Entropy of scalar fields in 3+1 dimensional constant curvature black hole background

TL;DR

This paper investigates the thermodynamics of a massive scalar field in a 3+1 dimensional constant curvature black hole background, revealing logarithmic divergences in free energy and entropy related to regularization parameters.

Contribution

It introduces the necessity of an angular cut-off parameter in addition to the radial one for regularizing scalar field solutions in this background.

Findings

Free energy and entropy diverge logarithmically with cut-off parameters.

An angular cut-off parameter is essential for regularization.

The study employs the brick wall model and WKB approximation.

Abstract

We consider the thermodynamics of minimally coupled massive scalar field in 3+1 dimensional constant curvature black hole background. The brick wall model of 't Hooft is used. When Scharzschild like coordinates are used it is found that apart from the usual radial brick wall cut-off parammeter an angular cut-off parameter is required to regularize the solution. Free energy of the scalar field is obtained through counting of states using the WKB approximation. It is found that the free energy and the entropy are logarithmically divergent in both the cut-off parameters.