SOME PROPERTIES OF THE NOETHER CHARGE AND A PROPOSAL FOR DYNAMICAL BLACK-HOLE ENTROPY

We consider a general, classical theory of gravity with arbitrary matt er fields in n dimensions, arising from a diffeomorphism-invariant Lag rangian L. We show that L always can be written in a ''manifestly cova riant'' form. We then show that the symplectic potential current (n-1) -form THETA and the symplectic current (n-1)-form omega 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 an d corresponding Noether charge (n - 2)-form Q. We derive a general ''d ecomposition formula'' for Q. Using this formula for the Noether charg e, we prove that the first law of black hole mechanics holds for arbit rary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for station ary perturbations.) Finally, we propose a local, geometrical prescript ion for the entropy S(dyn) of a dynamical black hole. This prescriptio n agrees with the Noether charge formula for stationary black holes an d their perturbations, and is independent of all ambiguities associate d with the choices of L, THETA, and Q. However, the issue of whether t his dynamical entropy in general obeys a ''second law'' of black hole mechanics remains open. In an appendix, we apply some of our results t o theories with a nondynamical metric and also briefly develop the the ory of stress-energy pseudotensors.