Uniqueness properties of the Kerr metric

TL;DR

This paper establishes a precise geometric condition under which vacuum, stationary, asymptotically flat spacetimes are locally identical to the Kerr metric, highlighting its fundamental role in general relativity.

Contribution

It proves a new characterization theorem for Kerr spacetime based on the eigenvector property of the Killing form relative to the Weyl tensor.

Findings

The theorem provides a necessary and sufficient condition for Kerr uniqueness.

Dropping asymptotic flatness leads to a family of different metrics.

The result aids in extending black hole uniqueness theorems beyond current limitations.

Abstract

We obtain a geometrical condition on vacuum, stationary, asymptotically flat spacetimes which is necessary and sufficient for the spacetime to be locally isometric to Kerr. Namely, we prove a theorem stating that an asymptotically flat, stationary, vacuum spacetime such that the so-called Killing form is an eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr. Asymptotic flatness is a fundamental hypothesis of the theorem, as we demonstrate by writing down the family of metrics obtained when this requirement is dropped. This result indicates why the Kerr metric plays such an important role in general relativity. It may also be of interest in order to extend the uniqueness theorems of black holes to the non-connected and to the non-analytic case.