Correspondence between the contracted BTZ solution of cosmological topological massive gravity and two-dimensional Galilean conformal algebra

Abstract

We show that BTZ black hole solution of Cosmological Topological Massive Gravity (CTMG) have a hidden conformal symmetry. In this regard, we consider the wave equation of a massless scalar field propagating in BTZ spacetime and find the wave equation could be written in terms of the $SL(2,R)$ quadratic Casimir. From the conformal coordinates, the temperatures of the dual CFTs could be read directly. Moreover, we compute the microscopic entropy of the dual CFT by Cardy formula and find a perfect match to Bekenstein-Hawking entropy of BTZ black hole. Then we consider Glilean conformal algebras (GCA), which arises as a contraction of relativistic conformal algebras ($x\rightarrow \epsilon x$, $t\rightarrow t$, $\epsilon \rightarrow 0$). We show that there is a correspondence between $GCA_2$ on the boundary and contracted BTZ in the bulk. For this purpose we obtain the central charges and temperatures of $GCA_2$. Then we compute the microscopic entropy of the $GCA_2$ by Cardy formula and find a perfect match to Bekenstein-Hawking entropy of BTZ black hole in non-relativistic limit. The absorption cross section of a near region scalar field also matches to microscopic absorption cross section of the dual $GCA_2$. So we find further evidence that show correspondence between contracted BTZ black hole and 2-dimensional Galilean conformal algebra.