An invariant measure of deviation from Petrov type D at the level of initial data

In this article, we describe a simple covariant characterisation of initial data sets which give rise to Petrov type D vacuum spacetime developments. As an application, we derive an integral invariant which, when restricted to the appropriate class of asymptotically Euclidean initial data sets, vanishes if and only if the initial dataset is isometric to initial data for the Kerr spacetime. As such, the invariant can be considered a measure of non-Kerrness on such initial data sets. In contrast with other similar invariants constructed through the notion of 'approximate Killing spinors', the present invariant is algebraic in the sense that it is algorithmically computable directly from initial data without having to solve any PDEs on the initial data hypersurface.