Solution Independent Analysis of Black Hole Entropy in Brick Wall Model

TL;DR

This paper analyzes black hole entropy using the brick wall model across various black hole types, classifying divergences and confirming the area law for non-extreme cases.

Contribution

It introduces a new index to classify entropy singularities and provides a general formula applicable to multiple black hole solutions.

Findings

Non-extreme black holes follow the area law with quadratic divergence.

Extreme black holes show logarithmic divergence or constant entropy.

The general formula is consistent with Reissner-Nordström, dilaton, and brane-world black holes.

Abstract

Using the brick wall regularization of 't Hooft, the entropy of non-extreme and extreme black holes is investigated in a general static, spherically symmetric spacetime. We classify the singularity in the entropy by introducing a {\it new} index $\delta $ with respect to the brick wall cut-off $\epsilon $. The leading contribution to entropy for non-extreme case $(\delta \neq 0)$ is shown to satisfy the area law with quadratic divergence with respect to the invariant cut-off $\epsilon_{{\rm inv}}$ while the extreme case $(\delta =0)$ exhibits logarithmic divergence or constant value with respect to $\epsilon $. The general formula is applied to Reissner-Nordstr\"{o}m, dilaton and brane-world black holes and we obtain consistent results.