Article,
Some properties of the Noether charge and a proposal for dynamical black hole entropy
Physical Review D, 50 (2): 846--864 (Jul 15, 1994)
DOI: 10.1103/physrevd.50.846
Abstract
We consider a general, classical theory of gravity with arbitrary matter
fields in \$n\$ dimensions, arising from a diffeomorphism invariant Lagrangian,
\$\bL\$. We first show that \$\bL\$ always can be written in a ``manifestly
covariant" form. We then show that the symplectic potential current
\$(n-1)\$-form, \$þ\$, and the symplectic current \$(n-1)\$-form, \$øm\$, for the
theory always can be globally defined in a covariant manner. Associated with
any infinitesimal diffeomorphism is a Noether current \$(n-1)\$-form, \$\bJ\$, and
corresponding Noether charge \$(n-2)\$-form, \$\bQ\$. We derive a general
``decomposition formula" for \$\bQ\$. Using this formula for the Noether charge,
we prove that the first law of black hole mechanics holds for arbitrary
perturbations of a stationary black hole. (For higher derivative theories,
previous arguments had established this law only for stationary perturbations.)
Finally, we propose a local, geometrical prescription for the entropy,
\$S\_dyn\$, of a dynamical black hole. This prescription agrees with the Noether
charge formula for stationary black holes and their perturbations, and is
independent of all ambiguities associated with the choices of \$\bL\$, \$þ\$, and
\$\bQ\$. However, the issue of whether this dynamical entropy in general obeys a
``second law" of black hole mechanics remains open. In an appendix, we apply
some of our results to theories with a nondynamical metric and also briefly
develop the theory of stress-energy pseudotensors.
