Covariant Schwarzschild perturbations I: Initial value formulation for scalars of spin-weight -+ 2

TL;DR

This paper develops a covariant, gauge-invariant framework for analyzing Schwarzschild perturbations, deriving wave equations for spin-weighted scalars and formulating an initial value problem with energy-momentum sources.

Contribution

It introduces a covariant, gauge-invariant formalism for Schwarzschild perturbations, deriving decoupled wave equations for spin-weighted scalars from Bianchi identities.

Findings

Derivation of covariant, gauge-invariant wave equations for spin-weighted scalars.

Explicit representation of energy-momentum perturbations on spacelike slices.

Construction of a Cauchy problem for perturbations with energy-momentum sources.

Abstract

We consider full perturbations to a covariantly defined Schwarzschild spacetime. By constructing complex quantities, we derive two decoupled, covariant and gauge-invariant, wave-like equations for spin-weighted scalars. These arise naturally from the Bianchi identities and comprise a covariant representation of the Bardeen-Press equations for scalars with spin-weight $\pm2$. Furthermore, the covariant and gauge-invariant 1+1+2 formalism is employed, and consequently, the physical interpretation of the energy-momentum perturbations is transparent. They are written explicitly in terms of the energy-momentum specified on spacelike three-slices. Ultimately, a Cauchy problem is constructed whereby, an initial three-slice may be perturbed by an energy-momentum source, which induces resultant gravitational fields.