On the Noether charge form of the first law of black hole mechanics

TL;DR

This paper discusses and justifies the Noether charge formulation of the first law of black hole mechanics, extending its validity to non-stationary variations within covariant gravity theories.

Contribution

It clarifies the gauge conditions and generalizes the Noether charge approach to non-stationary black hole variations, strengthening the theoretical foundation.

Findings

Justification of the gauge condition used in derivation

Extension of the first law to non-stationary variations

Reinforcement of the Noether charge formalism's validity

Abstract

The first law of black hole mechanics was derived by Wald in a general covariant theory of gravity for stationary variations around a stationary black hole. It is formulated in terms of Noether charges, and has many advantages. In this paper several issues are discussed to strengthen the validity of the Noether charge form of the first law. In particular, a gauge condition used in the derivation is justified. After that, we justify the generalization to non-stationary variations done by Iyer-Wald.