Master Functions and Equations for Perturbations of Vacuum Spherically-Symmetric Spacetimes

TL;DR

This paper derives the most general gauge-invariant master functions for perturbations of vacuum spherically-symmetric spacetimes, revealing new potential functions and extending known results like Regge-Wheeler and Zerilli functions.

Contribution

It identifies the complete set of gauge-invariant master functions satisfying wave equations with potentials, including new functions and an infinite family of potentials.

Findings

Two branches of solutions for master functions with similar features.

Known master functions are special cases within the first branch.

An infinite collection of potentials satisfying a nonlinear differential equation.

Abstract

Perturbation theory of vacuum spherically-symmetric spacetimes is a crucial tool to understand the dynamics of black hole perturbations. Spherical symmetry allows for an expansion of the perturbations in scalar, vector, and tensor harmonics. The resulting perturbative equations are decoupled for modes with different parity and different harmonic numbers. Moreover, for each harmonic and parity, the equations for the perturbations can be decoupled in terms of (gauge-invariant) master functions that satisfy 1+1 wave equations. By working in a completely general perturbative gauge, in this paper we study what is the most general master function that is linear in the metric perturbations and their first-order derivatives and satisfies a wave equation with a potential. The outcome of the study is that for each parity we have two branches of solutions with similar features. One of the branches includes the known results: In the odd-parity case, the most general master function is an arbitrary linear combination of the Regge-Wheeler and the Cunningham-Price-Moncrief master functions whereas in the even-parity case it is an arbitrary linear combination of the Zerilli master function and another master function that is new to our knowledge. The other branch is very different since it includes an infinite collection of potentials which in turn lead to an independent collection master of functions which depend on the potential. The allowed potentials satisfy a non-linear ordinary differential equation. Finally, all the allowed master functions are gauge invariant and can be written in a fully covariant form.