hep-th/0002091 MIT-CTP-2944 PUPT-1913

Linearized Gravity in Brane Backgrounds

Steven B. Giddings,^{†}^{†} Email address:
Emanuel Katz,^{†}^{†} Email
address:
and Lisa Randall^{†}^{†} Email address:

Department of Physics, University of California

Santa Barbara, CA 93106-9530

Center for Theoretical Physics, MIT

Cambridge, MA 02139

Joseph Henry Laboratories, Princeton University

Princeton, NJ 08543

and

Institute of Theoretical Physics, University of California

Santa Barbara, CA 93106

Abstract

A treatment of linearized gravity is given in the Randall-Sundrum background. The graviton propagator is found in terms of the scalar propagator, for which an explicit integral expression is provided. This reduces to the four-dimensional propagator at long distances along the brane, and provides estimates of subleading corrections. Asymptotics of the propagator off the brane yields exponential falloff of gravitational fields due to matter on the brane. This implies that black holes bound to the brane have a “pancake”-like shape in the extra dimension, and indicates validity of a perturbative treatment off the brane. Some connections with the AdS/CFT correspondence are described.

1. Introduction and summary

It is possible that the observed world is a brane embedded in
a space with more noncompact dimensions. This proposal was made more
concrete in the scenario advanced in [1,2], where the problem of
recovering four-dimensional gravity was addressed. (Earlier
work appears in refs. [3-5].)
Further exploration of this scenario has
included investigation of its
cosmology[6-14] and
phenomenology[15,16].^{†}^{†} For a
recent survey of some of these topics, see also [17].

In order to “localize” gravity to the brane, ref. [2] worked in an embedding space with a background cosmological constant, with total action of the form

and are the five-dimensional metric and coordinates, and and are the corresponding four-dimensional quantities with given as the pullback of the five-dimensional metric to the brane. is the five-dimensional Planck mass, and denotes the five-dimensional Ricci scalar. The bulk space is a piece of anti-de Sitter space, with radius , which has metric

The brane can be taken to reside at , or in scenarios [18] with both a probe (or “TeV”) brane and a Planck brane, this will be the location of the Planck brane. The horizon for observers on either brane is at .

There are a number of outstanding questions with this proposal.
One very interesting question is what black holes or more general
gravitational fields, e.g. due to sources on the brane,
look like, both on and off the brane.
For example, consider a black hole formed from matter on the brane.
From the
low-energy perspective of an observer on the brane it should appear like
a more-or-less standard four-dimensional black hole
but one expects a five-dimensional observer to measure a
non-zero transverse thickness. One can trivially find solutions that a
four-dimensional observer sees as a black hole by replacing with the
Schwarzschild metric in (1.2).^{†}^{†} Such metrics were independently found
in [19]. However, these “tubular”
solutions become singular at the horizon at
, suggesting that another solution be found.

Another related question concerns the dynamics of gravity. It was argued in [2] that four-dimensional gravitational dynamics arises from a graviton zero mode bound to the brane. Fluctuations in this zero mode correspond to perturbations of the form

where is a function only of ,

Computing the lagrangian of such a fluctuation yields

This and other measures of the curvature of the fluctuation generically grow without bound as . In particular, if we add a higher power of the curvature to the action, with small coefficient, as may be induced from some more fundamental theory of gravity, then generically divergences will be encountered. For example, one easily estimates

suggesting that Planck scale effects are important near the horizon. This would raise serious questions about the viability of the underlying scenario. These estimates are however incorrect as they neglect the non-zero modes.

Yet another question regards corrections to the 4d gravitational effective theory on the brane. We’d like to better understand the strength of corrections to Newton’s Law and other gravitational formulae; some of the leading corrections have already been examined via the mode sum[2,18]. Sufficiently large corrections could provide experimental tests of or constraints on these scenarios.

A final point addresses reinterpretation [20,21,22] of
these
scenarios within the context of duality
[23-25]. In this
picture, gravity in the bulk AdS off
the brane can be replaced by super-Yang Mills theory
on the brane. Witten[22]^{†}^{†} See also [26]. has suggested
that gravitational corrections from the bulk
can be reinterpreted in terms of the loop diagrams
in the SYM theory.

In order to address these questions, this paper will give an analysis of linearized gravity in the background of [2]. We begin in section two with a derivation of the propagator for a scalar field in the brane background of [2], generalized to +1 dimensions. This exhibits much of the physics with less complication than gravity. The propagator is the usual AdS propagator plus a correction term, and can be rewritten, for sources on the brane, in terms of a zero-mode contribution that produces -dimensional gravity on the brane plus a correction from the “Kaluza-Klein” modes. For even this term produces corrections of order at large distances from the source.

In section three we perform a linearization of gravity about the +1 dimensional brane background. For general matter source on the brane, the brane has non-zero extrinsic curvature, and a consistent linear analysis requires introduction of coordinates in which the bending of the brane is manifest, as exhibited in eq. (3.22). We outline the derivation of the graviton propagator, which can be written in terms of the scalar propagator of section 2. (For those readers interested primarily in the applications discussed in the subsequent sections, the results appear in eqs. (3.23), (3.24), and (3.26).) Special simplifying cases include treatment of sources restricted to the Planck brane, or living on a probe brane in the bulk.

In section four we discuss the asymptotics and physics of the resulting propagator. Linearized gravity on the Planck brane corresponds to -dimensional linearized gravity, plus correction terms from Kaluza-Klein modes. As in the scalar case, these yield large- corrections suppressed by for even and by for , in agreement with [27]. We also discuss corrections in the probe-brane scenario of [18]. We then find the falloff in the gravitational potential off the brane and thus deduce the shape in the extra dimensions of black holes bound to the brane or of more general gravitational fields. In particular, we find that black holes have a transverse size that grows with mass like , compared to the usual result along the brane. Thus black holes have a pancake-like shape. We also check consistency of the linearized approximation, and check that higher-order curvature terms in the action in fact do not lead to large corrections, as the naïve analysis of the zero mode would indicate.

Finally, section five contains some comments on the connection with the AdS/CFT correspondence. An extension of the Maldacena conjecture[23] may enable one to replace the five-dimensional physics off the brane by a suitably regulated large- gauge theory[20,21,22]. In particular, we discuss how this picture produces four-dimensional gravity plus correction terms like those mentioned above.

During the period during which this work has been completed, several related works have appeared. Ref. [27] has found exact solutions for black holes bound to a brane in a 2+1 dimensional version of this scenario, and in that case independently discovered the shape of black holes (which in fact easily follows from the earlier estimates of [19]). Ref. [28] has also outlined aspects of a linear analysis of gravity, and in particular emphasized the importance of the bending of the brane. Recent comments on the relation with the AdS/CFT correspondence were made by Witten [22], with further elaboration in [26]. Preliminary presentations of some of our results can be found in [29,30].

2. The massless scalar propagator

Much of the physics of linearized gravity in the scenario of [2] is actually found in the simpler case of a minimally coupled scalar field. Because of this, and because the scalar propagator is needed in order to compute the graviton propagator, this section will focus on computing the scalar Green function.

In much of this paper we will work with the generalization to a +1 dimensional theory with a brane of codimension one. The scalar action, with source terms, takes the form

Here we work in the brane background of [2]; for the metric is the -dimensional AdS metric,

Without loss of generality the brane can be located at . This problem serves as a toy-model for gravity; for a given source , the resulting field is analogous to the gravitational field of a fixed matter source. The field is given in terms of the scalar Green function, obeying

Analogously to the boundary conditions that we will find on the gravitational field, the scalar boundary conditions are taken to be Neumann,

these can be interpreted as resulting from the orbifold boundary conditions at the brane, or alternately as due to the energy density on the wall. The scalar field has a bound zero mode analogous to that of gravity.

In order to solve (2.3), we first reduce the problem to solving an ordinary differential equation via a Fourier transform in the dimensions along the wall,

The Fourier component must then satisfy the equation,

Making the definitions and

this becomes

For , the equation admits as its two independent solutions the Bessel functions and . Hence, the solution for and for must be linear combinations , of these functions. Eq. (2.8) then implies matching conditions at :

We begin with the Green function for . The boundary condition (2.4) translates to

This has solution

where is the first Hankel function.

Next, consider the region . The boundary conditions at the horizon are analogous to the Hartle-Hawking boundary conditions and are inferred by demanding that positive frequency waves be ingoing there, implying[31]

The matching conditions (2.9) between the regions then become

The solution to these gives

where () denotes the greater (lesser) of and . This leads to the final expression for the scalar propagator:

We note that the second term is nothing but the ordinary massless scalar propagator in .

A case that will be of particular interest in subsequent sections is that where one of the arguments of is on the Planck brane, at . In this case, the propagator is easily shown to reduce to

For both points at , a Bessel recursion relation gives a more suggestive result:

This can clearly be separated into the standard -dimensional scalar propagator , with

which is produced by the zero-mode, plus a piece due to exchange of Kaluza-Klein states:

Here

Note that for , this gives the very simple result

(One must however be careful in treating the gravitational field as a perturbation in this case; recall that in the potential is logarithmic, which is in a sense not a small perturbation on a Minkowski background.)

The effective action for exchange of fields between two sources as in (2.1) is given by the usual quadratic expression involving this propagator. While we’ll postpone general discussion of the asymptotics of these expressions for large or small or until we discuss the graviton propagator, it is worth noting that at large distances on the brane, , the zero mode piece dominates and we reproduce the standard effective action for -dimensional scalar exchange, plus subleading corrections from the Kaluza-Klein part.

3. Linearized gravity

3.1. General matter source

We next turn to a treatment of linearized gravity in the context of brane reduction. We again work with a -dimensional theory, with action

Here may include both matter in the bulk and restricted to the brane. We have explicitly separated out the brane tension from the matter lagrangian. Away from the Planck brane, the vacuum solution is -dimensional space, with metric (2.2). The radius is determined by solving Einstein’s equations. Denote the Einstein tensor by ; these then take the form

where is the projection operator parallel to the brane, given in terms of unit normal as

and gives the position of the brane in terms of its intrinsic coordinates . Off the brane, (3.2) gives

As in [2], the brane tension is fine-tuned to give a Poincare-invariant solution with symmetric (orbifold) boundary conditions about the brane; this condition is

The location of the brane is arbitrary; we take it to be .

The rest of this subsection will focus on deriving the linearized gravitational field due to an arbitrary source; the results are presented in eqs. (3.23), (3.24), and (3.26) for readers not wishing to follow the details of the derivation. As we’ll see, maintaining the linearized approximation requires choosing a gauge in which the brane is bent, with displacement given in eq. (3.22).

It is often easier to work with the coordinate , defined by

in which the metric takes the form

the brane is at , and we have written the solution in a form valid for all .

It is convenient to describe fluctuations about (3.7) in Riemann normal (or hypersurface orthogonal) coordinates, which can be locally defined for an arbitrary spacetime metric which then takes the form

The coordinate picks out a preferred family of hypersurfaces, . Such coordinates are not unique; the choice of a base hypersurface on which they are constructed is arbitrary. This base hypersurface may be taken to be the brane, but later another choice will be convenient.

Fig. 1: A generic deformation of the base surface leads to a redefinition of gaussian normal coordinates.

In the case where the coordinates are based on the brane, small fluctuations in the metric can be represented as

Parameterize a deformation of the coordinates corresponding to changing the base hypersurface (see fig. 1) by

and consider a small deformation in the sense that is small.
Working at , the condition that the metric takes Gaussian normal form
(3.8) in the new coordinates is^{†}^{†} Here we have suppressed a subtlety
that
arises in trying to
write a similar expression valid on both sides of the brane due to
the discontinuity in the change of coordinates (3.10) across the brane

In the background metric (3.7), at this has general solution in terms of arbitrary small functions of :

The corresponding small gauge transformation in the fluctuation is

It is straightforward to expand Einstein’s equations (3.2) in the fluctuation ; at linear order the result is

where denotes the linear part of the Einstein tensor. In these and subsequent equations, indices on “small” quantities , , etc. are raised and lowered with . and represent the curvatures of the induced -dimensional metric of (3.8) (which include the conformal factor and will subsequently be expanded in ), and we have used the definitions

and .

Boundary conditions on at the brane are readily deduced by integrating the equation (3.16) from just below to just above the brane – resulting in the Israel matching conditions [32] – and enforcing symmetry under . If the energy momentum tensor includes a contribution from matter on the brane,

then we find

The first step in solving Einstein’s equations (3.14)-(3.16) is to eliminate between the and equations, resulting in an equation for alone, (working on the side of the brane)

Conservation of allows this to be rewritten

This can be integrated with initial condition supplied by the trace of (3.19). There is however an apparent problem if or : the resulting grows exponentially, leading to failure of the linear approximation.

Fig. 2: In the presence of matter on the brane, the brane is bent with respect to a coordinate system that is “straight” with respect to the horizon.

Fortunately this is a gauge artifact, resulting from basing coordinates on the brane. Indeed, non-vanishing produces extra extrinsic curvature on the brane; to avoid pathological growth in perturbations one should choose coordinates that are straight with respect to the horizon. In this coordinate system, the brane appears bent, as was pointed out in [28]. (See fig. 2). For simplicity consider the case where all matter is localized to , for some . First, the exponential growth in due to the initial condition (3.19) can be eliminated by a coordinate transformation of the form (3.12). We may then integrate up in until we encounter another source for this growth due to on the RHS of (3.21), and kill that by again performing a small deformation of the Gaussian-normal slicing. We can proceed iteratively at increasing in this fashion, with net result that the exponentially growing part of can be eliminated by a general slice deformation satisfying

in this equation and the remainder of the section, we work in the region on the side of the brane. In particular, consider the case ; the solution to (3.22) then explicitly gives the bending of the brane due to massive matter on the brane.

In order to solve Einstein’s equations we’ll therefore work with the metric fluctuation in this gauge, which for small coordinate transformations is given by (3.12), (3.13), and (3.22) (the spatial piece is still arbitrary). Eq. (3.21) has first integral

and can be solved by quadrature, up to the boundary conditions at . Eq. (3.15) is then

and can be integrated, again given the boundary conditions, to give the longitudinal piece of . Finally, linearizing in (3.16) and defining

gives

In this expression, is the scalar anti-de Sitter laplacian. All quantities on the right-hand side of (3.26) are known, so is determined in terms of the scalar Green function for the brane background, found in the preceding section. The metric deformation itself is given by trace-reversing,

Note, however, that (3.26) also suffers potential difficulty from exponentially growing sources. By (3.23) we see that for a bounded matter distribution, the trouble only lies in the terms involving . Note also that outside matter , from (3.24) and (3.23). Therefore, the remaining gauge invariance in (3.12) can be used to set these contributions to zero just outside the matter distribution,

and the same holds for all , eliminating the difficulty. Eq. (3.26) can then be solved for using the scalar Green function , given in eq. (2.15).

This will give an explicit (but somewhat complicated) formula for the gravitational Green function, defined in general by

and which can be read off in this gauge from (3.23), (3.24), and (3.26). In order to better understand these results, the following two subsections will treat two special cases.

3.2. Matter source on the brane

Consider the case where the only energy-momentum is on the brane. In terms of the flat-space Green function , (3.22) determines a brane-bending function of the form

Eq. (3.23) and (3.24) then imply

The gauge freedom can then be used to set

and the remaining equation (3.26) becomes

Boundary conditions for this are determined from the boundary condition (3.19) and the gauge shift induced by (3.30), and take the form

In terms of the scalar Neumann Green function of the preceding section, the solution is given by

where the first equality follows from tracelessness of ; the source on the RHS is clearly transverse and traceless as well. Recall that in this gauge the brane is located at .

The quantity is appropriate for discussing observations in the bulk, but a simpler gauge exists for observers on the brane. First note that integration by parts and translation invariance of implies

this, together with a gauge transformation using , can be used to eliminate the third term in (3.35). We then return to by inverting the gauge transformation (3.13); from the -dimensional perspective the only gauge non-trivial piece is the term, which we rewrite using (3.30). Thus, modulo -dimensional gauge transformations,

Note from (2.17) that the zero-mode piece cancels in the term multiplying . Writing the result in terms of the -dimensional propagator and Kaluza-Klein kernel given in (2.20) then yields

The first term is exactly what would be expected from standard
-dimensional gravity, with Planck mass given by^{†}^{†} Our conventions are
related to the standard ones for the gravitational coupling (see e.g. [33]) by for four dimensions.

The second term contains the corrections due to the Kaluza-Klein modes.

3.3. Matter source in bulk

As a second example of the general solution provided by (3.26), suppose that the matter source is only in the bulk. This in particular includes scenarios with matter distributions on a probe brane embedded at a fixed in the bulk[18].

By the Bianchi identities, Einstein’s equations are only consistent in the
presence of a conserved stress tensor. If we wish to consider
matter restricted to a brane at constant , a stabilization
mechanism^{†}^{†} See e.g. [34,35,14].
must be present to support the matter at this constant “elevation.”
Consider a stress tensor of the form

which is conserved on the brane,

For simplicity assume . Energy conservation in bulk then states

A solution to this with for is

We can think of this as arising from whatever physics is responsible for the stabilization.

Whether we consider matter confined to the brane in this way, or free to move about in the bulk, the results of this section give the linearized gravitational solution for a general conserved bulk stress tensor. Assuming for simplicity that for , and that , we can gauge fix such that (see (3.23),(3.24))

for . Thus outside of matter, we see from eq. (3.26) that again satisfies the scalar AdS wave equation. In particular, for matter concentrated on the probe brane at , eq. (3.26) gives

where arises from nonvanishing for on the RHS of (3.26), resulting from the stabilization mechanism. This has solution

Again the graviton Green function is given in terms of the scalar Green function of (2.15).

4. Asymptotics and physics of the graviton propagator

We now turn to exploration of various aspects of the asymptotic behavior of the propagators given in the preceding two sections, both on and off the brane. This will allow us to address questions involving the strength of gravitational corrections, the shape of black holes, etc.

4.1. Source on ‘‘Planck’’ brane

We begin by examining the gravitational field seen on the “Planck” brane by an observer on the same brane. The relevant linearized field was given in (3.38). This clearly exhibits the expected result from linearized -dimensional gravity, plus a correction term. The latter gives a subleading correction to gravity at long distances. This can be easily estimated: corresponds to , where (2.20) and the small argument formula for the Hankel functions yields

for , with . For , we need subleading terms in the expansion of the Hankel function. For even , this takes the general form (neglecting numerical coefficients)

Powers of in the integrand of (2.20) yield terms smaller than powers of (contact terms, exponentially supressed terms). The leading contribution to the propagator comes from the logarithm, with coefficient the smallest power of . This gives

for and even. For odd , the logarithm terms are not present in the expansion (4.2), and such corrections vanish. Thus for general even , the dominant correction terms are supressed by a factor of relative to the leading term; these are swamped by post-newtonian corrections. Note that in the special case , was exactly given by eq. (2.21), yielding a correction term of order for a static source, as noted in [27].

One can also examine the short-distance, , behavior of the propagator, which is governed by the large- behavior of the Fourier transform. In this case we find

Here clearly the Kaluza-Klein term dominates, and gives the expected dimensional behavior.

Next consider the asymptotics for and/or , with a source on the Planck brane. These are dominated by the region of the integral with . This means that a small argument expansion in can be made in the denominator of the propagator (2.16), and this gives

In particular gives , and we find a falloff

in the propagator at large .

Note that this means that in the physical case , for a static source, the potential falls like at large . This calculation can be taken further to determine the asymptotic shape of the Green function and potential as a function of and ; we do so only for the potential though the calculation may be extended to other . The static potential for a source at follows from (4.5) by integrating over time,

which then becomes