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We investigate the covariant Hamiltonian symplectic structure of GeneralRelativity for spatially bounded regions of spacetime with a fixed time-flowvector. For existence of a well-defined Hamiltonian variational principletaking into account a spatial boundary, it is necessary to modify the standardArnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form dependson the spatial boundary conditions for the gravitational field. The mostgeneral mathematically allowed boundary conditions and corresponding boundaryterms are shown to be determined by solving a certain equation obtained fromthe symplectic current pulled back to the hypersurface boundary of thespacetime region. A main result is that we obtain a covariant derivation ofDirichlet, Neumann, and mixed type boundary conditions on the gravitationalfield at a fixed boundary hypersurface, together with the associatedHamiltonian boundary terms. As well, we establish uniqueness of these boundaryconditions under certain assumptions motivated by the form of the symplecticcurrent. Our analysis uses a Noether charge method which extends and unifiesseveral results developed in recent literature for General Relativity. As anillustration of the method, we apply it to the Maxwell field equations toderive allowed boundary conditions and boundary terms for existence of awell-defined Hamiltonian variational principle for an electromagnetic field ina fixed spatially bounded region of Minkowski spacetime.Comment: 49 pages; published version with minor errata corrected in Eqs.  (3.55) and (3.69

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