Introduction to black hole thermodynamics

TL;DR

The notes connect classical black hole mechanics to quantum aspects by developing a coherent semiclassical framework in which horizon geometry fixes thermodynamic quantities: the Bekenstein–Hawking entropy via Wald’s construction and the Hawking temperature via Euclidean regularity at horizons. The gravitational path integral is used to compare competing saddles (e.g., thermal AdS vs AdS–Schwarzschild) and to derive the Hawking–Page transition, with holographic renormalization ensuring finite on-shell actions for asymptotically AdS spacetimes. The work also analyzes rotating and charged solutions, the role of complex metrics, and subtleties such as the conformal factor problem and large diffeomorphisms, highlighting both the successes of Euclidean gravity as a thermodynamic tool and its conceptual limits as a fundamental quantum gravity definition. Overall, the approach reveals deep links between geometry, topology, and thermodynamics, and underscores holographic interpretations (AdS/CFT) as a natural arena for black hole thermodynamics and quantum gravity questions.

Abstract

These are the lecture notes for a course at the "Roberto Salmeron School in Mathematical Physics" held at the University of Brasilia in September 2025, to be published in the proceedings book "Modern topics in mathematical physics." The course provides a concise and biased introduction to black hole thermodynamics. It covers the laws of black hole mechanics, Hawking radiation, Euclidean quantum gravity methods, and AdS black holes.

Figures (11)

• Figure 1: The worldline of an accelerated observer in flat space. The line $\{ x = t \}$ is the Rindler horizon.
• Figure 2: Minkowski spacetime divided in the wedges determined by the norm of the boost Killing vector $b^a$. In blue are two loci of constant $\xi$, or equivalently, $\rho$, as defined in \ref{['eq:1_Rindler_Coordinates']} and by $\rho = {\rm e}^{\alpha\xi}/\alpha$, which are trajectories of the Rindler observers following orbits of $b^a$. In purple are two loci of constant $\eta$.
• Figure 3: Maximal analytic extension of the Schwarzschild black hole, covered by the coordinates $(U,V)$ (each point represents a two-sphere). In blue is a locus of constant $r>2M$, corresponding to an orbit of $k=\partial_t$. In purple is a locus of constant $t$. The event horizon $\{ r = 2M\}$ is the union of the two axes $\{U=0\} \cup \{ V=0 \}$, and the singularity $r=0$ is in red.
• Figure 4: Thanks to the KMS condition \ref{['eq:2_KMS_v1']} and the causality requirements of relativistic quantum field theory, we can extend the correlators to a unique analytic function on the complex plane for $z=t+{\rm i} t_E$, except for the horizontal branch cuts from $t=\pm d$ to $t\to \pm \infty$ at $t_E = \ell \beta$ ($\ell \in \mathbb{Z}$).
• Figure 5: The topology of a static spherically symmetric black hole is $\mathbb{R}^2\times S^2$. The metric on the $\mathbb{R}^2$ factor parametrized by $(t_E, r)$, and the ranges of the coordinates are such that the space caps off smoothly at $r=r_+$ with flat metric in a neighbourhood. The asymptotic behaviour as $r\to \infty$ depends on the cosmological constant: here we represent an asymptotically flat solution, where the $\mathbb{R}^2$ factor, as $r\to \infty$, becomes a cylinder with metric ${\rm d} t_E^2 + {\rm d} r^2$, since the size of the circles at constant $r$ doesn't grow. This is the "cigar" geometry.