Thermodynamic curvature of the BTZ black hole

Abstract

In this paper we apply the concept of thermodynamic geometry to the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole. We find the thermodynamic curvature diverges at the extremal limit of the black hole, which means the extremal black hole is the critical point with the temperature zero. We also study the effective dimensionality of the underlying statistical model. Near the critical point, the picture is clear; the spatial dimension of the underlying statistical model is just one, which agrees with other results. However, far from the critical point, the dimension becomes less than one and even negative. In order to interpret this result, we resort to a qualitative analogy with the Takahashi gas model.