Two-parameter non-linear spacetime perturbations: gauge transformations and gauge invariance

TL;DR

This paper develops a general formalism for analyzing gauge transformations and invariance in two-parameter non-linear spacetime perturbations, with applications to rotating star models in relativity.

Contribution

It introduces explicit higher order gauge transformation rules and a framework for gauge invariance in two-parameter non-linear perturbation theory.

Findings

Derived explicit gauge transformation rules for second-order perturbations.

Defined gauge invariance criteria for multi-parameter perturbations.

Applicable to various spacetime theories and astrophysical models.

Abstract

An implicit fundamental assumption in relativistic perturbation theory is that there exists a parametric family of spacetimes that can be Taylor expanded around a background. The choice of the latter is crucial to obtain a manageable theory, so that it is sometime convenient to construct a perturbative formalism based on two (or more) parameters. The study of perturbations of rotating stars is a good example: in this case one can treat the stationary axisymmetric star using a slow rotation approximation (expansion in the angular velocity Omega), so that the background is spherical. Generic perturbations of the rotating star (say parametrized by lambda) are then built on top of the axisymmetric perturbations in Omega. Clearly, any interesting physics requires non-linear perturbations, as at least terms lambda Omega need to be considered. In this paper we analyse the gauge dependence of non-linear perturbations depending on two parameters, derive explicit higher order gauge transformation rules, and define gauge invariance. The formalism is completely general and can be used in different applications of general relativity or any other spacetime theory.