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1、arXiv:hep-th/9909205v2 21 Oct 1999Brane-World Black HolesA. Chamblin, S.W. Hawkingand H.S. ReallDAMTPUniversity of CambridgeSilver Street, Cambridge CB3 9EW, United Kingdom.Preprint DAMTP-1999-13329 September 1999AbstractGravitational collapse of matter trapped on a brane will produce a black hole o
2、n the brane. Wediscuss such black holes in the models of Randall and Sundrum where our universe is viewed asa domain wall in five dimensional anti-de Sitter space. We present evidence that a non-rotatinguncharged black hole on the domain wall is described by a “black cigar” solution in five dimensio
3、ns.1IntroductionThere has been much recent interest in the idea that our universe may be a brane embedded in somehigher dimensional space. It has been shown that the hierarchy problem can be solved if the higherdimensional Planck scale is low and the extra dimensions large 1, 2. An alternative solut
4、ion, proposedby Randall and Sundrum (RS), assumes that our universe is a negative tension domain wall separatedfrom a positive tension wall by a slab of anti-de Sitter (AdS) space 3. This does not require a largeextra dimension: the hierarchy problem is solved by the special properties of AdS. The d
5、rawback withthis model is the necessity of a negative tension object.In further work 4, RS suggested that it is possible to have an infinite extra dimension. In thismodel, we live on a positive tension domain wall inside anti-de Sitter space. There is a bound stateof the graviton confined to the wal
6、l as well as a continuum of Kaluza-Klein (KK) states. For non-relativistic processes on the wall, the bound state dominates over the KK states to give an inversesquare law if the AdS radius is sufficiently small. It appears therefore that four dimensional gravityis recovered on the domain wall.This
7、conclusion was based on perturbative calculations for zerothickness walls. Supergravity domain walls of finite thickness have recently been considered 5, 6, 7and a non-perturbative proof that the bound state exists for such walls was given in 8. It is importantto examine other non-perturbative gravi
8、tational effects in this scenario to see whether the predictionsof four dimensional general relativity are recovered on the domain wall.If matter trapped on a brane undergoes gravitational collapse then a black hole will form. Sucha black hole will have a horizon that extends into the dimensions tra
9、nsverse to the brane: it will bea higher dimensional object. Phenomenological properties of such black holes have been discussed inemail: H.A. Chamblindamtp.cam.ac.ukemail: S.W.Hawkingdamtp.cam.ac.ukemail: H.S.Realldamtp.cam.ac.uk19 for models with large extra dimensions. In this paper we discuss bl
10、ack holes in the RS models. Anatural candidate for such a hole is the Schwarzschild-AdS solution, describing a black hole localizedin the fifth dimension. We show in the Appendix that it is not possible to intersect such a hole witha vacuum domain wall so it is unlikely that it could be the final st
11、ate of gravitational collapse on thebrane. A second possibility is that what looks like a black hole on the brane is actually a black stringin the higher dimensional space. We give a simple solution describing such a string. The inducedmetric on the domain wall is simply Schwarzschild, as it has to
12、be if four dimensional general relativity(and therefore Birkhoffs theorem) are recovered on the wall. This means that the usual astrophysicalproperties of black holes (e.g. perihelion precession, light bending etc.) are recovered in this scenario.We find that the AdS horizon is singular for this bla
13、ck string solution. This is signalled by scalarcurvature invariants diverging if one approaches the horizon along the axis of the string. If one ap-proaches the horizon in a different direction then no scalar curvature invariant diverges. However, in aframe parallelly propagated along a timelike geo
14、desic, some curvature components do diverge. Further-more, the black string is unstable near the AdS horizon - this is the Gregory-Laflamme instability 10.However, the solution is stable far from the AdS horizon. We will argue that our solution evolves to a“black cigar” solution describing an object
15、 that looks like the black string far from the AdS horizon(so the metric on the domain wall is Schwarzschild) but has a horizon that closes offbefore reachingthe AdS horizon. In fact, we conjecture that this black cigar solution is the unique stable vacuumsolution in five dimensions which describes
16、the endpoint of gravitational collapse on the brane. Wesuspect that the AdS horizon will be non-singular for the cigar solution.2The Randall-Sundrum modelsBoth models considered by RS use five dimensional AdS. In horospherical coordinates the metric isds2= e2y/lijdxidxj+ dy2(2.1)where is the four di
17、mensional Minkowski metric and l the AdS radius. The global structure ofAdS is shown in figure 1. Horospherical coordinates break down at the horizon y = .In their first model 3, RS slice AdS along the horospheres at y = 0 and y = yc 0, retain theportion 0 y ycand assume Z2reflection symmetry at eac
18、h boundary plane. This gives a jump inextrinsic curvature at these planes, yielding two domain walls of equal and opposite tension = 62l(2.2)where 2= 8G and G is the five dimensional Newton constant. The wall at y = 0 has positive tensionand the wall at y = ychas negative tension. Mass scales on the
19、 negative tension wall are exponentiallysuppressed relative to those on the positive tension one. This provides a solution of the hierarchyproblem provided we live on the negative tension wall. The global structure is shown in figure 1.The second RS model 4 is obtained from the first by taking yc .
20、This makes the negativetension wall approach the AdS horizon, which includes a point at infinity. RS say that their modelcontains only one wall so presumably the idea is that the negative tension brane is viewed as an auxiliarydevice to set up boundary conditions. However, if the geometry makes sens
21、e then it should be possibleto discuss it without reference to this limiting procedure involving negative tension objects. If onesimply slices AdS along a positive tension wall at y = 0 and assumes reflection symmetry then thereare several ways to analytically continue the solution across the horizo
22、n. These have been discussedin 11, 12, 13, 14. There are two obvious choices of continuation. The first is simply to assume thatbeyond the horizon, the solution is pure AdS with no domain walls present. This is shown in figure2Figure 1: 1. Anti-de Sitter space. Two horospheres and a horizon are show
23、n. The vertical lines rep-resent timelike infinity. 2. Causal structure of Randall-Sundrum model with compact fifth dimension.The arrows denote identifications.Figure 2: Possible causal structures for Randall-Sundrum model with non-compact fifth dimension.The dots denote points at infinity.2. An alt
24、ernative, which seems closer in spirit to the geometry envisaged by RS, is to include furtherdomain walls beyond the horizon, as shown in figure 2. In this case, there are infinitely many domainwalls present.3Black string in AdSLet us first rewrite the AdS metric 2.1 by introducing the coordinate z
25、= ley/l. The metric is thenmanifestly conformally flat:ds2=l2z2(dz2+ ijdxidxj).(3.1)In these coordinates, the horizon lies at z = while the timelike infinity of AdS is at z = 0. Wenow note that if the Minkowski metric within the brackets is replaced by any Ricci flat metric thenthe Einstein equation
26、s (with negative cosmological constant) are still satisfied1. A natural choice for1This procedure was recently discussed for general p-brane solutions in 15.3a metric describing a black hole on a domain wall at fixed z is to take this Ricci flat metric to be theSchwarzschild solution:ds2=l2z2(U(r)dt
27、2+ U(r)1dr2+ r2(d2+ sin2d2) + dz2)(3.2)where U(r) = 12M/r. This metric describes a black string in AdS. Including a reflection symmetricdomain wall in this spacetime is trivial: surfaces of constant z satisfy the Israel equations providedthe domain wall tension satisfies equation 2.2. For a domain w
28、all at z = z0, introduce the coordinatew = z z0. The metric on both sides of the wall can then be writtends2=l2(|w| + z0)2(U(r)dt2+ U(r)1dr2+ r2(d2+ sin2d2) + dw2)(3.3)with w 12M2, when it is possible to have bound states (i.e. orbits restricted to a finite range of r)outside the Schwarzschild horiz
29、on.The orbits that reach r = have late time behaviour r pE2 1 and hencer z1lqE2 1(3.12)as 0. Along such geodesics, the squared Riemann tensor does not diverge. The bound stategeodesics behave differently. These remain at finite r and therefore the square of the Riemann tensordoes diverge as 0. They
30、orbit the black string infinitely many times before hitting the singularity,but do so in finite affine parameter.It appears that some geodesics encounter a curvature singularity at the AdS horizon whereas othersmight not because scalar curvature invariants do not diverge along them. It is possible t
31、hat only partof the surface z = is singular. To decide whether or not this is true, we turn to a calculation of theRiemann tensor in an orthonormal frame parallelly propagated along a timelike geodesic that reachesz = but for which the squared Riemann tensor does not diverge (i.e. a non-bound state
32、geodesics).The tangent vector to such a geodesic (with L = 0) can be writtenu= zlsz2z21 1,Ez2U(r)l2,z2l2sE2l2z21U(r),0,0!,(3.13)5where we have written the components in the order (z,t,r,). A unit normal to the geodesic isn= 0,zz1l2U(r)sE2l2z21U(r),Ez1zl2,0,0!.(3.14)It is straightforward to check tha
33、t this is parallelly propagated along the geodesic i.e. u n= 0.These two unit vectors can be supplemented by three other parallelly propagated vectors to form anorthonormal set. However the divergence can be exhibited using just these two vectors. One of thecurvature components in this parallelly pr
34、opagated frame isR(u)(n)(u)(n) Runun=1l2 1 2Mz4z21r3!,(3.15)which diverges along the geodesic as 0.The black string solution therefore has a curvaturesingularity at the AdS horizon.It is well known that black string solutions in asymptotically flat space are unstable to long wave-length perturbation
35、s 10. A black hole is entropically preferred to a sufficiently long segment of string.The strings horizon therefore has a tendency to “pinch off” and form a line of black holes. One mightthink that a similar instability occurs for our solution. However, AdS acts like a confining box whichprevents fl
36、uctuations with wavelengths much greater than l from developing. If an instability occursthen it must do so at smaller wavelengths.If the radius of curvature of the strings horizon is sufficiently small then the AdS curvature will benegligible there and the string will behave as if it were in asympt
37、otically flat space. This means thatit will be unstable to perturbations with wavelengths of the order of the horizon radius 2M= 2Ml/z.At sufficiently large z, such perturbations will fit into the AdS box, i.e. 2M l, so an instability canoccur near the AdS horizon. However for M/z 1, the potential i
38、nstability occurs at wavelengthsmuch greater than l and is therefore not possible in AdS. Therefore the black string solution is unstablenear the AdS horizon but stable far from it.We conclude that, near the AdS horizon, the black string has a tendency to “pinch off” but furtheraway it is stable. Af
39、ter pinching off, the string becomes a stable “black cigar” which would extend toinfinity in AdS if the domain wall were not present, but not to the AdS horizon. The cigars horizonacts as if it has a tension which balances the force pulling it towards the centre of AdS. We showedabove that if our do
40、main wall is at z = z0then a black hole of astrophysical mass has M/z0 1,corresponding to the part of the black cigar far from the AdS horizon. Here, the metric will be wellapproximated by the black string metric so the induced metric on the wall will be Schwarzschild andthe predictions of four dime
41、nsional general relativity will be recovered.4DiscussionAny phenomenologically successful theory in which our universe is viewed as a brane must reproducethe large-scale predictions of general relativity on the brane. This implies that gravitational collapse ofuncharged non-rotating matter trapped o
42、n the brane ultimately settles down to a steady state in whichthe induced metric on the brane is Schwarzschild. In the higher dimensional theory, such a solutioncould be a localized black hole or an extended object intersecting the brane. We have investigatedthese alternatives in the models proposed
43、 by Randall and Sundrum (RS). The obvious choice of fivedimensional solution in the first case is Schwarzschild-AdS. However we have shown (in the Appendix)that it is not possible to intersect this with a vacuum domain wall so it cannot be the final state ofgravitational collapse on the wall.6We hav
44、e presented a solution that describes a black string in AdS. It is possible to intersect thissolution with a vacuum domain wall and the induced metric is Schwarzschild. The solution can thereforebe interpreted as a black hole on the wall. The AdS horizon is singular. Scalar curvature invariants only
45、diverge if this singularity is approached along the axis of the string. However, curvature componentsdiverge in a frame parallelly propagated along any timelike geodesic that reaches the horizon. Thissingularity can be removed if we use the first RS model in which there are two domain walls presenta
46、nd we live on a negative tension wall.However if we wish to use the second RS model (with anon-compact fifth dimension) then the singularity will be visible from our domain wall. In 8, it wasargued that anything emerging from a singularity at the AdS horizon would be heavily red-shiftedbefore reachi
47、ng us and that this might ensure that physics on the wall remains predictable in spiteof the singularity. However we regard singularities as a pathology of the theory since, in principle,arbitrarily large fluctuations can emerge from the singularity and the red-shift is finite.Fortunately, it turns
48、out that our solution is unstable near the AdS horizon. We have suggestedthat it will decay to a stable configuration resembling a cigar that extends out to infinity in AdS butdoes not reach the AdS horizon. The solution becomes finite in extent when the gravitational effectof the domain wall is inc
49、luded. Our domain wall is situated far from the AdS horizon so the inducedmetric on the wall will be very nearly Schwarzschild. Since the cigar does not extend as far as the AdShorizon, it does not seem likely that there will be a singularity there. Similar behaviour was recentlyfound in a non-linea
50、r treatment of the RS model 8. It was shown that pp-waves corresponding toKaluza-Klein modes are singular at the AdS horizon. These pp-waves are not localized to the domainwall. The only pp-waves regular at the horizon are the ones corresponding to gravitons confined tothe wall. We suspect that pert