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

TL;DR

This paper rigorously constructs a geometric invariant that uniquely characterizes Kerr initial data in vacuum Einstein equations, using approximate Killing spinors and analyzing their properties.

Contribution

It introduces a novel invariant based on approximate Killing spinors that distinguishes Kerr data from other initial data sets in general relativity.

Findings

Invariant vanishes only for Kerr data

Existence of solutions to approximate Killing spinor equation proven

Asymptotic analysis of spinor solutions conducted

Abstract

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.