Thermodynamic geometry of charged rotating BTZ black holes

We study the thermodynamics and the thermodynamic geometries of charged rotating Bañados-Teitelboim-Zanelli black holes in ($2+1$)-gravity. We investigate the thermodynamics of these systems within the context of the Weinhold and Ruppeiner thermodynamic geometries and the recently developed formalism of geometrothermodynamics. Considering the behavior of the heat capacity and the Hawking temperature, we show that Weinhold and Ruppeiner geometries cannot describe completely the thermodynamics of these black holes and of their limiting case of vanishing electric charge. In contrast, the Legendre invariance imposed on the metric in geometrothermodynamics allows one to describe the charged rotating Bañados-Teitelboim-Zanelli black holes and their limiting cases in a consistent and invariant manner.

Figure 1

Behavior of the heat capacity and temperature as functions of the event horizon radius $r_{+}$ of a CR-BTZ black hole with $Q=2$, $J=1$ and $l=1$. Temperature and heat capacity vanish at $r_{+}\approx 1.79$. The unstable region $(C_{J,Q}<0)$ corresponds to an unphysical region with negative temperature. To illustrate the contribution of the charge we include the plots (dotted curves) for the case of the nonstatic BTZ black hole with $Q=0$ [23, 44].

Figure 2

The mass $M$ of an extreme CR-BTZ black hole with horizon radius $r_{*}=\pi l^2{Q}^2(1+\eta )/8$ as a function of the charge $Q$ and the angular parameter $\eta >1$. Here, we chose $l=1$ as a representative value. The plane $M=0$ is plotted to visualize the region where $M>0$.

Figure 3

Thermodynamic curvature for the Weinhold ($R_W$) geometry as a function of the event horizon radius, $r_{+}$, of the CR-BTZ black hole. Here, the free parameters are chosen as $Q=2$, $J=1$ and $l=1$. The curvature is completely regular in the entire domain of $r_{+}$.

Figure 4

Thermodynamic curvature, $R_R$, of the Ruppeiner geometry as a function of the event horizon, $r_{+}$, for a CR-BTZ black hole with $J=l=1$, $Q=2$. The only singularity is located at $r_{+}\approx 1.79$.

Figure 5

Thermodynamic curvature of the CR-BTZ black hole ($R_{\mathrm{GTD}}$) as a function of event horizon $r_{+}$. A typical behavior is depicted for the specific values $Q=2$, $J=l$ and $l=1$.

Figure 6

Behavior of the scalar curvature in GTD and temperature as functions of the event horizon radius $r_{+}=S/4\pi$ of a neutral rotating BTZ black hole with $J=1$ and $l=1$. The curvature singularity coincides with the point of zero temperature.