A spacetime characterization of the Kerr metric

TL;DR

This paper introduces a new spacetime tensor that characterizes the Kerr metric among stationary, asymptotically flat vacuum spacetimes, extending previous tensor-based characterizations and providing a geometric interpretation involving the Weyl tensor.

Contribution

It defines a three-index tensor that vanishes for Kerr and extends the Simon tensor to the entire spacetime, offering a new characterization of Kerr.

Findings

A tensor vanishes for Kerr and characterizes it among stationary vacuum spacetimes.

Spacetime Simon tensor's extension to the whole spacetime is achieved.

A geometric interpretation involving principal null directions of the Weyl tensor is provided.

Abstract

We obtain a characterization of the Kerr metric among stationary, asymptotically flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the whole spacetime. More precisely, we define a three index tensor on any spacetime with a Killing field, which vanishes identically for Kerr and which coincides in the strictly stationary region with the Simon tensor when projected down into the manifold of trajectories. We prove that a stationary asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is locally isometric to Kerr. A geometrical interpretation of this characterization in terms of the Weyl tensor is also given. Namely, a stationary, asymptotically flat vacuum spacetime such that each principal null direction of the Killing form is a repeated principal null direction of the Weyl tensor is locally isometric to Kerr.