Topology of Event Horizons and Topological Censorship

We prove that, under certain conditions, the topology of the event horizon of a four dimensional asymptotically flat black hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let M be a four dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication CK to the future of a cut K of I is globally hyperbolic. Suppose further that a Cauchy surface Σ for CK is a topological 3-manifold with compact boundary ∂Σ in M , and Σ is a compact submanifold of Σ̄ with spherical boundary in Σ (and possibly other boundary components in M/Σ). Then we prove that the homology group H1(Σ , Z) must be finite. This implies that either ∂Σ consists of a disjoint union of 2-spheres, or Σ is nonorientable and ∂Σ contains a projective plane. Further, ∂Σ = ∂I[K]∩∂I[I], and ∂Σ will be a cross section of the horizon as long as no generator of ∂I[K] becomes a generator of ∂I[I]. In this case, if Σ is orientable, the horizon cross section must consist of a disjoint union of 2-spheres. [email protected] [email protected]