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This article contains a detailed and rigorous proof of the construction of ageometric invariant for initial data sets for the Einstein vacuum fieldequations. This geometric invariant vanishes if and only if the initial dataset corresponds to data for the Kerr spacetime, and thus, it characterises thistype of data. The construction presented is valid for boosted and non-boostedinitial data sets which are, in a sense, asymptotically Schwarzschildean. As apreliminary step to the construction of the geometric invariant, an analysis ofa characterisation of the Kerr spacetime in terms of Killing spinors is carriedout. A space spinor split of the (spacetime) Killing spinor equation isperformed, to obtain a set of three conditions ensuring the existence of aKilling spinor of the development of the initial data set. In order toconstruct the geometric invariant, we introduce the notion of approximateKilling spinors. These spinors are symmetric valence 2 spinors intrinsic to theinitial hypersurface and satisfy a certain second order elliptic equation---the approximate Killing spinor equation. This equation arises as theEuler-Lagrange equation of a non-negative integral functional. This functionalconstitutes part of our geometric invariant ---however, the whole functionaldoes not come from a variational principle. The asymptotic behaviour ofsolutions to the approximate Killing spinor equation is studied and anexistence theorem is presented.

On the construction of a geometric invariant measuring the deviation from Kerr data