1+1+2 gravitational perturbations on LRS class II space-times: GEM vector harmonic amplitudes

TL;DR

This paper develops a method to decouple and analyze gravitational perturbations in LRS class II space-times using harmonic expansions and eigen-analysis, leading to four real decoupled equations for GEM vector and shear tensor amplitudes.

Contribution

It introduces a novel decoupling approach for complex GEM vector and tensor perturbations in LRS class II space-times using harmonic and eigen-analysis techniques.

Findings

Derivation of four real decoupled equations for GEM and shear perturbations.

Classification of perturbations into polar and axial types.

Framework for analyzing gravitational perturbations in symmetric space-times.

Abstract

This is the second in a series of papers which considers first-order gauge-invariant and covariant gravitational perturbations to locally rotationally symmetric (LRS) class II space-times. This paper shows how to decouple a complex combination of the gravito-electromagnetic (GEM) 2-vectors with the 2-tensors describing the shear of the 2/3-sheets. An arbitrary harmonic expansion is then used along with an eigen-vector/value analysis of the first-order GEM system, analogous to the first paper in this series. This results in four real decoupled equations governing four real combinations of the harmonic amplitudes of the GEM 2-vectors and the (2/3-sheet) shear 2-tensors. Finally, these are categorized into polar and axial perturbations.