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

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.