P-V criticality of charged AdS black holes

TL;DR

By treating the cosmological constant as a thermodynamic pressure and introducing a conjugate volume, the paper demonstrates that charged RN–AdS black holes exhibit P–V criticality with a small–large black hole transition, analogous to the liquid–gas transition. Using a fixed-charge ensemble in the extended phase space, it derives the P(V,T) equation of state, computes the Gibbs free energy, and identifies a coexistence line ending at a critical point with Van der Waals–like critical exponents. The results show a precise correspondence between black hole thermodynamics and fluid behaviour in the extended space, including the universal ratio PcVc/Tc=3/8. The work clarifies limitations of non-extended analogies and suggests extensions to other black hole configurations and higher dimensions.

Abstract

Treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the critical behaviour of charged AdS black holes. We complete the analogy of this system with the liquid-gas system and study its critical point, which occurs at the point of divergence of specific heat at constant pressure. We calculate the critical exponents and show that they coincide with those of the Van der Waals system.

Figures (15)

• Figure 1: $P-V$ diagram of Van der Waals fluid. The temperature of isotherms decreases from top to bottom. The two upper dashed lines correspond to the "ideal gas" phase for $T>T_c$, the critical isotherm $T=T_c$ is denoted by the thick solid line, lower solid lines correspond to temperatures smaller than the critical temperature; $T=T_0$ isotherm is also displayed. The constants $a$ and $b$ in Eq. (\ref{['vdwA']}) were set equal to one.
• Figure 2: Maxwell's equal area law. The 'oscillating' (dashed) part of the isotherm $T<T_c$ is replaced by an isobar, such that the areas above and below the isobar are equal one another.
• Figure 3: Gibbs free energy of Van der Waals fluid. This picture shows the characteristic swallowtail behaviour of the Gibbs free energy as a function of pressure and temperature. This corresponds to a first-order liquid--gas phase transition which occurs at the intersection of $G$ surfaces. The corresponding curve is called the coexistence line. We have set $\Phi=1$.
• Figure 4: Gibbs free energy of Van der Waals fluid. This figure depicts the qualitative behaviour of the Gibbs free energy as a function of temperature for various pressures. The pressure decreases from right to left. The dashed line corresponds to $P=2P_c$, the thick solid line to $P=P_c$, and the remaining solid lines display $P=0.6 P_c$ and $P=0.09 P_c$, respectively. For $P<P_c$ there is a first-order transition in the system.
• Figure 5: Coexistence line of liquid--gas phase. Fig. displays the coexistence line of liquid and gas phases of the Van der Waals fluid in $(P, T)$-plane. The critical point is highlighted by a small circle at the end of the coexistence curve.