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

TL;DR

The paper proves a 5D stationary vacuum black hole uniqueness theorem for spacetimes with two commuting axial symmetries and no discrete isotropy, showing that solutions are fully determined by mass, two angular momenta, and a rod-structure datum. It combines Maison's sigma-model reduction with a detailed analysis of the orbit space hat M, and employs the Mazur identity to equate the relevant target-space data when rod structure and charges coincide. The horizon topology is tightly constrained to $S^3$, $S^2\times S^1$, or Lens spaces $L(p,q)$, with rod data encoding topology and horizon location. This work clarifies the parameter space necessary for 5D black hole classification and highlights the rod structure as a crucial invariant in the higher-dimensional uniqueness problem; extensions to include matter fields and scenarios with orbifold points remain important directions for future research.

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.