Gauge invariant perturbations of static spatially compact LRS II spacetimes

TL;DR

This paper develops a covariant, gauge-invariant framework for analyzing linear perturbations of static, spatially compact LRS II spacetimes, enabling analytical study of their stability and oscillation modes.

Contribution

It introduces a new covariant, gauge-invariant approach to perturbations in LRS II spacetimes, including a reduction to a Sturm-Liouville problem for isotropic, adiabatic cases.

Findings

Perturbation equations can be simplified by choosing appropriate frames.

Isotropic, adiabatic perturbations reduce to a Sturm-Liouville eigenvalue problem.

Lower bounds for eigenfrequencies are derived, aiding stability analysis.

Abstract

We present a framework to describe completely general first-order perturbations of static, spatially compact, and locally rotationally symmetric class II spacetimes within the theory of general relativity. The perturbation variables are by construction covariant and identification gauge invariant and encompass the geometry and the thermodynamics of the fluid sources. The new equations are then applied to the study of isotropic, adiabatic perturbations. We discuss how the choice of frame in which perturbations are described can significantly simplify the mathematical analysis of the problem and show that it is possible to change frames directly from the linear level equations. We find explicitly that the case of isotropic, adiabatic perturbations can be reduced to a singular Sturm-Liouville eigenvalue problem, and lower bounds for the values of the eigenfrequencies can be derived. These results lay the theoretical groundwork to analytically describe linear, isotropic, and adiabatic perturbations of static, spherically symmetric spacetimes.