Gauge dependence in the theory of non-linear spacetime perturbations

TL;DR

This paper develops a mathematical framework to understand how gauge choices affect non-linear spacetime perturbations, providing a compact formula for gauge transformations applicable in general relativity and other theories.

Contribution

It introduces the concept of knight diffeomorphisms and derives a general Taylor expansion formula, enabling a unified treatment of gauge transformations in perturbation theory.

Findings

Derived a compact gauge transformation formula for arbitrary order perturbations

Introduced the notion of knight diffeomorphisms and proved Taylor's formula for them

Applicable to any spacetime theory, not just general relativity

Abstract

Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we introduce the notion of knight diffeomorphism, that generalises the usual concept of flow, and prove a Taylor's formula for the action of a knight on a general tensor field. Then, we show that any one-parameter family of diffeomorphisms can be approximated by a family of suitable knights. Since in perturbation theory the gauge freedom is given by a one-parameter family of diffeomorphisms, the expansion of knights is used to derive our transformation formula. The problem of gauge dependence is a purely kinematical one, therefore our treatment is valid not only in general relativity, but in any spacetime theory.