BTZ Black Hole with Chern-Simons and Higher Derivative Terms

TL;DR

The paper advances BTZ black hole entropy calculations in the presence of higher-derivative gravity and gravitational Chern-Simons terms by performing a two-dimensional reduction of the three-dimensional theory and applying Wald's Noether-charge method alongside the entropy-function formalism. It demonstrates that the entropy decomposes into left- and right-moving sectors with central charges $c_L = 24\pi(C+K)$ and $c_R = 24\pi(C-K)$, yielding $E = 2\pi\sqrt{c_R q/6}$ for positive extremal charge $q$ (or $E = 2\pi\sqrt{c_L |q|/6}$ for negative $q$), while the CS term sets the difference $c_L - c_R = 48\pi K$ and leaves the AdS$_3$ background intact. For non-extremal BTZ, the entropy splits additively as $E = 2\pi\sqrt{c_L q_L/6} + 2\pi\sqrt{c_R q_R/6}$ with $q_L=(M-J)/2$ and $q_R=(M+J)/2$, and a corresponding two-variable entropy function in terms of $e_L,e_R$ is presented. The results agree with previous Euclidean-action analyses and illuminate how higher-derivative and CS terms modify the microscopic and macroscopic entropy via the left-right sector structure.

Abstract

The entropy of a BTZ black hole in the presence of gravitational Chern-Simons terms has previously been analyzed using Euclidean action formalism. In this paper we treat the BTZ solution as a two dimensional black hole by regarding the angular coordinate as a compact direction, and use Wald's Noether charge method to calculate the entropy of this black hole in the presence of higher derivative and gravitational Chern-Simons terms. The parameters labelling the black hole solution can be determined by extremizing an entropy function whose value at the extremum gives the entropy of the black hole.