Coexistent physics and microstructure of the regular Bardeen black hole in anti-de Sitter spacetime

Abstract

We study the phase structure and the microscopic interactions of a regular Bardeen–AdS black hole. The stable and metastable phases in the black hole are analysed through coexistence and spinodal curves. The solutions are obtained numerically as the analytic solution to the coexistence curve is not feasible. The $P_r-T_r$ coexistence equation is obtained using a fitting formula. The coexistence and spinodal curves are plotted in $P_r-T_r$ and $T_r-V_r$ planes to explore the phase structure of the black hole. In the second part of our study, we were able to probe the microscopic interactions of the regular Bardeen–AdS black hole using the novel Ruppeiner geometry proposed by Wei et al. (2019). It is found that the microscopic interactions are not the same in small black hole (SBH) and large black hole (LBH) phases. In the SBH phase, there exists a repulsive interaction in the microstructure in the low temperature regime. In contrast, the microstructure associated with the LBH phase has an attractive interaction throughout the parameter space. We found that, along the coexistence temperature, both the SBH and LBH branches diverge to negative infinity with a critical exponent equal to $1∕2$.

Introduction

Black hole thermodynamics is a subject that helps a theoretical physicist to unveil the deep connection between gravitational, quantum and statistical physics. Even after 50 years of Bekenstein and Hawking’s discovery, the problem of black hole entropy and the temperature is not well understood [1], [2], [3], [4]. During the past three decades, in quest to find a quantum gravity theory the focus is directed towards the black holes in asymptotically anti-de Sitter (AdS) spacetimes. It is mainly due to the pioneering work of Hawking and Page, which explored the phase transition between the radiation and a large black hole [5]. Black holes in AdS cavity provided necessary thermal stability to this thermodynamic system. But the Smarr relation for AdS black hole is found inconsistent with the first law [6]. To rectify this problem, the cosmological constant $\Lambda$ was considered as a thermodynamical variable. In the first law, it was interpreted as the thermodynamic pressure, and its conjugate quantity was found to be the geometrical volume. This has extended the first law of black hole thermodynamics with the necessary $VdP$ term [6], [7]. In this extended phase space, thermodynamics of charged AdS black hole was found to be analogous to a van der Waals fluid system [8], [9], [10]. This has led to a new arena in the black hole physics called black hole chemistry.

This macroscopic picture of the black hole is used to propose a phenomenological model for the black hole microstructure [11]. Even though the microscopic information is not a requirement for the thermodynamics, it may be used for quantum gravity studies. Prominent model for drawing microscopic information from thermodynamics is the Ruppeiner’s thermodynamic geometry [12]. It is constructed in the equilibrium state space in the context of thermodynamic fluctuation theory, but can be useful in studying black holes too [11]. Through Gaussian fluctuation moments, a Riemannian geometry is constructed in the thermodynamic equilibrium space, whose metric tells us about the fluctuations between the states. This method is applied to van der Waals fluids and to a variety of other statistical systems [12], [13], [14], [15], [16]. These studies show that the thermodynamic geometry encodes the information about the microscopic interaction. The thermodynamic scalar curvature $R$ is proportional to the correlation volume of the underlying system. The sign of $R$ indicates the type of interaction in the microstructure, positive for repulsive and negative for attractive interactions. In recent times, there has been a lot of interest in the thermodynamic geometry, to investigate critical phenomenon and microstructure of various black holes in AdS spacetime [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].

Recently, a novel approach in Ruppeiner geometry was developed to explore the missing information due to the singularity in the scalar curvature [34]. This is mainly due to the vanishing of heat capacity at constant volume. The new normalised scalar curvature takes care of this problem. A metric can be defined by Taylor expanding the Boltzmann entropy around the equilibrium value. The thermodynamical coordinates were chosen to be the temperature and volume, and the Helmholtz free energy was chosen as the thermodynamic potential. Applying this method to the van der Waals (vdW) fluid, it was found that dominant interaction in the microstructure is attractive throughout the parameter space. Utilising this analogy with vdW fluid, thermodynamic geometry of a charged AdS black hole is analysed. In contrast to the vdW fluid, the interaction is not attractive over the entire parameter space. Even though the interaction is attractive for the large black hole (LBH) every where, and small black hole (SBH) for most of the parameter space, there exists a weak repulsive interaction in the SBH phase at very low temperatures [34], [35]. Interestingly, this behaviour is not universal for all asymptotically AdS black holes. In the case of a five-dimensional neutral Gauss–Bonnet black hole, interaction similar to vdW fluid is observed, with a dominant attractive interaction throughout the SBH and LBH phases [22]. Later, the work was extended to 4-dimensional Gauss–Bonnet black holes [36]. Subsequently, the microscopic interactions for $4-D$ AdS topological black holes in dRGT massive gravity were studied [37], [38]. Microstructure was found to be distinct, with the presence of both repulsive and attractive interactions in both the SBH and LBH phases. In our recent paper, we have investigated the microstructure of regular Hayward and Born–Infeld AdS black holes [39], [40]. In regular Hayward case, the microscopic interactions observed are similar to that of charged AdS black holes. Where as, Born–Infeld AdS black holes show a reentrant phase transition, which has a distinct microstructure. Apart from these studies, the study of microstructure using this novel method is limited to a few black holes. Motivated by the recent progress, here we explore the phase structure and microstructure of a regular Bardeen–AdS black hole.

Regular black holes are the ones which do not possess a singularity at the centre. Even though it requires quantum theory of gravitation to obtain a singularity free solution, a phenomenological model can be constructed in the classical gravity. Firstly such a regular solution was derived by Bardeen [4]. Later many have found regular black holes that can be an exact solution to gravity coupled with a non-linear electromagnetic source [41], [42], [43]. We have studied phase transitions and thermodynamic geometry of regular black holes in our recent papers [44], [45], [46]. It is noticed that the presence of magnetic monopole charge imparts a phase structure to the regular black holes similar to an electric charge. So we find it interesting to probe the microstructure corresponding to the magnetically charged Bardeen black holes in asymptotically AdS spacetimes.

The paper is organised as follows: In Section 2, we review the action and derivation of the regular Bardeen black hole in AdS spacetime. In Section 3, we mainly focus on the thermodynamics and phase structure of the black hole. Then the Ruppeiner geometry and analysis of critical features are discussed in Section 4. Section 5 is dedicated for summary and conclusions.