1+1+2 Electromagnetic perturbations on non-vacuum LRS class II space-times: Decoupling scalar and 2-vector harmonic amplitudes

TL;DR

This paper develops a covariant, gauge-invariant framework to analyze electromagnetic perturbations in non-vacuum LRS class II space-times, deriving new decoupled equations and classifying perturbation types with source considerations.

Contribution

It introduces six decoupled equations for EM harmonic amplitudes in LRS class II space-times, including four new equations from complex EM 2-vector expansion and a full classification of energy-momentum sources.

Findings

Derived six decoupled equations for EM perturbations.

Identified four new decoupled equations from harmonic amplitude expansion.

Generalized Regge-Wheeler equations to include energy-momentum sources.

Abstract

We use the covariant and gauge-invariant 1+1+2 formalism of Clarkson and Barrett \cite{Clarkson2003} to analyze electromagnetic (EM) perturbations on non-vacuum {\it locally rotationally symmetric} (LRS) class II space-times. Ultimately, we show how to derive six real decoupled equations governing the total of six EM scalar and 2-vector harmonic amplitudes. Four of these are new, and result from expanding the complex EM 2-vector which we defined in \cite{Burston2007} in terms of EM 2-vector harmonic amplitudes. We are then able to show that there are four precise combinations of the amplitudes that decouple, two of these are polar perturbations whereas the remaining two are axial. The remaining two decoupled equations are the generalized Regge-Wheeler equations which were developed previously in \cite{Betschart2004}, and these govern the two EM scalar harmonic amplitudes. However, our analysis generalizes this by including a full description and classification of energy-momentum sources, such as charges and currents.