Thermal Fields, Entropy, and Black Holes

TL;DR

This work analyzes the statistical mechanics of quantum fields in spacetimes with Killing horizons and their connection to black hole entropy. It develops and compares canonical and covariant Euclidean formulations, elucidating how horizon-induced infrared and ultraviolet divergences arise and how they renormalize gravitational couplings. A key result is that, in standard gravity, a finite observable entropy requires accounting for bare geometric contributions and may not coincide with the statistical-mechanical entropy; in induced gravity, however, the black hole entropy can be identified with the regulated difference $S^{C}-Q$, where $Q$ is the Noether charge. The findings illuminate the role of conical singularities, the Noether charge, and the local horizon geometry in linking microscopic quantum fluctuations to macroscopic black hole thermodynamics, with potential implications for quantum gravity frameworks such as induced gravity and string theory.

Abstract

In this review we describe statistical mechanics of quantum systems in the presence of a Killing horizon and compare statistical-mechanical and one-loop contributions to black hole entropy. Studying these questions was motivated by attempts to explain the entropy of black holes as a statistical-mechanical entropy of quantum fields propagating near the black hole horizon. We provide an introduction to this field of research and review its results. In particular, we discuss the relation between the statistical-mechanical entropy of quantum fields and the Bekenstein-Hawking entropy in the standard scheme with renormalization of gravitational coupling constants and in the theories of induced gravity.