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

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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.