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

Abstract

In this paper, we construct a geometric invariant for initial datasets for the vacuum Einstein field equations , such that is a three-dimensional manifold with an asymptotically Euclidean end and an inner boundary with the topology of the 2-sphere. The hypersurface can be thought 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 is an initial dataset for the Kerr spacetime. The construction makes use of the notion of Killing spinors and of an expression for a Killing spinor candidate , which can be constructed out of concomitants of the Weyl tensor.