The generalization of the Regge-Wheeler equation for self-gravitating matter fields

TL;DR

This paper extends the Regge-Wheeler equation to self-gravitating matter fields using a curvature-based approach, enabling spectral analysis of perturbations in static spacetimes, including non-Abelian gauge fields.

Contribution

It introduces a generalized pulsation equation for self-gravitating matter fields that maintains self-adjointness, facilitating spectral analysis of gravitational perturbations.

Findings

The wave operator is self-adjoint in the curvature-based formulation.

Spectral theory applies to odd-parity perturbations of spherically symmetric backgrounds.

Explicit symmetric pulsation equations are derived for non-Abelian gauge fields.

Abstract

It is shown that the dynamical evolution of perturbations on a static spacetime is governed by a standard pulsation equation for the extrinsic curvature tensor. The centerpiece of the pulsation equation is a wave operator whose spatial part is manifestly self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. For a certain relevant subspace of perturbations the pulsation operator is symmetric with respect to a positive inner product and therefore allows spectral theory to be applied. In particular, this is the case for odd-parity perturbations of spherically symmetric background configurations. As an example, the pulsation equations for self-gravitating, non-Abelian gauge fields are explicitly shown to be symmetric in the gravitational, the Yang Mills, and the off-diagonal sector.