Perturbations Around Backgrounds with One Non-Homogeneous Dimension
Barak Kol
TL;DR
The paper addresses the problem of decoupling perturbations about backgrounds with a single non-homogeneous dimension (co-homogeneity 1) while preserving locality along that dimension. It introduces a canonical procedure for any 1d quadratic action with gauge symmetry, identifies derivatively-gauged (DG) fields as the image of $G_1$, and proves that, under mild invertibility and non-degeneracy conditions, the DG sector is algebraic and decouples after a shift to gauge-invariant coordinates, leaving a dynamic sector on $\dim D = n_F - 2 n_G$ gauge-invariant fields. The central theorem clarifies the algebraic mechanism behind gauge-invariant perturbation theory and applies to both spherically symmetric and cosmological backgrounds in GR or gauge theories, recovering known master equations (e.g., Zerilli and Regge-Wheeler) in Schwarzschild and yielding tensor-mode dynamics in cosmology. The work provides a rigorous, local, action-based framework for reducing perturbations to a minimal, gauge-invariant set of master variables, with explicit dimension counting and sector decomposition, and outlines open directions for higher-order perturbations and higher co-dimension cases. These results strengthen the theoretical foundation of perturbation analysis and offer practical tools for analyzing stability and dynamics in gravitational and gauge systems.
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.