A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

TL;DR

This paper proves that higher-dimensional stationary rotating black holes must be axisymmetric, extending a key 4D black hole uniqueness result to higher dimensions without assuming specific horizon topology.

Contribution

It establishes axisymmetry of higher-dimensional rotating black holes under minimal assumptions, generalizing the 4D uniqueness theorem to higher dimensions.

Findings

Higher-dimensional rotating black holes are necessarily axisymmetric.

The proof does not depend on horizon topology beyond compactness.

Analyticity and non-degeneracy are key assumptions.

Abstract

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is ``rotating''--i.e., is such that the stationary Killing field is not everywhere normal to the horizon--must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, $P$. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.