A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

TL;DR

This work extends the classical black hole rigidity theorem to higher-dimensional spacetimes by replacing the 4D reliance on closed horizon generator orbits with ergodic methods. Using Gaussian null coordinates and a careful foliation of the horizon, the authors prove the existence of a horizon-normal Killing field $K^a$ commuting with the stationary field $t^a$, under analytic and non-degenerate horizon assumptions. They then establish the existence of additional rotational Killing fields, yielding a decomposition of the stationary field into a sum of horizon-driven and rotational components, with either closed-orbit rotations or, if necessary, multiple rotations with irrational velocity ratios. The results hold in the presence of matter fields and a negative cosmological constant, thereby reinforcing the rigidity and axisymmetry structure of higher-dimensional black holes and contributing to higher-dimensional uniqueness programs.

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.