Energy, Hamiltonian, Noether Charge, and Black Holes

TL;DR

This paper explores the distinction between energy and Hamiltonian in black hole physics, linking the difference to a Noether charge and extending the first law of thermodynamics to include matter fields.

Contribution

It introduces a general framework for understanding the difference between energy and Hamiltonian in black hole spacetimes, relating it to Noether charges and extending thermodynamic laws.

Findings

The Hamiltonian generates time evolution, while energy enters the first law.

The difference between Hamiltonian and energy is a Noether charge Q.

Explicit expression for Q is derived for diffeomorphism invariant theories.

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.