DAMTP February 4, 2008 University of Cambridge The Einstein Equations on the 3-Brane World Tetsuya Shiromizu1,3,4, Kei-ichi Maeda2,5 and Misao Sasaki2,3,6 1DAMTP, University of Cambridge Silver Street, Cambridge CB3 9EW, United Kingdom 2Isaac Newton Institute, University of Cambridge, 0 0 20 ClarksonRoad, Cambridge CB3 0EH, United Kingdom 0 2 3Department of Physics, The University of Tokyo, Tokyo 113-0033,Japan n a 4Research Centre for the Early Universe(RESCEU), J The University of Tokyo, Tokyo 113-0033,Japan 7 1 5Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555,Japan 3 v 6Department of Earth and Space Science, Graduate School of Science, 6 Osaka University, Toyonaka 560-0043,Japan 7 0 0 We carefully investigate the gravitational equations of the brane world, in which all the matter 1 9 forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with Z2 symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional 9 / Einstein equations in the low energy limit. From our general argument we conclude that the first c Randall& Sundrum-typetheory (RS1)[hep-ph/9905221] predictsthatthebranewith thenegative q tension is an anti-gravity world and hence should be excluded from the physical point of view. - r Theirsecond-typetheory(RS2)[hep-th/9906064] wherethebranehasthepositivetensionprovides g the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti-de Sitter, : v generically thematteron thebraneisrequired tobespatially homogeneous becauseof theBianchi i identities. By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and X the Bianchi identities give just the relation between the Weyl tensor and the energy momentum r tensor. Inthepresentbraneworldscenario,theeffectiveEinsteinequationsceasetobevalidduring a an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by thepotential energy of thescalar field. PACS: 04.50.+h; 98.80.Cq DAMTP-1999-150; NI99018-SFU; UTAP-349; RESCEU-40/99; WU-AP/85/99;OUTAP-103 Recent progress in the superstring theory tells us that we are living in 11 dimensions [1], and different string theories are connected with each other via dualities. Among string theories, the 10-dimensional E ×E heterotic 8 8 string theory is a strong candidate for our real world because the theory may contain the standard model. Recently Horava & Witten showed that the 10-dimensionalE ×E heterotic string is related to an 11-dimensional theory on 8 8 the orbifold R10×S1/Z [2]. Therein the standard model particles are confined to the 4-dimensional spacetime. On 2 the other hand, gravitons propagate in the full spacetime. This situation can be simplified to a 5-dimensional problem where matter fields are confined to the 4-dimensional spacetime while gravity acts in 5 dimensions. In this category much work has been done. Among of all, the pioneer workinspacetimewithoneextradimensionhasbeendonebyRandall&Sundrum[3,4][5]whereourbraneisidentical to a domain wall in 5-dimensional anti-de Sitter spacetime. In their first paper [3], they proposed a mechanism to solve the hierarchy problem by a small extra dimension. In their second paper [4], the brane world with a positive tension was investigated. Then a non-perturbative aspect of the theory was investigated [6]. The final fate of gravitationalcollapse was discussed in the brane world picture [7]. The inflation solution has been discovered[8–10]. Inthesetreatments,however,thecontributionfrommatterexcitationshasnotbeenseriouslyconsidered. Suchworkis partiallyperformedinacosmologicalcontextlinkedtotheconventionalFriedmanequation[11–14]. Thecosmological solution associated with the heterotic string theory also has been constructed [15]. We mention work on the brane worldmotivated by the hierarchyproblem. Before Randall& Sundrum’s work,largeextra dimensions were proposed to solve the hierarchy problem [16]. The related cosmology also has been actively investigated [17]. Inthispaper,wederivetheeffectiveEinsteinequationsonthe3-brane. Forsimplicitythebulkspacetimeisassumed tohave5dimensions. Inthebeginningwedonotassumeanyconditionsonthebulkspacetime. Later,weassumethe Z -symmetry and confinement of the matter energy momentum tensor on the brane, in accordance with the brane 2 world scenario based on the Horava & Witten theory [2]. The notation basically follows Wald’s text [18]. In the brane world scenario, our 4-dimensional world is described by a domain wall (3-brane) (M,q ) in 5- µν dimensional spacetime (V,g ). We denote the vector unit normal to M by nα and the induced metric on M by µν q =g −n n . We start with the Gauss equation, µν µν µ ν (4)Rα =(5)Rµ q αq νq ρq σ+KαK −KαK , (1) βγδ νρσ µ β γ δ γ βδ δ βγ and the Codacci equation, D K ν −D K =(5)R nσq ρ, (2) ν µ µ ρσ µ where the extrinsic curvature of M is denoted by K = q αq β∇ n , K = Kµ is its trace, and D is the covariant µν µ ν α β µ µ differentiation with respect to q . Contracting the Gauss equation (1) on α and γ, we find µν (4)R =(5)R q ρq σ−(5)Rα n q βnγq δ+KK −K αK . (3) µν ρσ µ ν βγδ α µ ν µν µ να This readily gives 1 1 (4)G = (5)R − g (5)R q ρq σ +(5)R nρnσq +KK −K ρK − q (K2−KαβK )−E˜ , (4) µν (cid:20) ρσ 2 ρσ (cid:21) µ ν ρσ µν µν µ νρ 2 µν αβ µν where E˜ ≡(5)Rα n nρq βq σ. (5) µν βρσ α µ ν Using the 5-dimensional Einstein equations, 1 (5)R − g (5)R=κ2T , (6) αβ 2 αβ 5 αβ where T is the 5-dimensional energy-momentum tensor, together with the decomposition of the Riemann tensor µν into the Weyl curvature, the Ricci tensor and the scalar curvature; 2 1 (5)R = (g (5)R −g (5)R )− g g (5)R+(5)C , (7) µανβ 3 µ[ν β]α α[ν β]µ 6 µ[ν β]α µανβ we obtain the 4-dimensional equations as 2κ2 1 1 (4)G = 5 T q ρq σ+ T nρnσ− Tρ q +KK −K σK − q K2−KαβK −E , (8) µν 3 (cid:18) ρσ µ ν (cid:18) ρσ 4 ρ(cid:19) µν(cid:19) µν µ νσ 2 µν αβ µν (cid:0) (cid:1) where E ≡(5)Cα n nρq βq σ. (9) µν βρσ α µ ν Note that E is traceless. From the Codacci equation (2) and the 5-dimensional Einstein equations (6), we find µν D K ν −D K =κ2T nσq ρ. (10) ν µ µ 5 ρσ µ So far we have not assumed any particular symmetry nor particular form of the energy momentum tensor. From now on, we take a brane worldscenario. For convenience,we choose a coordinate χ such that the hypersurfaceχ=0 coincides with the brane world and n dxµ =dχ, which implies µ 2 aµ =nν∇ nµ =0. (11) ν This is a condition on the coordinate in the direction of the extra dimension. We assume this choice is possible at least in the neighbourhood of the brane, (M,q ). In more explicit terms, we assume the 5-dimensional metric to µν have the form, ds2 =dχ2+q dxµdxν. (12) µν Bearing brane world spirit in mind, we assume that the 5-dimensional energy-momentum tensor has the form T =−Λg +S δ(χ), (13) µν µν µν where S =−λq +τ , (14) µν µν µν with τ nν = 0. Λ is the cosmological constant of the bulk spacetime. λ and τ are the vacuum energy and the µν µν energy-momentum tensor, respectively, in the brane world. Note that λ is the tension of the brane in 5 dimensions. Properly speaking S should be evaluated by the variational principle of the 4-dimensional Lagrangian for matter µν fields because the normal matter except for gravity is assumed to be living only in the χ = 0 brane. It should be noted that the decomposition of S into λq and τ can be ambiguous, particularly in cosmologicalcontexts. µν µν µν The singular behaviour in the energy-momentum tensor leads us to the so-called Israel’s junction condition [19], [q ] =0, µν 1 [K ] =−κ2 S − q S , (15) µν 5 µν 3 µν (cid:16) (cid:17) where [X]:=lim X −lim X =X+−X−. χ→+0 χ→−0 Now we impose the Z -symmetry on this spacetime, with the brane as the fixed point. Interestingly the symmetry 2 uniquely determines the extrinsic curvature of the brane in terms of the energy momentum tensor, 1 1 K+ =−K− =− κ2 S − q S . (16) µν µν 2 5 µν 3 µν (cid:16) (cid:17) Hereafterwe focus ourattentiononquantities evaluatedonthe brane. Becauseofthe Z -symmetry,wemay evaluate 2 quantities either on the + or − side of the brane. Hence we omit the indices ± below for brevity. Substituting Eq. (16) into Eq. (8), we obtain the gravitationalequations on the 3-brane in the form, (4)G =−Λ q +8πG τ +κ4π −E , (17) µν 4 µν N µν 5 µν µν where 1 1 Λ = κ2 Λ+ κ2λ2 , (18) 4 2 5(cid:18) 6 5 (cid:19) κ4λ G = 5 , (19) N 48π 1 1 1 1 π =− τ τ α+ ττ + q τ ταβ − q τ2, (20) µν 4 µα ν 12 µν 8 µν αβ 24 µν and E is the part of the 5-dimensional Weyl tensor defined in Eq. (9). It should be noted that E in the above µν µν is the limiting value at χ = +0 or −0 but not the value exactly on the brane. This is our main result. It resembles the conventional Einstein equations in 4 dimensions. In fact, the Einstein equations can be recovered by taking the limit κ →0 while keeping G finite. Nevertheless there are some important differences. As can be easily seen, the 5 N existence of Newton’s gravitational constant G strongly relies on the presence of the vacuum energy λ. In other N words, it becomes impossible to define Newton’s gravitational constant during an era when the distinction between the vacuumenergyandthe normalmatter energyisambiguous. Furthermore,wewouldhavethe wrongsignofG if N λ<0 [12]. The π term, which is quadratic in τ could play a very important role, especially in the early universe µν µν when the matter energy scale is high [11–13]. Inadditiontothesefeaturesthathavebeenpointedoutpreviously,Eq.(17)containsanewterm,E . Itisapartof µν the 5-dimensionalWeyltensorandcarriesinformationofthe gravitationalfield outsidethe brane. It isnon-vanishing 3 if the bulk spacetime is not purely anti-de Sitter. At the same time, it is not freely specifiable but is constrained by the motion of the matter on the brane. Let us show this feature now. Together with Eq. (16), Eq. (10) implies the conservation law for the matter, D K ν −D K ∝D τ ν =0. (21) ν µ µ ν µ Therefore the contracted Bianchi identities Dµ(4)G =0 imply the relation between E and τ as µν µν µν DµE =Kαβ(D K −D K ) µν ν αβ β να 1 1 = κ4 ταβ(D τ −D τ )+ (τ −q τ)Dµτ . (22) 4 5 ν αβ β να 3 µν µν h i Thus E is not freely specifiable but its divergence is constrained by the matter term. If one further decomposes µν E into the transverse-traceless part, ETT, and the longitudinal part, EL , the latter is determined completely by µν µν µν the matter. Hence if the ETT part is absent, the equations will be closed solely with quantities that reside in the µν brane. However, as usually the case in the conventional gravity, the ETT part corresponds to gravitational waves or µν gravitons in 5 dimensions, and they will be inevitably excited by matter motions and their excitations affect matter motions in return. This implies the effective gravitational equations on the brane are not closed but one must solve the gravitational field in the bulk at the same time in general. Since the derivation of equations that govern the evolution of ETT is technically complicated, we defer it to Appendix A. µν Letus nowestimate the effect ofeachtermonthe right-handside ofEq.(17). We setκ−2 =M3 andλ=M4,and 5 G λ assume Λ = O(κ2λ2). It should be noted that these do not have to be planck scale quantities. One can scale them 5 as M → f2M and M → f3M , where f is an arbitrary constant, while keeping the gravitational constant G G G λ λ N unchaged. Nevertheless, here we assume M and M to be sufficiently large compared to the characteristic energy G λ scale of the matter which we denote by M. The first term on the right-hand side of Eq. (17) is the net cosmological constant in 4 dimensions. It is assumed that Λ < 0. Hence Λ may take arbitrary value as one may wish by appropriately specifying the values of Λ and λ. 4 The second term is the contribution from normal matters which should satisfy the local energy condition (assuming the decomposition of S into λ and τ is well-defined). The π term which is quadratic in τ is expected to be µν µν µν µν negligible in the low energy limit. In fact, the ratio of these terms to the third term is approximately given by κ4|π | κ4|τ τ α+···| M4 5 µν ∼ 5 µα ν ∼ . (23) G |τ | G |τ | M4 N µν N µν λ We now turn to the Weyl tensor part. First let us consider the longitudinal part EL . Since it is determined by µν τ through Eq. (22), we have µν |EL | κ4|τ τ α+···| M4 µν ∼ 5 µα ν ∼ . (24) G |τ | G |τ | M4 N µν N µν λ This is the same order of magnitude as the π term. Second, we consider the ETT part. We focus on the effect due µν µν to matter excitations on the brane. Here we borrow the discussion of [4] to evaluate the leading order of magnitude of its effect. The gravitationalpotential between two bodies onthe brane is modified via exchangeof gravitonsliving in 5 dimensions as [4] G m m 1 N 1 2 V(r)∼ 1+ , (25) r r2k2 (cid:16) (cid:17) where r is the distance between the two bodies and k = κ2λ/6. Since this effect must be contained in Eq. (17), it 5 should appear in the E term. Therefore as a conservative estimate, we obtain µν |E | M2 M6M2 µν ∼ ∼ G . (26) G |τ | k2 M8 N µν λ Thus E is also negligible in the low energy world. It is, however, worth noting that this term is larger than µν the terms quadratic in τ . The deviation from the ordinary Einstein equations in 4-dimensions first appears from µν gravitationalexcitationsinthebulkspacetime. Fromtheaboveestimationsweconcludethattheeffectivegravitational 4 equation (17) on the brane reduce to the 4-dimensional conventional Einstein gravity, (4)G ≃−Λ q +8πG τ , µν 4 µν N µν in the low energy limit. The presence of a well-defined cosmologicalconstant λ is obviously essential here.∗ Finally, we note an outcome of the constraint (22). We consider the case when the bulk spacetime is pure anti-de Sitter with E =0 and investigate the condition on the matter on the brane. For simplicity, we assume the perfect µν fluid form for the energy momentum tensor: τµν =ρtµtν +Phµν, (27) where hµν = qµν +tµtν. The quadratic term πµν in the 4-dimensional effective gravitational equations (17) then becomes 1 πµν = ρ(ρtµtν +(ρ+2P)hµν) . (28) 12 The normal conservation law D τµν =0 implies ν tµD ρ+(ρ+P)D tµ =0 and (ρ+P)tνD tµ+hµνD P =0. (29) µ µ ν ν If E =0, the 4-dimensional Bianchi identities imply D πµν =0, which gives µν ν 1 D πµν = (ρ+P)hµνD ρ=0. (30) ν ν 6 This means ∂ ρ=0. Hence an inhomogeneous perfect fluid is rejected. i We briefly summarize the present work. We first derived the effective 4-dimensional gravitational equations in 5 dimensions, Eq. (8), without any particular assumptions specific to the brane world scenario. Then based on the brane world scenario, we introduced the Z symmetry and assumed that the matter lives only on the brane, and 2 derivedthe4-dimensionaleffectivegravitationalequationsonthebrane,Eq.(17). Theequationtellsusthatanormal gravitational theory can be obtained on the brane only if the tension is positive, while an RS1-type theory [3] in which the brane has negative tension is rejected from the physical point of view (see also [12] for Friedmann cases). In the case of the brane with positive tension, the Einstein gravity is recovered in the low energy limit. Placing the brane in the 5-dimensional exact anti-de Sitter spacetime imposes a strong condition on the matter in 4-dimensions. Inparticular,ifthe matterenergy-momentumtensorhastheperfectfluidform,onlyspatiallyhomogeneousuniverses are allowed. Conversely, this means that the deviation of the bulk spacetime from the exact anti-de Sitter spacetime is essential to describe our real world with matter fields. ACKNOWLEDGEMENTS TS would like to thank Gary W. Gibbons and DAMTP relativity group for their hospitality at Cambridge. Sub- stantial part of this work was done while KM and MS were participating the program, “Structure Formation in the Universe”, at the Newton Institute, University of Cambridge. We are grateful to the Newton Institute for their hospitality. TS’s work is supported by JSPS Postdoctal Fellowship for Research Abroad. KM’s work is supported in part by Monbusho Grant-in Aid for Specially Promoted Research No. 08102010. MS’s work is supported in part by Monbusho Grant-in-Aid for Scientific Research No. 09640355. APPENDIX A: We derive the evolution equation of E to make our system of equations closed. First, we write down the Weyl µν tensor formulas. The n-dimensional Riemann tensor is written in terms of the Weyl and Ricci tensors as 2 2 (n)R =(n)C + (n)R g −(n)R g − (n)Rg g . (A1) αβµν αβµν n−2 α[µ ν]β β[µ ν]α (n−1)(n−2) α[µ ν]β (cid:16) (cid:17) ∗ AswecanseefromthefirstterminEq.(8),thereductiontothenormalEinsteingravityisalsopossiblewiththeintroduction of non-trivial bulk energy-momentum tensor [20]. 5 We decompose the Weyl tensor into the ‘electric’ and ‘magnetic’ parts: E ≡(n)C nαnβ, (A2) µν µανβ and B ≡qρqσ(n)C nβ. (A3) µνα µ ν ρσαβ B and E have the symmetry, µνα µν B =−B , B =0, Bα =0. αβµ βαµ [αβµ] βα E =E , Eα =0. (A4) αβ βα α The algebraic degrees of freedom are n2(n2−1) (n−3)n(n+1)(n+2) (n)R ··· , (n)C ··· , αβµν αβµν 12 12 n(n+1) (n−4)(n−1)n(n+1) (n)R ··· , (n−1)C ··· , µν αβµν 2 12 (n−3)(n−1)(n+1) (n−2)(n+1) B ··· , E ··· . (A5) αβµ αβ 3 2 The n-dimensional Weyl tensor can be written in terms of (n−1)C , E , B and the extrinsic curvature K , αβµν αβ µνα µν (n)C =(n−1)C +2B n +2B n αβµν αβµν αβ[µ ν] µν[α β] 1 + 2E n n −2E n n − 2E q −2E q α[µ ν] β β[µ ν] α n−3 α[µ ν]β β[µ ν]α (cid:0) (cid:1) (cid:0) (cid:1) 2 2 −f + q f −q f − fσ q q , (A6) αβµν n−3 α[µ ν]β β[µ ν]α (n−2)(n−3) σ α[µ ν]β (cid:0) (cid:1) where q =g −n n and µν µν µ ν f ≡K K −K K , αβµν αµ βν αν βµ f ≡f σ =fσ =KK −K Kσ =f , µν µ νσ µσν µν µσ ν νµ fµ =fµν =K2−KµνK . (A7) µ µν µν Fromnowonwesetn=5andderivethe evolutionequationsofE fromthe 5-dimensionalBianchiidentities. We µν assume aµ =nα∇ nµ =0. For convenience, we define E˜ and B˜ from the Riemann tensor, α µν µνα E˜ ≡(5)R nαnβ =−£ K +K Kα, (A8) µν µανβ n µν µα ν B˜ ≡qβqσ(5)R nρ =2D K . (A9) µνα µ ν βσαρ [µ ν]α These are related to E and B as µν µνα 1 1 1 E =E˜ − q (5)R nαnβ − qαqβ(5)R + q (5)R µν µν 3 µν αβ 3 µ ν αβ 12 µν 1 1 2 1 1 1 1 =− (4)R − q (4)R − £ K − q K + K Kα+ q K Kαβ − K2 , (A10) 3 µν 4 µν 3 n µν 4 µν 3 µα ν 4 µν αβ 3 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 2 B =B˜ + (D Kβ −D K)qα µνα µνα 3 β [µ [µ ν] 2 =2D K + (D Kβ −D K)qα . (A11) [µ ν]α 3 β [µ [µ ν] The 5-dimensional Bianchi identities are ∇ (5)R =0, (A12) [µ να]βσ from which we obtain the following four sets of identities: 6 D B˜ β +Kσ(4)R β =0, (A13) [µ να] [µ να]σ £ B˜ +2D E˜ −KσB˜ +2B˜ Kσ =0, (A14) n µνα [µ ν]α α µνσ ασ[µ ν] £ (4)R +2(4)R Kσ +2D B˜ =0, (A15) n µναβ µνσ[α β] [µ |αβ|ν] D (4)R =0. (A16) [µ να]βσ From Eq. (A11) and the Israel’s junction condition, we obtain 1 [B˜ ]=2D [K ]=−2κ2D τ − q τ . (A17) µνα [µ ν]α 5 [µ ν]α 3 ν]α (cid:16) (cid:17) Thus the Z -symmetry uniquely determines the value of B on the brane as 2 µνα B˜+ =−B˜− µνα µνα 1 =−κ2D τ − q τ , 5 [µ ν]α 3 ν]α (cid:16) (cid:17) 2 B+ =2D K+ + D K+β −D K+ q µνα [µ ν]α 3 β [µ [µ ν]α (cid:16) (cid:17) =B˜+ . (A18) µνα These equations give the boundary conditions on the brane when one solves the evolution of E in 5 dimensions. µν The equations that governthe evolution of E in the bulk (i.e., in the spacetime region away from the brane) are µν obtained as follows. Using the 5-dimensional Einstein equations (6), Eq. (A14) yields £ B =−2D E +KσB −2B Kσ , (A19) n µνα [µ ν]α α µνσ ασ[µ ν] in the bulk. Also, using Eq. (6) and (8), Eq. (A15) gives 1 £ E =DµB + κ2Λ(K −q K)+Kµν(4)R n αβ µ(αβ) 6 5 αβ αβ µανβ +3Kµ E −KE +(K K −K K )Kµν, (A20) (α β)µ αβ αµ βν αβ µν in the bulk. Together with the 4-dimensional Einstein equations (8) in the bulk, Eqs. (A19) and (A20) form a closed system of equations. 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