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

TL;DR

This paper introduces a gauge-invariant, covariant formalism for analyzing metric perturbations of Schwarzschild spacetime, enabling calculation of gravitational waves from sources with clear physical interpretation.

Contribution

It develops a new covariant, gauge-invariant framework for Schwarzschild perturbations, expressing waveforms and radiation in terms of scalar functions satisfying wave equations.

Findings

Waveforms and radiated energy are expressed via two scalar functions.

The formalism applies to gravitational wave calculations from matter sources.

Covariant source terms are derived from stress-energy tensor.

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.