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

In this paper, we construct a geometric invariant for initial datasets for the vacuum Einstein field equations (S, h(ab), K-ab), such that S is a three-dimensional manifold with an asymptotically Euclidean end and an inner boundary partial derivative S with the topology of the 2-sphere. The hypersurface S 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 (S, h(ab), K-ab) 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.