Unifying the Regge-Wheeler-Zerilli and Bardeen-Press-Teukolsky formalisms on spherical backgrounds

TL;DR

This paper presents a unified formalism for perturbation theory on spherical backgrounds, combining the Regge-Wheeler-Zerilli and Bardeen-Press-Teukolsky approaches using self-dual variables.

Contribution

It unifies two key formalisms in gravitational perturbation theory into a single tensorial framework, highlighting advantages like manifest isospectrality and algebraic metric reconstruction.

Findings

RWZ and BPT equations are different components of a single curvature tensor.

The formalism allows algebraic metric reconstruction from master functions in the frequency domain.

A Mathematica implementation based on xAct is provided for practical computations.

Abstract

We develop a formulation of perturbation theory on spherically symmetric backgrounds based on self-dual curvature equations combined with spherical harmonic expansions. The resulting framework unifies the Regge-Wheeler-Zerilli (RWZ) and Bardeen-Press-Teukolsky (BPT) formalisms and is designed to combine key advantages of both. The use of self-dual variables is crucial, and makes quasinormal mode isospectrality manifest, when present. We present the formalism first for a general energy-momentum tensor, and then specialize to vacuum General Relativity with matter sources to illustrate its practical advantages. A central result is that the RWZ and BPT equations arise directly as different components of a single tensorial curvature equation. We also show that, in the frequency domain, the metric can be reconstructed algebraically from any of the proposed master functions and their derivatives, and we comment on possible obstructions to such a reconstruction in the time domain. A Mathematica notebook, based on xAct, that implements the formalism and was used in our computations is released alongside this work.