The "non-Kerrness" of domains of outer communication of black holes and exteriors of stars

TL;DR

This paper introduces a geometric invariant for initial data in vacuum Einstein equations that uniquely identifies Kerr black hole spacetimes, using Killing spinors and Weyl tensor properties.

Contribution

It constructs a new invariant that vanishes precisely for Kerr initial data, aiding in the characterization of black hole and star exteriors in general relativity.

Findings

Invariant vanishes iff the data corresponds to Kerr spacetime

Uses Killing spinors and Weyl tensor concomitants

Applicable to domains of outer communication and stellar exteriors

Abstract

In this article we construct a geometric invariant for initial data sets for the vacuum Einstein field equations $(\mathcal{S},h_{ab},K_{ab})$, such that $\mathcal{S}$ is a 3-dimensional manifold with an asymptotically Euclidean end and an inner boundary $\partial \mathcal{S}$ with the topology of the 2-sphere. The hypersurface $\mathcal{S}$ can be though of being in the domain of outer communication of a black hole or in the exterior of a star. The geometric invariant vanishes if and only if $(\mathcal{S},h_{ab},K_{ab})$ is an initial data set for the Kerr spacetime. The construction makes use of the notion of Killing spinors and of an expression for a \emph{Killing spinor candidate} which can be constructed out of concomitants of the Weyl tensor.