Phase space of general relativity revisited: A canonical choice of time and simplification of the Hamiltonian

Abstract

The phase space of general relativity is considered in the asymptotically flat context. Using spinorial techniques introduced by Witten, a prescription is given to transport rigidly the space-time translations at infinity to the interior of the (spatial) three-manifold. This yields a preferred four-parameter family of lapses and shifts and hence reduces the infinite-dimensional freedom in the choice of ‘‘time’’ to the restricted freedom available in special relativity. The corresponding Hamiltonians are computed and are shown to have an especially simple form: the Hamiltonians are ‘‘diagonal’’ in the (spatial) derivatives of variables which define ‘‘time.’’ Furthermore, the Hamiltonians (generating timelike translations) are shown to be positive in a neighborhood of the constraint submanifold of the phase space, even at points at which the ADM energy is negative.