Gauge-invariant Non-spherical Metric Perturbations of Schwarzschild Black-Hole Spacetimes

TL;DR

This paper reviews gauge-invariant non-spherical metric perturbations of Schwarzschild black holes, clarifying conventions, and providing a comprehensive reference for calculations involving matter sources and gravitational wave extraction.

Contribution

It consolidates and clarifies the expressions for Regge-Wheeler and Zerilli equations, addressing inconsistencies and aiding future research on black hole perturbations.

Findings

Clarified conventions for perturbation equations.

Provided asymptotic expressions and wave amplitudes.

Highlighted inconsistencies in existing literature.

Abstract

The theory of gauge-invariant non-spherical metric perturbations of Schwarzschild black hole spacetimes is now well established. Yet, as different notations and conventions have been used throughout the years, the literature on the subject is often confusing and sometimes confused. The purpose of this paper is to review and collect the relevant expressions related to the Regge-Wheeler and Zerilli equations for the odd and even-parity perturbations of a Schwarzschild spacetime. Special attention is paid to the form they assume in the presence of matter-sources and, for the two most popular conventions in the literature, to the asymptotic expressions and gravitational-wave amplitudes. Besides pointing out some inconsistencies in the literature, the expressions collected here could serve as a quick reference for the calculation of the perturbations of Schwarzschild black hole spacetimes driven by generic sources and for those approaches in which gravitational waves are extracted from numerically generated spacetimes.