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

Abstract

We show that a BTZ black hole solution of cosmological topological massive gravity has a hidden conformal symmetry. In this regard, we consider the wave equation of a massless scalar field propagating in BTZ spacetime and find that the wave equation could be written in terms of the SL(2, R) quadratic Casimir. From the conformal coordinates, the temperatures of the dual conformal field theories (CFTs) could be read directly. Moreover, we compute the microscopic entropy of the dual CFT by the Cardy formula and find a perfect match to the Bekenstein–Hawking entropy of a BTZ black hole. Then, we consider Galilean conformal algebras (GCA), which arises as a contraction of relativistic conformal algebras (x → ϵx, t → t, ϵ → 0). We show that there is a correspondence between GCA2 on the boundary and contracted BTZ in the bulk. For this purpose we obtain the central charges and temperatures of GCA2. Then, we compute the microscopic entropy of the GCA2 by the Cardy formula and find a perfect match to the Bekenstein–Hawking entropy of a BTZ black hole in a non-relativistic limit. The absorption cross section of a near-region scalar field also matches the microscopic absorption cross section of the dual GCA2. So we find further evidence that shows correspondence between a contracted BTZ black hole and two-dimensional GCA.