Supergravity on the Brane

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Supergravity on the Brane 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 propa- gating 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 ij so that ¨ (3.1) ij = Hij .Thequantity is 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.

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