P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space

TL;DR

This work analyzes P-V criticality in the extended phase space of Gauss-Bonnet-AdS black holes across horizon topologies and spacetime dimensions. By treating the cosmological constant as pressure and incorporating the Gauss-Bonnet coupling as a thermodynamic variable, the authors derive the equation of state and Smarr relations, and identify when van der Waals-like phase transitions occur. They find that P-V criticality and small/large black-hole transitions arise only for spherical horizons with specific dimensional constraints, and that the presence of charge introduces dimension- and topology-dependent bounds on Gauss-Bonnet effects, while the critical exponents always match the van der Waals values. Overall, the results reinforce the van der Waals–like thermodynamic analogy for AdS black holes, while revealing the crucial influence of higher-curvature corrections and horizon topology on the phase structure.

Abstract

We study the $P-V$ criticality and phase transition in the extended phase space of charged Gauss-Bonnet black holes in anti-de Sitter space, where the cosmological constant appears as a dynamical pressure of the system and its conjugate quantity is the thermodynamic volume of the black hole. The black holes can have a Ricci flat ($k=0$), spherical ($k=1$), or hyperbolic ($k=-1$) horizon. We find that for the Ricci flat and hyperbolic Gauss-Bonnet black holes, no $P-V$ criticality and phase transition appear, while for the black holes with a spherical horizon, even when the charge of the black hole is absent, the $P-V$ criticality and the small black hole/large black hole phase transition will appear, but it happens only in $d=5$ dimensions; when the charge does not vanish, the $P-V$ criticality and the small black hole/large phase transition always appear in $d=5$ dimensions; in the case of $d\ge 6$, to have the $P-V$ criticality and the small black hole/large black hole phase transition, there exists an upper bound for the parameter $b=\widetildeα|Q|^{-2/(d-3)}$, where $\tilde α$ is the Gauss-Bonnet coefficient and $Q$ is the charge of the black hole. We calculate the critical exponents at the critical point and find that for all cases, they are the same as those in the van der Waals liquid-gas system.

Figures (6)

• Figure 1: The $P-V(r_h)$ diagram for $d=5$ (left plot) and $d=6$ (right plot) near the critical temperature, respectively. Without loss of generality, we consider the case $\widetilde{\alpha}=1$. In both plots, the temperature of isotherms decreases from top to bottom and the pressure satisfies the relation \ref{['al']}.
• Figure 2: The Gibbs free energy as a function of temperature for different pressures. The left plot is for the case with $d=5$, where the "swallow tail" appears and a critical point exists. The right plot is for the case with $d=6$, where there is no phase transition. In both plots we set $\widetilde{\alpha}=1$.
• Figure 3: The $p-x$ diagram (left plot) and Gibbs free energy (right plot) near the critical temperature and critical pressure for the case of $k=1$ in five dimensions. In this figure we take $b=0.2$. $p_c$ and $t_c$ are obtained by Eqs. (\ref{['pq3']}),(\ref{['crai']}), and (\ref{['Tc2']}) with $k=1$ and $b=0.2$.
• Figure 4: The range of the parameter $b$ in the cases with different dimensions. In this range, a physical critical point exists.
• Figure 5: The values of $a_{11}$ and $a_{03}$ when $d=5$, $k=1$ and $Q\neq0$. They are both negative. So the phase transition can occur when $\tau<0$.