Properties of the symplectic structure of general relativity for spatially bounded space–time regions

We continue a previous analysis of the covariant Hamiltonian symplectic structure of general relativity for spatially bounded regions of space–time. To allow for wide generality, the Hamiltonian is formulated using any fixed hypersurface, with a boundary given by a closed spacelike two-surface. A main result is that we obtain Hamiltonians associated with Dirichlet and Neumann boundary conditions on the gravitational field coupled to matter sources, in particular a Klein–Gordon field, an electromagnetic field, and a set of Yang–Mills–Higgs fields. The Hamiltonians are given by a covariant form of the Arnowitt–Deser–Misner (ADM) Hamiltonian modified by a surface integral term that depends on the particular boundary conditions. The general form of this surface integral involves an underlying “energy-momentum” vector in the space–time tangent space at the spatial boundary two-surface. We give examples of the resulting Dirichlet and Neumann vectors for topologically spherical two-surfaces in Minkowski space–ti...