Noncommutative correction to the entropy of charged BTZ black hole

TL;DR

This paper explores how noncommutative geometry affects the entropy of charged BTZ black holes, deriving corrections that include higher powers of Planck's constant and logarithmic terms.

Contribution

It introduces a method to compute noncommutative corrections to black hole entropy using the brick-wall approach and WKB approximation, specifically applied to charged BTZ black holes.

Findings

Entropy contains higher powers of  in lowest WKB order

Logarithmic and polynomial logarithmic corrections to entropy

Method applicable to higher-order WKB corrections

Abstract

Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outline the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist we deform spacetime symmetries, algebra of vector fields and differential forms leading to a formulation of noncommutative Einstein equations. We study a concrete example of charged BTZ spacetime and deformations steaming from the so called angular twist. The entropy of the noncommutative charged BTZ black hole is obtained using the brick-wall method. We used a charged scalar field as a probe and obtained its spectrum and density of states via WKB approximation. We provide the method to calculate corrections to the Bekenstein-Hawking entropy in higher orders in WKB, but we present the final result in the lowest WKB order. The result is that even in the lowest order in WKB, the entropy, in general, contains higher powers in $\hbar$, and it has logarithmic corrections, and polynomials of logarithms of the black hole area.