Canonical variables and quasilocal energy in general relativity

Abstract

Recently Brown and York(1993) have devised a new method for defining quasilocal energy in general relativity. Their method employs a Hamilton-Jacobi analysis of an action functional for a spatially bounded spacetime M, and this analysis yields expressions for the quasilocal energy and momentum surface densities associated with the two-boundary B of a spacelike slice of such a spacetime. These expressions are essentially Amowitt-Deser-Misner variables, but with canonical conjugacy defined with respect to the time history 3B of the two-boundary. These boundary ADM variables match previous variables which have arisen directly in the study of black hole thermodynamics. Further, a 'microcanonical' action which features fixed-energy boundary conditions has been introduced in related work concerning the functional-integral representation of the density of quantum states for such a bounded gravitational system. This paper introduces Ashtekar-type variables on the time history 3B of the two-boundary and shows that these variables lead to elegant alternative expressions for the quasilocal surface densities. In addition, it is demonstrated here that both the boundary ADM variables and the boundary Ashtekar-type variables can be incorporated into a larger framework by appealing to the tetrad-dependent Sparling differential forms. Finally, using these results and a tetrad action principle employed by Goldberg(1988), this paper constructs two new tetrad versions of the microcanonical action.