Self-adjoint wave equations for dynamical perturbations of self-gravitating fields

TL;DR

This paper develops a self-adjoint wave equation framework for analyzing linear perturbations in self-gravitating fields, unifying various matter models and reproducing classical results like Regge-Wheeler and Zerilli equations.

Contribution

It introduces a curvature-based, self-adjoint wave equation approach for perturbations that naturally extends to self-gravitating matter fields, unlike traditional metric methods.

Findings

Wave operator is elliptic and self-adjoint.

Framework applies to gauge fields, Higgs fields, and fluids.

Re-derivation of Regge-Wheeler and Zerilli equations.

Abstract

It is shown that the dynamical evolution of linear perturbations on a static space-time is governed by a constrained wave equation for the extrinsic curvature tensor. The spatial part of the wave operator is manifestly elliptic and 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. It is also demonstrated how to obtain symmetric pulsation equations for self-gravitating non-Abelian gauge fields, Higgs fields and perfect fluids. For vacuum fluctuations on a vacuum space-time, the Regge-Wheeler and Zerilli equations are rederived.