Energy, Hamiltonian, Noether charge, and black holes

It is shown that in general the energy $\mathcal{E}$ and the Hamiltonian $\mathcal{H}$ of matter fields on the black hole exterior play different roles. $\mathcal{H}$ is a generator of the time evolution along the Killing time while $\mathcal{E}$ enters the first law of black hole thermodynamics. For nonminimally coupled fields the difference $\mathcal{H}-\mathcal{E}$ 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 invariant theories. As an extension of the results by Wald and co-workers, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.