Gauge invariant perturbations of Schwarzschild black holes in horizon-penetrating coordinates

TL;DR

This paper develops a geometrical framework for analyzing gravitational perturbations of Schwarzschild black holes in horizon-penetrating coordinates, providing gauge-invariant equations and energy expressions useful for various black hole studies.

Contribution

It introduces a geometrical formulation of the Regge-Wheeler and Zerilli equations applicable to arbitrary spherically symmetric slices of Schwarzschild black holes, linking to the Teukolsky formalism.

Findings

Derived gauge-invariant perturbation equations

Provided a general expression for radiated energy

Linked the new equations to existing formalisms

Abstract

We derive a geometrical version of the Regge-Wheeler and Zerilli equations, which allows us to study gravitational perturbations on an arbitrary spherically symmetric slicing of a Schwarzschild black hole. We explain how to obtain the gauge-invariant part of the metric perturbations from the amplitudes obeying our generalized Regge-Wheeler and Zerilli equations and vice-versa. We also give a general expression for the radiated energy at infinity, and establish the relation between our geometrical equations and the Teukolsky formalism. The results presented in this paper are expected to be useful for the close-limit approximation to black hole collisions, for the Cauchy perturbative matching problem, and for the study of isolated horizons.