Perturbations Around Backgrounds with One Non-Homogeneous Dimension

Abstract

This paper describes and proves a canonical procedure to decouple perturbations and optimize their gauge around backgrounds with one non-homogeneous dimension, namely of co-homogeneity 1, while preserving locality in this dimension. Derivatively-gauged fields are shown to have a purely algebraic action; they can be decoupled from the other fields through gauge-invariant re-definitions; a potential for the other fields is generated in the process; in the remaining action each gauge function eliminates one field without residual gauge. The procedure applies to spherically symmetric and to cosmological backgrounds in either General Relativity or gauge theories. The widely used ``gauge invariant perturbation theory'' is closely related. The supplied general proof elucidates the algebraic mechanism behind it as well as the method's domain of validity and its assumptions.