# Observables and gauge invariance in the theory of nonlinear spacetime perturbations

**ABSTRACT** We discuss the issue of observables in general-relativistic perturbation theory, adopting the view that any observable in general relativity is represented by a scalar field on spacetime. In the context of perturbation theory, an observable is therefore a scalar field on the perturbed spacetime, and as such is gauge invariant in an exact sense (to all orders), as one would expect. However, perturbations are usually represented by fields on the background spacetime, and expanded at different orders into contributions that may or may not be gauge independent. We show that perturbations of scalar quantities are observable if they are first-order gauge invariant, even if they are gauge dependent at higher order. Gauge invariance to first order therefore plays an important conceptual role in the theory, for it selects the perturbations with direct physical meaning from those having only a mathematical status. The so-called `gauge problem', and the relationship between measured fluctuations and gauge-dependent perturbations that are computed in the theory are also clarified.

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Page 1

Class. Quantum Grav. 16 (1999) L29–L36. Printed in the UKPII: S0264-9381(99)03494-2

LETTER TO THE EDITOR

Observables and gauge invariance in the theory of nonlinear

spacetime perturbations

Marco Bruni† and Sebastiano Sonego‡

† Department of Physics and Astronomy, Cardiff University, Cardiff CF2 3YB, UK

‡ Universit` a di Udine, DIC, Via delle Scienze 208, 33100 Udine, Italy

E-mail: bruni@astro.cf.ac.uk and sebastiano.sonego@dic.uniud.it

Received 13 April 1999

Abstract.

theviewthatanyobservableingeneralrelativityisrepresentedbyascalarfieldonspacetime. Inthe

context of perturbation theory, an observable is therefore a scalar field on the perturbed spacetime,

and as such is gauge invariant in an exact sense (to all orders), as one would expect. However,

perturbations are usually represented by fields on the background spacetime, and expanded at

different orders into contributions that may or may not be gauge independent. We show that

perturbations of scalar quantities are observable if they are first-order gauge invariant, even if they

are gauge dependent at higher order. Gauge invariance to first order therefore plays an important

conceptualroleinthetheory,foritselectstheperturbationswithdirectphysicalmeaningfromthose

having only a mathematical status. The so-called ‘gauge problem’, and the relationship between

measured fluctuations and gauge-dependent perturbations that are computed in the theory are also

clarified.

Wediscusstheissueofobservablesingeneral-relativisticperturbationtheory,adopting

PACS numbers: 0425N, 0490, 9880H

Theissueofwhatisobservableingeneralrelativityisstillacontroversialone[1]. Herewewant

toaddressarelatedproblem,whichisfundamentalforpracticalpurposes: whichperturbations

are observable in general relativity? Clearly, the answer to this question is largely conditioned

on one’s attitude towards the more basic issue mentioned above. However, we believe that this

questiondeservesananalysisofitsown,giventhepeculiarnatureofgaugeissuesinthecontext

of relativistic perturbation theory, and the fact that the comparison of Einstein’s theory with

observations is almost entirely based on approximation methods (see, e.g., [2] for a discussion

of this point).

In the following, we adopt the view that an observable quantity in general relativity§ is

simply represented by a scalar field on spacetime. This definition may seem naive, but it

corresponds exactly to what most physicists have in mind when thinking about observable

quantities. At a more sophisticated level, it can be argued that such a notion of observables is

practically viable even if one considers the issues related to the invariance of general relativity

under spacetime diffeomorphisms.

In the relativistic theory of perturbations one is always dealing with two spacetimes, the

physical (perturbed) one, and an idealized (unperturbed) background. In this context, given

the above definition, an observable is a scalar on the perturbed spacetime. Thus, physical

§ Our discussion is, however, valid in any spacetime theory.

0264-9381/99/070029+08$19.50© 1999 IOP Publishing Ltd

L29

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Letter to the Editor

quantities such as, for example, the energy density in a cosmological model, trivially satisfy

our definition of observables. However, in perturbation theory one formally decomposes

quantities of the physical space into the sum of a background quantity and a perturbation.

The question therefore arises as to whether perturbations themselves can be regarded as

observables. Indeed, it is certainly useful to express the perturbative formalism directly

in terms of variables (the perturbations) that are observable quantities, or at least to have

clear in mind how to relate mathematical variables with physically meaningful ones. Given

the recent development of second-order perturbation theory in cosmology (see [3,4] and

referencestherein)andblackholephysics(see[5,6]andreferencestherein),theissuebecomes

even more important, because of the further complications that arise when nonlinearities are

considered.

As we shall discuss shortly, the perturbations are represented by fields on the background,

and expanded at different orders into contributions that may or may not be gauge independent.

On the other hand, one would expect that observables are described by gauge-invariant

quantities. Recently, explicit calculations have shown that fields which are usually regarded as

representing observables are not gauge invariant at second order [3,4,6], although it has long

been known that they are gauge invariant at first order (see, e.g., [7,8,14]). In this letter we

pointoutthattheperturbationofascalardescribesanobservableifandonlyifitsrepresentation

on the background is gauge invariant at first order, even when it is gauge dependent at higher

orders. We shall also discuss how this seemingly paradoxical result corresponds to having

observable perturbations that are gauge invariant in the exact sense (i.e. at all orders) in the

physicalspacetime, asexpected. Finally, weshallbrieflycommentonhowafirst-ordergauge-

dependent perturbation can acquire physical meaning in a specific gauge.

Let us begin by reviewing some general ideas about the perturbative approach in general

relativity†. Suppose that the physical and the background spacetimes are represented by

the Lorentzian manifolds (M,g) and (M0,g0), respectively (as manifolds, M0 = M,

but it is nevertheless convenient to label them differently). The perturbation of a quantity

should obviously be defined as the difference between the values that the quantity takes in

M and M0, evaluated at points which correspond to the same physical event. However,

there is nothing intrinsic to (M,g) and (M0,g0) that allows us to establish a one-to-one

correspondence between the two manifolds.

spacetime structure is assigned a priori in general relativity, in contrast to what happens,

for example, in the Newtonian theory, where M0and M are simply identified, thus making

possible a straightforward formulation of Eulerian perturbation theory [12] (see, however,

[13] and references therein for a Lagrangian formulation in Newtonian cosmology). Hence, in

general relativity, points of M are unrelated to points of M0, and if we want to compare

a physical quantity in the two spacetimes, we need an additional prescription about the

pairwise identification of points between M and M0. Mathematically, this corresponds to

the assignment of a diffeomorphism ϕ : M0→ M, sometimes called a point identification

map [14]. Such a diffeomorphism can be given directly as a mapping between M0and M

or—perhaps more commonly—by choosing charts X in M0and Y in M (with coordinates

{xµ} and {yµ}, respectively), and identifying points having the same value of the coordinates

[2,15], so that ϕ is defined implicitly through the relation xµ(p) = yµ(ϕ(p)), ∀p ∈ M0.

In any case, the point identification map is completely arbitrary; this freedom is peculiar to

general relativistic perturbation theory, and has no counterpart in theories that are formulated

on a fixed background, where no ambiguity arises in comparing fields. Following Sachs

This follows directly from the fact that no

† Hereafter, we shall adopt the conventions and notations of [9,10]. See also [11] for the more general mathematical

definitions.

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Letter to the Editor

L31

[16] (p 556), one may refer to it as gauge freedom ‘of the second kind’, in order to

distinguish it from the usual gauge freedom of general relativity. However, in the following

we shall never consider ordinary gauge transformations of the perturbed and background

spacetimes, and we shall therefore use the term ‘gauge’ as a synonym of ‘gauge of the second

kind’.

Once a gauge choice has been made (i.e. a point identification map ϕ has been assigned),

perturbations can be defined unambiguously. Let T be a tensor field on M. If T0is the

background tensor field corresponding to T, the total perturbation of T is simply given by

?ϕT := T − ϕ∗T0[9,10]. By definition, ?ϕT is a field on M. On the other hand, the aim

of perturbation theory in general relativity is to construct, through an iterative scheme, the

geometry on M (see, e.g., [17]). To this end, it is customary to work with fields on M0(see

[9,10] and references therein). Thus, to start with, the representation on M0of an arbitrary

tensorfieldT isdefinedasthepull-backϕ∗T. Then,therepresentationonthebackgroundofthe

perturbation is ?ϕ

expanded to obtain the contributions at different orders that are used in the above-mentioned

iteration scheme.

0T := ϕ∗?ϕT = ϕ∗T −T0. These are the background fields that are Taylor

Figure 1.

a curve in M0 and its ϕ-transformed in M have the same representation in Rm. Therefore,

the components of the tangent vectors V and ϕ∗V at the points ϕ(p) and p are the same:

(ϕ∗V)µ(x) = (ϕ∗V)(xµ)|p= V?xµ◦ ϕ−1?|ϕ(p)= V(yµ)|ϕ(p)= Vµ(x).

We would like to point out here that, for practical purposes, it is often very convenient to

use, on M0and M, coordinates {xµ} and {yµ} ‘adapted’ to ϕ, such that xµ(p) = yµ(ϕ(p)),

∀p ∈ M0. In this way the components of T at the point ϕ(p) ∈ M coincide with those of

ϕ∗T at p ∈ M0(see figure 1 for a pictorial explanation in the case of a vector). Obviously,

the same is true for ϕ∗T0and T0, so that one has, for a tensor of type (r,s),

??ϕT?µ1...µr

where x = X(p) = Y(ϕ(p)). This choice is, however, rather confusing from a conceptual

point of view because, once the above identification has been made, it is hard to distinguish

between ?ϕT and ?ϕ

By choosing the coordinates on M0 and M in such a way that yµ◦ ϕ = xµ,

ν1...νs(x) =??ϕ

0T?µ1...µr

ν1...νs(x) = Tµ1...µrν1...νs(x) − T0µ1...µrν1...νs(x),

(1)

0T. The point is that, as we shall show, ?ϕT may be gauge invariant in

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Letter to the Editor

an exact sense (at all orders) and thus, if T is a scalar, it may correspond to an observable, even

when ?ϕ

Under a gauge transformation ϕ → ψ, where ψ is another point identification map,

the representation on M0of tensor fields defined on M changes as under the action of a

diffeomorphism. This can easily be seen by noticing that a point p ∈ M corresponds, in two

gauges ϕ and ψ, to the points ϕ−1(p) and ψ−1(p) in M0. Defining a map ? : M0→ M0

as ? := ϕ−1◦ ψ, we have ϕ−1(p) = ?(ψ−1(p)). Then the two representations on M0of

a tensor T of M are related by ψ∗T = (ϕ ◦ ?)∗T = ?∗?ϕ∗T?. It follows that the gauge

?ψ

0T is gauge invariant only at first order.

transformation between the two representations of the perturbation of T in the two gauges is

0T + ?∗T0− T0= ?∗?ϕ∗T?− T0.

We want to point out here that from the first equality in this equation it would seem that a

fundamental fact about gauge transformations is that ?∗acts on T0. However, it is clear from

the second equality that this is not the case, as all that is really needed is the action of ?∗on

the representation on M0of the field T of the physical spacetime. In other words, it is only

the fact that we insist on looking at the gauge transformation of ?ϕ

about the ?∗T0term in the first equality. In this sense, it would be an improper extrapolation

to say that ‘a gauge transformation is equivalent to a diffeomorphism of the background’.

It is useful to classify fields on M, according to whether they are intrinsically gauge

independent (IGI) or intrinsically gauge dependent (IGD). We say that a field is IGI iff its

value at any point of M does not depend on the gauge choice; otherwise, we say that it is

IGD. As obvious examples of IGI and IGD quantities we mention, respectively, a tensor field

T defined on M, and the push-forward ϕ∗T0on M of a non-trivial† tensor field T0defined on

M0. It follows that, given our identification of observables with scalar functions on M and

the arbitrariness in the choice of the gauge ϕ, a scalar describes an observable only if it is IGI.

On the other hand, considering the representation on M0of the perturbations, one is led to

define a corresponding idea in the background, saying that a quantity on M0is identification

gauge-invariant (i.g.i.) iff its value at any point of M0does not depend on the gauge choice

[14]. An example of an i.g.i. quantity is a tensor field T0defined on M0, while the pull-back

ϕ∗T of a tensor field T defined on M is not i.g.i. unless T is trivial. It is then clear that the

representation on the background of an IGI quantity is not i.g.i. in general, but this is totally

irrelevant as far as the issue of observability is concerned, because measurements are always

performed in the physical spacetime M, whereas the background M0has merely the status of

a useful mathematical artifice.

Let us now turn our attention to perturbations, asking whether they are IGI or IGD, and

how this relates to the gauge dependence of their representation on M0. In order to answer

these questions, we consider a gauge transformation ϕ → ψ. Correspondingly, perturbations

transform as ?ϕT → ?ψT, where

?ψT = ?ϕT + (ϕ∗T0− ψ∗T0).

Similarly, their representations on the background transform as ?ϕ

0T +?ψ∗T − ϕ∗T?.

Therefore, in general, both the perturbations on M and their representations on M0change

under the action of a gauge transformation. This gauge dependence of perturbations does not

appear in theories that admit a canonical identification between M and M0, and is due to

the arbitrariness in the choice of a point identification map, which is additional to the usual

0T = ?∗?ϕ

(2)

0T as a whole that brings

(3)

0T → ?ψ

0T, with

?ψ

0T = ?ϕ

(4)

† Hereafter by a trivial tensor we mean one that is either vanishing or a constant multiple of the identity [14].

Page 5

Letter to the Editor

L33

gauge freedom of general relativity. In a sense, general relativistic perturbations ‘have a gauge

freedom of their own’, i.e. the freedom ‘of the second kind’ [16] mentioned before, even when

the full quantities have not. We have to note at this point that, once the Taylor expansion into

different order contributions has been carried out, it turns out that a perturbation on M0may

be gauge invariant at first order, and not at higher orders [9,10]. Now, it follows from (3) that

the perturbation is IGI iff ϕ∗T0= ψ∗T0, ∀ϕ,ψ. This can be rewritten as T0= ?∗T0, where

? := ϕ−1◦ ψ : M0→ M0, and is satisfied only if T0is trivial. However, this is precisely

the condition for first-order i.g.i. derived by Stewart and Walker [14]. Thus, the perturbation

of T is IGI iff its representation on the background is gauge invariant to first order. For the

particular case in which T is a scalar physical quantity, we obtain the main result of this letter:

the perturbation of T is observable iff its representation on M0is first-order i.g.i., even when

it is gauge dependent to higher orders.

This result may sound trivial, but only because we have approached the question of

observability of perturbations from the side of the physical spacetime M. Focusing attention

only on perturbations as fields on M0rather than on M, as is usually done in the iterative

scheme,thenotionofIGIquantitiesdoesnotnaturallyarise,andonewouldaskinsteadwhether

the representation of a perturbation is i.g.i., i.e. whether ?ϕ

maps [14]. Because of (4), this condition is equivalent to the requirement that ϕ∗T = ψ∗T,

∀ϕ,ψ, which is satisfied iff T is trivial. Thus, the perturbation of a quantity is IGI iff the

quantity itself is trivial in the background M0, while its representation is i.g.i. iff the quantity

is trivial on M. The important point is that the physically interesting condition is not i.g.i.,

but IGI, as one can clearly see by considering the case in which T is a scalar. The requirement

that the perturbation of T be i.g.i. amounts to saying that T must be a constant on M, which is

too strong a constraint to be fulfilled by most observables of physical interest. In contrast, IGI

requires only that T0be constant. As we have already pointed out, the physical content of the

theory resides in the quantities on M, not in their representations on M0. From the physical

point of view, there is nothing bad in having a gauge-dependent pull-back on M0, provided

that the quantity on M be uniquely defined. Of course, to the first order in a perturbative

approach, the perturbation of a quantity is IGI iff its representation is i.g.i.

The gauge dependence of perturbations has often been referred to in the past as the gauge

problem [8,15]†. It is now clear that, for IGI perturbations, such a gauge dependence is only

an artefact of focusing attention on the representation, and there is no real problem in the

physical spacetime M. The situation is different for IGD perturbations. Their very definition

requiresagaugetobedefined: thisisadifficulty,becausethenthesequantitiescannotrepresent

observables directly. However, especially in cosmology, it is often the case that astronomers

do measure quantities that they call perturbations, or fluctuations, which they define with

respect to an averaged quantity. It is obvious that such fluctuations must be IGI, as their very

definitiondoesnotinvolveanybackgroundorgaugechoice. Aparadigmaticexampleisthatof

thecosmicmicrowavebackground(CMB)temperatureanisotropy(see[19,20]andreferences

therein). We are then facing a paradox: on the one hand, theoretical perturbations that are

commonly considered are IGD (see, e.g., [8]), whereas their observational counterparts are

IGI. It is then necessary to understand the relationship between them.

Theresolutionoftheproblemliesinrecognizingthemisleadingidentificationthatisoften

made between averages in the physical spacetime and background quantities. Supposing that

0T = ?ψ

0T, ∀ϕ,ψ identification

† A secondary issue that follows from this gauge dependence is that of gauge modes. The latter arise when the

prescription of point identification has not been given, or when it is not sharp enough to select a single ϕ : M0→ M,

and defines instead a class of diffeomorphisms [18]. Of course, for gauge-invariant perturbations both problems

disappear.

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Letter to the Editor

an averaging procedure has been adopted†, the measured fluctuation of a quantity T is defined

by the observer simply as ?T := T −?T?. Clearly, such a definition is valid in any spacetime

and, by itself, has no relation whatsoever with the adoption of a background model and/or of

a perturbative formalism (of which the observer can be happily totally unaware). However, if

we want to consider this spacetime from a perturbative point of view, then such a relation is

easily established by rewriting ?T as

?T = T − ?T? = (T − ϕ∗T0) − (?T? − ϕ∗T0) = ?ϕT − ?ϕ?T?,

where the same background quantity T0 has been used for both T and ?T?.

theoretically computed perturbation, ?ϕT, differs from the measured one, ?T, by the term

?ϕ?T?. The apparent paradox outlined above arises therefore when ?ϕT is identified with

?T, which happens if one thinks of the average ?T? as the push-forward ϕ∗T0of some T0

defined in the background spacetime. This is generally wrong, as one can see by considering

the example of the CMB anisotropy, where ?T? is obtained by an angular average [19,20]

and depends therefore on the spatial position as well as on time, while T0can depend only

on time, being the temperature in a spatially homogeneous cosmological model. The only

case in which the identification is meaningful is when ?T? and T0are constant on spacetime.

Indeed, ?T? is IGI by definition, while in general ϕ∗T0is IGD. If we want ?T? = ϕ∗T0, it is

clear that ϕ∗T0must also be IGI, which can be the case only if T0is a constant. Instead, of

particular interest in cosmology is the case when ?T?0= T0, i.e. when we consider a sky or

spatial average, so that ?T has a vanishing background value, and therefore is IGI (and, of

course, first-order i.g.i.).

Scalars that deserve specific attention are those built by projecting a tensor over a tetrad:

a well known example is that of the Weyl scalars in the Newman–Penrose (NP) formalism,

which are often used in the study of black hole perturbations [6,7,14]. Let us consider, in

order to fix the ideas, the NP scalar ?4 := C(l,m,l,m), where C is the Weyl tensor and

{l,n,m,m} is the usual NP null tetrad. The perturbation of ?4is

?ϕ?4= ?4− ϕ∗?4,0,

where ?4,0 = C0(l0,m0,l0,m0) is constructed only from quantities defined on M0. It

may seem that ?ϕ?4contains more arbitrariness than the perturbations we have considered

so far, because its definition requires not only choosing a gauge ϕ, but also specifying a

tetrad {l0,n0,m0,m0} in the background and another one, {l,n,m,m}, in the physical space.

Actually,thetwotetradsarenotindependent,becauseifwerequirethat?4→ ?4,0inthelimit

of vanishing perturbations, we must have, for example, l = ϕ∗l0+ ?ϕl, so that the perturbed

tetrad can be constructed iteratively from the unperturbed one, by imposing that it be null

with respect to the perturbed metric, order by order. Nevertheless, we are still left with a

dependence of ?ϕ?4on the background tetrad. However, it is easy to understand that this

tetrad dependence is not a problem as far as observability is concerned. Indeed, changing the

tetrad from {l0,n0,m0,m0} to {l?

a different physical/geometrical interpretation. Tetrad dependence thus corresponds to the

possibility of constructing different measurable quantities starting from the Weyl tensor, and

must not be regarded, conceptually, on the same footing of the gauge dependence that we have

considered in the rest of this letter. Also, in practice one is guided by the Petrov algebraic type

of the background in choosing on it a specific tetrad, such that some of the Weyl scalars vanish

and thus are first-order i.g.i. For example, Schwarzschild and Kerr spacetimes are Petrov type

(5)

Thus, the

(6)

0,n?

0,m?

0,m?

0}, say, ?4will change to another scalar ??

4with

† Such an average can be performed on data taken on suitable subspaces of the whole spacetime. A particularly

interesting case is that of averages on the sky made by each observer at his spacetime point. Obviously, in general a

sky-averaged quantity is point dependent, an important fact for the following discussion.

Page 7

Letter to the Editor

L35

D, and the tetrad can be aligned with the principal null directions, so that only ?2?= 0, which

results in ?0and ?4describing gravitational wave perturbations in a gauge-invariant way

[6,7,14]. At second order, the construction of a gauge and tetrad invariant NP perturbative

formalism is computationally very useful, e.g. in order to compare results with those of a fully

numerical treatment. For Kerr, this can be achieved [6] by implementing a procedure that

strongly reminds one of that used at first order in cosmology [15]. One can construct infinitely

manysuchgauge-invariantquantities, butagainphysicalmeaningguidesthechoice: in[6]the

chosen second-order i.g.i. quantity is the one reducing to ?4in an asymptotically flat gauge.

Finally, one may wonder what gauge-dependent variables, which are commonly used in

perturbation theory, have to do with observables. The answer to this question is obvious:

provided that calculations are free from gauge modes, even a first-order gauge-dependent

perturbation acquires physical meaning in a specific gauge when, in that gauge, it can be

identified with a gauge-invariant quantity. This is the case, for example, for the density

perturbation δρ/ρ in cosmology: its value in the comoving gauge coincides with the value

in that gauge of one of Bardeen’s gauge-invariant variables [15], and in turn the latter is the

first-order expansion of a covariantly defined density gradient [21]. It is our opinion that there

is more appeal in working directly with observable variables which are automatically IGI,

such as the fluctuation ?T considered above. However, it may often be the case that one

is not able to find a complete set of such quantities, so that one will either resort to gauge-

dependent variables, or to appropriate combinations of gauge-dependent quantities that are

gauge invariant at the desired order [6,15].

We thank Manuela Campanelli and Carlos Lousto for valuable discussions. We are grateful

for hospitality to the Department of Astronomy and Astrophysics of Chalmers University,

G¨ oteborg, and to the Astrophysics Sector of SISSA, Trieste, where part of this work was

carried out.

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