Linear perturbations of matched spacetimes: the
gauge problem and background symmetries
Marc Mars∗, Filipe C. Mena♭ and Raül Vera‡
∗Facultad de Ciencias, Universidad de Salamanca,
Plaza de la Merced s/n, 37008 Salamanca, Spain
♭Departamento de Matemática, Universidade do Minho,
Campus de Gualtar, 4710 Braga, Portugal
♭Mathematical Institute, University of Oxford,
24-29 St Giles’, Oxford OX1 3LB, UK
‡Fisika Teorikoaren Saila, Euskal Herriko
Unibertsitatea, Apt. 644, Bilbao 48080, Basque Country, Spain
June 12, 2018
Abstract
We present a critical review about the study of linear perturbations of matched
spacetimes including gauge problems. We analyse the freedom introduced in the
perturbed matching by the presence of background symmetries and revisit the par-
ticular case of spherically symmetry in n-dimensions. This analysis includes settings
with boundary layers such as brane world models and shell cosmologies.
PACS numbers: 0420, 0240
1 Introduction
An important aspect in any geometric gravitational theory is the analysis of how to match
two spacetimes. This is true in particular for General Relativity and its perturbation
theory. Despite the relevance and maturity of the matching theory one often finds papers
where the matching conditions are not properly used. Most of the difficulties arise from
the fact that the matching conditions are imposed in specific coordinate systems in a
manner which is not completely coordinate independent. More specifically, matching two
spacetimes requires identifying the boundaries pointwise, and sometimes this identification
is done implicitly by fixing spacetime coordinates, without paying enough attention to the
fact that solving the matching involves finding an identification of the boundary and that
this should not be fixed a priori.
In perturbation theory this problem also arises, and it gets complicated by the fact
that the fields to be matched (as the perturbed metric) are gauge dependent. So, in
addition to a priori choices of identifications of the boundary, there is also the problem
http://arxiv.org/abs/0704.0078v1
that particular gauges are often used. It may be argued that the matching theory must
be gauge independent and therefore it can be performed in any gauge. This is true, but
only when due care is taken to ensure that the choice of gauge does not restrict, a priori,
the perturbed identification of the boundaries.
A complete description of the linearized matching conditions has been achieved only
recently by Carter and Battye [5] and independently by Mukohyama [6]. To second order,
the matching conditions have been recently found in [7]. Despite these papers, we believe
that some confusion still lingers in the field, in particular with respect to the existing gauge
invariant formulations. The aim of this paper is to try to clarify these issues. In order
to do that, we will critically discuss some of the approaches proposed in the literature
trying to make clear which are the implicit assumptions made and to what extent are
they justified.
The first papers discussing the perturbed matching theory are, as far as we know,
the classic papers by Gerlach and Sengupta [2, 3]. However, as explained below, their
description of the perturbed matching theory contains imprecisions, and we will therefore
start discussing their approach pointing out the difficulties they encounter. A first attempt
to justify the claims in [2, 3] is due to Mart́ın-Garćıa and Gundlach [4], who propose a
different but nevertheless closely related set of linearized matching conditions. Pointing
out the implicit assumption made by these authors will also help us to try to explain the
subtleties inherent to the perturbed matching theory.
In [6] the linearized matching conditions are described for arbitrary backgrounds,
perturbations and matching hypersurfaces, and then applied to the case of two background
spacetimes with a high degree of symmetry, namely those which admit a maximal group of
isometries acting on codimension two spacelike submanifolds (e.g. spherically symmetric
spacetimes). In order to simplify the matching conditions, Mukohyama derives a set of
matching conditions for so-called doubly gauge invariants. However, a gap arises in his
final conclusions as the presented set of conditions for doubly gauge invariant quantities
for the linearized matching of spacetimes are only shown to be necessary conditions.
Analysing sufficiency touches directly on the issue we are trying to emphasize in this
paper, so we devote one section to clarify this point, where we show how these conditions
are, stricktly speaking, not sufficient. Since the matching conditions in terms of doubly
gauge invariants are widely used in the literature, we consider important to close this
gap. Moreover, the constructions of gauge invariant quantities using spherical harmonic
decompositons leaves out the l = 0 and l = 1 sectors. We will discuss this issue and its
consequences.
The paper is organized as follows. We start by summarising the perturbed matching
conditions in Section 2, where we also describe the gauge freedom involved. Then, the
procedures used in the classic papers [2, 3] together with the justifications and further
developments in [4] are reviewed in Section 3. Section 4 focuses on the consequences of
the existence of symmetries in the background configuration, which will have relevance
in our final discussion. Section 5 has three subsections. The first one is devoted to
present briefly the procedure and results discussed in [6] particularised to the case of
spherically symmetric backgrounds. In the second subsection we analyse the sufficiency
of the doubly gauge invariant matching conditions in [6]. The last subsection is devoted
to the study of the freedom left in the perturbation of the matching hypersurface once
the metric perturbations have been fixed at both sides. We finish with an appendix where
explicit expressions for the discontinuities of the perturbed second fundamental forms in
the spherical case are given. Some of these expressions are used in the main text.
2 Linearized matching
In this section we describe the gauge freedom involved in the linearised spacetime matching
and summarise the perturbed matching conditions.
2.1 Gauge freedom
The purpose of the matching theory is to construct a new spacetime out of two spacetimes
M± with boundary by finding a suitable diffeomorphism between the boundaries which
allows for their pointwise identification. In particular, the matched spacetime cannot
be thought to exist beforehand. Another aspect to bear in mind is that the matching
conditions involve exclusively tensors on the identified boundary Σ and hence any coordi-
nate system in M± is equally valid. This is well-known but it is still source of confusion
sometimes.
In perturbed matching theory, not only the metrics are perturbed but also the match-
ing hypersurfaces may be deformed. Furthermore, as for the metric, the “deviation” of the
matching hypersurface is also a gauge dependent quantity. This can be best understood
by viewing perturbations as ε-derivatives (at ε = 0) of a one-parameter family of space-
times (M+ε , g
ε ) with boundary Σ
ε . It is convenient to embed M
ε within a larger manifold
(without boundary) V +ε to clarify the discussion. A priori, the manifolds (M
ε , gε) are
completely distinct so it makes no direct sense to talk about ε-derivatives. It is necessary
to identify first the different manifolds so that a single point p refers to one point on each
of the manifolds. Obviously, there are infinite ways to identify the manifolds, all of them
equally valid a priori. This freedom leads to the gauge dependence of the perturbed met-
ric (and of any other geometrically defined tensor). The identification above may, or may
not, map the boundaries Σ+ε among themselves. A priori, a point in Σ
0 may be mapped,
for ε 6= 0, to a point on Σ+ε , to a point interior to M+ε or to a point exterior to M+ε (within
the extension V +ε ) which is not part of the manifold. How can we then take derivatives
with respect to ε at those later points? Since only derivatives at ε = 0 are needed, re-
stricting to infinitesimal values of ε entails no loss of generality. Then, if for some small
ε, a point q ∈ Σ+0 is mapped to the exterior of M+ε , it follows from differentiability with
respect to ε that q is mapped, for the reverse value −ε, to a point interior to M+ε . Thus,
perturbations can be defined at the boundary by taking one sided derivatives, i.e. to take
limits ε → 0, with a sign restriction on ε (c.f. [7] for an alternative discussion).
However, an important issue remains: How do we describe the deformation of the
boundary Σ+0 ? As a set of points each boundary Σ
ε maps, with the above identification,
into a hypersurface of the background spacetime, which we call Σ̂+ε . In general, this
hypersurface will not coincide with Σ+0 and may well touch it or cross it. This gives us
an idea of how the boundary is deformed, but only as a subset, not pointwise. In order
to know how the boundary actually moves within the background, we need to prescribe a
priori a pointwise identification of Σ+0 with Σ
ε . This identification is completely different
and independent from the one described above involving spacetime points, and involves
only the points on the boundaries. As before, there are infinitely many ways to identify the
boundaries, and this defines a second gauge freedom, which involves objects intrinsically
defined on the boundary. This gauge freedom will be referred as hypersurface gauge, as
opposed to the usual spacetime gauge described above.
With both identifications chosen, the deformation of the boundary within the back-
ground can already be described: Fix a point q on the background boundary Σ+0 . The
identification of the boundaries defines a point qε on Σ
ε , for each ε. The spacetime iden-
tification takes this point qε and maps it into a point q̂ε of the background M
0 (perhaps
after a sign restriction on ε). Obviously q̂ε belongs to the perturbed hypersurface Σ̂
ε . We
have therefore not only a deformation of the background hypersurface as a set of points,
but also pointwise information. It only remains to take the tangent vector of the curve
q̂ε at ε = 0, i.e. ~Z
+ = dq̂ε
|ε=0 which encodes completely the deformation of the matching
hypersurface as seen from the background spacetime. Two final remarks are in order:
(i) ~Z+ is defined exclusively on Σ+0 , no extension thereof is defined or required and (ii)
~Z+ depends on both the spacetime and hypersurface gauges, since its defining curve is
constructed using both identifications. However, decomposing ~Z+ = Q+~n 0+ +
~T+, where
~n 0+ is the unit normal of Σ
0 (assumed non-null anywhere) and
~T+ is tangent to it, it turns
out that Q+ depends on the spacetime gauge but not on the hypersurface gauge. This
is because changing the hypersurface gauge reorganizes the points within each Σ̂+ε , but
cannot modify any of them as a set of points.
Tensors defined intrinsically on the boundaries Σ+ε are completely unaffected by the
spacetime identification, and are therefore invariant under spacetime gauge transforma-
tions. Recall that the matching conditions involve only objects intrinsic to the match-
ing hypersurfaces. Since the perturbed matching conditions are, formally, just their ε-
derivatives, it follows by construction that the perturbed matching conditions must be
gauge invariant under spacetime gauge transformations. This may seem surprising at first
sight since the matching conditions must involve the perturbed metric, which is obviously
gauge dependent. However, the conditions turn out to be gauge independent because they
also involve the deformation vector ~Z+, which is spacetime gauge dependent. This vector
is therefore of fundamental importance and must be taken into account in any sensible
approach to the problem, as we shall see next.
2.2 Matching conditions
Let (M±0 , g
0 ) be n-dimensional spacetimes with non-null boundaries Σ
0 . Matching them
requires an identification of the boundaries, i.e. a pair of embeddings Φ± : Σ0 −→ M±0
with Φ±(Σ0) = Σ
0 , where Σ0 is an abstract copy of any of the boundaries. Let ξ
(i, j, . . . = 1, . . . , n−1) be a coordinate system on Σ0. Tangent vectors to Σ±0 are obtained
by e±αi =
(α, β, . . . = 0, . . . , n − 1). There are also unique (up to orientation) unit
normal vectors n
α to the boundaries. We choose them so that if n
α points towards
M+ then n
α points outside of M− or viceversa. The first and second fundamental are
simply q(0)ij
± ≡ e±αi e
, K(0)ij
± = −n(0)±αe
i ∇±β e
j . The matching conditions
(in the absence of shells) require the equality of the first and second fundamental forms
on Σ±0 , i.e.
q(0)ij
+ = q(0)ij
−, K(0)ij
+ = K(0)ij
−. (1)
Under a perturbation of the background metric g±pert = g
(0)± + g(1)± and of the matching
hypersurfaces via ~Z± = Q± ~n(0)± + ~T
±, the matching conditions will be perturbatively
satisfied if and only if [6]
q(1)+ij = q
ij , K
ij = K
ij, (2)
q(1)ij
± = L~T±q
± + 2Q±K(0)ij
± + e±αi e
±, (3)
K(1)ij
± = L~T±K
± − ǫDiDjQ± +Q±(−n(0)± µn
αµβνe
±K(0)
g(1)αβ
βK(0)ij
± − n(0)± µS
(1)±µ
j , (4)
where ǫ = n
(0)α, D is the covariant derivative of (Σ, q(0)±) and S(1)
βγ ≡ 12(∇
(1)±α
∇±γ g(1)±αβ −∇±α g(1)
In these equations, Q± and ~T± are a priori unknown quantities and fulfilling the
matching conditions requires showing that two vectors ~Z± exist such that (2) are satisfied.
The spacetime gauge freedom can be exploited to fix either or both vectors ~Z± a priori,
but this should be avoided (or at least carefully analysed) if additional spacetime gauge
choices are made, in order not to restrict a priori the possible matchings. Regarding the
hypersurface gauge, this can be used to fix one of the vectors ~T+ or ~T−, but not both.
As already stressed the linearized matching conditions are by construction spacetime
gauge invariant (in fact each of the tensors q(1)ij
±, K(1)ij
± is). Moreover, the set of
conditions (2) are hypersurface gauge invariant, provided the background is properly
matched, since [6] under such a gauge transformation given by the vector ~ζ on Σ0, q
transforms as q(1)ij + L~ζ q(0)ij , and similarly for K(1)ij .
3 On previous spacetime gauge invariant formalisms
The first attempt to derive a general formalism for the matching conditions in linearized
gravity is, to our knowledge, due to Gerlach and Sengupta [2]. Their approach is based
on the description of the matching hypersurface Σ as a level set of a function f defined
on the spacetime. Assuming the level sets {f = const} to be timelike, a field of spacelike
unit normals is defined as nµ = (g
αβf,αf,β)
−1/2f,µ. The unperturbed matching conditions
correspond to the continuity everywhere (in particular across Σ) of the tensors
qαβ ≡ gαβ − nαnβ, Kαβ ≡ qαµqβν∇µnν , (5)
which are the spacetime versions of the first and second fundamental forms introduced
above. Being f defined everywhere, it makes sense to perturb it in order to describe the
variation of the matching hypersurface. Obviously, by perturbing f one also perturbs nµ.
The perturbed matching conditions proposed in [2] read
β△(qαβ)+ = qµαqνβ△(qαβ)−, qµαqνβ△(Kαβ)+ = qµαqνβ△(Kαβ)−, (6)
where qβ
α is the projector onto Σ, △ stands for perturbation and + and − denote the
quantities as computed from either side of the matching hypersurface Σ. These expres-
sions involve the projections of the perturbations of qαβ and Kαβ onto Σ. The need of
1We will abuse slightly the notation and refer to vectors on Σ0 and their images on spacetime with
the same symbol. The meaning should be clear from the context.
considering only the projected components is justified in [2] since the matching condi-
tions need to be intrinsic to the matching hypersurfaces. However, Gerlach and Sengupta
themselves note that conditions (6) are not gauge2 invariant.
Since the main interest in [2, 3] refers to spherically symmetric backgrounds, this
“ambiguity” is fixed in that case by finding suitable gauge invariant combinations of
the linearized matching conditions, which turn out to give a correct set of necessary
perturbed matching conditions in spherical symmetry. It should be stressed however, that
the authors consider these gauge invariant subset to be sufficient also, with no further
justification.
We know from the discussion in Sect. 2.1 above that (6) cannot be correct as it leads
to a set of gauge dependent conditions. Since, on the other hand the proposal (6) may
look plausible, it is of interest to point out where, and in which sense, it fails to be correct.
The first source of problems comes from assuming that the matched spacetime is given
beforehand. Indeed, qαβ and Kαβ are spacetime tensors and they can only exist (and be
continuous) once the matched spacetime is constructed. But this is precisely the purpose
of the matching conditions, so the conditions become circular. Another aspect of the same
problem is that one can only talk about continuity once the pointwise identification of
the boundaries is chosen. But a level set of a function defines only a set of points and not
the way those points must be identified. A third instance of the same issue is that tensor
components must be expressed in some basis, e.g. a common coordinate system covering
both sides of Σ. But again this cannot be assumed a priori. It needs to be constructed.
Let us however mention that once the pointwise identification of the boundaries is
chosen, the use of spacetime tensors is allowed provided they are finally projected onto
the hypersurface. In that sense, and when properly used, using spacetime indices may
simplify some calculations notably (see Carter and Battye, [5] where this notation is used
to derive the perturbed matching conditions).
Besides this aspect (which already affects the background matching) the perturbed
equations (6) suffer from one extra problem. The perturbations △(qαβ)(p) and△(Kαβ)(p)
at a point p in the background can be defined by taking ε-derivatives at fixed p and ε = 0
of the corresponding tensors (defined by gαβ(ε) and fε). For each value of ε, the matching
conditions impose the continuity of qαβ(ε) andKαβ(ε) everywhere (with the caveat already
mentioned regarding the identification of the boundaries). However, continuity of △(qαβ)
and △(Kαβ) at p would only follow if derivatives of continuous functions with respect
to an external parameter were necessarily continuous (in our case, the derivative with
respect to ε), which is not true in general. A trivial example is given by the function
u(ε, x) = |x + ε|, whith x ∈ R. For each ε this function is continuous. However, the
derivative with respect to ε does not even exist at x = 0, ε = 0. This reflects the fact that
subtracting continuous tensors at a fixed spacetime point p leads to objects that need not
be continuous. This is in fact the main problem of (6) as linearized matching conditions.
An immediate question arises: Why is the gauge invariant subset of matching condi-
tions found in [2, 3] for spherically symmetric backgrounds correct? In order to understand
this, let us rewrite (6) using the formalism of section 2.2. First of all, since △(nαnβ) will
contain, at least, one free n
α , we have
β△(qαβ)± = qµαqνβg(1)
αβ . (7)
2Throughout this section gauge will refer to spacetime gauge. Hypersurface gauges will only appear
briefly towards the end of the section.
Moreover, a simple calculation gives △(∇αnβ) = ∇α(△nβ) − S(1)µαβ n
µ and △(qαβ ) =
−g(1)αµn(0)µ n(0)β + g(0)αµ△(nµ)n
(0)α△(nβ). These, together with standard properties
of the projector, lead to
β△(Kαβ)± =
a(0)ν qµ
α△(nα) + qµαqνβ∇α(△nβ)− qµαqνβS(1)ραβ n
±, (8)
where a
ν ≡ n(0)α∇αn(0)ν . In general, these expressions do not agree with (3) and (4).
However, when the gauges are chosen so that ~Z± = 0, then △f ≡ 0 on Σ because
the matching hypersurface is unperturbed as seen from the background. Consequently
∂α(△f) ∝ n(0)α on Σ, which implies △(nα)
α for some function h. Imposing ~n(ε) to
be unit for all ε fixes h = ǫ
β . Inserting into (8) the matching conditions (6)
become
βg(1)αβ
βg(1)αβ
, (9)
n(0)α n
β Kµν − qµ
n(0)α n
β Kµν − qµ
which agree with (2) (with the exception that (9) refers to spacetime tensors and (2) are
defined on Σ). Since Gerlach and Sengupta derive a subset of gauge invariant matching
conditions out of (6) in the spherically symmetric case and their conditions are correct in
one gauge, it follows that the invariant subset is correct in any gauge. This is the reason
why the results in [2, 3] involving spherically symmetric backgrounds turn out to be fine.
Substantial progress in the linearized matching problem was made by Mart́ın-Garćıa
and Gundlach [4]. These authors pointed out the lack of justification in [2, 3] for the choice
of (6) as matching conditions. It was also argued that for spacetimes with boundary it
only makes sense to define perturbations by using gauges where the perturbed matching
hypersurface is mapped onto the background matching hypersurface. Perturbations in this
gauge, called “surface gauge” (not to be confused with hypersurface gauge) are denoted
by △̄, and its defining property is △̄f = 0. The idea was to write down the matching
conditions in this gauge and then transform into any other gauge if necessary. As noticed
by the authors, the surface gauge is not unique since there are still three degrees of freedom
left, which correspond to the three directions tangent to Σ.
A relevant observation made in [4] was that the continuity of tensorial perturbations
may depend on the index position in the tensors. The authors argue that the tensors
truly intrinsic to the hypersurfaces are qαβ, Kαβ (with indices upstairs) and propose the
following perturbed matching conditions
△̄(qαβ)+ = △̄(qαβ)−, △̄(Kαβ)+ = △̄(Kαβ)−, (10)
which are demonstrated to become exactly (9). This shows the equivalence of both pro-
posals in the surface gauge, as explicitly stated in [4]. This justifies partially the validity of
both approaches in the surface gauge. However, the justification is not complete because
of the issue we discuss next.
Indeed, conditions (10) still carry one implicit assumption that needs to be clarified.
As already stressed the perturbed matching conditions have two inherent and independent
degrees of gauge freedom. The approach by Mart́ın-Garćıa and Gundlach involves only
spacetime objects, and therefore can only notice the spacetime gauge freedom. This leads
to an incorrect statement in [4], as it is not true that the linearized matching conditions
read (10) in any surface gauge. Conditions (10) will only be valid when the spacetime
gauge maps pairs of background points (identified, via the background matching) to pairs
of points on the perturbed boundaries Σ±ε which are also identified through the matching.
Notice that not all surface gauges have this property. In explicit terms, this means that
the vectors ~Z± must (i) only have tangential components (so that we are in surface gauge)
and (ii) have the same components when written in terms of an intrinsic basis of Σ0. In
less precise, but more intuitive terms, condition (ii) states that ~Z+ and ~Z− are the same
vector, i.e. that the gauges in both regions are chosen such that the displacement of
one fixed point of the background hypersurface is identical in both regions (the displaced
point, of course, stays on the unperturbed hypersurface, due to the choice of surface
gauge). Observe finally that if Q± = 0 and ~T+ = ~T−, then the linearized matching
conditions (2) truly reduce to conditions (9), once the latter are projected on Σ. This
shows the correctness of the approaches by Gerlach and Sengupta and Mart́ın-Garćıa and
Gundlach in special gauges.
4 Freedom in matching due to symmetries
We devote this section to the study of the consequences of the existence of background
symmetries on perturbed spacetime matchings.
The existence of symmetries in the background configuration introduces two issues
which are important to take into consideration: the first corresponds to the freedom in-
troduced by the matching procedure, when preserving the symmetries, at the background
level [9], c.f. [10] for an application. The second issue corresponds to the consequences
that the symmetries in the background configuration may have on the perturbation of the
matching.
It must be stressed here that the arbitrariness introduced by the presence of symmetries
in the background configuration is completely independent from both the hypersurface and
spacetime gauge freedoms. However, that arbitrariness is gauge dependent and therefore
a gauge choice can be made to remove it. As we will show, an isometry in the background
implies that there is a direction along which the difference [~T ] ≡ ~T+ − ~T− cannot be
determined by the perturbed matching equations. But, as we have discussed at the end
of section 2, one could eventually choose part of the spacetime gauges (if there is any
freedom left) to fix [~T ]. Note, finally, that a change of hypersurface gauge leaves [~T ]
invariant.
4.1 Isometries
We shall now consider the presence of isometries in the background configuration. So,
let us assume that one of the sides, say (M+0 , g
(0)+), admits an isometry generated by
the Killing vector field ~ξ+ tangent to the boundary Σ+0 . The commutation of the Lie
derivative and the pull-back implies [9]
L~ξ+q
+ = e+αi e
j L~ξ+g
+|Σ0 = 0,
which means that ~ξ+ is a Killing vector of (Σ0, q
+). This implies from expression (3)
that q(1)ij
+ is invariant under the transformation ~T+ → ~T+ + ~ξ+|Σ0.
As for K(1)ij
+, from its expression (4), it is again clear that the previous transforma-
tions of ~T+ will leave K(1)ij
+ invariant provided L~ξ+K(0)ij+ = 0. But this is precisely the
case since ~ξ+ is a Killing vector orthogonal to n
+ , which implies L~ξ+n
+ β|Σ+
= 0, and
hence
L~ξ+K
+ = e+αi e
j L~ξ+(∇n
+ )αβ|Σ0 = e+αi e
j ∇αL~ξ+n
+ β|Σ0 = 0.
Of course, all this discussion also applies to the (−) side.
The combination of the invariance of q(1)ij
± and K(1)ij
± leads to the fact that the
first order perturbed matching conditions are invariant under a change of the vectors
~T± along the direction of any isometry of the background configuration (preserved by
the matching). Then, as expected, when symmetries are present the linearized matching
conditions cannot determine the difference [~T ] completely: they leave undetermined the
relative (between the two sides) deformation of the hypersurface along the direction of
the symmetry. Note that, still, the shape of the perturbed hypersurface is completely
determined, since that is driven by Q±.
The overall picture is as follows: at the background level we have the arbitrariness of
the identification of Σ+0 with Σ
0 [9], which can be seen as a “sliding” between Σ
0 and
Σ−0 . The perturbation adds to this an arbitrary shift of the deformation of the matching
hypersurface at each side along the orbits of the isometry group. As an example, in
the description of stationary and axisymmetric compact bodies discussed in [10, 9], the
background sliding corresponds to an arbitrary constant rotation of the interior with
respect to the exterior. Note that, in that case, this rotation is only relevant because
the exterior is taken to be asymptotically flat. As a result, two identical interiors can,
in principle, give rise to two exteriors that differ by a constant rate rotation [10]. The
shift of the surface deformation would, in principle, lead to an arbitrary constant rotation
along the axial coordinate of the surface deformation of the body. Likewise, two identical
perturbations in the interior of the body may produce two different perturbations in the
exterior, which may differ by a relative constant rate rotation. A choice of spacetime
gauge could be used to relate the deformations inside and outside. However, this may
interfere with other gauge fixings that may have been made.
5 n-dimensional spherically symmetric backgrounds
In this section we shall revisit Mukohyama’s theory for linearized matching in the special
case of spherical symmetry. Similar results [6] hold for backgrounds admitting isometry
groups of dimension (n−1)(n−2)/2 acting on non-null codimension-two orbits of arbitrary
topology (strictly speaking the orbits need to be compact).
5.1 The approach of Mukohyama
Concentrating on one of the two spacetimes to be matched, either + or −, we consider a
spherically symmetric background metric of the form
αdxβ = γabdx
adxb + r2ΩABdθ
AdθB, (11)
where γab (a, b, .. = 0, 1) is a Lorentzian two-dimensional metric (depending only on {xa}),
r > 0 is a function of {xa}, and ΩABdθAdθB is the n − 2 dimensional unit sphere metric
with coordinates {θA} (A,B, . . . = 2, 3, . . . , n− 1).
A general spherically symmetric background hypersurface can be given in parametric
form as
Σ0 := {x0 = Z(0)0(λ), x1 = Z(0)1(λ), θA = ϑA}, (12)
where {ξi} = {λ, ϑA} is a coordinate system in Σ0 adapted to the spherical symmetry.
The tangent vectors to Σ0 read
~eλ =
˙Z(0)0∂x0 +
˙Z(0)1∂x1
, ~eϑA = ∂θA |Σ0 , (13)
where dot is derivative w.r.t. λ. With N2 ≡ −ǫeλaeλbγab|Σ0, so that ǫ = 1 (ǫ = −1)
corresponds to a timelike (spacelike) hypersurface, the unit normal to Σ0 reads
(0) =
− det γ
− ˙Z(0)1dx0 + ˙Z(0)0dx1
, (14)
where the sign choice of N corresponds to the choice of orientation of the normal. The
background induced metric and second fundamental form on Σ0 read
q(0)ijdξ
idξj = −ǫN2dλ2 + r2|Σ0ΩAB|Σ0dϑAdϑB, (15)
K(0)ijdξ
idξj = N2Kdλ2 + r2K̄|Σ0ΩAB|Σ0dϑAdϑB, (16)
where
K ≡ N−2eλaeλb∇an(0)b , K̄ = n
(0)a∂xa ln r.
It follows that the background matching conditions (1) are
N2+ = N
+|Σ0 = r2−|Σ0, K+ = K−, K̄+ = K̄−. (17)
Using (3) and (4) we could now compute the first order perturbations q(1)ij and K
ij in
terms of the above quantities and ~Z (or equivalently Q and ~T ), c.f. Eqs. (45) and (46)
in [6]. Let us recall (see subsection 2.2) that while the individual tensors q(1)ij and K
are not hypersurface gauge invariant, their respective differences from the + and − sides
(i.e. the linearized matching conditions) are. Those tensors depend of the hypersurface
gauge through the tangent vectors ~T+ and ~T−, which under a gauge change transform
simply by adding the gauge vector. It follows that only their difference [~T ] can appear
in the linearized matching conditions. Consequently there are three degrees of freedom
that cannot be fixed by the equations, but can be fixed by choosing the hypersurface
gauge, for instance to set ~T+. Thus, the linearized matching conditions can be looked at
as equations for the difference [~T ] as well as for Q+ and Q−, i.e. for five objects. If these
equations admit solutions, then the linearized matching is possible and it is impossible
otherwise.
Mukohyama emphasizes the convenience to look for doubly gauge invariant quantities
to write down the linearized matching conditions, however the matching conditions are
already gauge invariant (both for the spacetime and hypersurfaces gauges). Looking for
gauge invariant combinations on each side amounts to writing equations where the dif-
ference vector [~T ] simply drops. Indeed, in many cases, knowing the value of such vector
in a specific matching is not interesting. In that sense, using doubly gauge invariant
quantities is useful as it lowers the number of equations to analyse. However, we want to
stress that this is not related to obtaining gauge invariant linearized matching equations.
It is just related to not solving for superfluous information. In fact, a set of equations
where also Q+ and Q− have disappeared would be even more convenient from this point
of view, provided one is not interested in knowing how the hypersurfaces are deformed in
the specific spacetime gauge being used.
Since the use of doubly gauge invariant matching conditions is used extensively, let us
recall its main ingredients in order to discuss if they really are equivalent to the full set
of linearized matching equations and in which sense.
To that aim Mukohyama [6], decomposes the perturbation tensors q(1)ij and K
ij in
terms of scalar Y , vector VA and tensor harmonics TAB on the sphere, as
q(1)ijdξ
idξj =
(σ00Y dλ
2 + σ(Y )T(Y )ABdϑ
AdϑB) +
2(σ(T )0V(T )A + σ(L)0V(L)A)dλdϑ
(σ(T )T(T )AB + σ(LT )T(LT )AB + σ(LL)T(LL)AB)dϑ
AdϑB, (18)
K(1)ijdξ
idξj =
(κ00Y dλ
2 + κ(Y )T(Y )ABdϑ
AdϑB) +
2(κ(T )0V(T )A + κ(L)0V(L)A)dλdϑ
(κ(T )T(T )AB + κ(LT )T(LT )AB + κ(LL)T(LL)AB)dϑ
AdϑB, (19)
where all the scalar coefficients depend only on λ. Each coefficient in the decomposition
has indices l and m which have been dropped for notational simplicity. Notice that
each coefficient σ and κ is defined in the range of l’s appearing in the corresponding
summatory. By construction, each of the σ and κ are spacetime-gauge invariant (but
not hypersurface-gauge invariant). For l ≥ 2 they can even be written down [6] explicity
in terms of spacetime-gauge invariant quantities. In a similar way, the doubly gauge-
invariant quantities presented in [6], are only defined for l ≥ 2 (except k(T )0, which is also
defined for l = 1), and read
l ≥ 2 : f00 ≡ σ00 − 2N∂λ
l ≥ 2 : f ≡ σ(Y ) + ǫN−2χ∂λ
r2|Σ0
k2l σ(LL),
l ≥ 2 : f0 ≡ σ(T )0 − r2|Σ0∂λ
r−2|Σ0σ(LT )
l ≥ 2 : f(T ) ≡ σ(T ),
l ≥ 2 : k00 ≡ κ00 + ǫKσ00 + ǫχ∂λK,
l ≥ 1 : k(T )0 ≡ κ(T )0 − K̄σ(T )0, (20)
l ≥ 2 : k(L)0 ≡ κ(L)0 +
(ǫK − K̄)σ(L)0 +
(ǫK + K̄)
χ− r2|Σ0∂λ(r−2|Σ0σ(LL))
l ≥ 2 : k(LT ) ≡ κ(LT ) − K̄σ(LT ),
l ≥ 2 : k(LL) ≡ κ(LL) − K̄σ(LL),
l ≥ 2 : k(Y ) ≡ κ(Y ) − K̄σ(Y ) + ǫN−2r2|Σ0χ∂λK̄,
l ≥ 2 : k(T ) ≡ κ(T ) − K̄σ(T ),
3The ranges of l’s are not made explicit in [6] in order to include also non-compact homogeneous
spaces, where the index l is continuous. However, to discuss sufficiency of the equations we need to be
precise on the range of validity of each equation.
where k2l = l(l + n− 3) and
l ≥ 2 : χ ≡ σ(L)0 − r2|Σ0∂λ(r−2|Σ0σ(LL)).
The orthogonality properties of the scalar, vector and tensor harmonics imply that
the equalities of the coefficients σ and κ for each l and m is equivalent to the equality of
the perturbation tensors (18) and (19) at both sides of Σ0. Thus, recalling the notation
[f ] ≡ f+|Σ0 − f−|Σ0 , the equations
l ≥ 0 : [σ00] = [σ(Y )] = 0
l ≥ 1 : [σ(L)0] = [σ(T )0] = 0
l ≥ 2 : [σ(T )] = [σ(LT )] = [σ(LL)] = 0
l ≥ 0 : [κ00] = [κ(Y )] = 0
l ≥ 1 : [κ(L)0] = [κ(T )0] = 0
l ≥ 2 : [κ(T )] = [κ(LT )] = [κ(LL)] = 0
are equivalent to (2) and therefore correspond exactly to the linearized matching condi-
tions in this setting. Notice that each of the equalities in (21) and (22) is in fact one
equation for each l and m in the appropriate range. We will however refer to them simply
as equations.
The full linearized matching conditions obviously imply the following equalities in
terms of the doubly-gauge invariant quantities (20),
l ≥ 2 : [f00] = [f ] = [f0] = [f(T )] = 0 (23)
l ≥ 1 : [k(T )0] = 0
l ≥ 2 : [k00] = [k(Y )] = [k(L)0] = [k(LL)] = [k(LT )] = [k(T )] = 0.
Whether these equations can be regarded as the full set of linearized matching conditions
or not requires studying their sufficiency, i.e. whether they imply (21)-(22) or not. This
point was not mentioned in [6] and in fact the answer turns out to be negative, although
in a mild way, as we discuss in the next subsection.
5.2 On the sufficiency of the continuity of the doubly-gauge in-
variants
Let us recall that fulfilling the matching conditions requires finding two ~Z± such that
(21)-(22) are satisfied. The key issue for the matching is therefore to show existence of
deformation vectors ~Z± so that all the equations hold.
A plausibility argument in favour of the sufficiency of (23)-(24) comes from simple
equation counting. Indeed, as already discussed, the linearized matching conditions are
spacetime and hypersurface gauge invariant and therefore can only involve the difference
vector [~T ], i.e. three quantities. Since constructing double gauge invariant quantities on
each side eliminates this vector, the number of equations should be reduced exactly by
three if they are to remain equivalent to the original set. This is precisely what happens
as we go from the original forteen equations in (21)-(22) down to eleven equations in
(23)-(24). This argument however is not conclusive, both because it is not rigorous and
because each equation in those expressions is, in fact, a set of equations depending on l
and m, and the range of l’s changes with the equations. Let us therefore analyse this issue
in detail. In particular we need to discuss what are the consequences of the non-existence
of doubly gauge-invariant variables for l = 0 and l = 1 (except for k(T )0 which exists for
l = 1), something not mentioned in [6].
Let us start by finding explicit expressions for σ’s valid in the whole range of l’s. As
in [6], we decompose g(1) in harmonics as
g(1)αβdx
αdxβ =
(habY dx
adxb + h(Y )T(Y )ABdθ
AdθB)
2(h(T )aV(T )A + h(L)aV(L)A)dx
(h(T )T(T )AB + h(LT )T(LT )AB + h(LL)T(LL)AB)dθ
AdθB, (25)
and ~Z as
zaY dx
(z(T )V(T )A + z(L)V(L)A)dθ
QY n(0) − ǫN−2zλY eλ
(z(T )V(T )A + z(L)V(L)A)dθ
A, (26)
which implies Tαdx
l=0(−ǫN−2zλY eλ) +
l=1(z(T )V(T )A + z(L)V(L)A)dθ
A. Inserting
these expressions into (2) and expanding in spherical harmonics it is straightforward to
l ≥ 0 : [σ00] = 0 ⇔ [hλλ] + 2[Q]N2K + 2N∂λ
N−1[zλ]
l ≥ 1 : [σ(L)0] = 0 ⇔ [zλ] + [h(L)λ] + r2|Σ0∂λ(r−2|Σ0[z(L)]) = 0, (27)
l ≥ 2 : [σ(LL)] = 0 ⇔ [z(L)] + [h(LL)] = 0, (28)
l ≥ 0 : [σ(Y )] = 0 ⇔ [h(Y )] + 2[Q]r2|Σ0K̄ − ǫN−2[zλ]∂λ(r2|Σ0)−
k2l [z(L)] = 0,
l ≥ 1 : [σ(T )0] = 0 ⇔ [h(T )λ] + r2|Σ0∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [σ(LT )] = 0 ⇔ [z(T )] + [h(LT )] = 0, (29)
l ≥ 2 : [σ(T )] = 0 ⇔ [h(T )] = 0,
where [hλλ], [h(L)λ], etc. denote eλ
b[hab], eλ
a[h(L)a], etc. Later on we will also write
down the explicit expressions for (22) but they are not needed in this subsection.
It is obvious by the form of f ’s and κ’s (20) that the set of equations (21)-(22) are
equivalent to (23)-(24) together with
l ≥ 2 : [σ(L)0] = [σ(LT )] = [σ(LL)] = 0 (30)
l = 0, 1 : [σ00] = [σ(Y )] = 0
l = 1 : [σ(L)0] = [σ(T )0] = 0
l = 0, 1 : [κ00] = [κ(Y )] = 0
l = 1 : [κ(L)0] = 0.
Sufficiency of Mukohyama’s doubly gauge invariant matching conditions would follow if
these equations serve exclusively to determine the discontinuity [~T ], i.e. [zλ] for l ≥ 0
and [z(T )], [z(L)] for l ≥ 1. Now, the explicit expressions (27), (29), (28) show that (30)
determine uniquely [zλ], [z(T )] and [z(L)] for l ≥ 2. So, restricted to the sector l ≥ 2
Mukoyama’s doubly gauge invariant matching conditions can be regarded as equivalent
to the full set of matching conditions. Taking all l’s into account, however, the equations
turn out not to be sufficient. To show this, it is enough to display one equation involving
the discontinuity of the background metric perturbations and [Q] (but not [~T ]) which holds
as a consequence of the full set of matching conditions (21)-(22) but not as a consequence
of (23)-(24). Using the fact that each l = 1 expression refers to n − 1 objects (one for
each m), the number of equations in (31)-(32) is 7n− 3, while the number of unkowns in
[~T ] not yet determined by (30), i.e. [zλ] for l = 0, 1 and [z(T )], [z(L)] for l = 1 is 3n − 2,
which is smaller. It is to be expected, therefore, that (31), (32) imply conditions where
these variables do not appear. This can be made explicit, for instance, by combining
[σ00]l=0 = 0 with [σ(Y )]l=0 = 0 which yields
l = 0 : [hλλ] + 2[Q]N
2K + 2N∂λ
∂λ(r2|Σ0)
[h(Y )] + 2[Q]r
2|Σ0K̄
whenever ∂λ(r
2|Σ0) 6= 0. (If ∂λ(r2|Σ0) = 0 it is enough to consider [σ(Y )]l=0 = 0.) This
relation is enough to show that the continuity of the doubly-gauge invariant variables of
Mukohyama is not sufficient to ensure the existence of the perturbed matching. Of course,
this does not invalidate Mukohyama’s approach in any way, which remains interesting
and useful. It only means that, when using this approach to solve linearized matchings,
one still needs to look more carefully into the l = 0 and l = 1 sector to make sure that
the remaining equations (31) and (32) hold.
On the other hand, equations (31), (32) do not completely determine [~T ]. The variable
[z(T )]l=1 only appears in [σ(T )0]l=1 = 0, in the term ∂λ(r
−2|Σ0 [z(T )]). As a result, the
matching conditions do not fix [z(T )]l=1 completely, but up to a constant factor times
r2|Σ0 (for each m). Recalling that V(T )AdϑA for l = 1 correspond to the three Killing
vectors on the sphere, this arbitrary constant (for each m) accounts for the addition to
[~T ] of an arbitrary Killing vector of the sphere. This is in accordance with the discussion
in Section 4. We devote the following subsection to complete the study of the freedom
left in the matching.
5.3 Freedom in the matching
As already emphasized, solving the linearized matching amounts to finding perturbation
vectors ~Z+ and ~Z−. Assume now that a linearized matching between two given back-
grounds and perturbations has been done. It is natural to ask what is the most general
matching between those two spaces, i.e. what is the most general solution for ~Z+ and ~Z−
of the matching conditions. Geometrically, this means finding all the possible deforma-
tions of the matching hypersurface Σ0 which allow the two spaces to be matched.
Since this problem is of interest not only when the full matching conditions are imposed
but also in situations where layers of matter are present (e.g. in brane-world or shell
cosmologies) so that jumps in the second fundamental forms are allowed, we will analyse
this issue in two steps. First, we will study the equations involving the perturbed first
fundamental forms and will determine the freedom they admit. On a second step we
will write down the extra conditions coming from the equality of the second fundamental
forms.
Thus, let us consider two perturbation configurations of the same background matching
and denote their respective sets of difference variables on Σ0 as [f ] and [f ]
′ for any given
variable f . Now, we will define the difference between the two configurations as < f >≡
[f ]′ − [f ] for any variable f . The assumption that the perturbation on each side is fixed
once and for all implies < g(1) >= 0. We are assuming that the linearized matching
conditions are satisfied in each case, and so we can subtract them. Linearity implies that
the differences of the linearised matching equations become equations for the difference
vector < ~Z >. The general solution of these equations clearly determines the freedom in
the deformation of the hypesurface.
The difference of the equations in (21) for the two configurations using < g(1) >= 0
give the following set of equations
l ≥ 0 : < σ00 >= 0 → < Q > N2K +N∂λ
N−1 < zλ >
= 0, (33)
l ≥ 1 : < σ(L)0 >= 0 → < zλ > +r2|Σ0∂λ(r−2|Σ0 < z(L) >) = 0, (34)
l ≥ 2 : < σ(LL) >= 0 → < z(L) >= 0, (35)
l ≥ 0 : < σ(Y ) >= 0 → 2 < Q > r2|Σ0K̄ − ǫN−2 < zλ > ∂λ(r2|Σ0)
k2l < z(L) >= 0, (36)
l ≥ 1 : < σ(T )0 >= 0 → ∂λ(r−2|Σ0 < z(T ) >) = 0, (37)
l ≥ 2 : < σ(LT ) >= 0 → < z(T ) >= 0, (38)
l ≥ 2 : < σ(T ) >= 0 → 0 = 0.
Expressions (35) and (38) readily determine < z(L) >l≥2=< z(T ) >l≥2= 0, which sub-
stituted in (34) give < zλ >l≥2= 0. As a result, (36) for l ≥ 2 lead to < Q >l≥2= 0.
Clearly all the equations for l ≥ 2 are now satisfied. We now concentrate on the l = 1
equations. Equation (37) implies that < z(T ) >l=1= ar
2|Σ0, where a is a constant for each
m. Combining equations (33), (34) and (36) for l = 1 we obtain the following equation
for r−2|Σ0 < z(L) >l=1,
K̄∂2λ(r−2|Σ0 < z(L) >l=1) +
(2K̄ + ǫK)∂λ(ln r|Σ0)− K̄∂λ lnN
−2|Σ0 < z(L) >l=1)
r−2|Σ0 < z(L) >l=1= 0, (39)
while (34) and (33) determine < zλ >l=1 and < Q >l=1 respectively (provided K 6= 0,
which occurs generically). The two equations for l = 0 can be rearranged onto
K̄∂λ(N−1 < zλ >l=0) +N−1 < zλ >l=0 ǫK∂λ ln(r|Σ0) = 0 (40)
plus the equation (33) for l = 0, which determines < Q >l=0.
Summarizing, we have found that the freedom in the deformation of the hypersurface
compatible with the linearized matching conditions involving the first fundamental form
[~Z]′ − [~Z] =
< Q > Y ~n(0) − ǫN−2 < zλ > Y ~eλ
+am~V
(T ) + r
−2|Σ0 < z(L) >l=1,m ~V m(L),
where r−2|Σ0 < z(L) >l=1,m, satisfy (39), < zλ >l=0 satisfy (40) and the rest of the
variables are completely determined as described above. The term in am corresponds
to adding Killing vectors on the sphere, something already discussed in Section 4. The
rest of terms involve combinations (with functions) of the conformal Killing vectors on
the sphere and tangential vectors along λ. Notice that the coefficients of the conformal
Killing (i.e. < z(L) >l=1,m ) determine all the rest of the l = 1 coefficients. In particular
when < z(L) >l=1,m vanishes, then all the l = 1 terms vanish and the freedom becomes
radially symmetric.
We now add to the analysis the difference of the equations in (22). Due to the fact
that all coefficients in < ~Z > vanish for l ≥ 2 we only need to consider the equations for
l = 0, 1, i.e. (32). We refer the reader to Appendix A for the explicit expressions of (32)
in terms of the metric perturbations and ~Z. For the sake of completeness we also include
all the explicit expressions of (22) in Appendix A. The difference of equations (32), see
(44)-(46), whenever < g(1) >= 0 read
l = 0, 1 : < κ00 >= 0 ⇔ (41)
− < QR(γ)dbac > n
(0)dn(0)aeλ
c − ǫ∂2λ < Q > +
2∂λ < Q > −ǫ < Q > K2N2
−ǫKN2∂λ(N−2 < zλ >)− ǫ∂λ(K < zλ >) = 0,
l = 1 : < κ(L)0 >= 0 ⇔ (42)
−ǫ∂λ < Q > +ǫK < zλ > +ǫ < Q > ∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0 < z(L) >) = 0
l = 0, 1 : < κ(Y ) >= 0 ⇔ (43)
N−2∂λ(r
2|Σ0) (∂λ < Q > +K < zλ >) +
< Qn(0)an(0)b∇a∇br2 >
N−2eλ
a < zλn
(0)b∇b∇ar2 > +
l(l + n− 3)
ǫ < Q > −2K̄ < z(L) >
It can be checked that in general these equations overdetermine the previous equations,
i.e. (39) and (40), although there may be particular cases for which they are compatible.
Therefore, generically, they will imply that < z(L) >l=1,m= 0 and < zλ >l=0= 0, and
thence all the rest of the variables vanish, < zλ >l=1,m=< Q >l=1,m= 0, < zλ >l=0=<
Q >l=0= 0, so that the only freedom left is given by
[~Z]′ − [~Z] = am~V m(T ).
Finding in which particular cases equations (39)-(43) are compatible is straightforward
but tedious and will not be carried out explicitly here.
Acknowledgements
FM and MM thank CRUP(Portugal)/MCT(Spain) for grant E-113/04. FM thanks FCT
(Portugal) for grant SFRH/BPD/12137/2003 and CMAT, University of Minho, for sup-
port. MM was supported by the projects FIS2006-05319 of the Spanish Ministerio de
Educación y Tecnoloǵıa and SA010CO of the Junta de Castilla y León. RV was supported
by the Irish IRCSET, Ref. PD/2002/108, and now is funded by the Basque Government
Ref. BFI05.335.
A Appendix
For the sake of completeness we devote this appendix to present the explicit expressions
of (22) in terms of the metric perturbations and ~Z, which read
l ≥ 0 : [κ00] = 0 ⇔ (44)
N2K[hnn]−
n(0)aeλ
c(2∇c[hab]−∇a[hbc])− [QR(γ)dbac]n
(0)dn(0)aeλ
−ǫ∂2λ[Q] +
2∂λ[Q]− ǫ[Q]K2N2 − ǫKN2∂λ(N−2[zλ])− ǫ∂λ(K[zλ]) = 0,
l ≥ 1 : [κ(L)0] = 0 ⇔ (45)
[hnλ]−
n(0)aeλ
b(∂b[h(L)a]− ∂a[h(L)b])− ǫ∂λ[Q]− ǫK[zλ]
+(ǫ[Q] + [h(L)n])∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(L)]) = 0
l ≥ 0 : [κ(Y )] = 0 ⇔ (46)
r2|Σ0K̄[hnn] +
N−2∂λ(r
2|Σ0) (ǫ[hnλ] + ∂λ[Q] +K[zλ]) +
[n(0)a∂ah(Y )]
[Qn(0)an(0)b∇a∇br2]−
N−2eλ
a[zλn
(0)b∇b∇ar2]
l(l + n− 3)
[h(L)n] + ǫ[Q]− 2K̄[z(L)]
l ≥ 1 : [κ(T )0] = 0 ⇔
n(0)aeλ
b(∂b[h(T )a]− ∂a[h(T )b]) + [h(T )n]∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(T )n] +
n(0)a∂a[h(LT )] + K̄[z(T )] = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(L)n] +
n(0)a∂a[h(LL)] + K̄[z(L)]−
[Q] = 0,
l ≥ 2 : [κ(T ] = 0 ⇔
n(0)a∂a[h(T )] = 0.
References
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1509-1519
	Introduction
	Linearized matching
	Gauge freedom
	Matching conditions
	On previous spacetime gauge invariant formalisms
	Freedom in matching due to symmetries
	Isometries
	n-dimensional spherically symmetric backgrounds
	The approach of Mukohyama
	On the sufficiency of the continuity of the doubly-gauge invariants
	Freedom in the matching
	Appendix

  We present a critical review about the study of linear perturbations of
matched spacetimes including gauge problems. We analyse the freedom introduced
in the perturbed matching by the presence of background symmetries and revisit
the particular case of spherically symmetry in n-dimensions. This analysis
includes settings with boundary layers such as brane world models and shell
cosmologies.

Introduction
An important aspect in any geometric gravitational theory is the analysis of how to match
two spacetimes. This is true in particular for General Relativity and its perturbation
theory. Despite the relevance and maturity of the matching theory one often finds papers
where the matching conditions are not properly used. Most of the difficulties arise from
the fact that the matching conditions are imposed in specific coordinate systems in a
manner which is not completely coordinate independent. More specifically, matching two
spacetimes requires identifying the boundaries pointwise, and sometimes this identification
is done implicitly by fixing spacetime coordinates, without paying enough attention to the
fact that solving the matching involves finding an identification of the boundary and that
this should not be fixed a priori.
In perturbation theory this problem also arises, and it gets complicated by the fact
that the fields to be matched (as the perturbed metric) are gauge dependent. So, in
addition to a priori choices of identifications of the boundary, there is also the problem
http://arxiv.org/abs/0704.0078v1
that particular gauges are often used. It may be argued that the matching theory must
be gauge independent and therefore it can be performed in any gauge. This is true, but
only when due care is taken to ensure that the choice of gauge does not restrict, a priori,
the perturbed identification of the boundaries.
A complete description of the linearized matching conditions has been achieved only
recently by Carter and Battye [5] and independently by Mukohyama [6]. To second order,
the matching conditions have been recently found in [7]. Despite these papers, we believe
that some confusion still lingers in the field, in particular with respect to the existing gauge
invariant formulations. The aim of this paper is to try to clarify these issues. In order
to do that, we will critically discuss some of the approaches proposed in the literature
trying to make clear which are the implicit assumptions made and to what extent are
they justified.
The first papers discussing the perturbed matching theory are, as far as we know,
the classic papers by Gerlach and Sengupta [2, 3]. However, as explained below, their
description of the perturbed matching theory contains imprecisions, and we will therefore
start discussing their approach pointing out the difficulties they encounter. A first attempt
to justify the claims in [2, 3] is due to Mart́ın-Garćıa and Gundlach [4], who propose a
different but nevertheless closely related set of linearized matching conditions. Pointing
out the implicit assumption made by these authors will also help us to try to explain the
subtleties inherent to the perturbed matching theory.
In [6] the linearized matching conditions are described for arbitrary backgrounds,
perturbations and matching hypersurfaces, and then applied to the case of two background
spacetimes with a high degree of symmetry, namely those which admit a maximal group of
isometries acting on codimension two spacelike submanifolds (e.g. spherically symmetric
spacetimes). In order to simplify the matching conditions, Mukohyama derives a set of
matching conditions for so-called doubly gauge invariants. However, a gap arises in his
final conclusions as the presented set of conditions for doubly gauge invariant quantities
for the linearized matching of spacetimes are only shown to be necessary conditions.
Analysing sufficiency touches directly on the issue we are trying to emphasize in this
paper, so we devote one section to clarify this point, where we show how these conditions
are, stricktly speaking, not sufficient. Since the matching conditions in terms of doubly
gauge invariants are widely used in the literature, we consider important to close this
gap. Moreover, the constructions of gauge invariant quantities using spherical harmonic
decompositons leaves out the l = 0 and l = 1 sectors. We will discuss this issue and its
consequences.
The paper is organized as follows. We start by summarising the perturbed matching
conditions in Section 2, where we also describe the gauge freedom involved. Then, the
procedures used in the classic papers [2, 3] together with the justifications and further
developments in [4] are reviewed in Section 3. Section 4 focuses on the consequences of
the existence of symmetries in the background configuration, which will have relevance
in our final discussion. Section 5 has three subsections. The first one is devoted to
present briefly the procedure and results discussed in [6] particularised to the case of
spherically symmetric backgrounds. In the second subsection we analyse the sufficiency
of the doubly gauge invariant matching conditions in [6]. The last subsection is devoted
to the study of the freedom left in the perturbation of the matching hypersurface once
the metric perturbations have been fixed at both sides. We finish with an appendix where
explicit expressions for the discontinuities of the perturbed second fundamental forms in
the spherical case are given. Some of these expressions are used in the main text.
2 Linearized matching
In this section we describe the gauge freedom involved in the linearised spacetime matching
and summarise the perturbed matching conditions.
2.1 Gauge freedom
The purpose of the matching theory is to construct a new spacetime out of two spacetimes
M± with boundary by finding a suitable diffeomorphism between the boundaries which
allows for their pointwise identification. In particular, the matched spacetime cannot
be thought to exist beforehand. Another aspect to bear in mind is that the matching
conditions involve exclusively tensors on the identified boundary Σ and hence any coordi-
nate system in M± is equally valid. This is well-known but it is still source of confusion
sometimes.
In perturbed matching theory, not only the metrics are perturbed but also the match-
ing hypersurfaces may be deformed. Furthermore, as for the metric, the “deviation” of the
matching hypersurface is also a gauge dependent quantity. This can be best understood
by viewing perturbations as ε-derivatives (at ε = 0) of a one-parameter family of space-
times (M+ε , g
ε ) with boundary Σ
ε . It is convenient to embed M
ε within a larger manifold
(without boundary) V +ε to clarify the discussion. A priori, the manifolds (M
ε , gε) are
completely distinct so it makes no direct sense to talk about ε-derivatives. It is necessary
to identify first the different manifolds so that a single point p refers to one point on each
of the manifolds. Obviously, there are infinite ways to identify the manifolds, all of them
equally valid a priori. This freedom leads to the gauge dependence of the perturbed met-
ric (and of any other geometrically defined tensor). The identification above may, or may
not, map the boundaries Σ+ε among themselves. A priori, a point in Σ
0 may be mapped,
for ε 6= 0, to a point on Σ+ε , to a point interior to M+ε or to a point exterior to M+ε (within
the extension V +ε ) which is not part of the manifold. How can we then take derivatives
with respect to ε at those later points? Since only derivatives at ε = 0 are needed, re-
stricting to infinitesimal values of ε entails no loss of generality. Then, if for some small
ε, a point q ∈ Σ+0 is mapped to the exterior of M+ε , it follows from differentiability with
respect to ε that q is mapped, for the reverse value −ε, to a point interior to M+ε . Thus,
perturbations can be defined at the boundary by taking one sided derivatives, i.e. to take
limits ε → 0, with a sign restriction on ε (c.f. [7] for an alternative discussion).
However, an important issue remains: How do we describe the deformation of the
boundary Σ+0 ? As a set of points each boundary Σ
ε maps, with the above identification,
into a hypersurface of the background spacetime, which we call Σ̂+ε . In general, this
hypersurface will not coincide with Σ+0 and may well touch it or cross it. This gives us
an idea of how the boundary is deformed, but only as a subset, not pointwise. In order
to know how the boundary actually moves within the background, we need to prescribe a
priori a pointwise identification of Σ+0 with Σ
ε . This identification is completely different
and independent from the one described above involving spacetime points, and involves
only the points on the boundaries. As before, there are infinitely many ways to identify the
boundaries, and this defines a second gauge freedom, which involves objects intrinsically
defined on the boundary. This gauge freedom will be referred as hypersurface gauge, as
opposed to the usual spacetime gauge described above.
With both identifications chosen, the deformation of the boundary within the back-
ground can already be described: Fix a point q on the background boundary Σ+0 . The
identification of the boundaries defines a point qε on Σ
ε , for each ε. The spacetime iden-
tification takes this point qε and maps it into a point q̂ε of the background M
0 (perhaps
after a sign restriction on ε). Obviously q̂ε belongs to the perturbed hypersurface Σ̂
ε . We
have therefore not only a deformation of the background hypersurface as a set of points,
but also pointwise information. It only remains to take the tangent vector of the curve
q̂ε at ε = 0, i.e. ~Z
+ = dq̂ε
|ε=0 which encodes completely the deformation of the matching
hypersurface as seen from the background spacetime. Two final remarks are in order:
(i) ~Z+ is defined exclusively on Σ+0 , no extension thereof is defined or required and (ii)
~Z+ depends on both the spacetime and hypersurface gauges, since its defining curve is
constructed using both identifications. However, decomposing ~Z+ = Q+~n 0+ +
~T+, where
~n 0+ is the unit normal of Σ
0 (assumed non-null anywhere) and
~T+ is tangent to it, it turns
out that Q+ depends on the spacetime gauge but not on the hypersurface gauge. This
is because changing the hypersurface gauge reorganizes the points within each Σ̂+ε , but
cannot modify any of them as a set of points.
Tensors defined intrinsically on the boundaries Σ+ε are completely unaffected by the
spacetime identification, and are therefore invariant under spacetime gauge transforma-
tions. Recall that the matching conditions involve only objects intrinsic to the match-
ing hypersurfaces. Since the perturbed matching conditions are, formally, just their ε-
derivatives, it follows by construction that the perturbed matching conditions must be
gauge invariant under spacetime gauge transformations. This may seem surprising at first
sight since the matching conditions must involve the perturbed metric, which is obviously
gauge dependent. However, the conditions turn out to be gauge independent because they
also involve the deformation vector ~Z+, which is spacetime gauge dependent. This vector
is therefore of fundamental importance and must be taken into account in any sensible
approach to the problem, as we shall see next.
2.2 Matching conditions
Let (M±0 , g
0 ) be n-dimensional spacetimes with non-null boundaries Σ
0 . Matching them
requires an identification of the boundaries, i.e. a pair of embeddings Φ± : Σ0 −→ M±0
with Φ±(Σ0) = Σ
0 , where Σ0 is an abstract copy of any of the boundaries. Let ξ
(i, j, . . . = 1, . . . , n−1) be a coordinate system on Σ0. Tangent vectors to Σ±0 are obtained
by e±αi =
(α, β, . . . = 0, . . . , n − 1). There are also unique (up to orientation) unit
normal vectors n
α to the boundaries. We choose them so that if n
α points towards
M+ then n
α points outside of M− or viceversa. The first and second fundamental are
simply q(0)ij
± ≡ e±αi e
, K(0)ij
± = −n(0)±αe
i ∇±β e
j . The matching conditions
(in the absence of shells) require the equality of the first and second fundamental forms
on Σ±0 , i.e.
q(0)ij
+ = q(0)ij
−, K(0)ij
+ = K(0)ij
−. (1)
Under a perturbation of the background metric g±pert = g
(0)± + g(1)± and of the matching
hypersurfaces via ~Z± = Q± ~n(0)± + ~T
±, the matching conditions will be perturbatively
satisfied if and only if [6]
q(1)+ij = q
ij , K
ij = K
ij, (2)
q(1)ij
± = L~T±q
± + 2Q±K(0)ij
± + e±αi e
±, (3)
K(1)ij
± = L~T±K
± − ǫDiDjQ± +Q±(−n(0)± µn
αµβνe
±K(0)
g(1)αβ
βK(0)ij
± − n(0)± µS
(1)±µ
j , (4)
where ǫ = n
(0)α, D is the covariant derivative of (Σ, q(0)±) and S(1)
βγ ≡ 12(∇
(1)±α
∇±γ g(1)±αβ −∇±α g(1)
In these equations, Q± and ~T± are a priori unknown quantities and fulfilling the
matching conditions requires showing that two vectors ~Z± exist such that (2) are satisfied.
The spacetime gauge freedom can be exploited to fix either or both vectors ~Z± a priori,
but this should be avoided (or at least carefully analysed) if additional spacetime gauge
choices are made, in order not to restrict a priori the possible matchings. Regarding the
hypersurface gauge, this can be used to fix one of the vectors ~T+ or ~T−, but not both.
As already stressed the linearized matching conditions are by construction spacetime
gauge invariant (in fact each of the tensors q(1)ij
±, K(1)ij
± is). Moreover, the set of
conditions (2) are hypersurface gauge invariant, provided the background is properly
matched, since [6] under such a gauge transformation given by the vector ~ζ on Σ0, q
transforms as q(1)ij + L~ζ q(0)ij , and similarly for K(1)ij .
3 On previous spacetime gauge invariant formalisms
The first attempt to derive a general formalism for the matching conditions in linearized
gravity is, to our knowledge, due to Gerlach and Sengupta [2]. Their approach is based
on the description of the matching hypersurface Σ as a level set of a function f defined
on the spacetime. Assuming the level sets {f = const} to be timelike, a field of spacelike
unit normals is defined as nµ = (g
αβf,αf,β)
−1/2f,µ. The unperturbed matching conditions
correspond to the continuity everywhere (in particular across Σ) of the tensors
qαβ ≡ gαβ − nαnβ, Kαβ ≡ qαµqβν∇µnν , (5)
which are the spacetime versions of the first and second fundamental forms introduced
above. Being f defined everywhere, it makes sense to perturb it in order to describe the
variation of the matching hypersurface. Obviously, by perturbing f one also perturbs nµ.
The perturbed matching conditions proposed in [2] read
β△(qαβ)+ = qµαqνβ△(qαβ)−, qµαqνβ△(Kαβ)+ = qµαqνβ△(Kαβ)−, (6)
where qβ
α is the projector onto Σ, △ stands for perturbation and + and − denote the
quantities as computed from either side of the matching hypersurface Σ. These expres-
sions involve the projections of the perturbations of qαβ and Kαβ onto Σ. The need of
1We will abuse slightly the notation and refer to vectors on Σ0 and their images on spacetime with
the same symbol. The meaning should be clear from the context.
considering only the projected components is justified in [2] since the matching condi-
tions need to be intrinsic to the matching hypersurfaces. However, Gerlach and Sengupta
themselves note that conditions (6) are not gauge2 invariant.
Since the main interest in [2, 3] refers to spherically symmetric backgrounds, this
“ambiguity” is fixed in that case by finding suitable gauge invariant combinations of
the linearized matching conditions, which turn out to give a correct set of necessary
perturbed matching conditions in spherical symmetry. It should be stressed however, that
the authors consider these gauge invariant subset to be sufficient also, with no further
justification.
We know from the discussion in Sect. 2.1 above that (6) cannot be correct as it leads
to a set of gauge dependent conditions. Since, on the other hand the proposal (6) may
look plausible, it is of interest to point out where, and in which sense, it fails to be correct.
The first source of problems comes from assuming that the matched spacetime is given
beforehand. Indeed, qαβ and Kαβ are spacetime tensors and they can only exist (and be
continuous) once the matched spacetime is constructed. But this is precisely the purpose
of the matching conditions, so the conditions become circular. Another aspect of the same
problem is that one can only talk about continuity once the pointwise identification of
the boundaries is chosen. But a level set of a function defines only a set of points and not
the way those points must be identified. A third instance of the same issue is that tensor
components must be expressed in some basis, e.g. a common coordinate system covering
both sides of Σ. But again this cannot be assumed a priori. It needs to be constructed.
Let us however mention that once the pointwise identification of the boundaries is
chosen, the use of spacetime tensors is allowed provided they are finally projected onto
the hypersurface. In that sense, and when properly used, using spacetime indices may
simplify some calculations notably (see Carter and Battye, [5] where this notation is used
to derive the perturbed matching conditions).
Besides this aspect (which already affects the background matching) the perturbed
equations (6) suffer from one extra problem. The perturbations △(qαβ)(p) and△(Kαβ)(p)
at a point p in the background can be defined by taking ε-derivatives at fixed p and ε = 0
of the corresponding tensors (defined by gαβ(ε) and fε). For each value of ε, the matching
conditions impose the continuity of qαβ(ε) andKαβ(ε) everywhere (with the caveat already
mentioned regarding the identification of the boundaries). However, continuity of △(qαβ)
and △(Kαβ) at p would only follow if derivatives of continuous functions with respect
to an external parameter were necessarily continuous (in our case, the derivative with
respect to ε), which is not true in general. A trivial example is given by the function
u(ε, x) = |x + ε|, whith x ∈ R. For each ε this function is continuous. However, the
derivative with respect to ε does not even exist at x = 0, ε = 0. This reflects the fact that
subtracting continuous tensors at a fixed spacetime point p leads to objects that need not
be continuous. This is in fact the main problem of (6) as linearized matching conditions.
An immediate question arises: Why is the gauge invariant subset of matching condi-
tions found in [2, 3] for spherically symmetric backgrounds correct? In order to understand
this, let us rewrite (6) using the formalism of section 2.2. First of all, since △(nαnβ) will
contain, at least, one free n
α , we have
β△(qαβ)± = qµαqνβg(1)
αβ . (7)
2Throughout this section gauge will refer to spacetime gauge. Hypersurface gauges will only appear
briefly towards the end of the section.
Moreover, a simple calculation gives △(∇αnβ) = ∇α(△nβ) − S(1)µαβ n
µ and △(qαβ ) =
−g(1)αµn(0)µ n(0)β + g(0)αµ△(nµ)n
(0)α△(nβ). These, together with standard properties
of the projector, lead to
β△(Kαβ)± =
a(0)ν qµ
α△(nα) + qµαqνβ∇α(△nβ)− qµαqνβS(1)ραβ n
±, (8)
where a
ν ≡ n(0)α∇αn(0)ν . In general, these expressions do not agree with (3) and (4).
However, when the gauges are chosen so that ~Z± = 0, then △f ≡ 0 on Σ because
the matching hypersurface is unperturbed as seen from the background. Consequently
∂α(△f) ∝ n(0)α on Σ, which implies △(nα)
α for some function h. Imposing ~n(ε) to
be unit for all ε fixes h = ǫ
β . Inserting into (8) the matching conditions (6)
become
βg(1)αβ
βg(1)αβ
, (9)
n(0)α n
β Kµν − qµ
n(0)α n
β Kµν − qµ
which agree with (2) (with the exception that (9) refers to spacetime tensors and (2) are
defined on Σ). Since Gerlach and Sengupta derive a subset of gauge invariant matching
conditions out of (6) in the spherically symmetric case and their conditions are correct in
one gauge, it follows that the invariant subset is correct in any gauge. This is the reason
why the results in [2, 3] involving spherically symmetric backgrounds turn out to be fine.
Substantial progress in the linearized matching problem was made by Mart́ın-Garćıa
and Gundlach [4]. These authors pointed out the lack of justification in [2, 3] for the choice
of (6) as matching conditions. It was also argued that for spacetimes with boundary it
only makes sense to define perturbations by using gauges where the perturbed matching
hypersurface is mapped onto the background matching hypersurface. Perturbations in this
gauge, called “surface gauge” (not to be confused with hypersurface gauge) are denoted
by △̄, and its defining property is △̄f = 0. The idea was to write down the matching
conditions in this gauge and then transform into any other gauge if necessary. As noticed
by the authors, the surface gauge is not unique since there are still three degrees of freedom
left, which correspond to the three directions tangent to Σ.
A relevant observation made in [4] was that the continuity of tensorial perturbations
may depend on the index position in the tensors. The authors argue that the tensors
truly intrinsic to the hypersurfaces are qαβ, Kαβ (with indices upstairs) and propose the
following perturbed matching conditions
△̄(qαβ)+ = △̄(qαβ)−, △̄(Kαβ)+ = △̄(Kαβ)−, (10)
which are demonstrated to become exactly (9). This shows the equivalence of both pro-
posals in the surface gauge, as explicitly stated in [4]. This justifies partially the validity of
both approaches in the surface gauge. However, the justification is not complete because
of the issue we discuss next.
Indeed, conditions (10) still carry one implicit assumption that needs to be clarified.
As already stressed the perturbed matching conditions have two inherent and independent
degrees of gauge freedom. The approach by Mart́ın-Garćıa and Gundlach involves only
spacetime objects, and therefore can only notice the spacetime gauge freedom. This leads
to an incorrect statement in [4], as it is not true that the linearized matching conditions
read (10) in any surface gauge. Conditions (10) will only be valid when the spacetime
gauge maps pairs of background points (identified, via the background matching) to pairs
of points on the perturbed boundaries Σ±ε which are also identified through the matching.
Notice that not all surface gauges have this property. In explicit terms, this means that
the vectors ~Z± must (i) only have tangential components (so that we are in surface gauge)
and (ii) have the same components when written in terms of an intrinsic basis of Σ0. In
less precise, but more intuitive terms, condition (ii) states that ~Z+ and ~Z− are the same
vector, i.e. that the gauges in both regions are chosen such that the displacement of
one fixed point of the background hypersurface is identical in both regions (the displaced
point, of course, stays on the unperturbed hypersurface, due to the choice of surface
gauge). Observe finally that if Q± = 0 and ~T+ = ~T−, then the linearized matching
conditions (2) truly reduce to conditions (9), once the latter are projected on Σ. This
shows the correctness of the approaches by Gerlach and Sengupta and Mart́ın-Garćıa and
Gundlach in special gauges.
4 Freedom in matching due to symmetries
We devote this section to the study of the consequences of the existence of background
symmetries on perturbed spacetime matchings.
The existence of symmetries in the background configuration introduces two issues
which are important to take into consideration: the first corresponds to the freedom in-
troduced by the matching procedure, when preserving the symmetries, at the background
level [9], c.f. [10] for an application. The second issue corresponds to the consequences
that the symmetries in the background configuration may have on the perturbation of the
matching.
It must be stressed here that the arbitrariness introduced by the presence of symmetries
in the background configuration is completely independent from both the hypersurface and
spacetime gauge freedoms. However, that arbitrariness is gauge dependent and therefore
a gauge choice can be made to remove it. As we will show, an isometry in the background
implies that there is a direction along which the difference [~T ] ≡ ~T+ − ~T− cannot be
determined by the perturbed matching equations. But, as we have discussed at the end
of section 2, one could eventually choose part of the spacetime gauges (if there is any
freedom left) to fix [~T ]. Note, finally, that a change of hypersurface gauge leaves [~T ]
invariant.
4.1 Isometries
We shall now consider the presence of isometries in the background configuration. So,
let us assume that one of the sides, say (M+0 , g
(0)+), admits an isometry generated by
the Killing vector field ~ξ+ tangent to the boundary Σ+0 . The commutation of the Lie
derivative and the pull-back implies [9]
L~ξ+q
+ = e+αi e
j L~ξ+g
+|Σ0 = 0,
which means that ~ξ+ is a Killing vector of (Σ0, q
+). This implies from expression (3)
that q(1)ij
+ is invariant under the transformation ~T+ → ~T+ + ~ξ+|Σ0.
As for K(1)ij
+, from its expression (4), it is again clear that the previous transforma-
tions of ~T+ will leave K(1)ij
+ invariant provided L~ξ+K(0)ij+ = 0. But this is precisely the
case since ~ξ+ is a Killing vector orthogonal to n
+ , which implies L~ξ+n
+ β|Σ+
= 0, and
hence
L~ξ+K
+ = e+αi e
j L~ξ+(∇n
+ )αβ|Σ0 = e+αi e
j ∇αL~ξ+n
+ β|Σ0 = 0.
Of course, all this discussion also applies to the (−) side.
The combination of the invariance of q(1)ij
± and K(1)ij
± leads to the fact that the
first order perturbed matching conditions are invariant under a change of the vectors
~T± along the direction of any isometry of the background configuration (preserved by
the matching). Then, as expected, when symmetries are present the linearized matching
conditions cannot determine the difference [~T ] completely: they leave undetermined the
relative (between the two sides) deformation of the hypersurface along the direction of
the symmetry. Note that, still, the shape of the perturbed hypersurface is completely
determined, since that is driven by Q±.
The overall picture is as follows: at the background level we have the arbitrariness of
the identification of Σ+0 with Σ
0 [9], which can be seen as a “sliding” between Σ
0 and
Σ−0 . The perturbation adds to this an arbitrary shift of the deformation of the matching
hypersurface at each side along the orbits of the isometry group. As an example, in
the description of stationary and axisymmetric compact bodies discussed in [10, 9], the
background sliding corresponds to an arbitrary constant rotation of the interior with
respect to the exterior. Note that, in that case, this rotation is only relevant because
the exterior is taken to be asymptotically flat. As a result, two identical interiors can,
in principle, give rise to two exteriors that differ by a constant rate rotation [10]. The
shift of the surface deformation would, in principle, lead to an arbitrary constant rotation
along the axial coordinate of the surface deformation of the body. Likewise, two identical
perturbations in the interior of the body may produce two different perturbations in the
exterior, which may differ by a relative constant rate rotation. A choice of spacetime
gauge could be used to relate the deformations inside and outside. However, this may
interfere with other gauge fixings that may have been made.
5 n-dimensional spherically symmetric backgrounds
In this section we shall revisit Mukohyama’s theory for linearized matching in the special
case of spherical symmetry. Similar results [6] hold for backgrounds admitting isometry
groups of dimension (n−1)(n−2)/2 acting on non-null codimension-two orbits of arbitrary
topology (strictly speaking the orbits need to be compact).
5.1 The approach of Mukohyama
Concentrating on one of the two spacetimes to be matched, either + or −, we consider a
spherically symmetric background metric of the form
αdxβ = γabdx
adxb + r2ΩABdθ
AdθB, (11)
where γab (a, b, .. = 0, 1) is a Lorentzian two-dimensional metric (depending only on {xa}),
r > 0 is a function of {xa}, and ΩABdθAdθB is the n − 2 dimensional unit sphere metric
with coordinates {θA} (A,B, . . . = 2, 3, . . . , n− 1).
A general spherically symmetric background hypersurface can be given in parametric
form as
Σ0 := {x0 = Z(0)0(λ), x1 = Z(0)1(λ), θA = ϑA}, (12)
where {ξi} = {λ, ϑA} is a coordinate system in Σ0 adapted to the spherical symmetry.
The tangent vectors to Σ0 read
~eλ =
˙Z(0)0∂x0 +
˙Z(0)1∂x1
, ~eϑA = ∂θA |Σ0 , (13)
where dot is derivative w.r.t. λ. With N2 ≡ −ǫeλaeλbγab|Σ0, so that ǫ = 1 (ǫ = −1)
corresponds to a timelike (spacelike) hypersurface, the unit normal to Σ0 reads
(0) =
− det γ
− ˙Z(0)1dx0 + ˙Z(0)0dx1
, (14)
where the sign choice of N corresponds to the choice of orientation of the normal. The
background induced metric and second fundamental form on Σ0 read
q(0)ijdξ
idξj = −ǫN2dλ2 + r2|Σ0ΩAB|Σ0dϑAdϑB, (15)
K(0)ijdξ
idξj = N2Kdλ2 + r2K̄|Σ0ΩAB|Σ0dϑAdϑB, (16)
where
K ≡ N−2eλaeλb∇an(0)b , K̄ = n
(0)a∂xa ln r.
It follows that the background matching conditions (1) are
N2+ = N
+|Σ0 = r2−|Σ0, K+ = K−, K̄+ = K̄−. (17)
Using (3) and (4) we could now compute the first order perturbations q(1)ij and K
ij in
terms of the above quantities and ~Z (or equivalently Q and ~T ), c.f. Eqs. (45) and (46)
in [6]. Let us recall (see subsection 2.2) that while the individual tensors q(1)ij and K
are not hypersurface gauge invariant, their respective differences from the + and − sides
(i.e. the linearized matching conditions) are. Those tensors depend of the hypersurface
gauge through the tangent vectors ~T+ and ~T−, which under a gauge change transform
simply by adding the gauge vector. It follows that only their difference [~T ] can appear
in the linearized matching conditions. Consequently there are three degrees of freedom
that cannot be fixed by the equations, but can be fixed by choosing the hypersurface
gauge, for instance to set ~T+. Thus, the linearized matching conditions can be looked at
as equations for the difference [~T ] as well as for Q+ and Q−, i.e. for five objects. If these
equations admit solutions, then the linearized matching is possible and it is impossible
otherwise.
Mukohyama emphasizes the convenience to look for doubly gauge invariant quantities
to write down the linearized matching conditions, however the matching conditions are
already gauge invariant (both for the spacetime and hypersurfaces gauges). Looking for
gauge invariant combinations on each side amounts to writing equations where the dif-
ference vector [~T ] simply drops. Indeed, in many cases, knowing the value of such vector
in a specific matching is not interesting. In that sense, using doubly gauge invariant
quantities is useful as it lowers the number of equations to analyse. However, we want to
stress that this is not related to obtaining gauge invariant linearized matching equations.
It is just related to not solving for superfluous information. In fact, a set of equations
where also Q+ and Q− have disappeared would be even more convenient from this point
of view, provided one is not interested in knowing how the hypersurfaces are deformed in
the specific spacetime gauge being used.
Since the use of doubly gauge invariant matching conditions is used extensively, let us
recall its main ingredients in order to discuss if they really are equivalent to the full set
of linearized matching equations and in which sense.
To that aim Mukohyama [6], decomposes the perturbation tensors q(1)ij and K
ij in
terms of scalar Y , vector VA and tensor harmonics TAB on the sphere, as
q(1)ijdξ
idξj =
(σ00Y dλ
2 + σ(Y )T(Y )ABdϑ
AdϑB) +
2(σ(T )0V(T )A + σ(L)0V(L)A)dλdϑ
(σ(T )T(T )AB + σ(LT )T(LT )AB + σ(LL)T(LL)AB)dϑ
AdϑB, (18)
K(1)ijdξ
idξj =
(κ00Y dλ
2 + κ(Y )T(Y )ABdϑ
AdϑB) +
2(κ(T )0V(T )A + κ(L)0V(L)A)dλdϑ
(κ(T )T(T )AB + κ(LT )T(LT )AB + κ(LL)T(LL)AB)dϑ
AdϑB, (19)
where all the scalar coefficients depend only on λ. Each coefficient in the decomposition
has indices l and m which have been dropped for notational simplicity. Notice that
each coefficient σ and κ is defined in the range of l’s appearing in the corresponding
summatory. By construction, each of the σ and κ are spacetime-gauge invariant (but
not hypersurface-gauge invariant). For l ≥ 2 they can even be written down [6] explicity
in terms of spacetime-gauge invariant quantities. In a similar way, the doubly gauge-
invariant quantities presented in [6], are only defined for l ≥ 2 (except k(T )0, which is also
defined for l = 1), and read
l ≥ 2 : f00 ≡ σ00 − 2N∂λ
l ≥ 2 : f ≡ σ(Y ) + ǫN−2χ∂λ
r2|Σ0
k2l σ(LL),
l ≥ 2 : f0 ≡ σ(T )0 − r2|Σ0∂λ
r−2|Σ0σ(LT )
l ≥ 2 : f(T ) ≡ σ(T ),
l ≥ 2 : k00 ≡ κ00 + ǫKσ00 + ǫχ∂λK,
l ≥ 1 : k(T )0 ≡ κ(T )0 − K̄σ(T )0, (20)
l ≥ 2 : k(L)0 ≡ κ(L)0 +
(ǫK − K̄)σ(L)0 +
(ǫK + K̄)
χ− r2|Σ0∂λ(r−2|Σ0σ(LL))
l ≥ 2 : k(LT ) ≡ κ(LT ) − K̄σ(LT ),
l ≥ 2 : k(LL) ≡ κ(LL) − K̄σ(LL),
l ≥ 2 : k(Y ) ≡ κ(Y ) − K̄σ(Y ) + ǫN−2r2|Σ0χ∂λK̄,
l ≥ 2 : k(T ) ≡ κ(T ) − K̄σ(T ),
3The ranges of l’s are not made explicit in [6] in order to include also non-compact homogeneous
spaces, where the index l is continuous. However, to discuss sufficiency of the equations we need to be
precise on the range of validity of each equation.
where k2l = l(l + n− 3) and
l ≥ 2 : χ ≡ σ(L)0 − r2|Σ0∂λ(r−2|Σ0σ(LL)).
The orthogonality properties of the scalar, vector and tensor harmonics imply that
the equalities of the coefficients σ and κ for each l and m is equivalent to the equality of
the perturbation tensors (18) and (19) at both sides of Σ0. Thus, recalling the notation
[f ] ≡ f+|Σ0 − f−|Σ0 , the equations
l ≥ 0 : [σ00] = [σ(Y )] = 0
l ≥ 1 : [σ(L)0] = [σ(T )0] = 0
l ≥ 2 : [σ(T )] = [σ(LT )] = [σ(LL)] = 0
l ≥ 0 : [κ00] = [κ(Y )] = 0
l ≥ 1 : [κ(L)0] = [κ(T )0] = 0
l ≥ 2 : [κ(T )] = [κ(LT )] = [κ(LL)] = 0
are equivalent to (2) and therefore correspond exactly to the linearized matching condi-
tions in this setting. Notice that each of the equalities in (21) and (22) is in fact one
equation for each l and m in the appropriate range. We will however refer to them simply
as equations.
The full linearized matching conditions obviously imply the following equalities in
terms of the doubly-gauge invariant quantities (20),
l ≥ 2 : [f00] = [f ] = [f0] = [f(T )] = 0 (23)
l ≥ 1 : [k(T )0] = 0
l ≥ 2 : [k00] = [k(Y )] = [k(L)0] = [k(LL)] = [k(LT )] = [k(T )] = 0.
Whether these equations can be regarded as the full set of linearized matching conditions
or not requires studying their sufficiency, i.e. whether they imply (21)-(22) or not. This
point was not mentioned in [6] and in fact the answer turns out to be negative, although
in a mild way, as we discuss in the next subsection.
5.2 On the sufficiency of the continuity of the doubly-gauge in-
variants
Let us recall that fulfilling the matching conditions requires finding two ~Z± such that
(21)-(22) are satisfied. The key issue for the matching is therefore to show existence of
deformation vectors ~Z± so that all the equations hold.
A plausibility argument in favour of the sufficiency of (23)-(24) comes from simple
equation counting. Indeed, as already discussed, the linearized matching conditions are
spacetime and hypersurface gauge invariant and therefore can only involve the difference
vector [~T ], i.e. three quantities. Since constructing double gauge invariant quantities on
each side eliminates this vector, the number of equations should be reduced exactly by
three if they are to remain equivalent to the original set. This is precisely what happens
as we go from the original forteen equations in (21)-(22) down to eleven equations in
(23)-(24). This argument however is not conclusive, both because it is not rigorous and
because each equation in those expressions is, in fact, a set of equations depending on l
and m, and the range of l’s changes with the equations. Let us therefore analyse this issue
in detail. In particular we need to discuss what are the consequences of the non-existence
of doubly gauge-invariant variables for l = 0 and l = 1 (except for k(T )0 which exists for
l = 1), something not mentioned in [6].
Let us start by finding explicit expressions for σ’s valid in the whole range of l’s. As
in [6], we decompose g(1) in harmonics as
g(1)αβdx
αdxβ =
(habY dx
adxb + h(Y )T(Y )ABdθ
AdθB)
2(h(T )aV(T )A + h(L)aV(L)A)dx
(h(T )T(T )AB + h(LT )T(LT )AB + h(LL)T(LL)AB)dθ
AdθB, (25)
and ~Z as
zaY dx
(z(T )V(T )A + z(L)V(L)A)dθ
QY n(0) − ǫN−2zλY eλ
(z(T )V(T )A + z(L)V(L)A)dθ
A, (26)
which implies Tαdx
l=0(−ǫN−2zλY eλ) +
l=1(z(T )V(T )A + z(L)V(L)A)dθ
A. Inserting
these expressions into (2) and expanding in spherical harmonics it is straightforward to
l ≥ 0 : [σ00] = 0 ⇔ [hλλ] + 2[Q]N2K + 2N∂λ
N−1[zλ]
l ≥ 1 : [σ(L)0] = 0 ⇔ [zλ] + [h(L)λ] + r2|Σ0∂λ(r−2|Σ0[z(L)]) = 0, (27)
l ≥ 2 : [σ(LL)] = 0 ⇔ [z(L)] + [h(LL)] = 0, (28)
l ≥ 0 : [σ(Y )] = 0 ⇔ [h(Y )] + 2[Q]r2|Σ0K̄ − ǫN−2[zλ]∂λ(r2|Σ0)−
k2l [z(L)] = 0,
l ≥ 1 : [σ(T )0] = 0 ⇔ [h(T )λ] + r2|Σ0∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [σ(LT )] = 0 ⇔ [z(T )] + [h(LT )] = 0, (29)
l ≥ 2 : [σ(T )] = 0 ⇔ [h(T )] = 0,
where [hλλ], [h(L)λ], etc. denote eλ
b[hab], eλ
a[h(L)a], etc. Later on we will also write
down the explicit expressions for (22) but they are not needed in this subsection.
It is obvious by the form of f ’s and κ’s (20) that the set of equations (21)-(22) are
equivalent to (23)-(24) together with
l ≥ 2 : [σ(L)0] = [σ(LT )] = [σ(LL)] = 0 (30)
l = 0, 1 : [σ00] = [σ(Y )] = 0
l = 1 : [σ(L)0] = [σ(T )0] = 0
l = 0, 1 : [κ00] = [κ(Y )] = 0
l = 1 : [κ(L)0] = 0.
Sufficiency of Mukohyama’s doubly gauge invariant matching conditions would follow if
these equations serve exclusively to determine the discontinuity [~T ], i.e. [zλ] for l ≥ 0
and [z(T )], [z(L)] for l ≥ 1. Now, the explicit expressions (27), (29), (28) show that (30)
determine uniquely [zλ], [z(T )] and [z(L)] for l ≥ 2. So, restricted to the sector l ≥ 2
Mukoyama’s doubly gauge invariant matching conditions can be regarded as equivalent
to the full set of matching conditions. Taking all l’s into account, however, the equations
turn out not to be sufficient. To show this, it is enough to display one equation involving
the discontinuity of the background metric perturbations and [Q] (but not [~T ]) which holds
as a consequence of the full set of matching conditions (21)-(22) but not as a consequence
of (23)-(24). Using the fact that each l = 1 expression refers to n − 1 objects (one for
each m), the number of equations in (31)-(32) is 7n− 3, while the number of unkowns in
[~T ] not yet determined by (30), i.e. [zλ] for l = 0, 1 and [z(T )], [z(L)] for l = 1 is 3n − 2,
which is smaller. It is to be expected, therefore, that (31), (32) imply conditions where
these variables do not appear. This can be made explicit, for instance, by combining
[σ00]l=0 = 0 with [σ(Y )]l=0 = 0 which yields
l = 0 : [hλλ] + 2[Q]N
2K + 2N∂λ
∂λ(r2|Σ0)
[h(Y )] + 2[Q]r
2|Σ0K̄
whenever ∂λ(r
2|Σ0) 6= 0. (If ∂λ(r2|Σ0) = 0 it is enough to consider [σ(Y )]l=0 = 0.) This
relation is enough to show that the continuity of the doubly-gauge invariant variables of
Mukohyama is not sufficient to ensure the existence of the perturbed matching. Of course,
this does not invalidate Mukohyama’s approach in any way, which remains interesting
and useful. It only means that, when using this approach to solve linearized matchings,
one still needs to look more carefully into the l = 0 and l = 1 sector to make sure that
the remaining equations (31) and (32) hold.
On the other hand, equations (31), (32) do not completely determine [~T ]. The variable
[z(T )]l=1 only appears in [σ(T )0]l=1 = 0, in the term ∂λ(r
−2|Σ0 [z(T )]). As a result, the
matching conditions do not fix [z(T )]l=1 completely, but up to a constant factor times
r2|Σ0 (for each m). Recalling that V(T )AdϑA for l = 1 correspond to the three Killing
vectors on the sphere, this arbitrary constant (for each m) accounts for the addition to
[~T ] of an arbitrary Killing vector of the sphere. This is in accordance with the discussion
in Section 4. We devote the following subsection to complete the study of the freedom
left in the matching.
5.3 Freedom in the matching
As already emphasized, solving the linearized matching amounts to finding perturbation
vectors ~Z+ and ~Z−. Assume now that a linearized matching between two given back-
grounds and perturbations has been done. It is natural to ask what is the most general
matching between those two spaces, i.e. what is the most general solution for ~Z+ and ~Z−
of the matching conditions. Geometrically, this means finding all the possible deforma-
tions of the matching hypersurface Σ0 which allow the two spaces to be matched.
Since this problem is of interest not only when the full matching conditions are imposed
but also in situations where layers of matter are present (e.g. in brane-world or shell
cosmologies) so that jumps in the second fundamental forms are allowed, we will analyse
this issue in two steps. First, we will study the equations involving the perturbed first
fundamental forms and will determine the freedom they admit. On a second step we
will write down the extra conditions coming from the equality of the second fundamental
forms.
Thus, let us consider two perturbation configurations of the same background matching
and denote their respective sets of difference variables on Σ0 as [f ] and [f ]
′ for any given
variable f . Now, we will define the difference between the two configurations as < f >≡
[f ]′ − [f ] for any variable f . The assumption that the perturbation on each side is fixed
once and for all implies < g(1) >= 0. We are assuming that the linearized matching
conditions are satisfied in each case, and so we can subtract them. Linearity implies that
the differences of the linearised matching equations become equations for the difference
vector < ~Z >. The general solution of these equations clearly determines the freedom in
the deformation of the hypesurface.
The difference of the equations in (21) for the two configurations using < g(1) >= 0
give the following set of equations
l ≥ 0 : < σ00 >= 0 → < Q > N2K +N∂λ
N−1 < zλ >
= 0, (33)
l ≥ 1 : < σ(L)0 >= 0 → < zλ > +r2|Σ0∂λ(r−2|Σ0 < z(L) >) = 0, (34)
l ≥ 2 : < σ(LL) >= 0 → < z(L) >= 0, (35)
l ≥ 0 : < σ(Y ) >= 0 → 2 < Q > r2|Σ0K̄ − ǫN−2 < zλ > ∂λ(r2|Σ0)
k2l < z(L) >= 0, (36)
l ≥ 1 : < σ(T )0 >= 0 → ∂λ(r−2|Σ0 < z(T ) >) = 0, (37)
l ≥ 2 : < σ(LT ) >= 0 → < z(T ) >= 0, (38)
l ≥ 2 : < σ(T ) >= 0 → 0 = 0.
Expressions (35) and (38) readily determine < z(L) >l≥2=< z(T ) >l≥2= 0, which sub-
stituted in (34) give < zλ >l≥2= 0. As a result, (36) for l ≥ 2 lead to < Q >l≥2= 0.
Clearly all the equations for l ≥ 2 are now satisfied. We now concentrate on the l = 1
equations. Equation (37) implies that < z(T ) >l=1= ar
2|Σ0, where a is a constant for each
m. Combining equations (33), (34) and (36) for l = 1 we obtain the following equation
for r−2|Σ0 < z(L) >l=1,
K̄∂2λ(r−2|Σ0 < z(L) >l=1) +
(2K̄ + ǫK)∂λ(ln r|Σ0)− K̄∂λ lnN
−2|Σ0 < z(L) >l=1)
r−2|Σ0 < z(L) >l=1= 0, (39)
while (34) and (33) determine < zλ >l=1 and < Q >l=1 respectively (provided K 6= 0,
which occurs generically). The two equations for l = 0 can be rearranged onto
K̄∂λ(N−1 < zλ >l=0) +N−1 < zλ >l=0 ǫK∂λ ln(r|Σ0) = 0 (40)
plus the equation (33) for l = 0, which determines < Q >l=0.
Summarizing, we have found that the freedom in the deformation of the hypersurface
compatible with the linearized matching conditions involving the first fundamental form
[~Z]′ − [~Z] =
< Q > Y ~n(0) − ǫN−2 < zλ > Y ~eλ
+am~V
(T ) + r
−2|Σ0 < z(L) >l=1,m ~V m(L),
where r−2|Σ0 < z(L) >l=1,m, satisfy (39), < zλ >l=0 satisfy (40) and the rest of the
variables are completely determined as described above. The term in am corresponds
to adding Killing vectors on the sphere, something already discussed in Section 4. The
rest of terms involve combinations (with functions) of the conformal Killing vectors on
the sphere and tangential vectors along λ. Notice that the coefficients of the conformal
Killing (i.e. < z(L) >l=1,m ) determine all the rest of the l = 1 coefficients. In particular
when < z(L) >l=1,m vanishes, then all the l = 1 terms vanish and the freedom becomes
radially symmetric.
We now add to the analysis the difference of the equations in (22). Due to the fact
that all coefficients in < ~Z > vanish for l ≥ 2 we only need to consider the equations for
l = 0, 1, i.e. (32). We refer the reader to Appendix A for the explicit expressions of (32)
in terms of the metric perturbations and ~Z. For the sake of completeness we also include
all the explicit expressions of (22) in Appendix A. The difference of equations (32), see
(44)-(46), whenever < g(1) >= 0 read
l = 0, 1 : < κ00 >= 0 ⇔ (41)
− < QR(γ)dbac > n
(0)dn(0)aeλ
c − ǫ∂2λ < Q > +
2∂λ < Q > −ǫ < Q > K2N2
−ǫKN2∂λ(N−2 < zλ >)− ǫ∂λ(K < zλ >) = 0,
l = 1 : < κ(L)0 >= 0 ⇔ (42)
−ǫ∂λ < Q > +ǫK < zλ > +ǫ < Q > ∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0 < z(L) >) = 0
l = 0, 1 : < κ(Y ) >= 0 ⇔ (43)
N−2∂λ(r
2|Σ0) (∂λ < Q > +K < zλ >) +
< Qn(0)an(0)b∇a∇br2 >
N−2eλ
a < zλn
(0)b∇b∇ar2 > +
l(l + n− 3)
ǫ < Q > −2K̄ < z(L) >
It can be checked that in general these equations overdetermine the previous equations,
i.e. (39) and (40), although there may be particular cases for which they are compatible.
Therefore, generically, they will imply that < z(L) >l=1,m= 0 and < zλ >l=0= 0, and
thence all the rest of the variables vanish, < zλ >l=1,m=< Q >l=1,m= 0, < zλ >l=0=<
Q >l=0= 0, so that the only freedom left is given by
[~Z]′ − [~Z] = am~V m(T ).
Finding in which particular cases equations (39)-(43) are compatible is straightforward
but tedious and will not be carried out explicitly here.
Acknowledgements
FM and MM thank CRUP(Portugal)/MCT(Spain) for grant E-113/04. FM thanks FCT
(Portugal) for grant SFRH/BPD/12137/2003 and CMAT, University of Minho, for sup-
port. MM was supported by the projects FIS2006-05319 of the Spanish Ministerio de
Educación y Tecnoloǵıa and SA010CO of the Junta de Castilla y León. RV was supported
by the Irish IRCSET, Ref. PD/2002/108, and now is funded by the Basque Government
Ref. BFI05.335.
A Appendix
For the sake of completeness we devote this appendix to present the explicit expressions
of (22) in terms of the metric perturbations and ~Z, which read
l ≥ 0 : [κ00] = 0 ⇔ (44)
N2K[hnn]−
n(0)aeλ
c(2∇c[hab]−∇a[hbc])− [QR(γ)dbac]n
(0)dn(0)aeλ
−ǫ∂2λ[Q] +
2∂λ[Q]− ǫ[Q]K2N2 − ǫKN2∂λ(N−2[zλ])− ǫ∂λ(K[zλ]) = 0,
l ≥ 1 : [κ(L)0] = 0 ⇔ (45)
[hnλ]−
n(0)aeλ
b(∂b[h(L)a]− ∂a[h(L)b])− ǫ∂λ[Q]− ǫK[zλ]
+(ǫ[Q] + [h(L)n])∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(L)]) = 0
l ≥ 0 : [κ(Y )] = 0 ⇔ (46)
r2|Σ0K̄[hnn] +
N−2∂λ(r
2|Σ0) (ǫ[hnλ] + ∂λ[Q] +K[zλ]) +
[n(0)a∂ah(Y )]
[Qn(0)an(0)b∇a∇br2]−
N−2eλ
a[zλn
(0)b∇b∇ar2]
l(l + n− 3)
[h(L)n] + ǫ[Q]− 2K̄[z(L)]
l ≥ 1 : [κ(T )0] = 0 ⇔
n(0)aeλ
b(∂b[h(T )a]− ∂a[h(T )b]) + [h(T )n]∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(T )n] +
n(0)a∂a[h(LT )] + K̄[z(T )] = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(L)n] +
n(0)a∂a[h(LL)] + K̄[z(L)]−
[Q] = 0,
l ≥ 2 : [κ(T ] = 0 ⇔
n(0)a∂a[h(T )] = 0.
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1509-1519
	Introduction
	Linearized matching
	Gauge freedom
	Matching conditions
	On previous spacetime gauge invariant formalisms
	Freedom in matching due to symmetries
	Isometries
	n-dimensional spherically symmetric backgrounds
	The approach of Mukohyama
	On the sufficiency of the continuity of the doubly-gauge invariants
	Freedom in the matching
	Appendix
