Entropy function approach to charged BTZ black hole

TL;DR

This paper applies the entropy function and Wald's formalism to analyze the entropy of extremal charged BTZ black holes, providing new solutions and demonstrating the methods' effectiveness in this context.

Contribution

It introduces a novel solution for the metric function using Lambert functions and validates entropy calculations via two different formalisms.

Findings

Derived the extremal charged BTZ black hole solution using Lambert functions.

Showed the entropy function approach works with proper charge normalization.

Reproduced the black hole entropy using Wald's Noether formalism without normalization.

Abstract

We find solution to the metric function f(r)=0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M_e(Q)=Q^2(1-\ln Q^2). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS_2 x S^1 represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald's Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.