Action and Energy of the Gravitational Field

We present a detailed examination of the variational principle for metric general relativity as applied to a " quasilocal " spacetime region M (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of M into spacelike hypersurfaces Σ. We allow for near complete generality in the choice of foliation. Using a field–theoretic generalization of Hamilton–Jacobi theory, we define the quasilocal stress-energy-momentum of the gravitational field by varying the action with respect to the metric on the boundary ∂M. The gravitational stress-energy-momentum is defined for a two–surface B spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy-momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian.