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-time, spherically symmetric space-times, and stationary axisymmetric space-times. Moreover, we establish the relation between these vectors and the ADM energy-momentum vector for a two-surface taken in a limit to be spatial infinity in asymptotically flat space-times. We also discuss the geometrical properties of the Dirichlet and Neumann vectors and obtain several striking results relating these vectors to the mean curvature and normal curvature connection of the two-surface. Most significantly, the part of the Dirichlet vector normal to the two-surface depends only on the space-time metric at this surface and thereby defines a geometrical normal vector field on the two-surface. We show that this normal vector is orthogonal to the mean curvature vector, and its norm is the mean null extrinsic curvature, while its direction is such that there is zero expansion of the two-surface, i.e., the Lie derivative of the surface volume form in this direction vanishes. This leads to a direct relation between the Dirichlet vector and the condition for a spacelike two-surface to be (marginally) trapped. (C) 2002 American Institute of Physics.