Symmetry and internal time on the superspace of asymptotically flat geometries

A difficulty with the canonical approach to quantum gravity, leading to attempts at "third quantization," is the absence of symmetry vectors on the superspace of three-metrics: vector fields that generate transformations of superspace leaving the action invariant. We show that on the superspace of asymptotically flat three-metrics, such symmetry vectors exist. They correspond to diffeomorphisms of each three-geometry that behave asymptotically as elements of the symmetry group at spatial infinity. The conserved momentum associated with a symmetry vector has a conjugate variable which can be regarded as an internal time coordinate of an isolated system. In particular, for asymptotic translations, a corresponding internal time is a center-of-mass coordinate. An appendix considers the natural contravariant and covariant metrics on superspace. Because natural contravariant metrics are not everywhere invertible, the associated covariant metrics are not everywhere defined.