Moduli space of stationary vacuum black holes from integrability

TL;DR

This paper classifies stationary vacuum black hole spacetimes in four and five dimensions by integrating spectral equations, enabling a complete description of their moduli space and providing constructive proofs of uniqueness for known solutions.

Contribution

It introduces a method to fully determine black hole metrics from spectral equations, advancing the classification and uniqueness proofs of higher-dimensional black holes.

Findings

Derived the moduli space of solutions without conical singularities.

Provided constructive proofs for Kerr and Myers-Perry black holes.

Extended the integrability approach to classify black hole solutions.

Abstract

We consider the classification of asymptotically flat, stationary, vacuum black hole spacetimes in four and five dimensions, that admit one and two commuting axial Killing fields respectively. It is well known that the Einstein equations reduce to a harmonic map on the two-dimensional orbit space, which itself arises as the integrability condition for a linear system of spectral equations. We integrate the Belinski-Zakharov spectral equations along the boundary of the orbit space and use this to fully determine the metric and associated Ernst and twist potentials on the axes and horizons. This is sufficient to derive the moduli space of solutions that are free of conical singularities on the axes, for any given rod structure. As an illustration of this method we obtain constructive uniqueness proofs for the Kerr and Myers-Perry black holes and the known doubly spinning black rings.