Impact of Perfect Fluid Dark Matter on the Thermodynamics of $AdS$ Ayón--Beato--García Black Holes

Abstract: In this paper, we derive the black hole solution in the context of nonlinear electrodynamics (NLED) coupled to a perfect fluid dark matter (PFDM) field. The resulting black hole solution interpolates between the $AdS$ Ayón--Beato--García (ABG) black hole in the absence of the PFDM field and the Schwarzschild black hole devoid of magnetic monopole charges and PFDM influence. A numerical investigation of the horizon structure and thermodynamic properties, including both local and global stability, is conducted for the obtained black hole solution. The thermodynamic quantities are shown to be modified by the presence of the NLED and PFDM fields. We observe that the behaviour of thermodynamical quantities of black holes depends on these parameters significantly. We also discuss the stability and phase transition dependency on these parameters.

• The paper derives an exact AdS ABG black hole solution coupled to PFDM and NLED, revealing notable modifications in horizon structure and thermodynamic properties.
• It introduces a corrected first law of thermodynamics that reconciles the entropy-area relationship through detailed numerical and analytical analysis.
• The stability analysis identifies critical phase transitions marked by shifts from local to global stability using heat capacity and Gibbs free energy.

Impact of Perfect Fluid Dark Matter on the Thermodynamics of $AdS$ Ayón--Beato--García Black Holes

Introduction

This paper explores the impact of a perfect fluid dark matter (PFDM) field on the thermodynamics of the anti-de Sitter (AdS) Ayón--Beato--García (ABG) black holes within the framework of nonlinear electrodynamics (NLED). The authors derive a black hole solution in which the ABG configuration is coupled to both a PFDM field and NLED. The resultant solution interpolates between the ABG black hole and the Schwarzschild black hole in scenarios devoid of magnetic monopole charges and PFDM influence. The paper conducts a numerical and theoretical investigation of the horizon structure and thermodynamic properties of this solution, revealing insights into the stability and phase transitions influenced by these factors.

Black Hole Solution: PFDM and NLED Fields

The study presents a gravitational action that integrates the effects of a PFDM field and NLED. The derived action is described as:

$$\mathcal{S} =\int d^{4}x\sqrt{-g}\left[ R-2\Lambda +\frac{1}{2} \nabla_{a} \varphi\nabla^{a}\varphi - V(\varphi) + \mathcal{L}_{DM} + \mathcal{L}(F)\right]$$

Here, $R$ denotes the curvature scalar, $\Lambda$ is the cosmological constant, $\varphi$ is the phantom field, and $V(\varphi)$ is its potential. $\mathcal{L}_{DM}$ and $\mathcal{L}(F)$ represent the dark matter Lagrangian density and the NLED Lagrangian density, respectively.

The resulting metric, characterized by parameters $M$, $g$, $\lambda$, and $l$, provides an exact black hole solution that describes a complex horizon structure, with potential for three horizons: the Cauchy horizon ($r_-$), event horizon ($r_+$), and cosmological horizon ($r_\Lambda$). The horizon dynamics and existence are intricately dependent on the black hole parameters, as demonstrated through numerical solutions.

Thermodynamics and Phase Transitions

The paper meticulously derives the thermodynamic properties, including black hole entropy, temperature, and the modified first law of thermodynamics. A generalized expression for the Hawking temperature is developed, taking into account the presence of the PFDM and magnetic monopole charge.

The real advancement in this analysis is the treatment of entropy, where the authors propose a correction to the classical thermodynamic laws, addressing discrepancies related to the area law of black hole thermodynamics. The modified first law is expressed as:

$\mathcal{C}(M_+, r_+)\,dM_+ = T_+ dS_+$

Where $\mathcal{C}(M_+,r_+)$ is a correction factor based on the energy density $T^0_0$. This modification reconciles the entropy-area relationship with the principles of regular black holes.

Stability Analysis

The stability of the black holes is analyzed in terms of both local and global stability criteria. The heat capacity expression indicates sign changes that correspond to phase transitions at critical radii. A comprehensive analysis shows that the $AdS$ ABG black hole transitions from stability ($r_+ < r_{1+}$) to instability ($r_{1+} < r_+ < r_{2+}$) and back to stability ($r_+ > r_{2+}$), with critical points identified where second-order phase transitions occur.

The global stability is further evaluated using the Gibbs free energy, where negative values signify thermodynamic stability. These evaluations assert and quantify stability against parameter variations like magnetic monopole charge ($g$) and the scale parameter ($\lambda$).

Conclusion

The paper successfully demonstrates that the coupling of a PFDM field to an ABG black hole within the framework of NLED leads to substantial modifications in black hole thermodynamics. By introducing a corrected first law of thermodynamics, the singularity issues often associated with black hole entropy are addressed, aligning the modified law with the expected thermodynamic behavior of regular black holes. The dual evaluation of local stability (via heat capacity) and global stability (via Gibbs free energy) offers an elaborate understanding of phase transitions and the parameter space that governs these dynamics. This work lays a foundational understanding that bears significant implications for theoretical and computational explorations of exotic black hole configurations in generalized theories of gravity.