Energy, Hamiltonian, Noether charge, and black holes - art. no. 064020

It is shown that in general the energy E and the Hamiltonian H of matter fi elds on the black hole exterior play different roles. H is a generator of t he time evolution along the Killing time while E enters the first law of bl ack hole thermodynamics. For nonminimally coupled fields the difference H-E is not zero and is a Noether charge Q analogous to that introduced by Wald to define the black hole entropy. If fields vanish at the spatial boundary , Q is reduced to an integral over the horizon. The analysis is carried out and an explicit expression for Q is found for general diffeomorphism invar iant theories. As an extension of the results by Wald and co-workers, the f irst law of black hole thermodynamics is derived for arbitrary weak matter fields. [S0556-2821(99)00806-1].