On the Topology of Black Hole Event Horizons in Higher Dimensions

TL;DR

This paper investigates the possible topologies of black hole event horizons in higher dimensions, extending classical results and applying advanced geometric and topological classifications to identify allowed horizon shapes.

Contribution

It generalizes Hawking's topology theorem to higher dimensions and classifies possible event horizon topologies using geometric and topological tools.

Findings

In higher dimensions, event horizons can have topologies like $S^3$, $S^2 imes S^1$, $S^4$, and $S^2 imes ext{surface}$.

Allowed topologies depend on dimension and topological constraints such as cobordism and smooth structure.

The paper discusses classification of horizons beyond six dimensions following Smale's results.

Abstract

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere $S^2$. We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.