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arXiv · hep-th/9809049

Energy, Hamiltonian, Noether Charge, and Black Holes

D. V. Fursaev
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Abstract

It is shown that in general the energy ${\cal E}$ and the Hamiltonian ${\cal H}$ of matter fields on the black hole exterior play different roles. ${\cal H}$ is a generator of the time evolution along the Killing time while ${\cal E}$ enters the first law of black hole thermodynamics. For non-minimally coupled fields the difference ${\cal H}-{\cal E}$ 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 et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.

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D. V. Fursaev. 1998-09-08. Energy, Hamiltonian, Noether Charge, and Black Holes. https://doi.org/10.1103/physrevd.59.064020

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