Black Holes and Thermodynamics

TL;DR

The paper investigates the connection between black hole mechanics and thermodynamics, arguing that Hawking radiation provides a physical temperature $T_H = \kappa/(2\pi)$ and that, in general relativity, the black hole entropy is $S_{ m bh}=A/4$, thereby completing the thermodynamic analogy with $E \leftrightarrow M$, $T \leftrightarrow \kappa/(2\pi)$, and $S \leftrightarrow A/4$. It develops a covariant, theory-agnostic first-law framework via Noether charges, yielding $\delta M = \frac{\kappa}{2\pi}\,\delta S_{ m bh} + \Omega\,\delta J$ and identifying $S_{ m bh}$ with a local geometric functional $S_{ m bh} = -2\pi \int_{\cal C} \frac{\delta L}{\delta R_{abcd}} n_{ab} n_{cd}$ (reducing to $A/4$ in GR). The generalized second law $S' = S + S_{ m bh}$ with $\Delta S' \ge 0$ is discussed as a resolution to entropy accounting when black holes radiate; however, the microscopic origin of $S_{ m bh}$ and the role of the horizon's thermal atmosphere remain open questions. The paper also surveys string-theory results, where D-brane microstate counting agrees with $S_{ m bh}$ for certain extremal/near-extremal holes and reproduces low-energy emission, supporting a shared microscopic basis for black hole entropy and ordinary thermodynamics, while emphasizing the need for a full quantum gravity framework.

Abstract

We review the remarkable relationship between the laws of black hole mechanics and the ordinary laws of thermodynamics. It is emphasized that - in analogy with the laws of thermodynamics - the validity the laws of black hole mechanics does not appear to depend upon the details of the underlying dynamical theory (i.e., upon the particular field equations of general relativity). It also is emphasized that a number of unresolved issues arise in ``ordinary thermodynamics'' in the context of general relativity. Thus, a deeper understanding of the relationship between black holes and thermodynamics may provide us with an opportunity not only to gain a better understanding of the nature of black holes in quantum gravity, but also to better understand some aspects of the fundamental nature of thermodynamics itself.