Black hole entropy in the Chern-Simons-like theories of gravity and Lorentz-diffeomorphism Noether charge

TL;DR

This paper develops a method to calculate black hole entropy in non-Lorentz-diffeomorphism covariant gravity theories, specifically Chern-Simons-like theories, by extending existing conserved charge techniques.

Contribution

It introduces a new approach to compute conserved charges and black hole entropy in Lorentz-diffeomorphism non-covariant gravity theories, extending Tachikawa's method.

Findings

Derived a general entropy formula for CSLTG black holes

Applied the method to compute BTZ black hole entropy

Demonstrated the approach with specific CSLTG examples

Abstract

In the first order formalism of gravity theories, there are some theories which are not Lorentz-diffeomorphism covariant. In the framework of such theories we cannot apply the method of conserved charge calculation used in Lorentz-diffeomorphism covariant theories. In this paper we firstly introduce the total variation of a quantity due to an infinitesimal Lorentz-diffeomorphism transformation. Secondly, in order to obtain the conserved charges of Lorentz-diffeomorphism non-covariant theories, we extend the Tachikawa method \cite{3}. This extension includes not only Lorentz gauge transformation but also the diffeomorphism. We apply this method to the Chern-Simons-like theories of gravity (CSLTG) and obtain a general formula for the entropy of black holes in those theories. Finally, some examples on CSLTG are provided and the entropy of the BTZ black hole is calculated in the context of the examples.