Observables and gauge invariance in the theory of nonlinear spacetime perturbations

We discuss the issue of observables in general-relativistic perturbation th eory, adopting the view that any observable in general relativity is repres ented 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 ba ckground spacetime, and expanded at different orders into contributions tha t 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 therefore plays an important conceptual role in the theory, for it selects the perturbations with direct physical meaning from those having only a mat hematical status. The so-called 'gauge problem', and the relationship betwe en measured fluctuations and gauge-dependent perturbations that are compute d in the theory are also clarified.