Free Energy and Entropy for Semi-classical Black Holes in the Canonical Ensemble

TL;DR

This paper analyzes the thermodynamics of semi-classical black holes in a finite spherical region, calculating quantum corrections to key thermodynamic quantities and exploring conditions for black hole nucleation.

Contribution

It provides the first computation of $O( abla)$ quantum corrections to black hole thermodynamics in the canonical ensemble using recent semi-classical solutions.

Findings

Quantum corrections to black hole mass, entropy, and free energy are derived.

Conditions for black hole nucleation from hot flat space are identified.

Thermodynamic stability criteria are discussed.

Abstract

We consider the thermodynamics of a black hole coupled to thermal radiation in a spatially finite (spherical) region. Thermodynamic state functions are derived in the canonical ensemble, defined by elements of radius $r_o$ and boundary temperature $T(r_o)$. Using recent solutions of the semi-classical back reaction problem, we compute the $O(\hbar)$ corrections to the mass of the black hole, thermal energy, the entropy and free energy due to the presence of hot conformal scalars, massless spinors and U(1) gauge quantum fields in the vicinity of the hole. The free energy is particularly important for assessing under what conditions the nucleation of black holes from hot flat space is likely to occur.