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

TL;DR

This work consolidates the theory of gauge-invariant, non-spherical metric perturbations of Schwarzschild black holes, focusing on the odd-parity Regge-Wheeler and even-parity Zerilli formalisms for multipoles with $\ell \ge 2$ and generic matter sources. It clarifies the definitions and relations among master functions $\Psi^{({\rm o})}$, $\Psi^{({\rm e})}$, and alternative Moncrief variables, derives their wave equations, and connects them to observable gravitational waves via asymptotic amplitudes $h_+$ and $h_\times$ in two common conventions (CPM/ RW and Abrahams-Price). The paper provides detailed initial-data constraints, frequency-domain quasi-normal-mode analyses, and explicit expressions for energy and angular-momentum fluxes, offering a practical reference for GW extraction in numerical spacetimes and highlighting potential literature inconsistencies. It also outlines clear pathways to extend these gauge-invariant methods to Kerr spacetimes and to include non-vacuum perturbations, broadening applicability to realistic astrophysical scenarios.

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.