Black hole thermodynamics

TL;DR

These notes establish the thermodynamic interpretation of stationary black holes by linking horizon area to entropy and surface gravity to temperature, grounded in both classical and quantum analyses. They unify area/entropy and $\,\kappa$-temperature via the zeroth and first laws, then derive thermal behavior of quantum fields in curved spacetime through the Unruh and Hawking effects, with the Euclidean path integral providing a robust bridge to entropy calculations. The discussion extends to higher-curvature gravity (Wald entropy) and the AdS/CFT framework, illustrating how black hole thermodynamics maps to thermal physics in dual field theories and highlighting the role of horizon regularity and Euclidean regularity in defining equilibrium states. Overall, the work emphasizes the universality of horizon thermodynamics and its deep connections to quantum gravity and holography.

Abstract

These notes introduce basic aspects of black hole thermodynamics. I review the classical laws of black hole mechanics, give a brief introduction to the essential concepts of quantum field theory in curved spacetime, and derive the Unruh and Hawking effects. I conclude with a discussion of entropy from the Euclidean path integral point of view and in the context of the AdS/CFT correspondence. Originally delivered as a set of lectures at the PIMS summer school ``Strings, Gravity and Cosmology'' in August 2004.

Figures (12)

• Figure 1: Kruskal diagram for Schwarzschild.
• Figure 2: Penrose diagram for Schwarzschild.
• Figure 3: Kruskal and Penrose diagrams for Schwarzschild-AdS.
• Figure 4: Rindler horizon in flat space.
• Figure 5: A family of null geodesics with $\theta <0$ initially will form a caustic; the dotted curve connecting $p$ and $q$ lies within the local light cone, so these points are timelike separated.