Action Functional of General Relativity as a Path Length. I. Closed Empty Universes

Abstract

For the case of closed empty universes it is established (up to a uniqueness conjecture) that the action functional of general relativity is a Riemannian path length in superspace, an infinite-dimensional manifold whose points represent three-dimensional geometries. Each procedure for representing spacetimes by trajectories in superspace yields a corresponding superspace metric. These metrics are just the "geodesic sheaf" metrics found by DeWitt. A general expression for all of these metrics is obtained in terms of arbitrary coordinates on superspace. The expression requires an explicit and unique solution to the spacelike constraints ($G^{0i}=0\mathrm{}$, $i=1, 2, 3$) of general relativity. Supertrajectories are separated into "timelike" ones which are permitted because they have real actions and "spacelike" ones which are forbidden because their actions are imaginary. By varying the path-length action, one finds that solutions of Einstein's sourcefree field equations correspond to a class of timelike geodesics in superspace. This class of geodesics is subject to conserved constraints which are homogeneous and quadratic in time derivatives. The truncated superspace which contains the empty "mixmaster" universes studied by Misner is discussed as an application of the supergeometric approach. The approach is particularly useful for analyzing the maximum expansion stage of the universe. Universes at this stage of their evolution encounter a real singularity of superspace. Time-symmetric universes end their supertrajectories on this singularity, while all other universes penetrate it in a well-defined way. The singularity, together with the causal structure of superspace, limits the anisotropy of empty mixmaster universes which are a finite number of $e$-foldings from the stage of maximum expansion.