Perturbations of the Kerr spacetime in horizon penetrating coordinates

TL;DR

This paper develops a formalism for analyzing perturbations of Kerr black holes in horizon-penetrating coordinates, facilitating initial data setup and evolution studies relevant for numerical relativity and black hole mergers.

Contribution

It derives the Teukolsky equation in Kerr-Schild coordinates and provides explicit initial data formulas for time-domain evolution.

Findings

Formulated Teukolsky equation in ingoing/outgoing Kerr-Schild form

Provided explicit initial data expressions for numerical simulations

Facilitated studies of black hole perturbations in horizon-penetrating coordinates

Abstract

We derive the Teukolsky equation for perturbations of a Kerr spacetime when the spacetime metric is written in either ingoing or outgoing Kerr-Schild form. We also write explicit formulae for setting up the initial data for the Teukolsky equation in the time domain in terms of a three metric and an extrinsic curvature. The motivation of this work is to have in place a formalism to study the evolution in the ``close limit'' of two recently proposed solutions to the initial value problem in general relativity that are based on Kerr-Schild slicings. A perturbative formalism in horizon penetrating coordinates is also very desirable in connection with numerical relativity simulations using black hole ``excision''.