Lorentz-diffeomorphism Conserved Charges And Virasoro Algebra In Chern-Simons-Like Theories Of Gravity

TL;DR

This paper develops a formalism for calculating off-shell, quasi-local conserved charges in Chern-Simons-like theories of gravity, enabling the derivation of Virasoro algebra and black hole entropy consistent with the Cardy formula.

Contribution

It introduces a new off-shell conserved charge formalism using total variation and Lorentz-Lie derivatives for CSLTG, applicable to any diffeomorphism generator.

Findings

Derived an off-shell conserved charge expression for CSLTG.

Applied the formalism to BTZ black holes in Einstein and massive gravity.

Confirmed the entropy matches the Cardy formula results.

Abstract

The Chern-Simons-like theories of gravity (CSLTG) are formulated at first order formalism. In this formalism, the derivation of the entropy of a black hole on bifurcation surface, as a quasi-local conserved charge is problematic. In this paper we overcome to these problems by considering the concept of total variation and the Lorentz-Lie derivative. We firstly find an expression for the ADT conserved current in the context of the CSLTG which is based on the concept of the Killing vector fields. Then, we generalize it to be conserved for all diffeomorphism generators. Thus, we can extract an off-shell conserved charge for any vector field which generates a diffeomorphism. The formalism presented here is based on the concept of quasi-local conserved charges which are off-shell. The charges can be calculated on any codimension two space-like surface surrounding a black hole and the results are independent of the chosen surface. By using the off-shell quasi-local conserved charge, we investigate the Virasoro algebra and find a formula to calculate the central extension term. We apply the formalism to the BTZ black hole solution in the context of the Einstein gravity and the Generalized massive gravity, then we find the eigenvalues of their Virasoro generators as well as the corresponding central charges. Eventually, we calculate the entropy of the BTZ black hole by the Cardy formula and we show that the result exactly matches with the one obtained by the concept of the off-shell conserved charges.