On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data

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.

A Geometric Invariant Characterising Initial Data for the Kerr–Newman Spacetime

“…Theorem 6 is the electrovacuum generalisation of the characterisation of initial data for the Kerr spacetime given in Theorem 28 in [4].…”

“…In this section, we introduce the necessary terminology and conventions to follow the discussion. The required properties of these objects for the present analysis are discussed in detail in Section 6.2 of [4] to which the reader is directed for further reference. Given u a scalar function over S and δ ∈ R, let u δ denote the weighted L 2 Sobolev norm of u.…”

“…These conditions are, like the KID equations, an overdetermined system and so do not necessarily admit a solution for an arbitrary initial surface. However, in [3,4] it has been shown that given an asymptotically Euclidean hypersurface it is always possible to construct a Killing spinor candidate which, whenever there exists a Killing spinor in the development, coincides with the restriction of the Killing spinor to the initial hypersurface. This approximate Killing spinor is obtained by solving a linear second-order elliptic equation which is the Euler-Lagrange equation of a certain functional over S. The approximate Killing spinor can be used to construct a geometric invariant, which in some way parametrises the deviation of the initial data set from Kerr initial data.…”

“…This approximate Killing spinor is obtained by solving a linear second-order elliptic equation which is the Euler-Lagrange equation of a certain functional over S. The approximate Killing spinor can be used to construct a geometric invariant, which in some way parametrises the deviation of the initial data set from Kerr initial data. Variants of the basic construction in [4] have been given in [5,6].…”

“…The purpose of this article is to extend the analysis of [4] to the electrovacuum case. In doing so, we rely on the characterisation of the Kerr-Newman spacetime given in [14] which, in turn, builds upon the characterisation provided in [18] for the vacuum case and [24] for the electrovacuum case.…”