A perturbative approach to Dirac observables and their spacetime algebra

We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non-trivial correction. Furthermore, we will make a first investigation into the Poisson algebra between observables corresponding to fields at different spacetime points and consider the locality properties of the observables.