All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from J ( − to J ( + is homotopic to a topologically trivial curve from J ( − to J ( + . (If the Poincaré conjecture is false, the theorem does not prevent one from probing fake 3-spheres).
Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ''scri.'' We show that the principle of topological censorship ͑PTC͒, which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces ͑Cauchy surfaces or analogues thereof͒ in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes.
We reexamine the thermodynamics of anti-de Sitter (adS) black holes with Ricci flat horizons using the adS soliton as the thermal background. We find that there is a phase transition which is dependent not only on the temperature but also on the black hole area, which is an independent parameter. As in the spherical adS black hole, this phase transition is related via the adS/conformal-field-theory correspondence to a confinement-deconfinement transition in the large- N gauge theory on the conformal boundary at infinity.
All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from J ( − to J ( + is homotopic to a topologically trivial curve from J ( − to J ( + . (If the Poincaré conjecture is false, the theorem does not prevent one from probing fake 3-spheres).
In [1] it was shown that (n + 1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, I, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result [2], but is independent of the scalar curvature of I. Furthermore, as shown in [1], the topology of V ′ , the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of I; the homomorphism Π 1 (Σ 0 ) → Π 1 (V ′ ) induced by the inclusion map is onto where Σ 0 is the intersection of V ′ with I. In 3 + 1 dimensions, the homology of V ′ can be completely determined from this as shown in [1]. In this paper, we prove in arbitrary dimension that H n−1 (V ; Z) = Z k where V is the closure of V ′ and k is the number of boundaries Σ i interior to Σ 0 . As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n = 2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B 2 or I × S 1 .
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