14 September 2000

Ž .Physics Letters B 489 2000 203–206www.elsevier.nlrlocaternpe

Gravity and antigravity in a brane worldwith metastable gravitons

Ruth Gregory a, Valery A. Rubakov b, Sergei M. Sibiryakov b

a Centre for Particle Theory, Durham UniÕersity, South Road, Durham, DH1 3LE, UKb Institute for Nuclear Research of the Russian Academy of Sciences, 60th October AnniÕersary prospect, 7a, Moscow 117312, Russia

Received 26 July 2000; accepted 11 August 2000Editor: P.V. Landshoff

Abstract

In the framework of a five-dimensional three-brane model with quasi-localized gravitons we evaluate metric perturba-tions induced on the positive tension brane by matter residing thereon. We find that at intermediate distances, the effectivefour-dimensional theory coincides, up to small corrections, with General Relativity. This is in accord with Csaki, Erlich andHollowood and in contrast to Dvali, Gabadadze and Porrati. We show, however, that at ultra-large distances this effectivefour-dimensional theory becomes dramatically different: conventional tensor gravity changes into scalar anti-gravity. q 2000Published by Elsevier Science B.V.

w xThe papers by Dvali, Gabadadze and Porrati, 1 ,w xand Csaki, Erlich and Hollowood, 2 , address the

issue of whether four-dimensional gravity is phe-nomenologically acceptable in a class of brane mod-els with infinite extra dimensions in which the five-dimensional gravitons have a metastable ‘boundstate’, rather than a genuine zero mode. A model of

w xthis sort has been proposed in Refs. 3,4 and is aŽ .variation of the Randall–Sundrum RS scenario for

w xa non-compact fifth dimension 5 . The constructionwith metastable gravitons has been put in a more

w xgeneral setting in Refs. 1,6 . It has been argued inw xRef. 1 that models with metastable gravitons are

not viable: from the four-dimensional point of view,gravitons are effectively massive and hence appear

w xto suffer from a van Dam–Veltman–Zakharov 7

Ž .E-mail address: r.h.w.gregory@durham.ac.uk R. Gregory .

discontinuity in the propagator in the massless limit.w xIn particular, it has been claimed 1 that the predic-

tion for the deflection of light by massive bodieswould be considerably different from that of GeneralRelativity. The issue has recently been analyzed in

w xmore detail in Ref. 2 , where explicit calculations offour dimensional gravity have been performed along

w xthe lines of Garriga and Tanaka 8 , and Giddings,w xKatz and Randall 9 . The outcome of that analysis is

that four-dimensional gravity has been claimed to bein fact Einsteinian, despite the peculiarity of appar-ently massive gravitons.

In this letter we also apply the Garriga–TanakaŽ .GT technique to obtain effective four-dimensionalgravity at the linearized level, considering as an

w xexample the model of Refs. 3,4 . We find that atŽintermediate distances which should extend from

microscopic to very large scales in a phenomenologi-.cally acceptable model four dimensional gravity is

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0370-2693 00 00917-5

( )R. Gregory et al.rPhysics Letters B 489 2000 203–206204

w xindeed Einsteinian, in accord with Ref. 2 and inw xcontrast to Ref. 1 . However, at ultra-large scales we

find a new phenomenon: four-dimensional gravitychanges dramatically, becoming scalar anti-graÕityrather than tensor gravity. This may or may notsignal an internal inconsistency of the models underdiscussion.

w xTo recapitulate, the set up of Refs. 3,4 is asfollows. The model has five dimensions and containsone brane with tension s)0 and two branes withequal tensions ysr2 placed at equal distances tothe right and to the left of the positive tension branein the fifth direction. There is a reflection symmetry,z™yz, which enables one to consider explicitlyonly the region to the right of the positive tension

Ž .brane hereafter z denotes the fifth coordinate . Con-ventional matter resides on the central positive ten-sion brane. The bulk cosmological constant betweenthe branes, L, is negative, whereas it is equal to zeroto the right of the negative tension brane. Withappropriately tuned L, there exists a solution to thefive-dimensional Einstein equations for which bothpositive and negative tension branes are at rest atzs0 and zsz respectively, z being an arbitraryc c

constant. The metric of this solution is

ds2 sa2 z h dx mdxn ydz 2 1Ž . Ž .mn

where

eyk z , 0-z-zca z s 2Ž . Ž .yk z½ ce 'a , z)zy c

The constant k is related to s and L. The four-di-mensional hypersurfaces zsconst. are flat, thefive-dimensional space-time is flat to the right of thenegative-tension brane and anti-de Sitter between thebranes. The spacetime to the left of the positivetension brane is a mirror image of this set-up.

This background has two different length scales,ky1 and

r sky1e3k zc 3Ž .c

These are assumed to be well separated, r 4ky1. Itcw xhas been argued in Ref. 4 that the extra dimension

‘opens up’ both at short distances, r<ky1 andultra-long ones, r4r .c

To find the four-dimensional gravity experiencedby matter residing on the positive tension brane, we

Ž .follow GT and consider a Gaussian–Normal GNgauge

g sy1 g s0 4Ž .z z zm

In the bulk, one can further restrict the gauge to beŽ .transverse-tracefree TTF

h m sh m s0 5Ž .m n ,m

Hereafter h are metric perturbations; indices aremn

raised and lowered by the four-dimensionalMinkowski metric. The linearized Einstein equationsin the bulk take one and the same simple form for allcomponents of h ,mn

1° XX Ž .2 4h y4k hy I hs0, 0-z-zc2a~1

XX Ž .4h y I hs0, z)zc2¢ ay

6Ž .

It is convenient, however, to formulate the junctionconditions on the positive tension brane in the localGN frame. In this frame, metric perturbations hmn

are not transverse-tracefree, so the two sets of pertur-bations are related in the bulk between the twobranes by a five-dimensional gauge transformation

Ž .preserving 4 ,

15 2 5 2ˆ ˆh sh q j y2ka h j qa j qjŽ .mn mn ,mn mn m ,n n ,mk

7Ž .

ˆ5Ž . Ž .where j x and j x are the gauge parameters.m

Notice that if j 5 is not zero, there is a ‘shift’ in thelocation of the wall relative to an observer at infinity,

Ži.e. the wall appears bent to such an observer asw x.discussed in 8,9 . Physically, this simply repre-

sents the fact that the wall GN frame is construc-ted by integrating normal geodesics from the wall,and in the presence of matter these geodesics willbe distorted, thereby altering the proper distancebetween the wall and infinity. In fact, one findsa similar ‘bending’ of the equatorial plane in theSchwarzschild spacetime if one tries to impose alocal GN frame away from the horizon.

( )R. Gregory et al.rPhysics Letters B 489 2000 203–206 205

In the presence of additional matter on the posi-tive tension brane with energy momentum T , themn

junction conditions on this brane read

X 1 lh q2kh s8p G T y h T 8Ž .Ž .mn mn 5 mn mn l3

where G is the five-dimensional gravitational con-5Ž . Ž .stant. The solution to Eqs. 5 – 8 has been obtained

ˆ5by Garriga and Tanaka. They found that j obeys

4pŽ4. 5 lˆ

I j sy G T 9Ž .5 l3

We will need the expression for the induced metricon the positive tension brane. Up to terms that can be

w xgauged away on this brane, the induced metric is 8

Žm. 5ˆh zs0 sh y2kh j 10Ž . Ž .mn mn mn

where

hŽm.s16p G dxX GŽ5. x , xX ; zszX s0Ž .Hmn 5 R

= 1 XlT y h T x 11Ž . Ž .Ž .mn mn l3

Ž5. Ž .Here G is the retarded Green’s function of Eq. 6RŽ .with appropriate source-free junction conditions on

the two branes. This Green’s function is mirror-sym-metric and obeys

1Ž .2 2 4E y4k u z yz y I q4kd zŽ . Ž .z c 2a

X XŽ .5y2kd zyz G x , x ; z , zŽ . Ž .c R

sd xyxXd zyzX 12Ž . Ž . Ž .

Let us consider the case of the static source first.w x y1It has been found in Ref. 4 that for k <r<r ,c

the leading behavior of the static Green’s functionŽ Ž5.Ž X ..given by H dt G zsz s0 is the same as in theR

Ž .RS model up to small corrections , and correspondsto a 1rr potential. Hence, at intermediate distancesthe analysis is identical to GT, and the inducedmetric is the same as in the linearized four-dimen-sional General Relativity. This is in accord with Ref.w x2 .

w xOn the other hand, it follows from Ref. 4 that atŽ .ultra-large distances, r4r , the contribution 11c

2 Ž .behaves like 1rr the fifth dimension ‘opens up’ .Ž .There remains, however, the second term in Eq. 10 .

Ž .Since Eq. 9 has a four-dimensional form, this term

Ž w x.gives rise to a 1rr potential missed in Ref. 4even at ultra-large distances. For a point-like staticsource of unit mass, the corresponding gravitationalpotential is

11 1V r ' h r sq G 13Ž . Ž . Ž .00 42 3 r

where G skG is the four-dimensional Newton’s4 5

constant entering also into the conventional Newton’slaw at intermediate distances. We see that at r4r ,c

four-dimensional gravity is induced by the trace ofenergy-momentum tensor and has a repulsive 1rrpotential. At ultra-large distances tensor gravitychanges to scalar anti-gravity.

Likewise, the four-dimensional gravitationalwaves emitted by non-static sources are conventionaltensor ones at intermediate distances and transform

Žinto scalar waves at ultra-large distances the rele-vant distance scale being different from r due toc

w x.relativistic effects, see Ref. 4 . Indeed, the firstŽ . w xterm in Eq. 10 dissipates 4 , whereas the second

term survives, again due to the four-dimensionalŽ .structure of Eq. 9 .

These two cases illustrate the general property ofŽ .Eq. 10 : the first term becomes irrelevant at ultra-

Žlarge distances the physical reason being themetastability of the five-dimensional graviton bound

. Žstate , so the four-dimensional gravity in effect,.anti-gravity is entirely due to the second, scalar

term.This bizarre feature of models with a metastable

graviton bound state obviously deserves further in-vestigation. In particular, it will be interesting toidentify the four-dimensional massless scalar mode,which is present at ultra-large distances, among thefree sourceless perturbations. This mode is unlikely

w xto be the radion 10,11 , studied in this model in Ref.w x3 : the radion would show up at intermediate dis-tances, as well as at ultra-large ones1; furthermore,

w xthe experience 13 with models where the distancew xbetween the branes is stabilized 14 suggests that the

1 The radion presumably couples exponentially weakly to thew xmatter on the positive tension brane, as it does 3,8 in the

w xtwo-brane model of Randall and Sundrum 12 . This may be thereason why the radion effects have not been revealed by the

w xanalyses made in Ref. 2 and this note.

( )R. Gregory et al.rPhysics Letters B 489 2000 203–206206

ˆ5massless four-dimensional mode parametrized by j

exists even if the radion is made massive.More importantly, one would like to understand

whether anti-gravity at ultra-large distances is a sig-nal of an intrinsic inconsistency of this class ofmodels, or simply a signal that physics is intrinsi-cally five-dimensional at these scales. In four dimen-sions, scalar antigravity requires either negative ki-netic and gradient energy or a ghost. Whether or nota similar feature is inherent in models with extradimensions remains an open question. If it is, therestill would remain a possibility that fields with nega-tive energy might be acceptable, as their effect mightshow up at ultra-large distances only.

We note finally, that anti-gravity may not be aspecial feature of models with quasi-localized gravi-tons. It is also possible that this phenomenon may bepresent in models of the type suggested by Kogan et.

w xal. 15 , where some Kaluza–Klein graviton excita-tions are extremely light. The same question aboutinternal consistency then would apply to these mod-els as well.

Acknowledgements

We would like to thank Sergei Dubovsky, DmitryGorbunov, Maxim Libanov and Sergei Troitsky foruseful discussions. We are indebted to C. Csaki, J.Erlich and T. J. Hollowood for sending their paperw x2 prior to publication. R.G. was supported in partby the Royal Society, and V.R. and S.S. by the

Russian Foundation for Basic Research, grant990218410.

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