Supergravity on the Brane

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A. Chamblin∗∗ & G.W. Gibbons

DAMTP, Silver Street, Cambridge, CB3 9EW, England (November 23, 1999) We show that smooth domain wall spacetimes supported by a scalar ﬁeld separating two anti- de-Sitter like regions admit a single graviton bound state. Our analysis yields a fully non-linear supergravity treatment of the Randall-Sundrum model. Our solutions describe a pp-wave propagating in the domain wall background spacetime. If the latter is BPS, our solutions retain some supersymmetry. Nevertheless, the Kaluza-Klein modes generate “pp curvature” singularities in the bulk located where the horizon of AdS would ordinarily be.

12.10.-g, 11.10.Kk, 11.25.M, 04.50.+h DAMTP-1999-126

I. INTRODUCTION full dynamics of the domain wall is not treated in detail in the Randall-Sundrum model. In fact gravitating domain walls have a drastic effect on the curvature of the ambi- It has long been thought that any attempt to model ent spacetime and it is not obvious that a simple model the Universe as a single brane embedded in a higher- involving a single collective coordinate representing the dimensional bulk spacetime must inevitably fail because transverse displacement of the domain wall is valid. the gravitational forces experienced by matter on the For these reasons it seems desirable to have a simple brane, being mediated by gravitons travelling in the non-singular model which is exactly solvable. It is the bulk, are those appropriate to the higher dimensional purpose of this note to provide that. spacetime rather than the lower dimensional brane. Re- cently however, Randall and Sundrum have argued that there are circumstances under which this need not be II. THICK DOMAIN WALLS IN ADS so. Their model involves a thin “distributional” static flat domain wall or three-brane separating two regions of We first seek a static domain wall solution of of the five-dimensional anti-de-Sitter spacetime. They solve for d-dimensional Einsten equations the linearized graviton perturbations and find a square integrable bound state representing a gravitational wave 1 Rmn Rgmn = ∂mΦ ∂nΦ (2.1) confined to the domain wall. They also found the lin- − 2 · earized bulk or “Kaluza-Klein” graviton modes. They 1 ab gmn ∂aΦ ∂bΦ g + V (Φ) argue that the latter decouple from the brane and make − 2 · negligible contribution to the force beween two sources where a, b =0,1,2,...,d 1. The right hand side of (2.1) in the brane, so that this force is due primarily to the is the energy momentum− tensor of one or more scalar bound state. In this way we get an inverse square law fields Φ with potential V (Φ) whose kinetic energy term attraction rather than the inverse cube law one might may contain a non-trivial metric on the scalar field man- naively have anticipated (see [3] for a related discussion). ifold. The metric is assumed to be of the form: This result is rather striking and raises various ques- 2 2 2A(r) µ ν tions. For example one would like to know how general ds = dr + e ηµν dx dx , (2.2) the effect is. Is it just an effect of the linearized pertur- where µ, ν =0,1,2,...,d 2andη is the flat bations or does it persist when non-linearities are taken µν Minkowski metric. The scalar− field is assumed to depend into account? One would expect to get only one massless only on the transverse coordinate r and if denotes dif- spin two bound state if the effective theory on the brane ferentation with respect to r then the Einstein0 equations is to be general relativity. In their derivation a crucial require role is played by a delta-function in the linearized graviton equation of motion. This is responsible for the unique Φ0 Φ0 =(d 2)A00, (2.3) bound state. It also seems that the effect will only work − · − 1 (d 2)(d 1) 2 Φ0 Φ0 V = − − (A0) . for domain walls and not for other branes. However the 2 · − 2

∗∗Address from 15 October, 1999: Center for Theoretical Physics, MIT, Bldg. 6-304, 77 Massachusetts Ave., Cambridge, MA 02139, USA

1 2A 2 These two equations imply the scalar field equation: H00 +(d 1)H0A0 + e− H =0, (3.2) − ∇⊥ ∂V 2 Φ00 +(d 1)Φ0A0 = . (2.4) where is the flat Laplace operator in the coordinates − ∂Φ xi . This∇⊥ will have half as much supersymmetry as the ⊥ If there is a non-trivial covariant metric on the scalar domain wall background. One may further generalize i i field manifold the right hand side of (2.4) includes the this solution by replacing the flat metric dx dx by an ⊥ ⊥ contravariant metric. arbitary (d 3)-dimensional Ricci flat metric g .Ifg − ⊥ ⊥ A domain wall solution separating two anti-de-Sitter admits covariantly constant spinors, then the background domains with the same cosmological constant would have will still admit some supersymmetry. If g is flat space, solutions of (3.2) propagate in sur- has A r/a as r . ⊥ If the≈−| potential| V| |→∞has the special form faces of constant r at the speed of light in the (arbi- trarily chosen) x1 direction with an amplitude depend- 1 ∂W ∂W d 1 2 ing upon r. Fourier analyzing in the x direction gives V = − W (2.5) ik x ⊥ 2 ∂Φ · ∂Φ − d 2 H e · ⊥ ,wherekcould in principle depend upon u.If − kis∝ real, solutions would propagate faster than light in a where W = W (Φ) is a suitable superpotential then Ein- given r =constant surface, and would appear as tachyons stein equations (2.3) and the scalar equation (2.4) are to an observer on the brane. On the other hand, solu- solved by solutions of the first order Bogomol’nyi equations for which k is pure imaginary propagate on the tions: brane like Kaluza-Klein modes. Thus, if k2 = m2, i.e., ∂W 1 2 H = m2H, we are led to the equation − Φ0 = ,A0= W, (2.6) ∇ ⊥ ∂Φ −d 2 2A 2 − H00 +(d 1)H0A0 + e− m H =0. (3.3) Note that the spacetime is uniquely specified by giv- − ing a solution of (2.6) which is the same as the equation Consider the zero modes, i.e., solutions with m2 =0. for a domain wall in the absence of gravity. One then i j We take H = F (r)Hij (u)x x and find that F = obtains A by quadratures. The vacua correspond to crit- r (d 1)A(s) ⊥ ⊥ C1 + C2 dse− − where C1 and C2 are constants. ical points of the superpotential W . At these points the 2A The graviton perturbation h = e− H will diverge ex- potential V is negative, and so one is in an anti-de-Sitter R ponentially for large values of r unless C2 = 0. We will phase. Recently, there has been a lot of interest in the return to this divergence in the| | next section. The mode possibility of obtaining such potentials within the con- for which C2 =0andC1 =1and text of d = 5 gauged supergravity models ( [5], [6], [12], i j [7]. At present no superpotential with the correct prop- H=Hij (u)x x (3.4) erties derived from a supergravity model has yet been ⊥ ⊥ found. However a solution was exhibited in [4] which is may be identified as a fully non-linear version of the zero not derived from a supergravity model. We will return mode of Randall and Sundrum on a general domain wall to this point in the last section. We will now show, with- background. Here, Hij (u) is an arbitrary trace free sym- out assuming that it is supersymmetric or satisfies the metric matrix which determines the polarization state first order equations, how to superpose a smooth domain of the graviton. The choice (3.4) is made so that the wall background with plane-fronted gravitational waves solution has a d-dimensional isometry group acting on moving in the anti-de-Sitter background. the surfaces r =constant, u =constant. This invariance is not manifest in the coordinates (r, u, v, x ), but is in Rosen coordinates [14] (˜u, v,˜ x˜ ), in which⊥ (3.1), given III. PP-WAVES ON THE BRANE: THE BOUND (3.4), assumes the form ⊥ STATE 2 2 2A i j ds = dr + e ( dudv˜ + Aij (u)dx˜ dx˜ ) (3.5) An exact solution of Einstein’s equations representing − ⊥ ⊥ 1 1˙ i j i a gravitional wave moving at the speed of light in the x where u =˜u,v=˜v+ Aij (u)˜x x˜ ,andx = 2 ⊥ direction is given by retaining the form Φ(r)andA(r) i j m ⊥m⊥ Pj(u)˜x . Here, Aij (u)=P i(u)P j(u), denotes but modifying the metric (2.2) to take the form: ⊥ · i differentiation with respect to u and the matrix P j(u) ï k 2 2 2A(r) i 2 i i is a solution of P j = HikP j . To make contact with ds = dr + e dudv + H(u, r, x )du + dx dx , 1 − ⊥ ⊥ ⊥ ( [2], [1]), we linearize, setting Pij = δij + 2 ψij so that ¨ (3.1) ψij = Hij .Thequantityψis essentially the perturbation considered in ( [1]). Rosen coordinates are in general with u = t x1, v = t + x1, i =2,...,d 3andwhere rather pathological at the non-linear level and awkward the u dependence− of H is arbitrary but it’s− dependence to use. In our non-linear analysis we shall, from now on, upon r and xi is governed by only use the coordinates (r, u, v, x ). ⊥ ⊥

2 IV. PP-WAVES IN THE BULK: BLUESHIFT AND In order to get a better feel for the singular nature of CURVATURE SINGULARITIES these spacetimes, it is useful to focus on a specific example of a Siklos-type metric where the z-dependence is Our spacetimes are timelike and lightlike geodesically non-trivial. The simplest example is the higher dimen- incomplete as r . In the absence of gravita- sional generalization [18] of Kaigorodov’s spacetime [19], tional waves, i.e.,| |−→∞H =0,r= corresponds to a reg- for which H is ular Cauchy horizon, and the solution∞ may be extended d 1 through the horizon (see for example [17]). If H=0 how- H(z)=z − . ever, the solutions will generically become singular6 as r , and will not admit an extension. The nature The Kaigorodov metric is | |−→∞ of this singularity is most easily studied when the back- 2 r/a 2 a d 1 2 d 1 1 ground is taken to be exactly AdSd.Ifweletz=ae ds = ( (1 z )dt 2z dtdx (4.4) 2 − − then the metric (3.1) can be recast in so-called ‘Siklos’ z − − − d 1 1 2 2 2 +(1 + z − )(dx ) + dz + dx ). coordinates [15]: ⊥ a2 This is the AdS analogue of the simplest vacuum pp- ds2 = (dz2 dudv + Hdu2 + dxi dxi ), (4.1) d z2 − ⊥ ⊥ wave, namely, the homogeneous pp-wave in flat space. It has d 1 obvious translational Killing vectors, and is also where H now satisfies the generalized Siklos equation invariant− under the R+-action:

(d 2) ∂ 1 ∂H 2 3 d d+1 z [ ]+ H =0. − − d 2 (z,u,v) ( λz, λ 2 u, λ 2 v). ∂z z − ∂z ∇⊥ −→ Because all invariants formed from the Weyl tensor of This action, combined with translations in u and v,gen- (4.1) necessarily vanish, it is not possible to detect cur- erates a three-dimensional group of Bianchi Type VIh, vature singularities directly by calculating invariants. 1 where h = − 2 . Therefore, the Kaigorodov isometry (d 1)− However, the necessary condition that one may extend group contains− a simply transitive subgroup which takes through the singularity in the metric at z = is that ∞ every point with z positive to any other point with z pos- the components of the Riemann tensor in an orthonor- itive. A similar d-dimensional simply transitive group ex- mal frame which has been parallelly propagated along ists in the AdS case, for which the R+ action is simply every timelike geodesic are finite. This requirement d z λz.IntheAdSd case, we can extend beyond the arises because freely falling observers move along timelike reach−→ of the group, in the Kaigorodov case we cannot. geodesics, and the components of the curvature tensor Clearly, freely falling timelike observers (who can cross will measure the tidal forces which these observers expe- the surface z = after a finite period of affine parame- rience. Following the demonstration in [15], one may cal- ter time [15]) will∞ see infinite tidal forces in this region. culate these terms explicitly for the Siklos metrics. One This shows that there are naked curvature singularities at finds that certain frame components of the Riemann ten- the points z = . Given our discussion in the previous sor generically assume the form section, where we∞ saw that generic z-dependent graviton Λ 1 ∂H perturbations will diverge at large z, it is clear that we R = z5( ) (4.2) (a)(b)(a)(b) d 1± z ∂z ,z should regard these singularities as a generic feature of − Siklos spacetimes. where we have suppressed various constants which are irrelevant to this discussion. It follows that any solution with z-dependence cannot be extended, and hence is sin- V. DISCUSSION gular. One sees that the z-dependent piece of (4.2) is the contribution from the Weyl tensor. It would therefore seem that the gravitons will be heavily ‘blueshifted’ as We have shown that it is possible to include a non- we move towards large values of z. linear gravitational wave on a thick domain wall back- If 2 H = m2H, the Siklos equation has solutions of ground, in such a way that one may recover the Randall- the form∇⊥ Sundrum bound state. Given the formal Witten style stability proofs in [5], which work as long as one has a solu- d 1 − ik x H = z 2 e ⊥ [D J d 1 (mz)+D Yd 1(mz)], (4.3) tion of the first order equations, one might have thought · 1 − 2 − 2 2 that this would ensure that the Randall-Sundrum sce- where Jn(x)andYn(x) are Bessel functions, and D1, D2 nario could be perturbed in this way without problems. are some constants. The z-dependence of H has the same However somewhat to our surprise, we have found that form as the Kaluza-Klein modes of ( [2], [1]). The be- generically gravitons propagating in the bulk become sin- haviour near z = shows that these are singular on the gular on what is a Cauchy horizon in the unperturbed Cauchy horizon. ∞ spacetime. These singularities are somewhat unusual, in

3 that scalar invariants formed from the curvature tensor Extra Dimension, hep-ph/9908347. do not blow up but rather the components of the cur- [4] O DeWolfe, D Z Freedman, S.S. Gubser and A Karch, vature in a parallelly propagated frame along a timelike Modelling the fifth dimension with scalars and gravity geodesic do blow up. Such singularities are called “pp hep-th/9909134 curvature singularities” [15]. [5] Klaus Behrndt and Mirjam Cvetic, Supersymmetric One might worry that these singularities signal a Domain-Wall World from D=5 Simple Gauged Super- gravity, hep-th/9909058. breakdown in our ability to make unitary predictions. [6] D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. However, any statements about unitarity should be re- Warner, Renormalization Group Flows from Holography– stricted to physics on the brane at z = constant. Any Supersymmetry and a c-Theorem, hep-th/9904017. pathological effects which may emerge from the singular- [7] R. Kallosh, A. Linde and M. Shmakova, Supersymmetric ity will be heavily red-shifted by the time they reach the Multiple Basin Attractors, hep-th/9910021. brane. Consequently, the extent to which these singulari- [8] L. Girardello, M. Petrini, M. Poratti and A Zaffaroni ties signal a pathology of the theory is at present unclear. The Supergravity Dual of N=1 Super Yang-Mills theory , Interestingly, if one considers massless z-independent pp- hep-th/9909047 waves (these would correspond to the Randall-Sundrum [9] S.T.C. Siklos, in: Galaxies, axisymmetric systems and zero mode bound state), one finds that the components relativity, ed. M.A.H. MacCallum, Cambridge University of the curvature do not blow up, and presumably the Press, Cambridge (1985). [10] G.W. Gibbons and P.J. Ruback, Classical gravitons and spacetime has a non-singular extension. their stability in higher dimensions, Phys. Lett. 171B, To conclude we would like to return to the question 390-395, (1986). of whether a suitable super-potential exists which can be [11] G.W. Gibbons Vacuua and Solitons in Exteded Super- derived from a supergravity model. The results of [5] and gravity, in: Relativity, Cosmology, Topological Mass and [7] show that for the simplest case of a single scalar field Supergravity, ed. C. Aragone, pages 163 - 177, World in models of the type studied in [13] they do not. In fact Scientific (1983). one may show quite generally that for the models in [13] [12] K. Skenderis and P.K. Townsend, Gravitational Stability with an arbitrary number scalar fields they do not. The and Renormalization-Group Flow, hep-th/9909070. same is true for the models considered in [8]. It therefore [13] M. Gunaydin, G. Sierra, and P.K. Townsend, More on remains an important open problem to find a suitable D=5 Maxwell-Einstein supergravity: Symmetric spaces supergravity model or prove that no such model exists. and kinks, Class. Quant. Grav. 3: 763, (1986). [14] G.W. Gibbons, Quantized fields propagating in plane- The authors thank M. Bañados, J. Harvey, S. Hawk- wave spacetimes, Comm. Math. Phys. 45, 191-202 (1975). ing, N. Lambert, R. Myers, M. Porrati, H. Reall and S. [15] J. Podolsky, Interpretation of the Siklos solutions as exact Siklos for useful conversations and correspondence. A.C. gravitational waves in the anti-de Sitter universe, Class. was supported by Pembroke College, Cambridge. Quant. Grav. 15, 719-733, (1998); gr-qc/9801052. [16] J. Bicak and J. Podolsky, Gravitational waves in vacuum spacetimes with cosmological constant. II. Deviation of geodesics and interpretation of non-twisting type N solutions, gr-qc/9907049. [17] G.W. Gibbons, Global structure of supergravity domain walls spacetimes,Nucl.Phys.B394, 3 (1993). [1] Lisa Randall and Raman Sundrum, A Large Mass Hier- [18] M. Cvetic, H. Lu and C.N. Pope, Spacetimes of Boosted archy from a Small Extra Dimension, hep-ph/9905221; p-branes, and CFT in Infinite-momentum Frame,Nucl. Joseph Lykken and Lisa Randall, The Shape of Gravity, Phys. B545, 309-339, (1999); hep-th/9810123. hep-th/9908076. [19] V.R. Kaigorodov, Sov. Phys. Doklady. 7, 893 (1963). [2] Lisa Randall and Raman Sundrum, An Alternative to [20] J. Ehlers and W. Kundt, Gravitation: An Introduction Compactification, hep-th/9906064. to Current Research, ed. L. Witten (New York: Wiley) [3] M. Gogberashvili, Gravitational Trapping for Extended (1962).