A local non-negative initial data scalar characterisation of the Kerr solution

TL;DR

This paper introduces a local scalar quantity for vacuum initial data that vanishes precisely for Kerr initial data, enabling numerical and stability analyses of Kerr solutions.

Contribution

It defines a new local, non-negative scalar characterizing Kerr initial data, computable from initial data sets, aiding numerical and stability studies.

Findings

Scalar vanishes only for Kerr initial data

Scalar is computable via algebraic and differential operations

Useful for numerical simulations and stability analysis

Abstract

For any vacuum initial data set, we define a local, non-negative scalar quantity which vanishes at every point of the data hypersurface if and only if the data are {\em Kerr initial} data. Our scalar quantity only depends on the quantities used to construct the vacuum initial data set which are the Riemannian metric defined on the initial data hypersurface and a symmetric tensor which plays the role of the second fundamental form of the embedded initial data hypersurface. The dependency is {\em algorithmic} in the sense that given the initial data one can compute the scalar quantity by algebraic and differential manipulations, being thus suitable for an implementation in a numerical code. The scalar could also be useful in studies of the non-linear stability of the Kerr solution because it serves to measure the deviation of a vacuum initial data set from the Kerr initial data in a local and algorithmic way.