Invariant states and quantized gravitational perturbations

We study the problem of quantizing the gravitational fluctuations about a symmetric vacuum background spacetime with compact Cauchy surfaces. In the context of lowest-order perturbation theory we show that the allowed physical states must all be invariant under the symmetry transformations of the background spacetime. This constraint does not unduly restrict the range of allowed states and is consistent with temporal evolution (in the presence of a timelike symmetry) or spacial localization (for a spacelike symmetry) provided the evolution or localization is interpreted intrinsically rather than with reference to the background spacetime.