∂xα1∂xα1 Px∂=f (x, p)pα1 . . . pαp θp ,α1
Px ∂x
but when the Christoﬀel symbols are zero,


∂
α1
αp
f
(x,
p)p
.
.
.
p
θ
=
LX f (x, p)pα2 . . . pαp θp .
p
α1
∂x
Px
Px

(X.5.5)16 For the ﬁrst moment, this expresses
the conservation law of the macroscopic
momentum.

252

Relativistic kinetic theory

We have obtained a relation between tensors on V at an arbitrary point
in particular coordinates; in arbitrary coordinates, it reads
∇α1 M α1 α2 ...αp ≡LX f (x, p)pα2 . . . pαp θp .
(X.5.6)PxX.6

17 Choquet-Bruhat(1971b), for particles with a given rest mass. For particles with rest mass in a range of
positive numbers, see YCB-OUP2009,
Chapter X.

The Cauchy problem

In a Cauchy problem, one looks for solutions of a system of diﬀerential or
integrodiﬀerential equations taking given data on an initial hypersurface.
A local-in-time existence theorem for a solution of the Cauchy problem
for the coupled Einstein–Vlasov system was proved long ago;17 here wewill only give the idea of the proof and state the main results.
As in the general case of Einstein equations with source a stress–
energy tensor satisfying the conservation laws, the solution of the
Cauchy problem for the Einstein equations with a kinetic source splits
into an elliptic system called the constraints for Cauchy data on theinitial hypersurface and a hyperbolic evolution system for the metricin a suitably chosen gauge, for example a wave gauge. This evolution
system is coupled with an evolution system for the source, here a
Vlasov equation. The solution in wave gauge is proved, as always, to bea solution of the full Einstein equations by using the conservation law
satisﬁed by the stress–energy tensor—here it is a distribution function
solution of the GR–Vlasov equation.

X.6.1

Cauchy data and constraints

An initial data set for the Einstein–Vlasov system (see Chapter VIII)
is a quadruplet (M, ḡ, K, f¯), where M is a hypersurface of a diﬀerentiable manifold V, ḡ and K are respectively a properly Riemannian
and a symmetric 2-tensor on M, while f¯ is a function on the bundle
PM := ∪x∈M Px (V ) corresponding to the kind of particles we consider.
As always, the data must satisfy on M the Hamiltonian constraint
R(ḡ) − |K|2ḡ + (trḡ K)2 = 2N 2 T̄ 00(X.6.1)and the momentum constraint¯¯ K = N̄ T̄ 0i ,
∇.K
− ∇tr
T̄ αβ (x) =Pxf¯(x, p)pα pβ θp ,(X.6.2)x ∈ M, p ∈ Px (V ).

A general property of the Einstein equations is that the conformallyformulated constraints decouple if the initial manifold has constant mean
extrinsic curvature and the source has zero momentum, i.e. T̄ 0i = 0.

X.6

The Cauchy problem

Exercise X.6.1 Show that T̄ 0i (x) = 0 if f¯(x, p0 , pi ) = f¯(x, p0 , −pi ).
In the Maxwell–Einstein–Vlasov case, the Cauchy data on M0 are,in addition to ḡ, K, and f¯, the data F̄αβ of Fαβ on M0 .
Exercise X.6.2 Write the system of constraints satisﬁed by ḡ, K, F̄ .

X.6.2

Evolution

As is often the case for a coupled system, a general method to solve theEinstein equations in wave gauge coupled with a GR–Vlasov equation
is by iteration: ﬁnd a solution f1 of the Vlasov equation for a given
Lorentzian metric g1 ; then look for a solution g2 of the reduced Einstein equations with source f1 ; then iterate and study the convergenceof the iteration, either numerically or mathematically. The mathematical method used by Choquet-Bruhat18 to prove the existence of these
iterates and their convergence is through classical energy estimates forwave equations for the metric and weighted-in-p0 energy estimates for
the Vlasov equation to be satisﬁed by the function f . It is probably possible to use instead constructive methods, relying on the parametrix inthe Einstein case and the formula (4.3) for the Vlasov part.

X.6.3

18 Choquet-Bruhat (1971b).

Local existence and uniqueness theorem

We state the theorem, leaving the formulation of relevant functional
spaces to the interested reader:
Theorem X.6.1 The Cauchy problem for the Einstein–Vlasov systemwith initial data (ḡ, K) on M, f¯, on PM , satisfying the constraints, admits a solution (g, f ) on V × PV , with V := M × [0, ]), and smallenough.

X.6.4

Global theorems

The collisionless kinetic theory having no global problems of its own,
one may hope to extend to the Einstein–Vlasov system global results
obtained for the vacuum Einstein equations. Such results are alreadyhard to prove. Global existence theorems, or proofs of the cosmic censorship conjectures, for the Einstein–Vlasov system have been obtained
only in the presence of an isometry group. The ﬁrst global existence
theorem, for small initial data, was proved by Rein and Rendall19 in thecase of spherical symmetry. A paper by Dafermos and Rendall20 proves
cosmic censorship in the case of surface-symmetric compactly supported
initial data. References to other global results concerning the Einstein–
Vlasov system can be found in that paper and in a review by Rendall.21

19 Rein and Rendall (1992).
20 Dafermos and Rendall (2006).

21 Rendall (2005).

253

254

Relativistic kinetic theory

X.7

The Maxwell–Einstein–Vlasov system

Charged particles in a kinetic model generate by their motion an average
electric current J, and hence an electromagnetic ﬁeld, a 2-form F.

X.7.1

Particles with given rest mass and charge

If all particles have the same charge e and rest mass m, the current isf (x, p)pα θm,p .
(X.7.1)J α (x) = m−1 e
Pm,xThe electromagnetic ﬁeld satisﬁes on V the Maxwell equations
dF = 0,22 The Maxwell–Vlasov system in Spe-cial Relativity has been studied extensively.

δF = J,i.e.

∇α F αβ = J β .

(X.7.2)The Maxwell–Einstein–Vlasov system22 is the coupled system of theseMaxwell equations with the EM–GR–Vlasov equation together with the
Einstein equations with source a stress–energy tensor that is the sum of
the second moment of the distribution function f and the stress–energy
tensor τ of the electromagnetic ﬁeld:

f (x, p, e)pα pβ θp de + τ αβ ,S αβ = T αβ ≡Px ×Rτ23 Recallthat ∇α S αβ ≡ 0 and∇α ∇β F αβ ≡ 0 for any metric g and
2-form F.

αβ

1≡ F α λ F βλ − g αβ Fλμ F λμ .
4

The following theorem makes the system coherent:23
Theorem X.7.1
1. If the distribution function f satisﬁes the EM–GR–Vlasov equation,then the current J has zero divergence.
2. If, in addition, F satisﬁes the Maxwell equations, then the stress–
energy tensor T is divergence-free.
Proof. 1. The electric current is given by the product with m−1 e of theﬁrst moment of f :pα f (x, p)θp .
(X.7.3)J α ≡ m−1 ePxHence, by (5.6),α∇α J = m

−1


e
Px

LX f (x, p)θp ,(X.7.4)

which implies, if LY f (x, p) ≡ LX f (x, p) + Φα ∂f (x, p)/∂pα = 0,
 
∂f (x, p)∇α J α = −m−1 eΦαθp .
(X.7.5)
∂pα
Px Px

X.7

The Maxwell–Einstein–Vlasov system

Using integration by parts and the equality ∂Φα /∂pα = 0 givesPx∂f (x, p)Φαθp = −∂pαf (x, p)Px∂Φα
θp = 0,∂pα(X.7.6)and hence∇α J α = 0.
2. Analogous reasoning gives for the second moment when LY f = 0∂f∇α M αβ =−Φα α pβ θp .
(X.7.7)∂p
Px
The calculus identity
Φα

∂f β
∂
∂Φα β
α
β
p
≡
(Φ
f
p
)
−
f p − Φβ f,
∂pα
∂pα
∂pα

(X.7.8)integration by parts, the property ∂Φα /∂pα = 0, and antisymmetry of
F show thatαββ∇α M = −Φ f θp ≡ −eFλ β pλ θ p .
(X.7.9)Px

Px

We have seen in Chapter III that the divergence of the Maxwell tensor
is, if F satisﬁes the Maxwell equations,∇α τ αβ = J λ Fλ β ≡ e
Fλ β pλ f θ p .
(X.7.10)Px

We have proved that∇α (M αβ + τ αβ ) = 0

if

LY f = 0.

The local existence theorem (Theorem X.6.1) extends to the Maxwell–
Einstein–Vlasov system modulo hypotheses on the electromagnetic data
(see the ﬂuid case).

X.7.2

Particles with random masses and charges

If the particles have random masses and charges, then the electric current
is given, at each point x ∈ V , byf (x, p, e)pα θp de.
(X.7.11)J α (x) =Px ×R

See Problem X.14.2 for the local existence theorem in this case.

255

256

Relativistic kinetic theory

X.8

Boltzmann equation. Deﬁnitions

When the particles undergo collisions, their trajectories in phase space
are no longer connected integral curves of the vector ﬁeld X: their momenta undergo jumps on crossing other trajectories. Consequently, thederivative of the distribution function f along X is no longer zero. In
the Boltzmann model, this derivative is equal to the so-called collision
operator If :LX f = If.

24 Thislaw assumes that there is nochange in internal properties of theparticles.

(X.8.1)

If is a quadratically nonlinear integral operator that is being interpreted
as being linked with the probability that two particles of momenta respectively p and q  collide at x and give, after the shock, two particles,
one with momentum p and the other with momentum q. One says thatthe shock is elastic when the following law of conservation of momentum
holds:24
p + q  = p + q.

(X.8.2)This equation deﬁnes a submanifold Σ in the ﬁbre (×Px )4 . For ﬁxed
p, q, one denotes by Σpq the submanifold of (×Px )2 deﬁned by (8.2).
The volume element ξ  (Leray form) in Σpq is such thatξ  ∧ (∧(d(pα + q α )) = θp ∧ θq .
αThe collision operator is(If )(x, p) ≡Px (q)[f (x, p )f (x, q  ) − f (x, p)f (x, q)]

Σpq




(X.8.3)(X.8.4)

× A(x, p, q, p , q )ξ ∧ θq .

25 Fora mathematical study of the
Boltzmann equation in classical mechanics, one may consult the review article by Villani (2002).

The function A(x, p, q, p , q  ) is called the shock cross-section. It is aphenomenological quantity that depends on the physics of the shocks. No
explicit expression is known for it in Relativity.25 A generally admitted
property is the reversibility of elastic shocks, namelyA(x, p, q, p , q  ) = A(x, p , q  , p, q).

(X.8.5)

Lemma X.8.1 When the particles have all the same, non-zero, propermass m, the integral on Σpq can be written by using a formula of thetype, with θ, ϕ canonical angular parameters on the sphere S 2 ,A(x, p, q, p , q  )ξ  = S(x, p, q, θ, ϕ) sin θ dθ ∧ dϕ.
Proof. Introduce at the point x, for a given pair p, q of timelike vectors,
an orthonormal Lorentz frame with time axis e0 in the direction of p + q.
Set p + q = 2λe0 , 2λ = [−g(p + q)]1/2 . Then also p + q  = 2λe0 , andsince p, q, p , q  are timelike vectors with length m,pα qα = pα qα = m2 − 2λ2 ≤ −m2 .
In such a frame, p0 + q 0 = p0 + q 0 = 2λ, while pi + q i = pi +q i = 0; hence Σ(pi )2 = Σ(q i )2 , and therefore p0 = q 0 = λ, whileX.9

Moments and conservation lawsΣ(pi )2 = λ2 − m2 =: α2 . The same properties hold for the primed
variables; in this frame, Σpq is represented by α2-sphere, Σ(pi )2 = α2of radius α in the plane p0 = λ . It holds thatα=11(−pα qα − m2 )1/2 = g(p − q, p − q)1/2 .
2
2

Take a vector parallel to p − q (which is orthogonal to p + q = 2λe0 ) as
axis for polar coordinates θ, ϕ on Σpq , denoting by θ the angle between
the space vectors p − q and p − q  ; the deﬁnition of ξ  then givesξ  = (2λ)−1 α sin θ dθ ∧ dϕ.
The given relation holds withS(p, q, θ, ϕ) = A(x, p, q, p , q  )(2λ)−1 α,(X.8.6)

where on the sphere Σpq both p and q  are given by the θ-, ϕ-dependent
4-vectorspΣpq = qΣpq
11
= − g(p + q)1/2 , g(p − q, p − q)1/2 (cos θ, sin θ cos ϕ, sin θ sin ϕ) .
2
2
(X.8.7)


X.9

Moments and conservation laws

The moments of a distribution function f have been deﬁned in Section X.2 by integrals over a ﬁbre in phase space. The moment of ordern is
α1 ...αn(x) =:
pα1 . . . pαn f (x, p)θp .
(X.9.1)TPx

It satisﬁes the identity∇α T

αα2 ...αn≡Pxpα2 . . . pαn LX f (x, p)θp .

(X.9.2)

The right-hand side is zero if f satisﬁes the Vlasov equation.
We have interpreted the ﬁrst and second moments as giving respectively the proper rest-mass–momentum vector and the stress–energy
tensor of the macroscopic matter corresponding to the distributionfunction f . We prove a lemma, important for the coherence with the Einstein equations, when f satisﬁes a Boltzmann equation with reversible
collisions:Lemma X.9.1 The ﬁrst and second moments of a distribution f satisfying a Boltzmann equation on a spacetime (V, g) have zero divergencein the spacetime metric if the collisions are reversible.

257

258

Relativistic kinetic theory

Proof. The ﬁrst moment of f is the vector ﬁeld on V, interpreted as
proper rest-mass–energy–momentum density, given by
pα f (x, p)θp .
(X.9.3)P α :=Px

The equation (9.2) reads in this case∇α P α ≡LX f (x, p)θp =Px

Px(If )(x, p)θp .

(X.9.4)The second moment of f, interpreted as the stress–energy tensor, isdeﬁned bypα pβ f (x, p)θp .
(X.9.5)T αβ :=Px

For f satisfying a Boltzmann equation, we haveαβ∇α T ≡pβ (If )(x, p)ωp .

(X.9.6)

Px

Standard calculus, which we leave to the reader as an exercise, showsthat, for reversible collisions, (9.4) and (9.6) have zero right-hand sides.
In the case of particles with non-zero rest mass, we have set P α = ruα ,with u the unit ﬂow vector of the macroscopic matter corresponding to
the distribution function f. Equation (9.4) is identical to the matter
conservation law found for ﬂuids:∇α (ruα ) = 0.

(X.9.7)∇α T αβ = 0,

(X.9.8)Equation (9.6),

is to be satisﬁed by all stress–energy tensors in General Relativity.

The higher moments do not satisfy conservation laws, but form an
inﬁnite hierarchy; see Section X.11.

X.10

26 Bancel(1973) for the Boltzmann
equation and Bancel and ChoquetBruhat (1973) for the coupled system.

Einstein–Boltzmann system

The Einstein equations with source the stress–energy tensor of a distribution function f satisfying a Boltzmann equation form a coherent
system if the collision operator is such that the stress–energy tensor isconservative. We have said that this property holds if the collisions arereversible.
It has been proved26 that the local Cauchy problem, with initial dataset (M, ḡ, K, f¯), is well posed for such a Einstein–Boltzmann system.
The same kinds of results hold for an Einstein–Maxwell–Boltzmannsystem obtained on replacing the GR–Vlasov operator by the EM–
Vlasov operator.

X.11

X.11

Thermodynamics

259

Thermodynamics

One of the main reasons for the interest in relativistic kinetic theory is
the possibility of obtaining the laws of thermodynamics in a relativistic
context. The problem is already diﬃcult in Special Relativity, and newchallenges arise in General Relativity because of the non-existence of an
equilibrium distribution function in a non-stationary universe.

X.11.1

Entropy and the H theorem

The entropy density of a one-particle distribution f was computed by
Boltzmann to be in phase space the function27 −k(f log f )(x, p). The
entropy ﬂux in spacetime is the future-directed timelike vector28
H α (x) := −kPxpα (f log f )(x, p)θp .

(X.11.1)

The following theorem is the relativistic formulation of a theorem wellknown in non-relativistic thermodynamics:
Theorem X.11.1 (H theorem) If collisions are reversible and satisfy
the symmetry property (8.5), then the entropy ﬂux H α is such that∇α H α ≥ 0.

(X.11.2)

Proof. The divergence of H is found to be, by a computation similar toa previous one,
(LX [f log f ])(x, p)θp(∇α H α )(x) ≡ −kPx(X.11.3)≡ −k(LX f [log f + 1])(x, p)θp .
Px

Using the Boltzmann equation and the reversibility of collisions, which
implies that[f (x, p )f (x, q  )−f (x, p)f (x, q)]A(x, p, q, p , q  )ξ  ∧θq ∧θp = 0,
Px (q)

Σpq(X.11.4)
we ﬁnd that(∇α H α )(x) = −kPx (q)

[f (x, p )f (x, q  ) − f (x, p)f (x, q)]Σpq

×(log f )(x, p)A(x, p, q, p , q  )ξ  ∧ θq ∧ θp .

(X.11.5)

Making, moreover, the natural assumption of symmetry,A(x, p, q, p , q  ) = A(x, q, p, p , q  ),(X.11.6)

27 Boltzmann’s constant k is the quotient of the perfect gas constant andAvogadro’s number, with the value
1.3806568 J K−1 .
28 From its physical deﬁnition, it holds
that 0 ≤ f ≤ 1; hence −f log f ≥ 0.

260

Relativistic kinetic theory

we rewrite the above divergence as follows (the dependence on x has
been made implicit for brevity):k[f (p )f (q  ) − f (p)f (q)]∇α H α = −4 Px (q) Σpq

f (p)f (q)× log
A(p, q, p , q  )ξ  ∧ ωq ∧ ωp , (X.11.7)
f (p )f (q  )29 Because (a − b)(log a − log b) ≥ 0.

which is non-negative29 when A is non-negative.
α

The gas is said to be in thermal equilibrium if ∇α H = 0. A generalrelativistic gas undergoing collisions will not in general attain in a ﬁnite
time thermal equilibrium (see Section X.11.2).
We can make a simple link between the microscopic and macroscopic
properties of the gas when the distribution f is isotropic in phase space,
namely when there exists a timelike vector v on spacetime such that f
depends only on vα pα . We have seen (Section X.2) that the macroscopic
gas is then a perfect ﬂuid with momentum ﬂow P collinear with V . The
same type of proof gives the following lemma:
Lemma X.11.1 Assume there exists on spacetime a timelike vector V
such that the function f is symmetric in the spaces orthogonal to V, i.e.
in each ﬁbre,
f (x, q) = f (x, p)if Vα pα = Vα q α andp̄ = −q̄,(X.11.8)

where p̄ and q̄ denote respectively the projections of p and q on the subspace orthogonal to V. Then the ﬁrst moment P = rU as well as theentropy vector H are collinear with V.
Lemma X.11.2 When the vectors H and P are collinear, one deﬁnes
a positive speciﬁc scalar entropy S by setting
H α = SP α .

(X.11.9)

This speciﬁc entropy S satisﬁes on spacetime the inequality
P α ∂α S ≥ 0.

(X.11.10)

Proof. The lemma follows from inequality ∇α H α ≥ 0 and the conser
vation law ∇α P α = 0.
When f does not have the property (11.8), the entropy and matterﬂux are not collinear.
In all cases, integrating (11.2) on a spacelike slice VT , with compact
space or appropriate boundary conditions at spacelike inﬁnity, we ﬁnd
that


0
H N μḡ ≥H 0 N μḡ .
(X.11.11)
MT

M0

Exercise X.11.1 Prove this inequality.
The inequality (11.11) leads some cosmologists to think that it is the
expansion of the universe that permits its ever-increasing organizationfrom an initial anisotropy of f.

X.11

X.11.2

Thermodynamics

261

Maxwell–Jüttner equilibrium distribution

A gas is considered to be in thermal equilibrium if its entropy isconserved in the sense that∇α H α = 0.

(X.11.12)

Equation (11.3) shows that a suﬃcient condition for this equality to holdis that the distribution function f be conserved along the trajectories
of X in phase space, i.e. LX f = 0. This condition is, in a Boltzmann
framework,I(f ) =0.

(X.11.13)

A suﬃcient condition for this equality to hold is that at each point x ofspacetime, the function f is such thatf (p )f (q  ) − f (p)f (q) = 0, if

p + q = p + q  .

(X.11.14)

This condition is also necessary if A(p, q, p , q  ) is strictly positive. It canalso be proved to hold under weaker assumptions.30 It can immediately
be checked that a solution of the above functional equation isf (x, p) = a(x) exp[bα (x)pα ],30 Marle (1969), p. 94.

(X.11.15)

with a a positive scalar function and b a covariant vector on spacetime. It
can be proved that all continuous solutions of (11.14) are of this form. If
a is non-negative, the same is true of f ; if, moreover, b is future timelike,then bα (x)pα < 0 and f (x, .) is integrable on Px . Such functions arecalled Maxwell–Jüttner distributions.
For such a Maxwell–Jüttner distribution f , we ﬁnd by straightforward
computation that


λ
∂a
1
(X.11.16)
LX f ≡ ebλ p pα α + pα pβ (∇α bβ + ∇β bα ) .
∂x
2
A function f given by (11.15) will satisfy LX f = 0 on phase space if andonly if on spacetime it holds that ∂a/∂xα = 0 and ∇α bβ + ∇β bα = 0,
that is, iﬀ a is a constant and b is a Killing vector ﬁeld: the spacetime
must therefore be stationary to admit a thermal equilibrium distributionfunction—an unsurprising result.
We deduce from Theorem X.2.1 that the rest-mass–momentum vector
and the stress–energy tensor on spacetime associated with a Maxwell–Jüttner distribution with a timelike vector b are those of a perfect ﬂuid
with ﬂow vector collinear to b. The same kind of proof shows that theentropy vector H is collinear to b, and hence to u. The integrals giving the moments can be expressed through modiﬁed Bessel functions of1
the second kind and index n.31 The scalar (−bα bα ) 2 is interpreted as
the inverse of the product of the absolute temperature by Boltzmann’sconstant:1(−bα bα ) 2 = (kT )−1 .

(X.11.17)

31 See Pichon (1967) and Marle (1969).

262

Relativistic kinetic theory

Remark X.11.1 If the particles have zero mass (e.g. photons), one
can satisfy the condition LX f = 0 only by imposing that b is a conformal
Killing vector ﬁeld, i.e. ∇α bβ + ∇β bα = λgαβ . This is the reason why
Robertson and Walker used the Planck distribution for photons in their
expanding spacetimes.

X.11.3

Dissipative ﬂuids

In classical mechanics, the Navier–Stokes equations can be obtained froma distribution function that is a ﬁrst-order perturbation of the Maxwell equilibrium distribution, either by the Chapman–Enskog method
or by Grad’s polynomial expansion method. Both methods have been
extended by Marle, who found the stress–energy tensor corresponding
to a ﬁrst-order perturbation of the Maxwell–Jüttner distribution. The
corresponding, very complex, system of equations have been shown to beof hyperbolic type only in some simpliﬁed cases. The fact that thermal
equilibrium is not compatible with non-stationary spacetimes limits thevalidity of such equations in General Relativity.

X.12
32 See, for a summary, YCB-OUP 2009,

Chapter X, Section 11 (contributed byTommaso Ruggeri) and, for a complete
exposition, Müller and Ruggeri (1998).

Extended thermodynamics

The objective of extended thermodynamics32 is to obtain equations forﬂuids in General Relativity that respect fundamental physical laws andalso the relativistic causality principle. It uses for general relativistic
ﬂuids, in addition to the usual ﬁrst moment P α (x) and second moment
T αβ (x), higher moments and relations between them that are physicallyand mathematically meaningful.

X.13

Solutions of selected exercises

Exercise X.3.1
Let xα be another coordinate system in V. Set




Aα
α :=

∂xα
.
∂xα

The corresponding change of components of a vector in Tx V is


pα =




∂xα α
α
p = Aα
α p .
α
∂x




Therefore, the change of coordinates (xα , pα ) → (xα , pα ) in T V implies



∂pα
= Aα
α ,
∂pα




∂pα
α
= ∂β (Aαα p ).
∂xβ(X.13.1)

X.13




Hence, for the components (pα , Gα ) of X in the new coordinates on T V,αpα = Aα
α p ,







α
α α λ μGα = pβ ∂β (Aαα p ) − Aα Γλμ p p .

We know that (see Chapter I)






β
γ α
α
α
α
Γα
β  γ  = Aα ∂β  Aγ  + Aα Aβ  Aγ  Γβγ .

(X.13.2)

Some straightforward manipulations shows that








β γ
Gα ≡ −Γαβ γ p p .

(X.13.3)

Exercise X.3.2∂gλμλ μ− 2gαρ pρ Γαλμ p p∂xα∂gλμ≡ pα p λ p μ− gαρ pρ g αν [λμ, ν]pλ pμ
α
∂x




∂gλμ
∂gλμ
∂gλα
α λ μ
≡ pα p λ p μ
−
[λμ,
α]
≡
2p
p
p
−
≡ 0.
∂xα
∂xα
∂xμ

LX g(p, p) ≡ pα pλ pμExercise X.3.3
gαβ pα pβ = constant implies
2g00 p0

0
∂p0
∂p0
j ∂p
0
j
+
2g
p
+
2g
p
+
2g
p
≡
2p
+ 2pi = 0;
0j
0i
ij
0
∂pi
∂pi
∂pi

hence
pi
∂p0
=− .
∂pi
p0
We have fm (x, p) := f (x, p0 , pi ), with gλμ pλ pμ = −m2 , and hence∂fm∂fpλ pμ ∂gλμ ∂f
∂fm
∂f
pi ∂f
=
−
and
= i−
,
∂xα
∂xα
2p0 ∂xα ∂p0∂pi∂pp0 ∂p0from which the result follows.

Exercise X.6.1
For an arbitrary function such that φ(t) = φ(−t), it holds that
 0
 +∞
 +∞
φ(t)t dt =φ(t)t dt +φ(t)t dt = 0,−∞

because

 0

−∞

 0
φ(t)t dt =

−∞

∞

0

φ(−t )t dt = − ∞0φ(−t )t dt .

Solutions of selected exercises

263

264

Relativistic kinetic theory

X.14

Problems

X.14.1

Liouville’s theorem and generalization

We have shown in Problem I.14.1 in Chapter I that the volume form θ ofthe tangent bundle T V of a Lorentzian spacetime is invariant under thegeodesic ﬂow; that is, with LX denoting the Lie derivative with respect
to the vector X = (pα , Gα ), it holds that LX θ = 0.
Using the identity LY ≡ d(iY θ)+iY dθ, show that the volume form θ isinvariant under the ﬂow of a vector Y = (pα , Gα + Φα ) if Φ is orthogonal
to p and has a vanishing divergence.

Solution
The volume form on T V is
θ := |Det g|dx0 ∧ . . . ∧ dxn ∧ dp0 ∧ . . . ∧ dpn .
Recall that LY θ ≡ d(iY θ) + iY dθ. Since the volume form is of maximum
degree on T V , we have dθ ≡ 0, and hence LY θ = diY θ. The linearity of
the interior product gives
iY θ = iX θ + i(0,Φ) θ.
We already know that d(iX θ) = 0. We set θ = θx + θp ; theni(0,Φ) θ = θx ∧ iΦ θp ,(X.14.1)with(iΦ θp )α1 ...αn = Φα (θp )αα1 ...αn ,and hence, because of antisymmetry,d(iΦ θp ) =∂Φαθp = 0
∂pα

if

∂Φα
= 0.
∂pα

When Φα = F α β pβ , it holds that∂Φα
= F α α = 0.
∂pαX.14.2

Vlasov equation for particles with randomchargesAssume that the considered ‘particles’ have, like momenta, random electric charges. The phase space FV is the bundle over V whose ﬁbre Fx V
is the product Px V × I, with I an interval of R. The volume form in FV
is the product, with θ given by (2.2),θe := θ ∧ de.

X.14

The particles create an average electromagnetic ﬁeld F and are subjectedto the Lorentz forceΦα := F αβ jβ ,where jβ is the electric current associated to a particle of momentum p
and charge e located at x, i.e. with u denoting its unit velocity,1jβ = euβ ≡ e|g(p, p|− 2 pβ .
We denote by Ye the vector ﬁeld on T V × R with components
pα , Gα + Φα , 1.
The volume form θe is clearly invariant under the ﬂow of Ye , as θ wasunder the ﬂow of Y ; that is, LY θe = 0. Particle number conservation
gives for the distribution function f the EM–GR–Vlasov equation
with random charges:
LY f ≡ pα

X.14.3

∂fe
∂fe
λ μ
α
β ∂fe
= 0.
+ (−Γα+λμ p p + F β j )
α
α
∂x
∂p
∂e

(X.14.2)

Distribution function on a Robertson–Walker
spacetime with Vlasov source

Consider a spacetime M × R with a Robertson–Walker metric.
g ≡ −dt2 + R2 (t)σ 2 ,with

σ 2 ≡ γij dxi dxj .

σ is a given Riemannian metric on M.
1. Derive the Einstein–Vlasov equation for a general distribution function f .
2. Look for a solution fm depending only on t and p0 for particles of agiven rest mass m. Show that it satisﬁes the equationR 1 ∂fm∂fm  0 2
− (p ) − m2 0
= 0.
∂R
p ∂p0



3. Show that fm is an arbitrary function of the scalar R2 (p0 )2 − m2 .
4. Suppose that fm vanishes at time t0 for particles with momentum
such that(p0 )2 ≥ m2 + kR−2 (t0 ),k some constant.

Show that the maximum possible energy p0 of particles with a given
rest mass decreases with expansion, as foreseen physically.

Problems

265

266

Relativistic kinetic theory

Solutionλ μ
1. Straightforward computation of Gα ≡ −Γαλμ p p gives

LX f ≡ pα∂f∂f∂f− RR γij pi pj 0 − 2R−1 R p0 pi i .
α
∂x
∂p
∂p

For particles of a given rest mass m, i.e. such that
R2 γij pi pj = (p0 )2 − m2 ,the equation for a distribution fm depending only on t and p0 reduces
to

 ∂fm
∂fm
p0
− R−1 R (p0 )2 − m2
= 0.
∂t
∂p0
Taking R instead of t as a variable, the equation reads as the following
linear ﬁrst-order partial diﬀerential equation: 1 ∂fm∂fm  0 2
− (p ) − m2 0
= 0.
R
∂R
p ∂p0
The general solution is constant along the rays (bicharacteristics) thatsatisfy the diﬀerential systemp0 dp0dR=− 0 2= dλ.
R(p ) − m2
These rays are such that




1
log R+ log (p0 )2 − m2 = constant, i.e. R2 (p0 )2 − m2 = constant.
2
The distribution
 fm is therefore an arbitrary function of the scalar
R2 (p0 )2 − m2 . Suppose, for instance, that fm vanishes at time t0 forparticles with momentum such that(p0 )2 ≥ m2 + kR−2 (t0 ).
Then the function fm vanishes at time t for particles with momentum(p0 )2 ≥ m2 + kR−2 (t).
Hence the maximum possible energy p0 of particles with a given rest
mass decreases with t if R increases.

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IndexNotes: references to marginal notes are indicated by the suﬃx ‘n’ followed by the note number, forexample 188n33; page numbers in italics refer to ﬁgures.

1-form on V 6Aacceleration, relativistic dynamics 46–7
ADM composition 184
ADM mass 104
aether 37
age of the universe 164, 167, 170–1
Alfvén cone 230
Alfvén waves 230
algebraic Bianchi identity 21
algebraic gauge (densitizing thelapse) 188n33
Ampère–Maxwell law 37
Anthropic principle 58n11
anti-de Sitter spacetime 97–8
antisymmetry properties of tensors 8
astronomical distance estimation 163
astrophysics, equations of state 213–14
asymptotically Euclidean manifolds 74
atlases 3
axisymmetric spacetimes 141–2
Bbarotropic ﬂuids 213
Euler equations 223
Bel–Robinson energy 207
Bel–Robinson tensor 31–2
Berger–Ebin decomposition
theorem 182
Bianchi constraints 207
Bianchi identities 30–1, 71
algebraic 21
diﬀerential 22
linearization 24
big bang 164, 167, 197
binary pulsars 95n47, 115–16
Birkhoﬀ’s theorem 108
black bodies 158n12
black hole region 148

black holes 112, 136
classiﬁcation of 150
deﬁnitions 147–8
detection of 147, 149
Eddington–Finkelstein
extensions 137–9
interiors of 150
irreducible mass 152–3
Kerr spacetime 142–6
Kruskal spacetime 139–41
observations 149–50
Reissner–Nordström metric 152
Riemannian Penrose inequality 153–4
Schwarzschild black hole 136–7
stationary 141–2
uniqueness theorems 146
thermodynamics 148–9
weak cosmic censorshipconjecture 148
Boltzmann equation 241–2, 256–7
Boltzmann’s constant 259n27
boosted Schwarzschild metric 110
boundary conditions 73–4
Boyer–Linquist coordinates 142–3
Bruhat, Y., see Choquet-Bruhat, Y.
BSSN code 194
Ccaesium clock 43, 45n21
canonical aﬃne parameters 20
Cauchy adapted frames 184
Cauchy hypersurfaces 195
Cauchy problem 175–6, 218–19
for Einstein–Vlasov system 252–3
evolution in wave gauge 178–9
ﬁrst-order symmetric hyperbolic
systems 220–1
geometric n + 1 splitting 184
adapted frame and coframe 184–6
dynamical system with constraints
on ḡ and K, 187–8
regularity assumptions 189–90

Leray hyperbolic system 219–20
local existence and uniqueness 181
preservation of the wave
gauges 179–80
solution of the constraints by the
conformal method 190–3
solution of wave gauge
constraints 181–4
wave coordinates 176–8
causal cone 14
causal curves 14
length 15
causality 219
causal vectors 14
CGS system of units 59–60
characteristic cones 215
characteristic polynomials 175, 215
characteristic systems 249
charged dust 210–11
charged ﬂuids 61, 226–7
with ﬁnite conductivity 228–9
with inﬁnite conductivity 229
with zero conductivity 227–8
charts 3
Christodoulu, D., 127, 152n22, 196,
197–8, 235
Christoﬀel symbols 18–19
Choquet-Bruhat, Y., 97n54, 178n14,
181, 188, 190, 192n43, 192n45,
195, 196, 222n29, 229n38,
231n41, 252n17, 253
circular orbits, stability of 117–18
classical Navier–Stokes equations 232
Claudel–Newman metric 128
closed forms 9‘closed’ universes 160n15
codiﬀerential 29
cold dark matter 167
collision operator 256
comoving frames 50
complementary inertial force 36n5

274

Index

conformally formulated (CF)constraints 190–2
elliptic equations 192–3
conformal method 190–3
conformal metric, geodesics of 239
connection coeﬃcients 17
connection frame-change formula 27
connections
linear 16–18
Riemannian 18–19
conservation identity 22
conservation laws 71, 212
Einstein–Vlasov system 251–2
and moments 257–8
conservation of entropy 212–13
conservation of rest mass 212
constants 59–60
Anthropic principle 58
Boltzmann’s constant 259n27
cosmological constant 72, 157
ﬁne-structure constant 58
velocity of light 57
constraints for ḡ and K, 187, 188conformally formulated (CF)constraints 190–2
constraints on the initial data 180
continuous matter 49–50
dust model 51
perfect ﬂuids 51–2
contracted Bianchi identities 71
contracted product of tensors 8
contravariant tensors 7
contravariant vectors 5
Copernican frame 36
Copernican principle 158
Coriolis (inertial) forces 36
cosmic censorship conjectures 127, 132,
196–7
cosmic microwave background (CMB)
radiation 168
isotropy 158
cosmic time 158
cosmological constant 72, 157
cosmological expansion 157, 162, 164,
167
cosmological models 155
ACDM model 167
de Sitter spacetime 157
Einstein static universe 156–7
Friedmann–Lemaı̂tre
universes 164–7, 171–2
general models 157
Milne universe 172
Robertson–Walker spacetimes 158–61
general properties 161–4
cosmological principle

assumptions 157–8
observational support 158
cosmological redshift 161–2
Hubble law 162–3
cosmology
age of the universe 170–1
equations of state 214–15, 240
primordial 167–8
cotangent spaces 5
Cotton tensor 33
Cotton–York tensor 33
Coulomb’s law 37
covariant derivatives 16–18
covariant tensors 7
covariant vectors 5
curvaturedeﬁnitions 20–1
and gravitation 62
symmetries and antisymmetries 21
Cygnus X1, 149Dd’Alembertian 29
damped wave coordinates 178
dark energy 156
dark matter 155–6, 167
Darmois, G., 146, 187n27
deceleration parameter 163
deﬂection of light rays
Fermat’s principle 120
observations 120
theoretical prediction 118–19
densitizing the lapse (algebraic
gauge) 188n33
density parameter 165
de Rham theorem 9
derivation operators 6derivative of f along v, 6de Sitter spacetime 96–7, 157
diﬀeomorphisms 3
diﬀerentiable manifolds 3–4
diﬀerentiable mappings 4
diﬀerential Bianchi identity 22
dimensional constants 59n14
dimensional equations 59n14
Dirichlet problem 175
dissipative ﬂuids 228
distance
Riemannian manifolds 13
see also length
distribution function
deﬁnition of 242
interpretation 242–3
moments of 243–5
particles of a given rest mass 245
dominant energy condition 212
Doppler–Fizeau eﬀect 54

dust (pure matter) 208–9
charged 210–11
stress–energy tensor 71
dust model of matter 51
dust shell collapse 130–2
dynamical velocity 222
Euler equations 223, 236–7
ﬂuid index 223
hyperbolic Leray system 237–9
and perfect ﬂuid enthalpy 224
vorticity tensor and Helmholtz
equations 223–4
dynamics of a pointlike mass
Newtonian approximation of therelativistic equation 47–8
Newtonian law 46
relativistic law 46–7

E
Earth, mass of 65–6
eclipses, observation of light ray
deﬂection 58, 63, 120
Eddington–Finkelstein (EF) blackhole 137–9
Eddington–Finkelstein (EF) whitehole 139, 140eﬀective one-body (EOB) method 194
eikonal solutions 215–16
Einstein–Boltzmann system 258‘Einstein cross’, 63
Einstein cylinder (Einstein static
universe) 95, 156–7
Einstein–de Sitter universe 166
Einstein eﬀect, veriﬁcation of 63
Einstein equations 68
cosmological constant 72
coupling with irrotational ﬂows 225
in dimension 2 + 1, 202–3
Minkowski approximation 79–80
Newtonian approximation 75–7
observations and experiments 94–5
post-Newtonian approximation 77–8
with sources 70–1
Lagrangians 92–4
static solutions with compact
spacelike sections 103
in a vacuum 69
see also Cauchy problem
Einstein equivalence principles 57–8
conﬁrmatory experiments 62
Einstein–Euler entropy system 221–2
Einstein–Hilbert Lagrangian invacuo 92
Einsteinian spacetimes 72–3
boundary conditions 73–4

Indexphysical comment 74–5
regularity 73
Einsteinian vacuum developments 189
Einstein tensor 21, 69
Einstein–Vlasov system
Cauchy problem 252–3
conservation law 251–2
equations 251
electromagnetic 2-form 40
electromagnetic ﬁelds
in general relativity 61
stress–energy tensor 71
electromagnetic potential 52
electrovac Einsteinian spacetimes
constraints 203–4
Lorenz gauge 204–5
elliptic equations 175, 182n21
embedded submanifolds 5
EM–GR–Vlasov equation 247–8
EM–GR–Vlasov equation with randomcharges 265
energy density 50
energy equality 198
energy equation of a perfect ﬂuid 212
energy–momentum 48–9
of a photon 49
energy–momentum vector 211
energy of u, 198
enthalpy 224
entropy 259–60
black holes 149
conservation of 212–13
Einstein–Euler entropy system 221–2
entropy equations, as a Leray hyperbolic
system 219–20
equations of motion 50, 58
Newtonian approximation 76–7
perfect ﬂuids 51, 212
equations of state 213–15, 240
equivalent atlases 4
ergosphere 145
escape velocity 116–17
Euclidean space 35
Euler equations 212
for a barotropic ﬂuid 223
for dynamic velocity 236–7
as a Leray hyperbolic system 219–20
event horizon 112, 138deﬁnition of 148
Kerr metric 144
evolution set of Maxwell equations 202
exact 1-forms 7
exact forms 9
extended thermodynamics 262
exterior derivatives, p-forms 9
exterior forms 9–10
Lie derivative 26

exterior products, p-forms 9
extrinsic curvature (second fundamental
form) 185
F
Faraday’s law 37
Fermat’s principle 120
Ferrarese, G., 226
ﬁeld sources, Einstein equations 71
Lagrangians 93
ﬁne-structure constant 58
ﬁnite conductivity, charged ﬂuids 228–9
ﬁrst moment of the distributionfunction 243, 244–5
ﬁrst-order symmetric hyperbolic
systems 220–1
ﬁrst variation (linearization) 23–4of the Riemann tensor 34
ﬂatness problem 165–6, 168
ﬂat spaces 12
ﬂow-adapted frames 226
ﬂow vector ﬁeld 50
ﬂuid index 223
ﬂuid matter sources 208
barotropic and polytropic
ﬂuids 213–14
charged dust 210–11
dust 208–9
ﬂuid index and Euler equations 223
incompressible (stiﬀ) ﬂuids 218
irrotational ﬂows 225
perfect ﬂuid enthalpy 224
perfect ﬂuids 211–12
shocks 226
vorticity tensor and Helmholtz
equations 223–4
ﬂuid model of matter 50
perfect ﬂuids 51–2
Foucault, J.B.L., 36
frames 5
moving 6–7
Friedmann equation 165
Friedmann–Lemaı̂tre universes 171–2
ACDM cosmological model 167
density parameter 165
Einstein–de Sitter universe 166
equations 164–5
general models with p = 0, 166–7
Friedrich, H., 196, 226
Friedrichs, K.O., 178n14, 220, 229n37
fundamental observers 157–8
future-oriented vectors 14
G
Galileo group 36
Galileo–Newton equivalenceprinciple 55–7

275

Galileo–Newton spacetime 35
Maxwell equations 37
Gauss’s law 37
relativistic 90–1
G E , 70, 76
generalized wave coordinates 177
general relativistic (GR) Vlasov
equation 246–7
general relativity 55, 57
Einstein equivalence principles 57–8
expression of classical ﬁelds 60–1
gravitation and curvature 62
observations and experiments 62–5
generic spacetimes 148
geodesic deviation 22–3
geodesic ﬂow 249
geodesicsof conformal metric 239
Lorentzian manifolds 15–16, 20
normal geodesic coordinates 30
and parallel transport 19–20
Riemannian manifolds 13, 20
Gevrey classes 188n31
global existence theorem 195–6
for Einstein–Vlasov system 253
for GR–Vlasov equation 250
global hyperbolicity 195
global uniqueness 195
G N , 60
gram, deﬁnition of 59
gravitational collapse 127
monotonically decreasing
density 130–2
Tolman, Gu, Hu, andClaudel–Newman metrics 127–9
gravitational constant 60
gravitational lensing 63
gravitational potential 56
gravitational radiation energy 82
observational proof 116
gravitational radiation energy loss,
observations and experiments 95
gravitational solitons 90
gravitational wave detectors 193
gravitational waves 80–1
high-frequency 82–4
observable displacements 87
phase and polarization 84–5
propagation and backreaction 85–7
from a spherically symmetricstar 101–3
observations and experiments 94–5
gravitation theory
general relativity 62
Newtonian 56, 65–6

276

Index

gravitons 95
Grönwall’s lemma 200n82
Gu, C.-H. 127, 129
H
Hamiltonian constraint 187, 190–1, 192
harmonic (wave) coordinates,
Schwarzschild metric 109–10
Hausdorﬀ manifolds 4
Hawking, S., 141n9, 196
Hawking area theorem 149, 152–3
Hawking–Klainerman theorem 146
Hawking radiation 149
heat equation 234–5
Helmholtz equations 223–4
higher moments of the distributionfunction 245
high-frequency waves
observable displacements 87
phase and polarization 84–5
propagation and backreaction 85–7
from a spherically symmetricstar 101–3
high-frequency waves (progressive
waves) 82–4
Hoﬀman, F., 229n38
homeomorphisms 3
homogeneity 170
cosmological principle 158
horizon problem 168
H theorem 259–60
Hu, H.-S., 127, 128
Hubble law 162–3
Hubble parameter 162
Huisken, G., 149, 154
hyperbolic equations 175, 219
hyperquasilinear partial diﬀerential
equations 69n6, 219n19I
identity notation, ix
Ilmanen, T., 149, 154
incoherent matter see dust
incompressible (stiﬀ) ﬂuids 218
inertial coordinates 40
inertial frames 36–7
inertial mass, in Newtonianmechanics 55
inﬁnite conductivity 229
Yang–Mills ﬂuids 231
inﬁnite-redshift eﬀect 121
inﬂation 168
initial data sets 189
integral energy inequality 198
interior products, p-forms 10
Ionescu, A.D., 141n10
irrotational ﬂows 225

Isenberg, J., 197
isolated bodies 209
motion of 235–6
isometry, pseudo-Riemannian
manifolds 11
isotropic coordinates, Schwarzschild
metric 109
isotropy 169–70
cosmological principle 158
Robertson–Walker spacetimes 159
Israel uniqueness theorem 146
K
Kerr black holestability 146–7
uniqueness 146
Kerr metric 142
absence of realistic interior
solutions 145–6
Boyer–Linquist coordinates 142–3
essential singularity 144
extended Kerr spacetime 145
horizons 144
limit of stationarity andergosphere 144–5
Kerr–Newman metric 152
Kerr–Schild spacetime 143
Killing vectors 12, 27
kinetic theory 241–2
distribution function
deﬁnition 242
interpretation 242–3
moments of 243–5
particles of a given rest mass 245
Vlasov equations
EM–GR–Vlasov equation 247–8
general relativistic equation 246–7
solution of 249–50
Yang–Mills plasmas 248–9
Klainerman, S., 141n10, 147, 196
Kretschman scalar 137, 144, 195
Kruskal spacetime 139–41
L
Lagrangians 91
for Einstein equations withsources 92–4
Einstein–Hilbert Lagrangian invacuo 92
Taub Lagrangian 104
Lanczos identity 32
Landau, L.D., 232n46
Landau–Lifshitz pseudotensor 100–1
Laplacian 29
lapse function 184–5
Leibniz rule 17–18
Lemaı̂tre coordinates 151

lengthof a causal curve 15in Euclidean space 35
Riemannian manifolds 13
Leray, J., 195
Leray hyperbolic systems 219
Euler and entropy equations
as 219–20
Leray–Ohya hyperbolic systems 187,
228–9, 231
for dynamical velocity 237–9
for viscous ﬂuids 233–4
Leray theory 178n14
Lichnerowicz, A., 222n29, 231n41
Lichnerowicz equation 192
Lie brackets 8, 10
Lie derivatives
components of 25–6
of an exterior form 10, 26
of a tensor ﬁeld 8–9
Lifshitz, E.M., 232n46
light, velocity of 37n8, 59–60
as a constant 57
Michelson–Morley experiment 45
light cone (null cone) 14
light cone of a spacetime 216, 235
light deﬂection 63
light rays, deﬂection of
Fermat’s principle 120
observations 58, 63, 120
theoretical prediction 118–19
light travel parameter time 120
light-year 57n10
limit of stationarity 145
Lindblad, H., 196
linear connections 16–18
linearization (ﬁrst variation) 23–4of the Riemann tensor 34
Liouville theorem 28, 264
local coordinates 3
locally ﬂat spaces 12
locally isometric pseudo-Riemannian
manifolds 12
locally static spacetimes 88
Lorentz contraction 39, 40
Lorentz dilation 40
Lorentz force 42
Lorentz groups 13, 38
Lorentzian metrics 13, 57
causal and null cones 14
future and past 14
length and geodesics 15–16, 20
spacelike submanifolds 15
Lorentz transformations 38
of Maxwell equations 54
lunar laser ranging (LLR) 62

IndexM
Mach, E., 36
magnetic 2-form 41, 42
magneto-acoustic cone 230
magnetoacoustic wave fronts 230–1
magnetohydrodynamics 229
manifolds 4
parallelizable 6
mass
relativistic and Newtonian,comparison 132
upper limit 124–5
mass–energy equivalence 48–9
mass hyperboloid (mass shell) 245
matched asymptotic expansions 115
matter, continuity of see continuous
mattermatter sources, Einstein equations 71
Lagrangians 93
matter wave fronts 218
Maxwell–Einstein–Vlasov system 253,
254–5
Maxwell equations 37, 61
in arbitrary dimensions 42–3
Lorentz transformation 54
in Minkowski spacetime 40–2
Maxwell–Jüttner equilibrium
distributions 261–2
Maxwell tensor 41–2, 43, 61, 71
mean curvature 185
metre, deﬁnition of 45, 59, 60
Michelson–Morley experiment 45
Milne universe 172
Minkowski metric 13
Minkowski spacetime
approximation of Einsteinequations 79–80
Maxwell equations 40–2
moments of a distributionfunction 243–5, 257–8
momentum constraint 187, 191
Moncrief, V., 197
monotonically decreasing density 130–2
Mossbauer resonant eﬀect 63
motion of a system of compact
bodies 193–4
eﬀective one-body method 194
Numerical Relativity 194
moving frames 6–7
orthonormal 11
structure coeﬃcients 10
Müller zum Hagen, H., 127
multichart approach 78, 115
N
naked singularities 148, 197
natural coframes 5

natural frames 5
Navier–Stokes equations 232–3
neutrinos, rest mass of 48n28
neutron stars 115
Newman, R., 127, 128
Newtonian dynamics 35–6, 46
approximation of Einsteinequations 75–7
approximation of the relativistic
equation 47–8
Newtonian mechanics
Galileo–Newton spacetime 35
gravitation theory 56, 65–6
Maxwell equations 37
Newtonian stress tensor of a viscous
ﬂuid 232
no naked singularity conjecture 148
norm, Riemannian manifolds 12
normal geodesic coordinates 30
notation, ix
null condition 196
null cone (light cone) 14
null vectors 14
Numerical Relativity 194
O
Olbers’ paradox 156n5‘open’ universes 160n15
Oppenheimer, R., 136
orbits see circular orbits; planetary
motionorientable manifolds 4
orthochronous Lorentz group 38
orthonormal frames 11
P
Painlevé–Gullstrand-like coordinates,
Schwarzschild metric 110
p-antisymmetrization operator 9parallelizable manifolds 6
parallel transport 19–20
parametrized post-Newtonian (PPN)
formalism 78
past-oriented vectors 14
Penrose, R., 132, 148, 154, 195, 196, 197
Penrose process 145
perfect ﬂuids 51–2
Euler equations 212
in general relativity 60–1
stress–energy tensor 71, 211–12
wave fronts 217–18
perihelion precession 114–15
Pfaﬀ derivatives 7, 10
p-forms 9–10
phase of high-frequency waves 84
phase space 242
photons, energy–momentum 49

277

Pichon, G., 233
planar gravitational waves 79–80
Planck satellite 155n1, 158n13, 167n24,
214n10Planck’s constant 60
Planck time 167
Planck units 67
planetary motion
equations 112–14
results of observations 114–16
plasmas 241
Poincaré groups 39
Poincaré lemma 9
Poincaré–Minkowski spacetime 40n10
Poisson equation 56
polarization conditions 80
polarization operator 84
polytropic ﬂuids 214
positive-mass theorem 111
post-Newtonian approximation,
Einstein equations 77–8
primordial cosmology 167–8
principal symbol 182n20
principle of general covariance 55
progressive waves see high-frequency
waves
propagation of gravitational ﬁeld 178
propagation of high-frequency
waves 85–7
propagation speed, wave fronts 216–17,
218
proper frames 44, 208, 211
of ﬂuids 50of an observer 41n13
proper Lorentz transformations 38
proper time 43, 57
cosmic time 158
experimental results 63–5
pseudo-Riemannian metrics 11–12
geodesics 19–20
pullback of a tensor ﬁeld 8
pulsars, binary 95n47, 115–16
pure matter see dustQ
quadrupole formula 99–100
quantum theory 149
quark–gluon plasmas (Yang–Mills
ﬂuids) 231
quasilinear, quasidiagonal wave
operators 177
quasilinear partial diﬀerential
equations 69n6
hyperbolicity 219

278

Index

R
Rankine–Hugoniot equations 226
recombinations 168
redshift 63, 121, 161–2
Hubble law 162–3
redshift parameter 161
reduced Einstein equations 177
reduced EM–GR–Vlasov equation 248
Regge–Wheeler coordinates,
Schwarzschild metric 111
regularity, Einsteinian spacetimes 73
Reissner–Nordström metric 132–3, 152
relative velocities 44–5
relativistic dynamics 46–7
Newtonian approximation 47–8
relativistic Gauss law 90–1
Rendall, A., 220n27
representative vectors 5
rest mass 46n23, 47n25
conservation of 212
of elementary particles 48
particles with zero rest mass 49
Ricci identity 21
Ricci tensor 21
linearization 23–4
second derivative 34
Riemann curvature tensor 21
and Bianchi identities 30–1
linearization 34
Riemannian connection 18–19
Riemannian manifolds
asymptotically Euclidean 74
geodesics 20
Riemannian metrics 12–13
Riemannian Penrose inequality 149,
153–4
Riemann tensor
ﬁrst-order symmetric hyperbolic
system 207
wave equation 206
Robertson–Walker cosmologies 160–1
Robertson–Walker spacetimes 158–9,
265–6
general properties
age and future of the universe 164
cosmological redshift 161–2
deceleration parameter 163
Hubble law 162–3
metric at given t, 159–60
Rodnianski, I., 196
S
scalar curvature 21
scalar products, pseudo-Riemannian
metrics 11
Schwarzschild black hole 136–7
Schwarzschild metric 107–8

non-standard coordinates 108
isotropic 109
Painlevé–Gullstrand-like 110
Regge–Wheeler 111
wave (harmonic) 109–10
Schwarzschild radius 111–12
Schwarzschild singularity 112, 136
Schwarzschild spacetime 105, 111–12
escape velocity 116–17
in isotropic coordinates 133–5(n + 1) dimensional 133
planetary motion and perihelion
precession 112–16
spherically symmetric interior
solutions 122–6
stability of circular orbits 117–18
wave coordinates 135
Schwarzschild throat 140, 141
Schwarzschild trapped surfaces 207
second, deﬁnition of 59
second derivative (second variation) 34
second fundamental form (extrinsic
curvature) 185
second moment of the distributionfunction 243–4
Segal cosmos 156–7
Seifert, H.J., 127
semilinear diﬀerential equations,
hyperbolicity 219
shell crossing 130n30
shift 184
shock cross-section 256
shocks 226
SI (Système International) units 59
signature of a quadratic form 11
Lorentzian metrics 13
singularities 127
cosmic censorship conjectures 196–8
Kerr metric 144
Schwarzschild singularity 112, 131–2,
136
shell-crossing 130n30
see also big bangsliced spacetimes 185
smooth manifolds 4
solar eclipses 58, 63, 120
solitons 90
sound wave fronts 218
spacelike submanifolds 15
spacelike vectors 14
spaces of constant curvature 159
spacetime causality structure 57
spacetime metric signature, ix
special Lorentz transformations 38
special relativity
proper frame and relative
velocities 44–5
proper time 43

speciﬁc volume 212, 235
spherically symmetric gravitational
collapse 127
monotonically decreasing
density 130–2
Tolman, Gu, Hu, andClaudel–Newman metrics 127–9
spherically symmetric interior
solutions 122–3
matching with an exterior
solution 125–6
non-static solutions 126
static solutions with compact
spacelike sections 123–5
spherically symmetric spacetimes 105–7
spin 95n51
stability of circular orbits 117–18
stability of Kerr black hole 146–7
standard choice 106
standard coordinates 107
standard cosmological models 157
standard model Lagrangian 94
static spacetimes 88
stationary axisymmetric spacetimes 142
stationary black holes 141–2
Kerr metric 142–6
uniqueness theorems 146
stationary spacetimes 87
equations 88–9
gravitational solitons 90
relativistic Gauss law 90–1
static spacetimes 88
stellar-mass black holes 150
stiﬀ (incompressible) ﬂuids 218
Stokes’s formula 89
stress–energy tensor 243–4, 250
string theory, cosmological models 167
strong causality 195
strong cosmic censorshipconjecture 196–7
strong energy condition 211
structure coeﬃcients 10
submanifolds 5
spacelike 15
supermassive black holes 150
surface gravity, black holes 149
symmetric ﬁrst-order partial diﬀerential
equations 220
symmetric hyperbolic ﬁrst-order partial
diﬀerential equations 220
symmetric hyperbolic systems 188n36,
198–200
evolution set of Maxwell equations
as 202

Indexfor Riemann tensor 207
wave equation as 200–1
symmetry properties of tensors 8Ttachyons 140n7
tangent bundles 5tangent spaces 5
tangent vectors 5
Taub Lagrangian 93, 104
Taub–NUT spacetime 98–9
Teller, E., 229n38
tensor ﬁelds 8
tensor notation, ix
tensor products 7
tensors 7–8
Bel–Robinson tensor 31–2
Cotton tensor 33
Cotton–York tensor 33
Einstein tensor 21, 69
Maxwell tensor 41–2, 43, 61, 71
Ricci tensor 21, 23–4, 34
stress–energy tensor 243–4, 250
TT (transverse, traceless) tensors 191
viscosity stress tensor 232
vorticity tensor 223–4
Weyl tensor 33
thermodynamics
black holes 148–9
conservation of entropy 212–13
conservation of rest mass 212
dissipative ﬂuids 262
entropy and the H theorem 259–60
equations of state 213–15
extended 262
Maxwell–Jüttner equilibrium
distributions 261–2
tidal force 22n19, 81–2
time delay 64
time dilation 121–2
timelike curves 14
timelike vectors 14
time-orientable manifolds 14
Tolman, R.C., 127
Tolman–Oppenheimer–Volkov
equation 124
topological manifolds 3transparency of the universe 168transverse-traceless waves 80
trapped surfaces 197–8
in Schwarzschild spacetime 207
TT (transverse, traceless) tensors 191
twin paradox, vii 43, 122
two-timing method 83
Uultrarelativistic ﬂuids 214
uniqueness theorems, stationary black
holes 146
units of measurement 59–60
Planck units 67
unit velocity 46
universe, age and future 164
Vvacuum 69n3
Einstein equations 69
vacuum spacetime 73
Vaidya metric 101
vanishing torsion 18
vector ﬁelds 6
vector notation, ix
velocities 44
addition of 44–5
unit velocity 46
viscosity stress tensor 232
viscous ﬂuids 231
generalized Navier–Stokes
equations 232–3
Leray–Ohya hyperbolic system 233–4
Vlasov models 246
EM–GR–Vlasov equation 247–8
equation for particles with randomcharges 264–5
general relativistic (GR) Vlasov
equation 246–7
solution of a Vlasov equation 249–50
Yang–Mills plasmas 248–9
volume element, pseudo-Riemannian
manifolds 12volume form, pseudo-Riemannian
metrics 12volume radius 189n39
vorticity tensor 223–4

279

W
wave (harmonic) coordinates 109–10,
176–7
damped 178
generalized 177
wave cones 216
wave equations 30–1
for F, 205–6
for Riemann tensor 206
as a symmetric hyperbolic
system 200–1
wave fronts 215
magnetoacoustic 230–1
in perfect ﬂuids 217–18
propagation speed 216–17
wave gauge constraints 180
solution of 181–3
asymptotically Euclidean
manifolds 183
compact manifolds 184
wave gauge preservation 179–80
wave maps, Lagrangians 94
wave operator 29
weak cosmic censorship conjecture 148,
197
weak Einstein equivalence principle 58
weak energy condition 211
well posed Cauchy problems 175,
218–19Weyl tensor 33
Wheeler, J.A., 136, 146n13
white holes 139, 140
WIMPS (weakly interacting massive
particles) 155–6, 167
WKB anzatz 82–3
Y
Yang–Mills ﬁelds 52–3
Lagrangians 94
Yang–Mills ﬂuids 231
Yang–Mills plasmas 248–9
Yodzis, P., 127Zzero conductivity, charged ﬂuids 227–8