Thermodynamics of Quantum Fields in Black Hole Backgrounds

TL;DR

This work analyzes quantum-field thermodynamics in black-hole backgrounds by relating the thermal partition function to an optical-space functional integral, highlighting a Liouville-type term that distinguishes it from the original metric. It provides general high-temperature expressions for scalar and fermion fields, and demonstrates that the optical method yields both corrections to black-hole entropy and the bulk entropy from radiation, whereas the conical-singularity approach captures only divergences. The study applies the framework to Rindler and Schwarzschild/Reissner–Nordström/dilaton black holes, revealing linear and logarithmic divergences that demand higher-derivative counterterms and exposing pathologies in extremal cases. The results have implications for the second law and the thermodynamic stability of black holes in thermal environments, while clarifying the relationship between geometric (entanglement) entropy and thermodynamic entropy.

Abstract

We discuss the relation between the micro-canonical and the canonical ensemble for black holes, and highlight some problems associated with extreme black holes already at the classical level. Then we discuss the contribution of quantum fields and demonstrate that the partition functions for scalar and Dirac (Majorana) fields in static space-time backgrounds, can be expressed as functional integrals in the corresponding optical space, and point out that the difference between this and the functional integrals in the original metric is a Liouville-type action. The optical method gives both the correction to the black hole entropy and the bulk contribution to the entropy due to the radiation, while (if the Liouville term is ignored) the conical singularity method just gives the divergent contribution to the black hole entropy. A simple derivation of a general formula for the free energy in the high-temperature approximation is given and applied to various cases. We conclude with a discussion of the second law.