THE UNIQUENESS THEOREM FOR ROTATING BLACK HOLE SOLUTIONS OF SELF-GRAVITATING HARMONIC MAPPINGS

TL;DR

This paper proves that the Kerr metric is the unique stationary, axisymmetric, asymptotically flat black hole solution with a regular horizon for self-gravitating harmonic mappings, extending the uniqueness theorems in general relativity.

Contribution

It establishes the uniqueness of the Kerr black hole solution within the class of self-gravitating harmonic mappings, including rotating configurations.

Findings

Proves the Kerr metric is the only solution under specified conditions.

Establishes integrability conditions for Killing fields in this context.

Extends the black hole uniqueness theorem to self-gravitating harmonic mappings.

Abstract

We consider rotating black hole configurations of self-gravitating maps from spacetime into arbitrary Riemannian manifolds. We first establish the integrability conditions for the Killing fields generating the stationary and the axisymmetric isometry (circularity theorem). Restricting ourselves to mappings with harmonic action, we subsequently prove that the only stationary and axisymmetric, asymptotically flat black hole solution with regular event horizon is the Kerr metric. Together with the uniqueness result for non-rotating configurations and the strong rigidity theorem, this establishes the uniqueness of the Kerr family amongst all stationary black hole solutions of self-gravitating harmonic mappings.