Gauge and motion in perturbation theory

TL;DR

This paper clarifies the mathematical treatment of perturbed motion in second-order perturbative general relativity, emphasizing gauge invariance and relationships between different approximation methods, with broad applicability to various descriptions.

Contribution

It explicitly derives the Gralla-Wald approximation from the self-consistent approach and develops a general, invariant framework for smooth gauge transformations in second-order perturbation theory.

Findings

Derived the Gralla-Wald approximation from the self-consistent approach.

Presented a gauge transformation framework that preserves invariance.

Applicable to multiple perturbation descriptions beyond the specific methods discussed.

Abstract

Through second order in perturbative general relativity, a small compact object in an external vacuum spacetime obeys a generalized equivalence principle: although it is accelerated with respect to the external background geometry, it is in free fall with respect to a certain \emph{effective} vacuum geometry. However, this single principle takes very different mathematical forms, with very different behaviors, depending on how one treats perturbed motion. Furthermore, any description of perturbed motion can be altered by a gauge transformation. In this paper, I clarify the relationship between two treatments of perturbed motion and the gauge freedom in each. I first show explicitly how one common treatment, called the Gralla-Wald approximation, can be derived from a second, called the self-consistent approximation. I next present a general treatment of smooth gauge transformations in both approximations, in which I emphasise that the approximations' governing equations can be formulated in an invariant manner. All of these analyses are carried through second perturbative order, but the methods are general enough to go to any order. Furthermore, the tools I develop, and many of the results, should have broad applicability to any description of perturbed motion, including osculating-geodesic and two-timescale descriptions.