Topology of Event Horizons and Topological Censorship

Abstract

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 $\C_K$ to the future of a cut $K$ of $\Sm$ is globally hyperbolic. Suppose further that a Cauchy surface $\Sigma$ for $\C_K$ is a topological 3-manifold with compact boundary $\partial\S$ in $M$, and $\S'$ is a compact submanifold of $\bS$ with spherical boundary in $\S$ (and possibly other boundary components in $M/\S$). Then we prove that the homology group $H_1(\Sigma',Z)$ must be finite. This implies that either $\partial\S'$ consists of a disjoint union of 2-spheres, or $\S'$ is nonorientable and $\partial\S'$ contains a projective plane. Further, $\partial\S=\partial\Ip[K]\cap\partial\Im[\Sp]$, and $\partial \Sigma$ will be a cross section of the horizon as long as no generator of $\partial\Ip[K]$ becomes a generator of $\partial\Im[\Sp]$. In this case, if $\S$ is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.}