Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded space–time regions

We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of space–time with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt–Deser–Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space–time region. A main result is that we obtain a covariant derivation of Dirichlet, Neumann, and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface, together with the associated Hamiltonian boundary terms. As well, we establish uniqueness of these boundary conditions under certain assumptions ...