Topology of Event Horizons and Topological Censorship

Ted Jacobson; Shankar Venkataramani

doi:10.1088/0264-9381/12/4/012

paper

Topology of Event Horizons and Topological Censorship

arXiv:gr-qc/9410023 · doi:10.1088/0264-9381/12/4/012

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 $Σ$ for $\C_K$ is a topological 3-manifold with compact boundary $\partial §$ in $M$ , and $§^{'}$ is a compact submanifold of $\bS$ with spherical boundary in $§$ (and possibly other boundary components in $M /§$ ). Then we prove that the homology group $H_{1} (Σ^{'}, Z)$ must be finite. This implies that either $\partial §^{'}$ consists of a disjoint union of 2-spheres, or $§^{'}$ is nonorientable and $\partial §^{'}$ contains a projective plane. Further, $\partial§=\partial\Ip[K]\cap\partial\Im[\Sp]$, and $\partial Σ$ 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 $§$ is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.}

11 pages, plain latex (minor revision: $Σ$ replaced by its closure in various places.)

Topology of Event Horizons and Topological Censorship

Abstract

References in corpus (2)

Cited by in corpus (57)