Observables and gauge invariance in the theory of non-linear spacetime perturbations

TL;DR

This paper clarifies the role of gauge invariance in general-relativistic perturbation theory, emphasizing that first-order gauge-invariant scalar perturbations are physically observable, thus resolving conceptual issues related to gauge dependence.

Contribution

It demonstrates that first-order gauge-invariant scalar perturbations are observable, clarifying the gauge problem and the physical interpretation of perturbations in general relativity.

Findings

First-order gauge invariance ensures observability of scalar perturbations.

Gauge dependence at higher orders does not negate the physical meaning of first-order gauge-invariant quantities.

The gauge problem is clarified by linking measured fluctuations to gauge-invariant perturbations.

Abstract

We discuss the issue of observables in general-relativistic perturbation theory, adopting the view that any observable in general relativity is represented by a scalar field on spacetime. In the context of perturbation theory, an observable is therefore a scalar field on the perturbed spacetime, and as such is gauge invariant in an exact sense (to all orders), as one would expect. However, perturbations are usually represented by fields on the background spacetime, and expanded at different orders into contributions that may or may not be gauge independent. We show that perturbations of scalar quantities are observable if they are first order gauge-invariant, even if they are gauge dependent at higher order. Gauge invariance to first order plays therefore an important conceptual role in the theory, for it selects the perturbations with direct physical meaning from those having only a mathematical status. The so-called ``gauge problem'', and the relationship between measured fluctuations and gauge dependent perturbations that are computed in the theory are also clarified.