Construction of gauge-invariant variables of linear metric perturbations on an arbitrary background spacetime

TL;DR

This paper demonstrates a method to decompose linear metric perturbations into gauge-invariant and gauge-variant parts on arbitrary backgrounds with ADM decomposition, facilitating higher-order gauge-invariant perturbation theory development.

Contribution

It explicitly constructs gauge-invariant variables for linear metric perturbations on general backgrounds, advancing the framework for higher-order gauge-invariant perturbation theories.

Findings

Explicit construction of gauge-invariant variables using Green functions

Confirmation of decomposition through alternative approach

Implications for developing higher-order perturbation theories

Abstract

An outline of a proof of the decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on the an arbitrary background spacetime which admits ADM decomposition is discussed. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations through the assumption of the existence of some Green functions. We also confirm the result through another approach. This implies that we can develop the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime. Remaining issues to complete the general-framework of the general-relativistic higher-order gauge-invariant perturbation theories are also discussed.