Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations

TL;DR

This paper develops a method to construct gauge invariant variables for two-parameter nonlinear perturbations in general relativity, enabling consistent analysis of higher order metric and physical variable perturbations.

Contribution

It introduces a systematic procedure to define gauge invariant variables at higher orders, extending linear perturbation techniques to second and third order in two-parameter settings.

Findings

Gauge invariant variables are constructed up to third order.

The method applies to metric and arbitrary physical variables.

Gauge transformations behave similarly to linear case at higher orders.

Abstract

The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. For each order metric perturbation, we define the variable which is defined by the appropriate combination with lower order metric perturbations. Under the gauge transformation, this variable is transformed in the manner similar to the gauge transformation of the linear order metric perturbation. We confirm this up to third order. This implies that gauge invariant variables for higher order metric perturbations can be found by using a procedure similar to that for linear order metric perturbations. We also derive gauge invariant combinations for the perturbation of an arbitrary physical variable, other than the spacetime metric, up to third order.