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

DOI10.1063/1.1505984zbMATH Open1060.83003arXivgr-qc/0109013OpenAlexW2075203614MaRDI QIDQ4832809

Author name not available (Why is that?)

Publication date: 14 December 2004

Published in: (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/gr-qc/0109013

Mathematics Subject Classification ID

No records found.

This page was built for publication: Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded space–time regions

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q4832809)