Boundary Effects on the Thermodynamics of Quantum Fields Near a Static Black Hole

TL;DR

This paper analyzes the thermodynamics of quantum fields near black hole horizons using the brick wall method, deriving high-temperature expansions and examining boundary effects across various black hole geometries.

Contribution

It provides a unified high-temperature expansion framework for different fields and black hole backgrounds, including explicit formulas for horizon divergences.

Findings

Leading horizon divergence is consistent across metrics for a given field.

High-temperature expansion terms become comparable at the Hawking temperature.

Explicit formula for sub-leading divergence in terms of physical parameters.

Abstract

We investigate thermodynamics of a non-interacting quantum field in a static black hole background. The horizon divergences are regulated by the brick wall method, which consists of subjecting the quantum field to Dirichlet boundary conditions on a surface (the brick wall) just outside the horizon. Using heat kernel and Mellin transform methods, we derive high-temperature expansions for the free energy and entropy and study the boundary and higher-order geometric effects on the horizon divergences induced by the brick wall. We consider real scalar, complex scalar, and Dirac fields in Schwarzschild, Reissner-Nordstr\"{o}m and dilatonic black hole backgrounds, as well as in their near-horizon geometries. By evaluating the high-temperature expansion of the entropy (up to a certain order) at the Hawking temperature, we show that, for a given field type, the leading horizon divergence is the same for all the metrics considered. Moreover, we show that the different orders in the high-temperature expansion become comparable at the Hawking temperature and compare our findings with existing results in the literature. We derive an explicit formula for the sub-leading horizon divergence expressed in terms of the field mass, surface gravity, horizon area, and dilaton parameter, which is applicable to all the exact metrics considered. We also consider the possibility of using Neumann boundary conditions to regularize the horizon divergences.