From singular to regular: revisiting thermodynamics of Bardeen-AdS black holes

TL;DR

This paper investigates the thermodynamics of the regular Bardeen-AdS black hole by constructing a singular 'mother' black hole with nonlinear electrodynamics and then imposing a constraint that yields the regular solution while reducing the thermodynamic phase space. In the singular case, the first law holds and the system exhibits rich phase behavior, including reentrant transitions analogous to higher-dimensional Kerr-AdS black holes. For the regular Bardeen-AdS black hole, the standard first law breaks due to constraint-induced phase-space reduction, and while $P$-$V$ criticality resembles RN-AdS, the Gibbs free energy lacks the usual swallow-tail; instead, it shows an 8-shaped or C-shaped structure indicating first-order and zeroth-order transitions between small and large black holes. The results highlight how constraints can profoundly affect critical phenomena in black-hole thermodynamics and motivate further exploration of constrained Hamiltonian formulations.

Abstract

We explore the thermodynamic properties of the regular Bardeen-AdS black hole obtained by imposing an additional constraint on a singular "mother" black hole. This constraint eliminates the physical singularity but leads to the breakdown of the standard first law of black hole thermodynamics. The singular black hole exhibits a reentrant phase transition similar to that of the higher-dimensional Kerr-AdS black hole. The Bardeen-AdS black hole exhibits $P-V$ criticalities similar to that of the RN-AdS black hole, however has striking differences in its Gibbs free energy behavior. In particular, the characteristic "swallow-tail" structure associated with first-order phase transitions disappears. Instead, an "8-shaped" or "c-shaped" structure occurs, signifying a first-order phase transition or a zeroth-order phase transition between the small black hole and the large black hole phases, respectively. Our analysis suggests that constraint-induced modifications in the thermodynamic phase space may have deep consequences for the critical behaviors of black holes.

Figures (8)

• Figure 1: The $P-r_{+}$ and the $T-r_{+}$ curves of the singular black hole for fixed $Q_m=0.1$ and $\alpha=2$.
• Figure 2: The region with $C>0$ for fixed $Q_m=0.1$ and $\alpha=2$. The white areas correspond to $C<0$.
• Figure 3: The behaviors of $C-r_{+}$ for different values of $P$ for fixed $Q_m=0.1$ and $\alpha=2$. Here $P_0=0.0354$.
• Figure 4: The $G-T$ curves for different values of $P$ for fixed $Q_m=0.1$ and $\alpha=2$. Here $P_t=0.00047,~P_z=0.00064$. The red dashed lines correspond to the phases with negative heat capacity.
• Figure 5: The $T-r_{+}$ and $P-r_{+}$ curves of the Bardeen-AdS black hole with fixed $Q_m=0.1$.