Thermodynamic geometry for charged Gauss-Bonnet black holes in AdS spacetimes

In this paper, we study the thermodynamic geometry of charged Gauss-Bonnet black holes (and Reissner-Nordström black holes, for the sake of comparison) in anti–de Sitter spacetimes in both ($T$, $V$) and ($S$, $P$) planes. The thermodynamic phase space is known to have an underlying contact and metric structure; Ruppeiner geometry then naturally arises in this framework. Sign of Ruppeiner curvature can be used to probe the nature of interactions between the black hole microstructures. It is found that there are both attraction and repulsion dominated regions which are in general determined by the electric charge, Gauss-Bonnet coupling, and horizon radius of the black hole. The results are physically explained by considering that these black hole systems consist of charged as well as neutral microstructures much like a binary mixture of fluids.

Figure 1

Curvature of Ruppeiner metric on ($T,V$) plane for RN-AdS black holes as a function of $V$ for fixed $T$. The case with $q=0$ corresponds to the Schwarzschild-AdS black holes.

Figure 2

Curvature of Ruppeiner metric on ($S,P$) plane for RN-AdS black holes as a function of $S$ for fixed $P$.

Figure 3

Curvature of Ruppeiner metric on ($S,P$) plane for RN-AdS black holes as a function of $P$ for fixed $S$.

Figure 4

Curvature of Ruppeiner metric on ($S,P$) plane for Schwarzschild-AdS black holes as a function of $P$ for fixed $S$.

Figure 5

Curvature of Ruppeiner metric on ($T,V$) plane for Gauss-Bonnet-AdS black holes as a function of $V$ for fixed $T$.

Figure 6

Curvature of Ruppeiner metric on ($S,P$) plane for Gauss-Bonnet-AdS black holes as a function of $S$ for fixed $P$.

Figure 7

Horizon radius $r_{+}$ plotted against charge $q$ and GB coupling $\alpha$ in $d=6$: for the points where the Ruppeiner curvature is zero—indicating the balance of attractive and repulsive interactions.

Figure 8

Temperature versus horizon radius for Gauss-Bonnet black holes.