Uniqueness theorem for 5-dimensional black holes with two axial Killing fields

TL;DR

This paper proves a uniqueness theorem for 5-dimensional vacuum black holes with two axial symmetries, showing they are identical if their physical parameters and structures match, and classifies possible horizon topologies.

Contribution

It generalizes previous spherical case results to include ring and Lens-space topologies, establishing a comprehensive uniqueness criterion for these black holes.

Findings

Black holes are uniquely determined by mass, angular momentum, and rod structure.

Horizon topology is restricted to 3-sphere, ring, or Lens-space.

The proof extends previous methods to more general topologies.

Abstract

We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their ``rod structures'' coincide. We also show that the horizon must be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits.