Critical behavior of Gauss-Bonnet black holes via an alternative phase space

Recently, it was argued that charged anti–de Sitter (AdS) black holes admit critical behavior, without extending phase space, similar to the Van der Waals fluid system in the $Q^2-\mathrm{\Psi }$ plans where $\mathrm{\Psi }=1/v$ (the conjugate of $Q^2$) is the inverse of the specific volume [Dehyadegari et al., Phys. Lett. B 768, 02064 (2017)]. In this picture, the square of the charge of the black hole, $Q^2$, is treated as a thermodynamic variable and the cosmological constant $\mathrm{\Lambda }$ is fixed. In this paper, we examine whether this new approach toward the critical behavior of AdS black holes can work in other gravities such as the Gauss-Bonnet (GB) gravity as well as in higher dimensional spacetime. We obtain the equation of state, $Q^2=Q^2(\mathrm{\Psi },T)$, the Gibbs free energy, and the critical quantities of the system, and we study the effects of the GB coupling $\tilde{\alpha }$ on their behavior. We find out that the critical quantities have reasonable values, provided the GB coupling constant, $\tilde{\alpha }$, is taken small and the horizon topology is assumed to be a ($d-2$)-sphere. Finally, we calculate the critical exponents and show that they are independent of the model parameters and have the same values as the Van der Waals system that is predicted by the mean field theory.

Figure 1

$Q^2-\mathrm{\Psi }$ diagram of higher dimensional charged AdS black holes.

Figure 2

$G-Q^2$ diagram of AdS black holes. The diagrams are shifted for more clarity.

Figure 3

$Q^2-\mathrm{\Psi }$ diagram of GB black holes for $k=1$, $d=5$, $l=1$, $\tilde{\alpha }=0.01$.

Figure 4

$G-Q^2$ diagram of GB black holes for $d=5$, $l=1$, $\tilde{\alpha }=0.01$. The diagrams are shifted for more clarity.

Figure 5

$Q^2-\mathrm{\Psi }$ diagram of Gauss-Bonnet black holes in higher dimensions.

Figure 6

$G-Q^2$ diagram of Gauss-Bonnet black holes that shifted for more clarity.

Figure 7

Critical values in terms of $\tilde{\alpha }$ for GB black holes.

Figure 8

Critical values in terms of $\tilde{\alpha }$ for GB black holes.

Figure 9

The effect of $\tilde{\alpha }$ on isotherm diagrams for $l=1$, $d=7$.

Figure 10

The effect of $d$ on isotherm diagrams in constant temperature $T=0.5$ and $l=1$.

Figure 11

$f(r)$ versus $r$ for different values of $\tilde{\alpha }$ with $k=P=Q=1$.

Figure 12

$G-Q^2$ in Gauss-Bonnet black holes with $k=1$, $l=1$, and different values of $\tilde{\alpha }$.