A characterization of 3+1 spacetimes via the Simon-Mars tensor

Abstract

We present the 3+1 decomposition of the Simon-Mars tensor, which has the property of being identically zero for a vacuum and asymptotically flat spacetime if and only if the latter is locally isometric to the Kerr spacetime. Using this decomposition we form two dimensionless scalar fields. Computing these scalars provides a simple way of comparing locally a generic (even non vacuum and non analytic) stationary spacetime to Kerr. As an illustration, we evaluate the Simon-Mars scalars for numerical solutions of the Einstein equations generated by boson stars and neutron stars, for analytic solutions of the Einstein equations such as Curzon-Chazy spacetime and $\delta=2$ Tomimatsu-Sato spacetime, and for an approximate solution of the Einstein equations : the modified Kerr metric, which is an example of a parametric deviation from Kerr spacetime.