Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

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, $\th$, and the symplectic current $(n-1)$-form, $\om$, 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 $\bL$, $\th$, 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.