gr-qc/0309006 Universes encircling 5-dimensional black holes Sanjeev S. Seahra and Paul S. Wesson ∗ † Department of Physics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (Dated: September 1, 2003) Abstract Weclarifythestatusoftwoknownsolutionstothe5-dimensionalvacuumEinsteinfieldequations derivedbyLiu,Mashhoon&Wesson(lmw)andFukui,Seahra&Wesson(fsw),respectively. Both 5-metrics explicitly embed 4-dimensional Friedman-Lemaˆıtre-Robertson-Walker cosmologies with 4 0 a wide range of characteristics. We show that both metrics are also equivalent to 5-dimensional 0 topological black hole (tbh) solutions, which is demonstrated by finding explicit coordinate trans- 2 formations from the tbh to lmw and fsw line elements. We argue that the equivalence is a direct n a consequence of Birkhoff’s theorem generalized to 5 dimensions. Finally, for a special choice of J parameters we plot constant coordinate surfaces of the lmw patch in a Penrose-Carter diagram. 7 This shows that the lmw coordinates are regular across the black and/or white hole horizons. 2 v 6 0 0 9 0 3 0 / c q - r g : v i X r a 1 I. INTRODUCTION Over the past few years, there has been a marked resurgence of interest in models with non-compact or large extra-dimensions. Three examples of such scenarios immediately come to mind — namely the braneworld models of Randall & Sundrum1,2 (henceforth rs) and Arkani-Hamed, Dimopoulos & Dvali3,4,5 (henceforth add), as well as the older Space-Time- Matter (stm) theory6. The rs model is motivated from certain ideas in string theory, which suggest that the particles and fields of the standard model are naturally confined to a lower-dimensional hypersurface living in a non-compact, higher-dimensional bulk manifold. The driving goal behind the add picture is to explain the discrepancy in scale between the observed strength of the gravitational interaction and the other fundamental forces. This is accomplished by noting that in generic higher-dimensional models with compact extra dimensions, the bulk Newton’s constant is related to the effective 4-dimensional constant by factors depending on the size and number of the extra dimensions. Finally, stm or induced matter theory proposes that our universe is an embedded 4-surface in a vacuum 5-manifold. In this picture, what we perceive to be the source in the 4-dimensional Einstein field equations is really just an artifact of the embedding; or in other words, conventional matter is induced from higher-dimensional geometry. Regardless of the motivation, if extra dimensions are to betaken seriously then it is useful to have as many solutions of the higher-dimensional Einstein equations at our disposal as possible. These metrics serve as both arenas in which to test the feasibility of extra dimensions, as well as guides as to where 4-dimensional general relativity may break down. This simplest type of higher-dimensional field equations that one might consider is the 5- dimensional vacuum field equations Rˆ = 0. (In this paper, uppercase Latin indices run AB 0...4 while lowercase Greek indices run 0...3, and 5-dimensional curvature tensors are distinguished from the 4-dimensional counterparts by hats. Also, commas in subscripts indicate partial differentiation.) This condition is most relevant to the stm scenario, but can also be applied to the rs or add pictures. The are a fair number of known solutions that embed 4-manifolds of cosmological or spherically-symmetric character; one can consult the book by Wesson6 for an accounting of these metrics. However, when searching for new solutions to vacuum field equations, one must keep in mindaknown perilfrom4-dimensionalwork; i.e., anynewsolutioncouldbeapreviously dis- covered metric written down in terms of strange coordinates. Our purpose in this paper is to demonstrate that two 5-dimensional vacuum solutions intheliterature areactuallyisometric to a generalized 5-dimensional Schwarzschild manifold. Both of these solutions have been previously analyzed in the context of 4-dimensional cosmology because they both embed submanifolds with line elements matching that of standard Friedman-Lemaˆıtre-Robertson- Walker (flrw) models withflat, spherical, or hyperbolic spatialsections. InSection IIA, we discuss the first of these 5-metrics, which was originally written down by Liu & Mashhoon7 and later rediscovered in a different form by Liu & Wesson8. We will see that this met- ric naturally embeds flrw models with fairly general, but not unrestricted, scale factor behaviour. Several different authors have considered this metric in a number of different contexts9,10,11,12, including the rs braneworld scenario. The second 5-metric — which was discovered by Fukui, Seahra & Wesson13 and is the subject of Section IIB — also embeds flrw models with all types of spatial curvature, but the scale factor is much more con- strained. We will pay special attention to the characteristics of the embedded cosmologies in each solution, as well as the coordinate invariant geometric properties of the associated 2 bulk manifolds. The latter discussion will reveal that not only do the Liu-Mashhoon-Wesson (lmw) and Fukui-Seahra-Wesson (fsw) metrics have a lot in common with one another, they also exhibit many properties similar to that of the topological black hole (tbh) solution of the 5-dimensional vacuum field equations, which we introduce in Section III. This prompts us to suspect that the lmw and fsw solutions are actually isometric to topological black hole manifolds. We confirm this explicitly by finding transformations from standard black hole to lmw and fsw coordinates in Sections IVA and IVB respectively. We argue that the equivalence of the three metrics is actually a consequence of a higher-dimensional version of Birkhoff’s theorem in Section IVC. In Section V, we discuss which portion of the extended 5-dimensionalKruskal manifoldis covered by thelmw coordinatepatch andobtainPenrose- Carter embedding diagrams for a particular case. Section VI summarizes and discusses our results. II. TWO 5-METRICS WITH FLRW SUBMANIFOLDS In this section, we introduce two 5-metrics that embed 4-dimensional flrw models. Both of these are solutions of the 5-dimensional vacuum field equations, and are hence suitable manifoldsforstmtheory. Our goalsaretoillustrate whatsubset ofallpossible flrw models can be realized as hypersurfaces contained within these manifolds, and to find out about any 5-dimensional curvature singularities or geometric features that may be present. A. The Liu-Mashhoon-Wesson metric Consider a 5-dimensional manifold (Mlmw,gAB). We define the lmw metric ansatz as: a2(t,ℓ) ds2 = ,t dt2 a2(t,ℓ)dσ2 dℓ2. (1) lmw µ2(t) − (k,3) − Here,a(t,ℓ)andµ(t)areundeterminedfunctions, anddσ2 isthelineelementonmaximally (k,3) symmetric 3-spaces S(k) with curvature index k = +1,0, 1: 3 − dσ2 = dψ2 +S2(ψ)(dθ2 +sin2θdϕ2), (2) (k,3) k where sinψ, k = +1, S (ψ) ψ, k = 0, (3) k  ≡ sinhψ, k = 1, − It is immediately obvious that the ℓ = constant hypersurfaces Σ associated with (1) have  ℓ the structure of flrw models: R S(k). We should note that the original papers (refs. 7 × 3 and 8) did not really begin with a metric ansatz like (1); rather, the g component of the tt metric was initially taken to be some general function of t and ℓ. But one rapidly closes in on the above line element by direct integration of one component of the vacuum field equations Rˆ = 0; namely, Rˆ = 0. The other components are satisfied if AB tℓ ν2(t)+ a2(t,ℓ) = [µ2(t)+k]ℓ2 +2ν(t)ℓ+ K, (4) µ2(t)+k 3 where is an integration constant. As far as the field equations are concerned, µ(t) and ν(t) K arecompletely arbitrary functions of time. However, we should constrain them by appending the condition a(t,ℓ) R+ a2(t,ℓ) > 0 (5) ∈ ⇒ to the system. This restriction ensures that the metric signature is (+ ) and t is the −−−− only timelike coordinate. Now, if a is taken to be real, then it follows that ν must be real as well. Regarding (4) as a quadratic equation in ν, we find that there are real solutions only if the quadratic discriminant is non-negative. This condition translates into a2(t,ℓ)[µ2(t)+k]. (6) K ≤ If is positive this inequality implies that we must choose µ(t) such that µ2 +k > 0. This K relation will be important shortly. The reason that this solution is of interest is that the induced metric on ℓ = constant hypersurfaces is isometric to the standard flrw line element. To see this explicitly, consider the line element on the ℓ = ℓ 4-surface: 0 a2(t,ℓ ) ds2 = ,t 0 dt2 a2(t,ℓ )dσ2 . (7) (Σℓ) µ2(t) − 0 (k,3) Let us perform the 4-dimensional coordinate transformation a (u,ℓ ) ,u 0 Θ(t) = du µ(t(Θ)) = (Θ), (8) ′ µ(u) ⇒ A Zt where (Θ) = a(t(Θ),ℓ ), (9) 0 A and we use a prime to denote the derivative of functions of a single argument. This puts the induced metric in the flrw form ds2 = dΘ2 2(Θ)dσ2 , (10) (Σℓ) −A (k,3) where Θ is the cosmic time and (Θ) is the scale factor. So, the geometry of each of thAe Σ hypersurfaces is indeed of the flrw-type. But what ℓ kind of cosmologies can be thus embedded? Well, if we rewrite the inequality (6) in terms of and we obtain ′ A A 2( 2 +k). (11) ′ K ≤ A A Since is to be interpreted as the scale factor of some cosmological model, it satisfies the A Friedman equation: 2 1κ2ρ 2 = k. (12) A′ − 3 4 A − Here, ρ is the total density of the matter-energy in the cosmological model characterized by (Θ) and κ2 = 8πG is the usual coupling constant in the 4-dimensional Einstein equations. A 4 This implies a relation between the density of the embedded cosmologies and the choice of µ: µ2 +k = 1κ2ρ 2. (13) 3 4 A This into the inequality (11) yields 1κ2ρ 4. (14) K ≤ 3 4 A 4 Therefore, we can successfully embed a given flrw model on a Σ 4-surface in the lmw ℓ solution if the total density of the model’s cosmological fluid and scale factor satisfy (14) for all Θ. An obvious corollary of this is that we can embed any flrw model with ρ > 0 if < 0. K There is one other point about the intrinsic geometry of the Σ hypersurfaces that needs ℓ to be made. Notice that our 4-dimension coordinate transformation (8) has dΘ a ,t = , (15) dt µ which means that the associated Jacobian vanishes whenever a = 0. Therefore, the trans- ,t formationisreally onlyvalid inbetween the turning pointsof a. Also noticethat the original 4-metric(7) is badly behaved when a = 0, but the transformed one(10) isnot when = 0. ,t ′ A We can confirm via direct calculation that the Ricci scalar for (7) is 1 6µdµ ∂a − 6 (4)R = (µ2 +k). (16) − a dt ∂t − a2 (cid:18) (cid:19) We see that (4)R diverges when a = 0, provided that µµ /a = 0. Therefore, there can be ,t ,t 6 genuine curvature singularities in the intrinsic 4-geometry at the turning points of a. These features are hidden in the altered line element (10) because the coordinate transformation (8) is not valid in the immediate vicinity of any singularities, hence the Θ-patch cannot cover those regions (if they exist). We mention that this 4-dimensional singularity in the lmw metric has been recently investigated by Xu, Liu and Wang14, who have interpreted it as a 4-dimensional event horizon. Now, let us turnourattention tosome of the5-dimensional geometric propertiesof Mlmw. We can test for curvature singularities in this 5-manifold by calculating the Kretschmann scalar: 72 2 Klmw RˆABCDRˆABCD = K . (17) ≡ a8(t,ℓ) We see there is a singularity in the 5-geometry along the hypersurface a(t,ℓ) = 0. (Of course, whether or not a(t,ℓ) = 0 for any (t,ℓ) R2 depends on the choice of µ and ν.) ∈ This singularity is essentially a line-like object because the radius a of the 3-dimensional S(k) 3 subspace vanishes there. Other tools for probing the 5-geometry are Killing vector fields on Mlmw. Now, there are by definition 6 Killing vectors associated with symmetry operations on S(k), but there is also at least one Killing vector that is orthogonal to that submanifold. 3 This vector field is given by a ξlmwdxA = ,t h(a)+µ2(t)dt+ a2 h(a)dℓ. (18) A µ ,ℓ − p q Here, we have defined h(x) k K. (19) ≡ − x2 Using the explicit form of a(t,ℓ) from equation (4), we can verify that ξ satisfies Killing’s equation lmw lmw ξ + ξ = 0, (20) ∇B A ∇A B 5 lmw via computer. Also using (4), we can calculate the norm of ξ , which is given by lmw lmw ξ ξ = h(a). (21) · This vanishes at ka2 = . So, if k > 0 the 5-manifold contains a Killing horizon. If the lmwK K horizon exists then ξ will be timelike for a > and spacelike for a < . To summarize, we have seen that flrw m|od|els sa|tKis|fying (14) can be e|m|bedded|Ko|n a Σ ℓ 4-surface within the lmw metric, but that there pare 4-dimensional curvature spingularities wherever a = 0. Thelmw5-geometryalsopossessesaline-likesingularitywherea(t,ℓ) = 0, ,t lmw as well as a Killing horizon across which the norm of ξ changes sign. B. The Fukui-Seahra-Wesson metric For the time being, let us set aside the lmw metric and concentrate on the fsw solution. On a certain 5-manifold (Mfsw,gAB), this is given by the line element b2 (τ,w) ds2 = dτ2 b2(τ,w)dσ2 ,w dw2, (22a) fsw − (k,3)− ζ2(w) χ2(w) b2(τ,w) = [ζ2(w) k]τ2 +2χ(w)τ + −K. (22b) − ζ2(w) k − This metric (22a) is a solution of the 5-dimensional vacuum field equations Rˆ = 0 with AB ζ(w) and χ(w) as arbitrary functions. Just as before, we call equation (22a) the fsw metric ansatz, even though it was not the technical starting point of the original paper13. We have written (22) in a form somewhat different from that of ref. 13; to make contact with their notation we need to make the correspondences [F(w)] k ζ2(w), (23a) fsw ≡ − [h(w)] [χ2(w)+ ]/[ζ2(w) k], (23b) fsw ≡ K − [g(w)] 2χ(w), (23c) fsw ≡ [ ] 4 , (23d) fsw K ≡ − K where [ ] indicates a quantity from the original fsw work. A cursory comparison be- fsw tween th··e·lmw andfswvacuumsolutionsreveals thatbothmetrics haveasimilar structure, which prompts us to wonder about any sort of fundamental connection between them. We defer this issue to the next section, and presently concern ourselves with the properties of the fsw solution in its own right. Just as for the lmw metric, we can identify hypersurfaces in the fsw solution with flrw models. Specifically, the induced metric on w = w hypersurfaces Σ is 0 w ds2 = dτ2 b2(τ,w )dσ2 . (24) (Σw) − 0 (k,3) We see that for the universes on Σ , τ is the cosmic time and b(τ,w ) is the scale factor. It w 0 is useful to perform the following linear transformation on τ: χ 0 τ(Θ) = Θ , (25) − ζ2 k 0 − 6 ζ2 k > 0 ζ2 k < 0 0 − 0 − > 0 big bang big bang and big crunch K = 0 big bang C for all Θ R K B ∈ ∈ < 0 no big bang/crunch C for all Θ R K B ∈ ∈ TABLE I: Characteristics of the 4-dimensional cosmologies embedded on the Σ hypersurfaces in w the fsw metric where we have defined ζ ζ(w ) and χ χ(w ). This puts the induced metric into the 0 0 0 0 ≡ ≡ form ds2 = dΘ2 2(Θ)dσ2 , (26a) (Σw) −B (k,3) (ζ2 k)2Θ2 (Θ) = 0 − −K. (26b) B s ζ2 k 0 − Unlike the lmw case, the cosmology on the Σ hypersurfaces has restrictive properties. If w ζ2 k > 0, the scale factor (Θ) has the shape of one arm of a hyperbola with a semi-major 0− B axis of length /(ζ2 k). Note that this length may be complex depending on the values of ζ , k and−K. Th0a−t is, the scale factor may not be defined for all Θ R. When this 0 p K ∈ is the case, the embedded cosmologies involve a big bang and/or a big crunch. Conversely, it is not hard to see if ζ2 k < 0 and > 0 then the cosmology is re-collapsing; i.e., there is 0− K a big bang and a big crunch. However, if ζ2 k < 0 and 0, then there is no Θ interval 0 − K ≤ where the scale factor is real. We have summarized the basic properties of the embedded cosmologies in Table I. Finally, we note that if ζ2 k > 0 then 0 − lim (Θ) = (ζ2 k)1/2Θ. (27) Θ B 0 − →∞ Hence, the late time behaviour of such models approaches that of the empty Milne universe. Lake15 has calculated the Kretschmann scalar for vacuum 5-metrics of the fsw type. When his formula is applied to (22), we obtain: 72 2 Kfsw ≡ RˆABCDRˆABCD = b8(τK,w). (28) Asforthelmwmanifold,thisimpliestheexistenceofaline-likesingularityinthe5-geometry at b(τ,w) = 0. We also find that there is a Killing vector on Mfsw, which is given by b ξfswdxA = b +h(b)dτ + ,w ζ2 h(b)dw (29a) A ,τ ζ − 0 = q ξfsw + ξfsw. p (29b) ∇A B ∇B A The norm of this Killing vector is relatively easily found by computer: fsw fsw ξ ξ = h(b). (30) · fsw Hence, there is a Killing horizon in Mfsw where h(b) = 0. Obviously, the ξ Killing vector changes from timelike to spacelike — or vice versa — as the horizon is traversed. In summary, we have seen how flrw models with scale factors of the type (22b) are embedded in the fsw solution. We found that there is a line-like curvature singularity in Mfsw at b(τ,w) = 0 and the bulk manifold has a Killing horizon where the magnitude of fsw ξ vanishes. 7 III. CONNECTION TO THE 5-DIMENSIONAL TOPOLOGICAL BLACK HOLE MANIFOLD When comparing equations (17) and (28), or (21) and (30), it is hard not to believe that there is some sort of fundamental connection between the lmw and fsw metrics. We see that Klmw = Kfsw, ξlmw ξlmw = ξfsw ξfsw, (31) · · if we identify a(t,ℓ) = b(τ,w). Also, we notice that the lmw solution can be converted into the fsw metric by the following set of transformations/Wick rotations16: ψ iψ, t w, → → ℓ τ, k k, (32) → → − , dslmw idsfsw. K → −K → These facts lead us to the strong suspicion that the lmw and fsw metrics actually describe the same 5-manifold. But which 5-manifold might this be? We established in the previous section that both the lmw and fsw metrics involve a 5-dimensional line-like curvature singularity and Killing horizon if k > 0. This reminds us of another familiar manifold: that of a black hole. Consider theKmetric of a “topological” black hole (tbh) on a 5-manifold (Mtbh,gAB): ds2 = h(R)dT2 h 1(R)dR2 R2dσ2 . (33) tbh − − − (k,3) Theadjective“topological”comesfromthefactthatthemanifoldhasthestructureR2 S(k), as opposed to the familiar R2 S structure commonly associated with spherical sym×met3ry 3 × in 5-dimensions. That is, the surfaces T = constant and R = constant are not necessarily 3-spheres for the topological black hole; it is possible that they have flat or hyperbolic geometry. One can confirm by direct calculation that (33) is a solution of Rˆ = 0 for any AB value of k, and that the constant that appears in h(R) is related to the mass of the central K object. The Kretschmann scalar on Mtbh is 72 2 Ktbh = RˆABCDRˆABCD = K , (34) R8 implying a line-like curvature singularity at R = 0. There is an obvious Killing vector in this manifold, given by ξtbhdxA = h(R)dT. (35) A The norm of this vector is trivially tbh tbh ξ ξ = h(R). (36) · There is therefore a Killing horizon in this space located at kR2 = . Now, equations (34) and (36) closely match their counterpartsKfor the lmw and fsw metrics, which inspires the hypothesis that not only are the lmw and fsw isometric to one another, they are also isometric to the metric describing topological black holes. However, while these coincidences provide fairly compelling circumstantial evidence that the lmw, fsw, and tbh metrics are equivalent, we do not have conclusive proof — that will come in the next section. 8 IV. COORDINATE TRANSFORMATIONS In this section, our goal is to prove the conjecture that the lmw, fsw, and tbh solutions and the 5-dimensional vacuum field equations are isometric to one another. We will do so by finding two explicit coordinate transformations that convert the tbh metric to the lmw and fsw metrics respectively. This is sufficient to prove the equality of all three solutions, since it implies that one can transform from the lmw to the fsw metric — or vice versa — via a two-stage procedure. A. Transformation from Schwarzschild to Liu-Mashhoon-Wesson coordinates We first search for a coordinate transformation that takes the tbh line element (33) to the lmw line element (1). We take this transformation to be R = (t,ℓ), T = (t,ℓ). (37) R T Notice that we have not assumed R = a(t,ℓ) — as may have been expected from the discussion of the previous section — in order to stress that we are starting with a general coordinate transformation. We will soon see that by demanding that this transformation forces the tbh metric into the form of the lmw metric ansatz, we can recover R = a(t,ℓ) with a(t,ℓ) given explicitly by (4). In other words, the coordinate transformation specified in this section will fix the functional form of a(t,ℓ) in a manner independent of the direct attack on the vacuum field equations found in refs. 7 and 8. When (37) is substituted into (33), we get 2 ds2 = h( ) 2 R,t dt2 +2 h( ) R,tR,ℓ dtdℓ+ tbh R T,t − h( ) R T,tT,ℓ − h( ) (cid:20) R (cid:21) (cid:20) R (cid:21) 2 h( ) 2 R,ℓ dℓ2 2(t,ℓ)dσ2 . (38) R T,ℓ − h( ) −R (k,3) (cid:20) R (cid:21) For this to match equation (1) with (t,ℓ) instead of a(t,ℓ) we must have R 2 2 R,t = h( ) 2 R,t , (39a) µ2(t) R T,t − h( ) R ,t ,ℓ 0 = h( ) R R , (39b) ,t ,ℓ R T T − h( ) R 2 1 = h( ) 2 R,ℓ , (39c) − R T,ℓ − h( ) R with µ(t) arbitrary. Under these conditions, we find 2(t,ℓ) ds2 = R,t dt2 2(t,ℓ)dσ2 dy2, (40) tbh µ2(t) −R (k,3) − which is obviously the same as the lmw metric ansatz (1). However, the precise functional form of (t,ℓ) has yet to be specified. R 9 To solve for (t,ℓ), we note equations (39a) and (39c) can be rearranged to give R h( ) ,t = ǫ R 1+ R , (41a) ,t t T h( )s µ2(t) R 1 = ǫ 2 h( ), (41b) T,ℓ ℓh( ) R,ℓ − R R q where ǫ = 1 and ǫ = 1. Using these in (39b) yields t ℓ ± ± = h( )+µ2(t). (42) ,ℓ R ± R Our task it to solve the system of pdes foprmed by equations (41) and (42) for (t,ℓ) and T (t,ℓ). Once we have accomplished this, the coordinate transformation from (1) to (33) is R found. Using the definition of h( ), we can expand equation (42) to get R ∂ 1 = R R. (43) ± (µ2 +k) 2 ∂ℓ R −K Integrating both sides with respect topℓ yields (µ2 +k) 2 = (µ2 +k)( ℓ+γ), (44) R −K ± p where γ = γ(t) is an arbitrary function of time. Solving for gives R ν2(t)+ R2 = 2(t,ℓ) = [µ2(t)+k]ℓ2 +2ν(t)ℓ+ K, (45) R µ2(t)+k where we have defined ν(t) = γ(t)[µ2(t)+k], (46) ± which can be thought of as just an another arbitrary function of time. We have hence see that the functional form of (t,ℓ) matches exactly the functional form of a(t,ℓ) in equation R (4). This is despite the fact that the two expressions were derived by different means: (45) from conditions placed on a coordinate transformation, and (4) from the direct solution of the 5-dimensional vacuum field equations. When our solution for (t,ℓ) is put into equations (41), we obtain a pair of pdes that R expresses the gradient of in the (t,ℓ)-plane as known functions of the coordinates. This is T analogous to a problem where one is presented with the components of a 2-dimensional force and is asked to find the associated potential. The condition for integrablity of the system is that the curl of the force vanishes, which in our case reads ∂ h( ) ∂ 1 0 =? ǫ R,t 1+ R ǫ 2 h( ) . (47) t∂ℓ h( )s µ2(t)!− ℓ∂t h( ) R,ℓ − R R (cid:18) R q (cid:19) We have confirmed via computer that this condition holds when (t,ℓ) is given by equation R (45), provide we choose ǫ = ǫ = 1. Without loss of generality, we can set ǫ = ǫ = 1. t ℓ t ℓ ± Hence, equations (41) are indeed solvable for (t,ℓ) and a coordinate transformation from T (33) to (1) exists. 10