A geometric invariant characterising initial data for the Kerr-Newman spacetime

TL;DR

This paper introduces a geometric invariant that uniquely identifies initial data sets corresponding to the Kerr-Newman spacetime, aiding in the characterization and analysis of such solutions in general relativity.

Contribution

It develops a new invariant based on Killing spinors and space spinor formalism that precisely characterizes Kerr-Newman initial data sets.

Findings

Invariant vanishes only for Kerr-Newman initial data

Derived four conditions ensuring existence of Killing spinors

Constructed the invariant using approximate Killing spinor solutions

Abstract

We describe the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr-Newman spacetime in terms of Killing spinors. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant.