Energy, Hamiltonian, Noether Charge, and Black Holes

Abstract

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 non-minimally 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 et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.

20 pages, latex, no figures

References in corpus (3)

• Mechanism of Generation of Black Hole Entropy in Sakharov's Induced Gravity
• Hamiltonian Thermodynamics of Black Holes in Generic 2-D Dilaton Gravity
• Black Hole Entropy in Induced Gravity: Reduction to 2D Quantum Field Theory on the Horizon

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• Brane realization of q-theory and the cosmological constant problem
• The Edge of Entanglement: Getting the Boundary Right for Non-Minimally Coupled Scalar Fields
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• Statistical Mechanics of Charged Black Holes in Induced Einstein-Maxwell Gravity
• Can One Understand Black Hole Entropy without Knowing Much about Quantum Gravity?
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• Generalized Second Law for Non-minimally Coupled Matter Theories
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