Thermodynamic Curvature of the BTZ Black Hole

TL;DR

This paper investigates the thermodynamic geometry of the BTZ black hole to determine the effective spatial dimension of its underlying statistical model, revealing a critical point at extremality and a dimension that varies with proximity to this point.

Contribution

It introduces a geometric thermodynamic analysis of the BTZ black hole, identifying the extremal limit as a critical point and estimating the effective dimension of the associated statistical system.

Findings

Extremal limit is identified as a critical point.

Effective dimension near extremality is one.

Dimension decreases below one far from extremality.

Abstract

Some thermodynamic properties of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole are studied to get the effective dimension of its corresponding statistical model. For this purpose, we make use of the geometrical approach to the thermodynamics: Considering the black hole as a thermodynamic system with two thermodynamic variables (the mass $M$ and the angular momemtum $J$), we obtain two-dimensional Riemannian thermodynamic geometry described by positive definite Ruppeiner metric. From the thermodynamic curvature we find that the extremal limit is the critical point. The effective spatial dimension of the statistical system corresponding to the near-extremal BTZ black holes is one. Far from the extremal point, the effective dimension becomes less than one, which leads to one possible speculation on the underlying structure for the corresponding statistical model.