Variational principles for nonsmooth metrics.

Abstract

We analyze the variational principle of general relativity for two classes of metrics that are not necessarily smooth but for which the Einstein action is still well defined. The allowed singularities in the metric are (1) a jump discontinuity in the extrinsic curvature at a three-dimensional hypersurface and (2) a conical singularity in the Euclidean Kantowski-Sachs metric on the manifold D¯×S2. In agreement with general expectations, we demonstrate that in both cases the extremizing vacuum metrics are the usual smooth solutions to the Einstein equations, the smoothness conditions coming themselves out of the variational principle as part of the equations of motion. In the presence of a singular matter distribution on a three-dimensional hypersurface, we also demonstrate that the usual junction conditions for the metric are directly obtained from the variational principle. We argue that variational principles with nonsmooth Kantowski-Sachs metrics are of interest in view of constructing minisuperspace path integrals on the manifold D¯×S2, in the context of both black-hole thermodynamics and quantum cosmology. The relation of nonsmooth Kantowski-Sachs variational principles to the issue of Lorentzian versus Euclidean path integrals in quantum gravity is briefly discussed.