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 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. Finally, we present several new examples of quasilocal energy momentum, including a novel discussion of quasilocal energy momentum in the large-sphere limit toward spatial infinity