Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism

TL;DR

This work provides a mature covariant and gauge-invariant framework for Schwarzschild metric perturbations, expressing the even- and odd-parity sectors through master functions (Zerilli-Moncrief and Cunningham-Price-Moncrief) that obey covariant wave equations with complete source terms. By constructing these master variables in a coordinate-independent way, the authors derive explicit expressions for radiative fields h_+ and h_×, and for the energy and angular-momentum fluxes both at future null infinity and across the event horizon. The formalism unifies the treatment across multiple coordinate systems, furnishes covariant source terms from matter stress-energy, and clarifies the low-multipole content (monopole and dipole) as either gauge modes or physical mass/rotation perturbations. The results provide a practical toolkit for computing gravitational radiation from matter in Schwarzschild backgrounds, with direct relevance to gravitational-wave modeling and interpretation in astrophysical and numerical contexts.

Abstract

We present a formalism to study the metric perturbations of the Schwarzschild spacetime. The formalism is gauge invariant, and it is also covariant under two-dimensional coordinate transformations that leave the angular coordinates unchanged. The formalism is applied to the typical problem of calculating the gravitational waves produced by material sources moving in the Schwarzschild spacetime. We examine the radiation escaping to future null infinity as well as the radiation crossing the event horizon. The waveforms, the energy radiated, and the angular-momentum radiated can all be expressed in terms of two gauge-invariant scalar functions that satisfy one-dimensional wave equations. The first is the Zerilli-Moncrief function, which satisfies the Zerilli equation, and which represents the even-parity sector of the perturbation. The second is the Cunningham-Price-Moncrief function, which satisfies the Regge-Wheeler equation, and which represents the odd-parity sector of the perturbation. The covariant forms of these wave equations are presented here, complete with covariant source terms that are derived from the stress-energy tensor of the matter responsible for the perturbation. Our presentation of the formalism is concluded with a separate examination of the monopole and dipole components of the metric perturbation.