Some properties of the 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 L. We first show that L alwasy can be written in a ‘‘manifestly covariant’’ form. We then show that the symplectic potential current (n-1)-form FTHETA and the symplectic current (n-1)-form ω for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n-1)-form J and corresponding Noether charge (n-2)-form Q. We derive a general ‘‘decomposition formula’’ for Q. 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 ${\mathit{S}}_{\mathrm{dyn}}$ 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 L, FTHETA, and Q. 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.