Bardeen black hole chemistry

TL;DR

The paper studies the Bardeen regular black hole in anti-de Sitter space within black hole chemistry, treating the cosmological constant as pressure $P = -\\frac{\\Lambda}{8\\pi}$ and mass as enthalpy $M = H = U + PV$. It derives the BAdS metric with $f(r)=1-\\frac{2M r^2}{(q^2+r^2)^{3/2}}+\\frac{r^2}{l^2}$, using dimensionless variables $x=r/l$, $m=M/l$, $Q=q/l$ to reveal a regular core and a horizon structure governed by $f(r_+)=0$. In the extended thermodynamics framework, it computes $T$, $S$, $V$, and conjugates $\\varphi$ and $P_q$, obtaining the equation of state $P(V,T)$ that exhibits Van der Waals–like instability, a small/large black hole phase transition, and a critical point $(P_c,V_c,T_c)\\approx (0.0012/q^2, 287.44 q^3, 0.0251/q)$ with exponents $(\\alpha,\\beta,\\gamma,\\delta)=(0,\\tfrac{1}{2},1,3)$ and ratio $\\varepsilon \\approx 0.31$. A generalized Smarr relation including a secondary pressure $P_q= -1/(8\\pi q^2)$ and volume $V_q$ is discussed, with notes on consistency of the 1st law for finite $q$ and the requirement $r_+ \\gg q$ to suppress $q$-dependent corrections. The results reinforce a close analogy between BAdS black holes and liquid–gas systems and illustrate how a minimal length scale can be incorporated into black hole thermodynamics.

Abstract

In the present paper we try to connect the Bardeen black hole with the concept of the recently proposed black hole chemistry. We study thermodynamic properties of the regular black hole with an anti-deSitter background. The negative cosmological constant $Λ$ plays the role of the positive thermodynamic pressure of the system. After studying the thermodynamic variables, we derive the corresponding equation of state and we show that a neutral Bardeen-anti-deSitter black hole has similar phenomenology to the chemical Van der Waals fluid. This is equivalent to saying that the system exhibits criticality and a first order small/large black hole phase transition reminiscent of the liquid/gas coexistence.

Figures (4)

• Figure 1: The function $f(x)$vs$x$ for $Q=0.3\,.$ The solid curves with colors red, blue and black, correspond to $m=0.6\,$, $m=m_0 \approx 0.448$ and $m=0.32$ respectively. The dotted lines correspond to $Q=0$ for each case and represent the usual Schwarzschild-anti-deSitter (SAdS) black hole.
• Figure 2: The temperature $\mathcal{T}=T l$vs the horizon $x_+=r_+/l$ for $Q=0.05$ (red solid curve) with a zero temperature remnant at $x_0=0.0702 \,$, for $Q=Q_c \approx 0.0987$ (blue dashed curve) with $x_0=0.136$ and for $Q=0.18$ (green solid curve) with $x_0 = 0.236$. The black dotted curve corresponds to the conventional SAdS solution ($Q=0$).
• Figure 3: The isotherms of BAdS black hole on the $P-V \,$ plane are displayed for $q=1\,$. The red solid curves stand for $T<T_c$, the black dashed curve for $T=T_c \approx 0.025$ and the gray solid curves for $T>T_c\,.$
• Figure 4: The plots are displayed in $l$ units for $Q=0.05$ (red solid curves), for $Q=Q_c \approx 0.0987$ (blue dashed curves) and for $Q=0.18$ (green solid curves).