Unified approach to the entropy of an extremal rotating BTZ black hole: Thin shells and horizon limits

TL;DR

This paper investigates the entropy of extremal rotating BTZ black holes using a thin shell approach, revealing that entropy can be either the Bekenstein-Hawking value or an arbitrary function, depending on the shell's approach to the horizon.

Contribution

It introduces a unified method to analyze the entropy of extremal BTZ black holes via thin shells, exploring different horizon-approach scenarios and their impact on entropy.

Findings

BTZ black hole entropy can be the Bekenstein-Hawking entropy or an arbitrary function.

Three distinct approaches to the horizon lead to different entropy results.

The contributions of mass, velocity, and temperature to entropy vary across cases.

Abstract

Using a thin shell, the first law of thermodynamics, and a unified approach, we study the thermodymanics and find the entropy of a (2+1)-dimensional extremal rotating Ba\~{n}ados-Teitelbom-Zanelli (BTZ) black hole. The shell in (2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and rotating, with the inner region being a ground state of the anti-de Sitter (AdS) spacetime and the outer region being the rotating BTZ spacetime. The extremal BTZ rotating black hole can be obtained in three different ways depending on the way the shell approaches its own gravitational or horizon radius. These ways are explicitly worked out. The resulting three cases give that the BTZ black hole entropy is either the Bekenstein-Hawking entropy, $S=\frac{A_+}{4G}$, or it is an arbitrary function of $A_+$, $S=S(A_+)$, where $A_+=2\pi r_+$ is the area, i.e., the perimeter, of the event horizon in (2+1) dimensions. We speculate that the entropy of an extremal black hole should obey $0\leq S(A_+)\leq\frac{A_+}{4G}$. We also show that the contributions from the various thermodynamic quantities, namely, the mass, the circular velocity, and the temperature, for the entropy in all three cases are distinct. This study complements the previous studies in thin shell thermodynamics and entropy for BTZ black holes. It also corroborates the results found for a (3+1)-dimensional extremal electrically charged Reissner-Nordstr\"om black hole.