We discuss the weak gravitational field created by isolated matter sources in
the Randall-Sundrum brane-world. In the case of two branes of opposite tension,
linearized Brans-Dicke (BD) gravity is recovered on either wall, with different
BD parameters. On the wall with positive tension the BD parameter is larger
than 3000 provided that the separation between walls is larger than 4 times the
AdS radius. For the wall of negative tension, the BD parameter is always
negative but greater than -3/2. In either case, shadow matter from the other
wall gravitates upon us. For equal Newtonian mass, light deflection from shadow
matter is 25 % weaker than from ordinary matter. Hence, the effective mass of a
clustered object containing shadow dark matter would be underestimated if
naively measured through its lensing effect. For the case of a single wall of
positive tension, Einstein gravity is recovered on the wall to leading order,
and if the source is stationary the field stays localized near the wall. We
calculate the leading Kaluza-Klein corrections to the linearized gravitational
field of a non-relativistic spherical object and find that the metric is
different from the Schwarzschild solution at large distances. We believe that
our linearized solution corresponds to the field far from the horizon after
gravitational collapse of matter on the brane.Comment: 8 pages, 1 figure. Replaced with revised version to be published in
  Phys. Rev. Lett. Some comments adde

C. M. Will

Jaume Garriga

L. Randall

Takahiro Tanaka

English

arXiv

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Gravity in the Randall-Sundrum Brane World

We discuss the weak gravitational field created by isolated matter sources in the Randall-Sundrum brane world. For the case of a single wall of positive tension, the field stays localized near the wall if the source is stationary. We calculate the leading Kaluza-Klein corrections to the linearized gravitational field of a nonrelativistic spherical object, which is different from the Schwarzschild solution at large distances. In the case of two branes of opposite tension, linearized Brans-Dicke (BD) gravity is recovered on either wall, with different BD parameters. On the wall with positive tension the BD parameter is larger than 3000 provided that the separation between walls is larger than 4 times the AdS radius. The gravitational field due to shadow matter is also considered

Garriga Torres, Jaume

Tanaka, Takahiro, 1958-

Diposit Digital de la Universitat de Barcelona

VOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 20002778Gravity in the Randall-Sundrum Brane WorldJaume Garriga1,2 and Takahiro Tanaka1,2,31IFAE, Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain2Isaac Newton Institute, University of Cambridge, 20 Clarkson Road, Cambridge, CB3 0EH, United Kingdom3Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan(Received 17 November 1999)We discuss the weak gravitational field created by isolated matter sources in the Randall-Sundrumbrane world. For the case of a single wall of positive tension, the field stays localized near the wall if thesource is stationary. We calculate the leading Kaluza-Klein corrections to the linearized gravitational fieldof a nonrelativistic spherical object, which is different from the Schwarzschild solution at large distances.In the case of two branes of opposite tension, linearized Brans-Dicke (BD) gravity is recovered on eitherwall, with different BD parameters. On the wall with positive tension the BD parameter is larger than3000 provided that the separation between walls is larger than 4 times the AdS radius. The gravitationalfield due to shadow matter is also considered.PACS numbers: 04.50.+h, 11.10.Kk, 98.80.CqIt has recently been shown [1] that something very simi-lar to four-dimensional Einstein gravity exists on a domainwall (or 3-brane) of positive tension which is embedded ina five-dimensional anti–de Sitter space (AdS). The strik-ing feature about this model is that an effective dimen-sional reduction occurs without the need of compactifyingthe fifth dimension. The reason is that “Kaluza-Klein”(KK) excitations, which have nonvanishing momentum inthe fifth direction, are suppressed near the brane. Thus,even though the KK modes are light, they almost decou-ple from matter fields—which are constrained to live onthe wall. Gravitational interactions among matter fieldsare mediated predominantly by the “zero mode,” which isoften described as a bound state of gravity on the wall.The case of two parallel domain walls, one with positivetension and another with negative tension, has also beendiscussed in an attempt to solve the much debated hierar-chy problem [2]. The possibility that we may be living ina brane is rather tantalizing, and many questions arise asto how gravity should look in such a world. What are thecorrections to Einstein gravity? How does the “shadow”matter living in the other brane gravitate upon us? Whatis the final state of gravitational collapse? Exploring someof these aspects will be the subject of the present paper.Some attention has been devoted to cosmological [3,4],as well as nonperturbative vacuum solutions [5,6] in thiscontext. In Randall and Sundrum’s solution the metric in-duced on the brane is flat. However, straightforward gen-eralizations can be obtained in which the induced metricis any vacuum solution of the four dimensional Einstein’sequations. Generalizations of this sort were given in [5]and [6], where the plane wave and the Schwarzschild so-lutions were considered. In these solutions, the metric onevery spacetime slice parallel to the brane is the same asthe metric on the brane, just rescaled by the AdS confor-mal factor. Thus, the gravitational field extends all the wayto the AdS horizon, at infinite distance from the brane. Inthe Schwarzschild case, the five-dimensional solution is a0031-90070084(13)2778(4)$15.00black string hidden behind a “cylindrical” horizon extend-ing to infinity. As shown by Chamblin et al. [6] tidal forcesfelt by freely falling observers actually become infinite asthe AdS horizon is approached, which is not very satisfac-tory from the physical point of view. However, it was ar-gued that since the infinite cylindrical horizon is unstable,the final state of collapse would perhaps have a horizon inthe shape of a cigar (rather than a full infinite cylinder).Although some intuition can be drawn from the previ-ous examples, it would be interesting to find physical solu-tions where the gravitational field stays localized near thesources. For this purpose, an analysis of the weak gravita-tional field created by isolated matter sources on the braneseems to be the best starting point. Let us begin with thecase of a single membrane of positive tension embeddedin five-dimensional AdS space. The metric is given byds2  gabdxadxb  dy2 1 a2 yhmndxmdxn . (1)Here, a y  e2jyj, where  is the curvature radius ofAdS, and hmn is the Minkowski metric in four dimensions.The cosmological constant on the bulk is given by L 2622 and the wall tension is given by s  34pG5,where G5 is Newton’s constant in five dimensions.Denoting the perturbed metric by g˜ab  gab 1 hab , theRandall-Sundrum (RS) gauge is defined byh55  hm5  0, hnm ,n  0, hmm  0 . (2)It is possible to show that these conditions can be choseneverywhere in the bulk [7]. In this gauge, the equations ofmotion take the simple forma224 1 ≠2y 2 422 hmn  0 . (3)The advantage of this gauge is that all components ofthe metric are decoupled. However, in general, when wechoose the gauge (2) in the bulk, the brane will not be lo-cated at y  0. Instead, as we shall see, its location willbe given by y  2jˆ5xm (see Fig. 1), where jˆ5 is the© 2000 The American Physical SocietyVOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000solution of the equation4jˆ5 k6T . (4)Here T  Tmm, and k  8pG5. In our definition of Tmnwe are not including the contribution from the wall it-self. To proceed, it will be convenient to go momentarilyto Gaussian normal coordinates, which we denote by x¯a.By definition, the wall is located at y¯  0, and we haveh¯55  h¯m5  0. Gaussian coordinates are also interestingfor us because h¯mn y¯  0 is the metric perturbation in-duced on the wall. We impose even parity under y¯ ! 2y¯,and we shall work on the positive side in the followingdiscussion. Then, the junction condition on the extrin-sic curvature at the wall requires that ≠ygmn 1 h¯mn 2k3 sgmn 1 h¯mn 1 3Tmn 2 Tgmn, which implies≠y 1 221h¯mn  2kµTmn 213gmnT∂, y¯  01 . (5)Here gmn  e22jyjhmn is the background spatial metric.The condition (5) can now be expressed in the RS gauge.Since h55 and h5m vanish in both gauges, the most generaltransformation between them must take the formj5  jˆ5xr ,jm 22gmnjˆ5xr,n 1 jˆmxr , (6)where jˆ5 and jˆm are independent of y, and for the momentwe are not assuming the condition (4). From the gaugetransformation equationshmn  h¯mn 2 jˆ5,mn 2 221gmnjˆ5 1 grmjˆr,n , (7)the junction condition (5) becomes≠y 1 221hmn  2kSmn ,  y  01 , (8)where we have introduced the combinationSmn µTmn 213gmnT∂1 2k21jˆ5,mn . (9)This combination, which in some sense includes the “bend-ing” of the wall jˆ5, will play the role of the source termin the RS gauge.Our solutions must be even under parity, and so from(8) the derivative of the metric perturbation will be discon-tinuous at the wall. Combining (8) with (3), the equationsof motion in the RS gauge becomea224 1 ≠2y 2 422 1 421dyhmn  22kSmndy , (10)where the delta function terms will enforce the discontinuities. Of course, in order to solve (10), we must first determinethe function jˆ5, which enters in the definition of the source term S. This function is given by (4), as we shall nowexplain. Let us define the 5D retarded Green’s function, which satisfiesa224 1 ≠2y 2 422 1 421dyGRx, x0  d5x 2 x0 . (11)The formal solution of (10) is then given byhmnx  22kZd4x0GRx, x0Smnx0 , (12)where integration is taken over the y  0 surface. Since hmm must vanish, we must impose Smm  0, which impliesthe “equation of motion” (4) for jˆ5. With this choice of jˆ5, it is easy to check that hmn given by Eq. (12) satisfies theharmonic condition h nm ,n  0.The behavior of hmn at infinity is determined by the form of GRx, x0. The Green’s function can be constructed froma complete set of eigenstates in the usual way. Following [1], we haveGRx, x0  2Z d4k2p4eikmxm2x0m"a y2a y0221k2 2 v 1 ie21Z `0dmum yum y0m2 1 k2 2 v 1 ie2#, (13)where the first term corresponds to the zero mode andthe rest corresponds to the continuum of KK modesum y pm2 J1mY2ma 2 Y1mJ2mapJ1m2 1 Y1m2. For the stationary case, it ismore illustrative to consider the Green’s function for theLaplacian operator, which is related to the previous onethroughGx, y,x0, y0 Z `2`dt0GRx, x0 . (14)Here x are spatial Cartesian coordinates on the wall. Whenboth points are taken on the wall ( y  y0  0), we haveGx, 0,x0, 0 214pr∑1 122r21 ...∏, (15)where r  jx 2 x0j. Also, when one of the points is onthe wall, the leading behavior for large separations in anydirection is given byGx, y,x0, 0  2a38p2a2r2 1 32a2r2 1 232. (16)This means that the metric perturbation decays rathersteeply towards the AdS horizon at y ! `, i.e., a ! 0(in fact even the relative metric perturbation hmna2 fallsto zero as we move from the source). The behavior isillustrated in Fig. 1.Since we are interested in the metric on the wall, it isconvenient to transform back to Gaussian coordinates.From (7), we have h¯mn  hmmn 1 hj,mn 1 jˆ5,mn 1221gmnjˆ5 2 jˆm,n, where we decomposed hmn into2779VOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000the part corresponding to the matter fields and the partcorresponding to the wall displacement,hmmn  22kZd4x0GRx, x0µTmn 213gmnT∂x0 ,(17)hj  24Zd4x0GRx, x0jˆ5x0 . (18)Setting y  0 and choosing jˆm appropriately, we end upwith the rather simple expressionh¯mn  hmmn 1 221gmnjˆ5 , (19)which gives the metric perturbation on the wall.(i) Spherical symmetry.—As a simple application, letus now consider the effect of the KK modes on the metric.We shall restrict attention to the most interesting case ofa static and spherically symmetric source, as this may berelated to the final stage of gravitational collapse. Withour assumptions, the energy momentum tensor can be writ-ten asTmn  rrumun .From (18) with the aid of (4), we obtainhj 43Z r0dr 0r 0 2Z r 00dr 00r 00 2V r 00 , (20)where V  k2RGx, x0rx0 d3x0. Then, from (12),we haveh00  283V r , hrr  283r3Z r0dr 0r 0 2V r 0 .(21)The remaining metric components can be found from therequirement that h  0 plus spherical symmetry. Noticethat the fall-off properties of the metric components aty ! ` are the same as those for the Green’s function (16).Hence the field decays quite steeply away from the wall.It can be checked that the perturbation of the square of theRiemann tensor behaves asFIG. 1. Gravitational field of a spherically symmetric staticsource in the Randall-Sundrum gauge.2780d°RmnrsRmnrs¢~ a2,at large y (uniformly for all values of r .) For comparison,the same quantity behaves as M2a2ar26 in the case ofthe Schwarzschild black string [6].In order to find the metric on the brane, we transformto Gaussian normal coordinates. When the point is outsidethe source, we haveV  2kM8prµ1 122r2∂, jˆ5 kM24pr, (22)where M Rd3xr is the total mass. Using (19), wearrive at the resulth¯00 2GMrµ1 1223r2∂,h¯ij 2GMrµ1 123r2∂dij .(23)It should also be stressed that the Newtonian potentialh¯002, which determines the attraction of neighboring bod-ies, is not the same as V —which is just proportional tothe Green’s function Gx, 0. The coefficient in front ofthe correction 2r2, due to the KK modes, is different inboth cases, because jˆ5 is in some sense four dimensionaland contributes only to the zero mode.Our solution differs from the weak field limit of the usualfour-dimensional Schwarzschild solution. This seems toindicate that gravitational collapse of matter on the wallwill not lead to a Schwarzschild black hole, but to a metricwhich has the asymptotic form of the weak field solution(23) [8].(ii) Zero mode truncation.—In general, in order to ob-tain the metric perturbation induced on the brane, we mustfirst solve Eq. (4) for jˆ5, feed the solution into Eq. (10)for hmn , and then use the gauge transformation (7) toobtain h¯mn . The expectation is that this should reproducethe results of linearized Einstein gravity with some smallcorrections. Let us now show that, indeed, in the caseof a single brane the zero mode truncation of the five-dimensional theory coincides with the usual linearizedfour-dimensional gravity.If both arguments of the two-point function are on thewall, then GRx, x0 is dominated by the zero mode con-tribution, GRx, x0  d4xm 2 xm04. Substitutingin (17), we find that the induced metric on the wall is givenbyh¯mn  216pG14µTmn 212gmnT∂, (24)where G  21G5 is the four-dimensional Newton’s con-stant. Thus, we recover the linearized Einstein’s equations.It should be noted, however, that a happy cancellationhas occurred: the factor 13 in Eq. (17) has turned into thefamiliar 12 in the process of going to Gaussian normalcoordinates (that is, absorbing jˆ5) through Eq. (19). Aswe shall see, this cancellation does not occur in the caseVOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000when we have two branes, which leads of course to someinteresting consequences.(iii) Two branes and light deflection.—In the case whenwe have two branes, one with positive tension at y  0 anda second one at y  d with negative tension, the previousarguments can be repeated without any basic formal alter-ations. The only differences are that the normalization ofthe zero mode changes by a factor of 1 2 e22d and,more important, the effect of the “Goldstone” mode jˆ5does not cancel out. Following the steps of the derivationgiven above, we find that in the zero mode approximationthe gravitational field on each of the branes satisfies [9]µ1a24h¯mn∂6 2Xs616pGsµTmn 213gmnT∂s616pG63sinhde6dgmnT6 ,(25)where the plus and minus refer to quantities on the wallwith positive and with negative tension, respectively. Here,we have introducedG6 G521e6d2 sinhdl , (26)which plays the role of Newton’s constant in a Brans-Dicke (BD) parametrization (we follow the conventions ofRef. [10]).Strictly speaking, this parametrization holds when “theother” wall (the one in which we do not live) is empty.Let us first consider this situation. In this case, the BDparameter is given byv6BD 32e62dl 2 1 . (27)Observations require that vBD . 3000 [10]. In the posi-tive tension brane, this is achieved with dl . 4, and wehave an acceptable gravity theory even without stabilizingthe dilaton. In the negative tension brane, we find that theBD parameter is always negative but greater than 232.In the Einstein frame, the kinetic term for the BD field hasthe usual sign for vBD . 232. This suggests that thesystem of two branes is well behaved in spite of the nega-tive tension in one of the branes.Now, let us consider the effect of “shadow” matter,which lives on the other membrane. This appears onlyin the first term in (25). Hence, for nonrelativistic matter,and assuming spherical symmetry, its contribution to theNewtonian potential h¯00 will be twice as large as its con-tribution to any of the diagonal spatial components, sayh¯zz . This is in contrast with the situation in Einstein’s the-ory, where the contribution to the Newtonian potential isthe same as the contribution to h¯zz . For a source in thex, y plane, the deflection of a light ray traveling in the ydirection is given by x¨  12 h¯00 1 h¯yy,x . Therefore,for the same Newtonian mass, the deflection of light rayscaused by shadow matter is 25% smaller than that in Ein-stein gravity. It would be interesting to investigate thispossible effect in an astrophysical context. This is left forfuture research.This work was done while participating in the work-shop on Structure Formation in the Universe, at the IsaacNewton Institute. We thank the organizers for their hos-pitality, and the participants for numerous conversations.We owe special thanks to M. Bucher, T. Chiba, S. W.Hawking, R. Gregory, K. Maeda, X. Montes, A. Pomarol,O. Pujolas, V. A. Rubakov, H. Reall, M. Sasaki, T. Shi-romizu, and N. Turok. J. G. acknowledges support fromCICYT, under Grant No. AEN98-1093. T. T. acknowl-edges support from Monbusho System to Send JapaneseResearchers Overseas.[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690(1999).[2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999).[3] P. Binétruy, C. Deffayet, U. Ellwanger, and D. Langlois,hep-th /9910219.[4] T. Shiromizu, K. Maeda, and M. Sasaki, gr-qc/9910076.[5] A. Chamblin and G. W. Gibbons, hep-th /9909130.[6] A. Chamblin, S. W. Hawking, and H. S. Reall, hep-th/9909205.[7] The RS gauge can be chosen in the bulk for almost the samereason that allows the choice of the synchronous tranverseand traceless gauge in the cosmological context (in theabsence of matter sources). In fact, there is some residualgauge freedom generated by (6) with jˆn ,m,n 1 2jˆm ,n,n  0and jˆ5  jˆm,m4.[8] For the 2 1 1 dimensional case, some discussion of thisissue is given in R. Emparan, G. T. Horowitz, and R. C.Myers, hep-th /9911043.[9] In an earlier version of this paper the sign of the last termof Eq. (25) in the case of a negative tension brane was notcorrect. This was pointed out to us by V. A. Rubakov.[10] C. M. Will, Theory and Experiment in GravitationalPhysics (Cambridge University Press, Cambridge, 1993),revised ed., p. 123; in Proceedings of the 1998 SlacSummer Institute on Particle Physics (gr-qc/9811036).2781

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Gravity in the Randall-Sundrum brane world

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Gravity in the Randall-Sundrum Brane World

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