Symmetries of Higher Dimensional Black Holes

TL;DR

This paper proves that higher-dimensional stationary vacuum black holes with certain horizon properties possess additional symmetries, extending previous results to cases with non-closed horizon generators.

Contribution

It establishes the existence of extra Killing fields for higher-dimensional black holes with non-degenerate horizons, broadening the understanding of their symmetry structures.

Findings

Existence of additional Killing fields for each horizon component.

Extension of symmetry results to horizons with non-closed generators.

Implication that rotating black holes have at least a two-dimensional isometry group.

Abstract

We prove that if a stationary, real analytic, asymptotically flat vacuum black hole spacetime of dimension $n\geq 4$ contains a non-degenerate horizon with compact cross sections that are transverse to the stationarity generating Killing vector field then, for each connected component of the black hole's horizon, there is a Killing field which is tangent to the generators of the horizon. For the case of rotating black holes, the stationarity generating Killing field is not tangent to the horizon generators and therefore the isometry group of the spacetime is at least two dimensional. Our proof relies on significant extensions of our earlier work on the symmetries of spacetimes containing a compact Cauchy horizon, allowing now for non closed generators of the horizon.