The First Law of Black Hole Mechanics

TL;DR

This paper reviews the development and formalism of the first law of black hole mechanics, highlighting its extensions to various theories and discussing open problems related to black hole entropy.

Contribution

It provides a detailed, graduate-level review of the generalizations of the first law of black hole mechanics across different gravitational theories.

Findings

Extended the first law to Einstein-Maxwell theory

Reviewed formalism for general diffeomorphism invariant theories

Discussed open problems in defining dynamical black hole entropy

Abstract

The first law of black hole mechanics has been the main motivation for investigating thermodynamic properties of black holes. The first version of this law was proved in \cite{Bardeen:1973gs} by considering perturbations of an asymptotically flat, stationary black hole spacetime to other stationary black hole spacetimes. This result was then extended to fully general perturbations, first in the context of Einstein-Maxwell theory in \cite{Sudarsky:1992ty},\cite{Wald:1993ki}, and then in the context of a general diffeomorphism invariant theory of gravity with an arbitrary number of matter fields in \cite{Wald:1993nt},\cite{Iyer:1994ys}. Here a review of these two generalizations of the first law is presented, with particular attention to outlining the necessary formalisms and calculations in an explicit and thorough way, understandable at a graduate level. The open problem of defining the entropy for a dynamical black hole that satisfies a form of the second law of black hole mechanics is briefly discussed.