Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded spacetime regions

TL;DR

This paper derives covariant Hamiltonian boundary conditions for General Relativity in spatially bounded regions, unifying various boundary conditions through a symplectic current approach and extending the method to electromagnetic fields.

Contribution

It provides a covariant derivation of boundary conditions and Hamiltonian boundary terms in General Relativity, including Dirichlet, Neumann, and mixed types, with a uniqueness proof.

Findings

Derived covariant boundary conditions for gravitational fields.

Unified treatment of boundary conditions using symplectic current.

Applied method to electromagnetic fields in Minkowski spacetime.

Abstract

We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime 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 spacetime 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 motivated by the form of the symplectic current. Our analysis uses a Noether charge method which extends and unifies several results developed in recent literature for General Relativity. As an illustration of the method, we apply it to the Maxwell field equations to derive allowed boundary conditions and boundary terms for existence of a well-defined Hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of Minkowski spacetime.