Axial perturbations of general spherically symmetric spacetimes

TL;DR

This paper derives a generalized wave equation for axial metric perturbations in general spherically symmetric spacetimes, extending the Regge-Wheeler framework to include non-static backgrounds and matter effects.

Contribution

It introduces a governing wave equation for axial perturbations applicable to dynamic, matter-filled spherically symmetric spacetimes, generalizing the Regge-Wheeler equation.

Findings

Derived a two-dimensional wave equation for axial perturbations.

Showed the equation reduces to Regge-Wheeler in static vacuum cases.

Analyzed the case of viscous fluid matter perturbations.

Abstract

The aim of this paper is to present a governing equation for first order axial metric perturbations of general, not necessarily static, spherically symmetric spacetimes. Under the non-restrictive assumption of axisymmetric perturbations, the governing equation is shown to be a two-dimensional wave equation where the wave function serves as a twist potential for the axisymmetry generating Killing vector. This wave equation can be written in a form which is formally a very simple generalization of the Regge-Wheeler equation governing the axial perturbations of a Schwarzschild black hole, but in general the equation is accompanied by a source term related to matter perturbations. The case of a viscous fluid is studied in particular detail.