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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We provide a simple, unified proof of Birkhoff's theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(-anti)-de Sitter and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and over-extremal Schwarzschild-de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff's theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized Birkhoff's theorems are local uniqueness theorems whose corollary is that locally spherically symmetric solutions of Einstein's equations exhibit an additional local Killing vector field.PACS numbers: 04.20.Cv, 04.20.Gz Birkhoff's theorem, that any spherically symmetric vacuum solution of Einstein's equations is locally isometric to a region in Schwarzschild spacetime, is not only a classic contribution to general relativity but is also an important tool in gravitational physics and cosmology. First proven for the vacuum Einstein equations, this theorem is traditionally credited to Birkhoff [1]; however it was recently rediscovered by Deser, when providing an alternate proof, that this theorem was published two years earlier by Jebsen [2,3,4]. (An english translation of Jebsen's paper with an introduction by Deser was published in General Relativity and Gravitation in 2005 [5, 6].) In older references, this theorem is credited as independently discovered by not only Birkhoff and Jebsen but also Alexandrow [7] and Eiesland [8].1 The multiple provenance of this theorem is a testament to its significance in general relativity. It is natural to consider its extension to other situations in gravitational physics. Birkhoff's theorem readily generalizes to the inclusion of electromagnetic fields [10], cosmological constant [8], with explicit exhibition of both the Schwarzschild-de Sitter and Nariai solutions in [11,12] (also in [13] without citation of previous references), and to other symmetry groups [14,15].2 Generalized Birkhoff's theorems have also been proven in lower and higher dimensions [16,17,18,19,20,21], certain alternate theories of gravity [22,23,24,25,26,27,28] including Lovelock gravity [29,30,31,32], and shown not to hold on Randall-Sundrum branes [33] and in some modified theories of gravity [34,35].Although Birkhoff's theorem is a classic result, many current textbooks and review articles on general relativity no longer provide a proof or even a careful statement of the theorem. Frequently it is cited as proving that the spherically symmetric vacuum solution is static. This is clearly not the case as recognized in many (but not all) proofs of the theorem (See, for example, the comment on Jebsen's proof in [36]). The interior region of the maximal analytic extension of Schwarzschild spacetime is not static; instead, it exhibits a spacelike Killing vector field. Of course, other regions of this spacetime do exhi...