Quasilocal energy and conserved charges derived from the gravitational action

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on ${}_{}{}^3{}_{}{}^{}$B, the history of the system’s boundary. Energy surface density, momentum surface density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate ${}_{}{}^3{}_{}{}^{}$B. The integral of the energy surface density over such a two-surface B is the quasilocal energy associated with a spacelike three-surface Σ whose orthogonal intersection with ${}_{}{}^3{}_{}{}^{}$B is the boundary B. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary B as a surface embedded in Σ. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper-time translations on ${}_{}{}^3{}_{}{}^{}$B in the timelike direction orthogonal to B. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on ${}_{}{}^3{}_{}{}^{}$B. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt, Deser, and Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.