Aspects of Finite Temperature Quantum Field Theory in a Black Hole Background

TL;DR

This paper investigates finite temperature quantum fields near black holes, revealing horizon-localized energy and entropy contributions, and discusses how thermodynamic assumptions influence divergence issues.

Contribution

It introduces a detailed analysis of quantum fields at finite temperature in black hole backgrounds with mixed boundary conditions, highlighting horizon effects and divergence origins.

Findings

Energy density is finite only at Hawking temperature.

Surface contributions to energy and entropy are localized on the horizon.

Divergences in entropy are linked to thermodynamic assumptions.

Abstract

We quantize a scalar field at finite temperature T in the background of a classical black hole, adopting 't Hooft's ``brick wall'' model with generic mixed boundary conditions at the brick wall boundary. We first focus on the exactly solvable case of two dimensional space-time. As expected, the energy density is integrable in the limit of vanishing brick wall thickness only for T=T_H - the Hawking temperature. Consistently with the most general stress energy tensor allowed in this background, the energy density shows a surface contribution localized on the horizon. We point out that the usual divergences occurring in the entropy of the thermal atmosphere are due to the assumption that the third law of thermodynamics holds for the quantum field in the black hole background. Such divergences can be avoided if we abandon this assumption. The entropy density also has a surface term localized on the horizon, which is open to various interpretations. The extension of these results to higher space-time dimensions is briefly discussed.