Critical phenomena and thermodynamic geometry of Reissner-Nordström-anti-de Sitter black holes

The phase transition of Reissner-Nordström black holes in ($n+1$)-dimensional anti–de Sitter spacetime is studied in detail using the thermodynamic analogy between a RN-AdS black hole and a van der Waals liquid-gas system. We first investigate critical phenomena of the RN-AdS black hole. The critical exponents of relevant thermodynamical quantities are evaluated. We find identical exponents for a RN-AdS black hole and a van der Waals liquid-gas system. This suggests a possible universality in the phase transitions of these systems. We finally study the thermodynamic behavior using the equilibrium thermodynamic state space geometry and find that the scalar curvature diverges exactly at the van der Waals–like critical point where the heat capacity at constant charge of the black hole diverges.

Figure 1

Heat capacity at constant charge with $r_{+}$ for $Q<Q_c$.

Figure 2

Heat capacity at constant charge with $r_{+}$ for $Q=Q_c$.

Figure 3

The isotherm of a RN-AdS black hole along which $T>T_c$. The local maxima and minima located, respectively, at ${\Phi }_1$ and ${\Phi }_2$ are critical points of $C_Q$. For $\Phi \in ({\Phi }_1,{\Phi }_2)$, the black hole is unstable with $(\frac{\partial Q}{\partial \Phi }{)}_T>0$.

Figure 4

The critical isotherm along which $T=T_c$. The point of inflection located at ${\Phi }_c$ is a critical point of $C_Q$, $C_Q>0$ along the critical isotherm.

Figure 5

A horizontal line is drawn which connects points $a$ and $b$ of the subcritical isotherm. The area bounded by the line segment $ad$ and the isotherm is equal to that bounded by the line segment $db$ and the isotherm.