Entropy of black holes, charged probes and noncommutative generalization

TL;DR

This paper extends the brick wall model to higher dimensions, charged black holes, and noncommutative spacetime, providing a systematic way to compute black hole entropy with new corrections and generalizations.

Contribution

It introduces a generalized brick wall formalism for charged and higher-dimensional black holes, including noncommutative spacetime effects, and calculates associated entropy corrections.

Findings

Entropy formulas for Reissner-Nordström and charged BTZ black holes.

Noncommutative corrections to black hole entropy involving logarithmic terms.

Higher order WKB corrections also produce logarithmic entropy corrections.

Abstract

The brick wall model is a semi-classical approach to understanding the microscopic origin of black hole entropy. We outline the formalism for the brick wall model in arbitrary number of dimensions and generalize it to include both charged spacetimes and charged probes in order to systematically show how to calculate the entropy for any black hole, in higher orders of the WKB approximation. We calculate the entropy for the Reissner-Nordstr\"{o}m and charged BTZ black holes, and by looking at the chargeless limits we recover the entropy of the Schwarzschild and neutral BTZ black holes. We also study noncommutative corrections to the black hole entropy by using a Drinfeld twist to deform spacetime symmetries. Using the noncommutative action for a charged scalar field, we derive the noncommutative Klein-Gordon equation and the radial equation in arbitrary dimension on which we generalize the brick wall method. We study case by case the noncommutative Reissner-Nordstr\"{o}m and noncommutative charged BTZ black holes, the entropy of which we calculate using the generalized brick wall method. Corrections due to higher order in the WKB approximation arise, and they are given by the logarithms of the black hole area. The corrections in the lowest order of WKB due to noncommutativity are also given by the logarithms of the black hole area.