THE CHERN-SIMONS INVARIANT AS THE NATURAL TIME VARIABLE FOR CLASSICALAND QUANTUM COSMOLOGY

We propose that the Chem-Simons invariant of the Ashtekar-Sen connecti on is the natural internal time coordinate for classical and quantum c osmology. The reasons for this are a number of interesting properties of this functional, which we describe here. (1) It is a function on th e gauge and diffeomorphism invariant configuration space, whose gradie nt is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. (2) The im aginary part of the Chem-Simons form reduces in the limit of small cos mological constant, Lambda, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. (3) Small matter-field excitat ions of the Chem-Simons state satisfy, by virtue of the quantum constr aints, a functional Schrodinger equation in which the matter fields ev olve on a DeSitter background in the Chern-Simons time. We then propos e this is the natural vacuum state of the theory for Lambda not equal 0. (4) This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, whic h means that physical expectation values for all states in non-perturb ative quantum gravity will satisfy the KMS condition, and may then be interpreted as thermal states. (5) Forms for the physical Hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on t he connection superspace, is found.