Brane-World Localized Gravity and the Ads/CFT Correspondence

Brane-world localized gravity and the AdS/CFT correspondence

Author: Miguel Frias Perez∗ Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.

Advisor: Jaume Garriga Torres

Abstract: In this report, corrections to the gravitational potential of a massive point particle are computed for observers located in a brane-world scenario with intrinsic de Sitter (dS4) geometry embedded in an Anti de Sitter (AdS5) bulk. The results are compared, using the AdS/CFT correspondence, to a description in terms of four dimensional general relativity coupled to a conformal ﬁeld theory.

I. INTRODUCTION Gibbons-Hawking boundary term, and the others are given by (in Euclidean signature) In 1999, Randall and Sundrum proposed a geometric Z Z mechanism to solve the hierarchy problem [1, 2]. In their 3 4 p 4 l 4 p 4 S1 = d x g ,S2 = d x g R, scenario, matter fields are confined in four dimensional 8πG(5)l 32πG(5) branes, embedded in a higher-dimensional Anti de Sitter 3 Z 2 l ln(R/ρ) 4 p 4 µν R spacetime. By making the brane separation arbitrarily S3 = d x g Rµν R − . (3) large, they realized that even in the case of single brane 64πG(5) 3 embedded in AdS5, something close to four dimensional Here, G(5) is Newton’s constant in five dimensions. gravity can be recovered [2, 3], despite the presence of The four dimensional integrals are to be taken over the an arbitrarily large extra dimension. Here we focus on boundary, and so, the curvature tensors which appear this particular scenario. are the intrinsic ones. The R appearing in the logarithm of the third term measures the radius of the boundary, Our background geometry of interest consists of two and ρ is a finite renormalization length scale. identical balls of AdS5 (B1 and B2) glued alongside their Keeping this in mind, the action for the Randall- dS4 boundaries. A useful tool to analyze the dynamics of the spacetime in that setting is the gauge/gravity du- Sundrum scenario is given by ality, between supergravity in AdS5 (or in our case of interest, the classical limit of it) and a conformal field SRS = SEH + SGH + 2S1 + SM = SG + 2S1 + SM , (4) theory (CFT) living on its boundary. As stated in [4], the duality can be also formulated for finite regions of where 2S1 accounts for the action of a brane, located at the AdS spacetime, which amounts to a UV cut-off in the boundary of the two balls B1 and B2, with tension the dual theory. The correspondence states that the par- σ0 = 3/(4πG(5)l), and SM is the action for the matter tition functions for both theories are related as (g4 is the on the brane (later, we will take it to be the action for a induced metric on the boundary) point particle plus a vacuum energy contribution in order to sustain inflation of the brane). The partition function is then given by: Z Z 4 4 Z[g4] = d[g]e−Sgrav [g] = d[φ]e−SCFT [φ;g ] = e−WCFT [g ]. Z Z 2 4 −SRS −2S1−SM −SEH −SGH (1) ZRS [g ] = d[g]e = e d[g]e . The bulk path integral is taken over all the metrics g B1∪B2 B1 (5) that induce a conformal equivalence class of metrics g 4 The second integral can be related via the AdS/CFT over the boundary, and φ are the degrees of freedom of correspondence to the generating functional of the dual the CFT. The classical gravitational action for AdS is 5 theory plus two of the counterterms. Thus, we obtain a divergent, but counterterms which depend on the geome- dual description in terms of the 4D action try of the boundary can be introduced in order to render it finite [5]. Using them, the gravitational action reads S4D = −2S2 − 2S3 + SM + 2WCFT . (6)

The term S2 has exactly the form of Einstein-Hilbert Sgrav = SEH + SGH + S1 + S2 + S3, (2) action for the 4D brane geometry, and S3 renormalizes where SEH is the usual Einstein-Hilbert term with neg- the generating functional WCFT . Thus, we can see that 2 ative cosmological constant Λ = −6/l , SGH is the studying corrections to the gravitational potential on the brane in a 5D setting is analogous to a 4D analysis of the gravitational potential in dS4 coupled to a CFT. In what follows we do the bulk computation and compare it to ∗Electronic address: [email protected] the boundary result which was found in [6]. Brane-world localized gravity and the AdS/CFT correspondence Miguel Frias Perez

II. ACTION AND EQUATIONS OF MOTION III. BACKGROUND GEOMETRY

The gravitational action with a negative cosmological As we have stated in the introduction, we are inter- constant reads ested in a geometry consisting of two balls of AdS5 glued alongside its dS4 boundaries. For this purpose, consider an ansatz of the form: Z Z 1 5 √ (5) 12 4 √ 2w SG = d x −g R + 2 + 2 d x −gK , 16πG(5) l gµν = e γµν . (13) (7) The justification of this form is perhaps clearer in Eu- where K is the extrinsic curvature scalar and the integral clidean signature. The Euclidean version of dS4 is simply of the Gibbons-Hawking boundary term is to performed a 4-sphere, and so, γµν would simply be the its metric. over the brane. For simplicity, we work in Gaussian nor- In GNC, the fifth coordinate y is simply the physical dis- mal coordinates (GNC) with respect to the brane, tance between the spheres and the factor ew(y) accounts for the change in the volume of the spheres. The Ricci 2 M N 2 4 µ ν ds = gMN dx dx = dy + gµν dx dx . (8) tensor for dS4 is proportional to the metric, with a factor related to the dS length, or alternatively, to its Hubble In what follows, we drop the 4 superscript on the induced 4 parameter, metric, as it is the only one appearing in our equations. It is useful to express the five-dimensional curvature scalar R = 3H2γ . (14) in terms of its four-dimensional counterpart and extra µν µν terms related with the extrinsic curvature. In our sign Using this ansatz, the differential equation for the warp convention, the contraction of the Gauss-Codazzi rela- factor w(y) is: tions reads: 00 0 2 2 2 −2w 0 8 (5) 2 µν w + 2(w ) − 2 − H e = δ(y) w − πG(5)σ . (15) R = R − K − Kµν K − 2LnK. (9) l 3 Here n is the normal vector to the brane. Using this The solution is given by (after imposing the boundary slicing and carefully integrating by parts, we obtain conditions on the brane) y0 − |y| Z w(y) = ln Hl sinh , (16) 5 √ 1 µν 2 1 µν 12 l SG ∼ d x −g R+ (g ∂ygµν ) + (∂ygµν )(∂yg )+ . 4 4 l2 and the different parameters are related via (10) The action for the matter source on the brane is −1 3 p h y0 i σ = 1 + H2l2,H = l sinh . (17) 4πG(5)l l Z √ 1 2S + S = d5xδ(y) −σ −g + g T µν , (11) The flat brane results [2] are recovered with H −→ 0, and 1 M 2 µν the normal coordinate ranges from y ∈ (−y0, y0). where σ is the brane tension. The explicit form of T µν for a point particle will be discussed later. Imposing that the IV. PERTURBATIONS variations of the total action vanish, we find the equation of motion Studying general perturbations in this background be- comes rather involved, as the brane would be moved from 1 6 1 1 y = 0 to somewhere else. For this matter, we introduce Rµν − gµν R − gµν + ∂2gµν + gµν gαβ ∂2g + 2 l2 4 y 2 y αβ the perturbations so that we are still working in GNC with respect to the brane

1 µα νβ 2 1 αβ µν 3 µν αβ 2 − g g ∂ gαβ + g (∂ygαβ )(∂yg ) + g (∂yg ) + 2 2w µ ν 2 4 y 4 8 ds = e (γµν + hµν )dx dx + dy . (18) " In general, a linearised change of coordinates will 1 1 + (gαβ ∂ g )2gµν = δ(y) ∂ gµν + gµν gαβ ∂ g transform the metric as 8 y αβ 2 y y αβ −2w 2w h −→ h + e L 2w (e γ ) # MN MN e ξ MN µν µν 2w h 0 y y i −8πG(5) (−σg + T ) . (12) = hMN + e 2∇(M ξN) + 2w (y) γMN ζ + 2δ(N ξM) , (19) −2w Integrating the equation over a pill-box around y = 0 with γyy = e , γyµ = 0 and ∇N the covariant deriva- gives a condition on the normal derivative of the metric tive related to the metric γMN . We are interested in at the brane. studying the residual gauge freedom, keeping in mind

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−2 (0) that we want to restrict our study to normal coordinates where we have redeﬁned a ζν = Xν , as the generator respect to the wall. The conditions that we obtain after of gauge transformations. At linear order, the equations imposing the components hyM to be zero in every gauge for the diﬀerent modes are decoupled, and so we need not are worry about the tensor and vector modes. Then, our four variables of interest are the two Bardeen potentials ΦA, 0 0 ∂yξy = −3w ξy, ∂yξµ = −∂µξy − 2w ξµ. (20) ΦH and the two gauge variables X, X0 (X is the potential Keeping the brane at y = 0, amounts to ξy = 0. Then, of the curl-free part of the gauge vector Xi = X,i + Si the solution for the second condition is with Si,i = 0).

−2w (0) ν ξµ = e ξµ (x ), (21) B. Stress energy tensor and we are left with the usual (four dimensional) gauge freedom A point particle located at the origin has energy mo- (0) mentum tensor hµν −→ hµν + 2∇(µξν) . (22) Once we have justified the particular form of our per- −1 0 0 (3) 1 −1 0 0 1 Tµν = ma δ δ δ (x) = − ma δ δ ∆ , (27) turbations, we can study its linearized equations of mo- µ ν 4π µ ν r tion. For that matter, it is useful to introduce a new which has indeed the perfect fluid form with zero coordinate orthogonal to the brane, ζ = e|y|/l. Under solenoidal velocity. This justifies our omission of the vec- this change, we are imposing symmetry under y −→−y, tor modes. The energy momentum tensor satisfies the as we cannot study any other case with it. To first order appropriate conservation law. in hµν , the equations read

2 ρ 2 ∇ hµν − 2∇ ∇(µhν)ρ + ∇µ∇ν h + 6H hµν (23) V. SOLUTION FOR THE SCALAR PART 1 8πG(5) +Dζ hµν + γµν h = −δ(ζ − 1) (2Tµν − γµν T ) , 2 l After decomposing the linearized version of Einstein’s where ∇µ is the covariant derivative associated with the equation, four differential equations for the scalar modes induced metric in the brane γµν (traces are also with re- are found. As we are studying the gravitational poten- gard to this metric). We have also introduced a derivative tial of a point mass, we would like to search for a static operator in the bulk coordinate, given by solution. The existence of one is clearer if we introduce cosmological time and physical distances via e2w D = ζ2∂2 + (ζ + 4ζ2∂ w)∂ + 2δ(ζ − 1)∂ . (24) ∂2 ζ l2 ζ ζ ζ ζ xî(t) = a(t)xi = eHtxi,P = −δkl . (28) ∂xˆk∂xˆl At this point, it is necessary to introduce some coordi- Then, the following equations are obtained: nates on the brane. For simplicity, we can study the dS4 geometry of the domain wall in terms of conformally flat 1 2P Φ − D 6Φ − 2P Ξ − 6H2Ξ + 3HΞ˙ − 3HΞ coordinates. These coordinates cover only half of the hy- H 2 ζ H 0 perboloid [7], but are suitable for our study as we are in- 2G m 1 = δ(ζ − 1) (5) P , terested in a local property, such as the gravitational po- l rˆ tential, and these coordinates can be constructed around every point of dS . They take the form of a flat FLRW 4 1 geometry, with scale factor a(η) = −1/(Hη), 2P Φ − D 6Φ − 3Ξ¨ + 6H2Ξ + 3HΞ˙ + 3Ξ˙ − 3HΞ + 4P Ξ A 4 ζ A 0 0 −1 2 2G(5)m 1 ds2 = γ dxµdxν = η dxµdxν . (25) = δ(ζ − 1) P , µν Hη µν l rˆ

2 Dζ 6Φ˙ H + 6HΦH + 6HΦA − P Ξ˙ + P Ξ0 − 6H Ξ0 = 0, A. Decomposition

As dS is maximally symmetric, we can classify the 2 4 D 4Φ − 2Φ + Ξ¨ − 6H Ξ + HΞ˙ − Ξ˙ − HΞ = 0. perturbations according to their behaviour under spatial ζ H A 0 0 rotations of the spacetime. The diﬀerent components of (29) the perturbations can transform either as a scalar, vector Overdots denote derivatives with regard to cosmological time, and we have redeﬁned the gauge variables as Ξ = or tensor. The metric decomposed in terms of the scalar 2 perturbations reads a X and Ξ0 = aX0. As there is no explicit dependence on time, both in the equation and in the source, we can search for a static solution. After some algebra, we can 2 0 0 hµν = a 2δµδν (ΦA + ΦH ) + 2ηµν ΦH + 2∇(µXν) , (26) write the equations as

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(regularity at the horizon) was not applicable, as it was 2 2G(5)m Dζ Σ = (P − 6H ) Σ − δ(ζ − 1) , (30) already regular. The third equation in (31) does depend lrˆ in ∆(0), and both boundary conditions apply to Ξ(0).

2 2 2G(5)m Luckily, imposing the boundary conditions for the latter D ∆ = 2H (Σ − ∆) − 3H δ(ζ − 1) , (0) ζ lrˆ case, determines the form of ∆ . Having cleared the first order, we can move on to the second order. 1 2G m D Ξ = (Σ + 2∆),D Ξ = −6HΣ + 6Hδ(ζ − 1) (5) , ζ 3 ζ 0 lrˆ If H 6= 0, the horizon is at finite coordinate distance. In terms of the old coordinate, we must now impose reg- where Σ = ΦA + ΦH and ∆ = ΦA − 2ΦH are simply ularity at y0. Using Eq. (17), we can see that in terms combinations of the Bardeen potentials. of the new coordinate, it is located at h i 1 p 2 2 ζ0 = 1 + 1 + H l . (34) A. Expansion in H2 Hl Problems arise right here. In our expansion in H2, Σ = (0) 2 (1) (i) Despite the simple form of the equation for the scalar Σ +H Σ +..., the Σ at every order do not depend modes, and the fact that we will need only the first three in H. Trying to impose regularity (at every order) at a to solve for the Bardeen potentials, obtaining an exact point whose location depends on H, would be inconsistent solution is a challenging task. with the expansion. But, it is interesting to recall that we are considering the first order where H 6= 0 but it is One possible alternative is to expand the solution in arbitrarily small. Thus, we should impose regularity at a 2 point at finite distance, but arbitrarily far away. Keeping powers of H , and try to find a perturbative result. At 2 first order, we would expect to recover the flat brane re- this in mind, the relevant second order in H differential sults [8], and the second order would introduce the first equations are corrections due to the intrinsic geometry of the domain 2G m wall. This expansion will enable us to solve the differ- D(0) − P Σ(1) = −D(1)Σ(0) − 6Σ(0) + 6δ(ζ − 1) (5) , ential equations, but will unfortunately complicate the ζ ζ lrˆ task of imposing the appropriate boundary conditions. 2G m At order 0, the equations read D(0)∆(1) = −D(1)∆(0) − 2∆(0) + 2Σ(0) − 3δ(ζ − 1) (5) , ζ ζ lrˆ 2G m 1 D(0)Σ(0) = P Σ(0) − δ(ζ − 1) (5) ,D(0)∆(0) = 0, D(0)Ξ(1) = −D(1)Ξ(0) + Σ(1) + 2∆(1) , (35) ζ lrˆ ζ ζ ζ 3 where the RHS are strictly source terms. The derivative (1) (0) (0) 1 (0) (0) (0) (0) operator D can be obtained by expanding in H2 Dζ Ξ = (Σ + 2∆ ),Dζ Ξ0 = 0, (31) ζ 3 Eq.(24) (as we have done for one of the combinations with the differential operator D(0) given by (1) ζ of the Bardeen potentials, we keep the terms Dζ 1 independent of H in the expansion). D(0) = ζ2∂2 − 3ζ∂ + 2δ(ζ − 1)∂ . (32) ζ l2ζ2 ζ ζ ζ The delta terms enforce discontinuity of the derivative of The solution to the first of the three equations is a lin- the metric (in this particular case, of a combination of the ear combination of two modified Bessel functions (both scalar modes appearing in the metric). This amounts to resemble exponentials, one decaying and the other grow- one of the boundary conditions needed to solve the linear ing) plus some terms which are solution of the inhomo- second order differential equation. The second one can be geneous part, which decay as we move away from the obtained by imposing that the horizons remain regular. brane. Taking a combination of the two solutions so that At order zero (H = 0) the horizons are located at infinite regularity is achieved at the horizon (which for our pur- coordinate distance ζ −→∞, and so, the second condition poses, can be taken as far away as we desire), would give amounts to cancelling all the divergent terms (terms that exponentially suppressed corrections to the result in the diverge including the exponential of the warp factor e2w brane. For that reason, we take the coefficient of the appearing on the metric) as ζ −→∞. The solution is then growing mode to be zero and impose the brane condition on the second one. Using again the equation for one of (with Kn a modified Bessel function of the second kind) the gauge potentials, as the equation for ∆(1) is uncon- √ √ strained by fall-off conditions, we obtain the second order 2 PG mζ K2( P lζ) G m Σ(0) = (5) √ , ∆(0) = − (5) . (33) solution. In order to analyze the behaviour on the brane, K1(l P )ˆr lrˆ we take ζ = 1:

To solve the equation for ∆(0) we need to solve the √ PG m √ G m equation at order zero for the gauge function Ξ(0). This Σ(0) = (5) K P l , ∆(0) = − (5) , √ 2 lrˆ is due to the fact that the second boundary condition K1 l P rˆ

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brane case studied by Garriga and Tanaka. The second lG m Σ(1) = l2P + 15 (5) order result can checked with those of [6] if we take into 6ˆr account the relation between the constants appearing in G(5)lm 2 2 1 the two cases: − (12 + l P )K2/K1 − (24 + 3l P ) √ K2/K1, 6ˆr l P 2 (1) G(5)m (4 + l P ) G(5)m 2 2 ∆ = 2 √ K2/K1 − , (36) G(5) = lG(4), l = 64πblPL. (40) P rˆ P lrˆ where K2/K1 is the ratio of Bessel functions evaluated at the same point as in the order 0 solution. In order to G(4) is the four dimensional gravitational constant (with obtain the solution, we still need to eliminate the Lapla- p 3 Planck length lPL = ~G(4)/c ) and b is the central cians appearing in the expression and recombine Σ√ and charge of the CFT, related to the number of free ﬁelds as ∆ into the Bardeen potentials. Expanding for small P l

√ G m l P G m N0 + 6N1/2 + 12N1 Σ(0) = (5) 1 − P l2 γ + ln , ∆(0) = − (5) , b = , (41) lrˆ 2 lrˆ 1920π2 √ G ml 17 l P Σ(1) = (5) + 4γ + 4 ln , rˆ 6 2 where N0 is the number of conformally coupled scalar √ fields, N1/2 massless spinor fields and N1 vectors. Their G ml l P ∆(1) = − (5) 1 + 2γ + 2 ln , (37) analysis was based in computing first the gravitational rˆ 2 potential of a test mass in dS4, then studying the back where γ is Euler’s constant, which appears in the expan- reaction to it from the non-zero expected value of the sion of the ratio of Bessel functions for small arguments. energy momentum tensor of conformal fields in the new The result contains no negative powers of the Laplacian, geometry. as they would introduce non-local terms in our analysis. In order to get rid of the Laplacians it is useful to recall Our first order results coincide with [6], but there is two results: some discrepancy involving the second order solution. √ 1 lnr ˆ + γ lnr ˆ 1 From our point of view, there are two possible reasons ln P = − ,P = . (38) rˆ rˆ rˆ rˆ3 for this difference, either the term S3 in the four dimen- Which leads to the final expression sional action plays a role that we have not considered, or our assumptions over the second order boundary conditions are wrong. This is left for further research. G m 2l2 14 l Φ = (5) 1 + + H2l2 + 2 ln , A lrˆ 3ˆr2 9 2ˆr

2 G(5)m l 2 2 23 l ΦH = 1 + + H l + 2 ln . (39) lrˆ 3ˆr2 18 2ˆr Acknowledgments

VI. DISCUSSION I would like to thank my advisor Dr. Jaume Garriga for a patient and very involved guidance, and Dr. Markus The order zero in H2 results agree from those in [8]. Fr¨obfor advice when we were stuck. I also thank my This is what we expected, as H = 0 is simply the ﬂat family for their support and implication in my education.

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