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

TL;DR

This paper develops a gauge-invariant, covariant framework to derive four decoupled equations for gravitational perturbations on LRS class II space-times, advancing the understanding of GEM tensor harmonic amplitudes.

Contribution

It introduces a new complex GEM tensor formalism and decouples the perturbation equations into four independent equations for the first time.

Findings

Derived four decoupled equations for GEM tensor harmonic amplitudes.

Expressed the GEM system in a new complex form for easier decoupling.

Expanded GEM tensors in tensor harmonics to achieve full decoupling.

Abstract

This paper considers gauge-invariant and covariant gravitational perturbations on arbitrary vacuum locally rotationally symmetric (LRS) class II space-times. Ultimately, we derive four decoupled equations governing four specific combinations of the gravito-electromagnetic (GEM) 2-tensor harmonic amplitudes. We use the gauge-invariant and covariant 1+1+2 formalism which Clarkson and Barrett developed for analysis of vacuum Schwarzschild perturbations. In particular we focus on the first-order 1+1+2 GEM system and use linear algebra techniques suitable for exploiting its structure. Consequently, we express the GEM system new 1+1+2 complex form by choosing new complex GEM tensors, which is conducive to decoupling. We then show how to derive a gauge-invariant and covariant decoupled equation governing a newly defined complex GEM 2-tensor. Finally, the GEM 2-tensor is expanded in terms of arbitrary tensor harmonics and linear algebra is used once again to decouple the system further into 4 real decoupled equations.