A Survey of Black Hole Thermodynamics

TL;DR

This review analyzes how black holes exemplify thermodynamics in curved spacetime, detailing the four laws, horizon definitions, and entropy origin including quantum and stringy corrections. It connects classical results (like the area/entropy relation and the Raychaudhuri equation) with quantum refinements (GSL, QFC, QNEC) and the role of holography (AdS/CFT, HRT formula, entanglement wedges) in understanding information preservation and microstates. The work synthesizes how higher-curvature corrections, induced gravity, and entanglement entropy shape black hole thermodynamics, and discusses how holography provides a powerful framework to study information flow, thermalization, and scrambling in quantum gravity. Overall, the paper argues that black hole horizons encode fundamental thermodynamic and quantum-information principles that transcend gravity, with holography offering concrete tools to reconstruct bulk data from boundary degrees of freedom and to quantify information processing in gravitational settings.

Abstract

This is an introductory, up-to-date review of the essentials of black hole thermodynamics. The main topics surveyed are: (i) the four laws of thermodynamics as applied to a black hole horizon, and the current status of their proofs; (ii) different definitions of horizons, and their unique properties; (iii) the nature of black hole entropy, its quantum and stringy corrections, and ultimate origin from quantum gravity microstates; (iv) the focusing law for the area/entropy; and finally (v) the holographic principle, and how we can use it to learn about the information inside black holes.

Figures (4)

• Figure 1: Penrose diagram of the Schwarzschild black hole, an example of a Killing horizon. Each point represents a sphere, light travels at $45\degree$, solid boundary edges are infinity, and jagged edges are singularities. The future horizon $H^+$, past horizon $H^-$, bifurcation surface $B$, and the action of the Killing vector $\chi^a$ are shown. (In the case of a black hole that forms from collapse and later becomes stationary, only the right and upper quadrants exist.)
• Figure 2: The lightrays generating the future Killing horizon $H^+$ are shown (vertical lines) with $(v,y)$ coordinates. $\rho_{HH}$ is a ground state with respect to translation along any lightray $\gamma$, but is thermal with respect to a "boost" in the (shaded) region above any cut $v_*(y)$.
• Figure 3: (a) The different kinds of horizons $C^\pm$, $T^\pm$ are plotted on the Penrose diagram of a wormhole going between two asymptotically AdS regions (with timelike boundaries). This is not a Killing spacetime due to the gravitational effects of matter (not shown), which tends to make the diagram wider. Also shown is the HRT extremal surface $X$ discussed in section \ref{['HEF']}, and the contracting null surfaces coming out from it. Arrows are drawn in the direction of increasing area. In holography, this spacetime is dual to two entangled CFT's on the left and right. The entanglement wedge dual to $\text{CFT}_L$ is shaded. (b) A black hole that forms from the collapse of a star. There is neither an HRT surface, nor past horizons. $C^+$ is always null, but $T^+$ may have timelike and spacelike segments (here a dot marks the transition). On the top-right corner of each diagram, after the black hole settles into a stationary state, $C^+$ and $T^+$ both coincide with the late-time Killing horizon.
• Figure 4: A black hole is formed from collapse by exciting a scalar field $\varphi$ using operators available in the boundary algebra $\mathcal{A}(t_1)$ at an early time $t_1$. But because the Hamiltonian $H$ is measurable using the gravitational field at infinity, the details of the collapse are still encoded in the boundary algebra at any later time $t_2 > t_1$, even before the black hole totally evaporates into Hawking radiation!