Decoupled Gravitational Wave Equations in Spherical Symmetry from Curvature Wave Equations

TL;DR

This paper introduces a new approach to black hole perturbation theory in spherical symmetry, deriving a decoupled wave equation for a complex variable that simplifies analysis and reveals isospectrality between sectors.

Contribution

The authors develop a method to derive a single decoupled wave equation from curvature equations, simplifying traditional perturbation techniques in Schwarzschild backgrounds.

Findings

Derived a decoupled wave equation for a complex curvature variable.

Established isospectrality between even and odd metric perturbations.

Provided a systematic framework for perturbation analysis in spherical symmetry.

Abstract

Black hole perturbation theory on spherically symmetric backgrounds has been instrumental in establishing various aspects about the gravitational dynamics close to black holes, and continues to be an interesting avenue to confront current challenges in gravitational physics. In this paper, we present an approach to perturbation theory in spherical symmetry that addresses simultaneously some conceivably inconvenient aspects of the traditional methods. In particular, focusing on Schwarzschild's background we are able to derive a decoupled wave equation, for a single complex variable, by simply computing one component of the curvature wave equation satisfied by a complex self-dual version of the Riemann tensor. The real and imaginary parts of the variable consist only of even and odd pieces of the metric fluctuation, respectively, and both satisfy the Regge-Wheeler equation. Besides providing a systematic derivation of decoupled equations, an immediate corollary of our results is the isospectrality between even and odd sectors. We conclude by discussing potential extensions of our formalism to include matter and higher orders in perturbation theory.