Making use of geometrical invariants in black hole collisions

TL;DR

This paper introduces a curvature invariant called the speciality index ${ m S}$, combining Weyl invariants I and J, to measure distortions in black hole collisions, aiding in understanding the transition from nonlinear to linear dynamics.

Contribution

It proposes a new invariant ${ m S}$ for analyzing black hole collisions, providing a tool to assess perturbative accuracy and interpret numerical results invariantly.

Findings

${ m S}$ effectively measures deviations from Kerr black holes.

${ m S}$ predicts when perturbative methods are accurate.

Application to axisymmetric collisions demonstrates practical utility.

Abstract

We consider curvature invariants in the context of black hole collision simulations. In particular, we propose a simple and elegant combination of the Weyl invariants I and J, the {\sl speciality index} ${\cal S}$. In the context of black hole perturbations $\cal S$ provides a measure of the size of the distortions from an ideal Kerr black hole spacetime. Explicit calculations in well-known examples of axisymmetric black hole collisions demonstrate that this quantity may serve as a useful tool for predicting in which cases perturbative dynamics provide an accurate estimate of the radiation waveform and energy. This makes ${\cal S}$ particularly suited to studying the transition from nonlinear to linear dynamics and for invariant interpretation of numerical results.