⚛️ general relativity

Topological analysis of novel charged-AdS black hole solutions embedded in dark matter halos with quintessence field

This paper investigates the geometric and thermodynamic properties of a novel charged anti-de Sitter black hole embedded in a NGC 4649-type dark matter halo and a quintessence field, utilizing topological analysis and generalized free energy methods to demonstrate how dark matter enhances gravitational attraction while quintessence induces repulsive effects that significantly alter the system's phase structure and critical behavior.

Original authors: Muhammad Yasir, Hafiza Bushra Mujahid, Faisal Javed, Shahid Qaisar

Published 2026-10-27

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CC BY 4.0

Original authors: Muhammad Yasir, Hafiza Bushra Mujahid, Faisal Javed, Shahid Qaisar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the heart of almost every massive galaxy lies a cosmic giant: a black hole. These are not just empty voids but regions where gravity is so intense that nothing, not even light, can escape. For decades, astronomers have known that these objects do not exist in isolation. They are surrounded by vast, invisible clouds of dark matter, a mysterious substance that holds galaxies together but cannot be seen. Furthermore, the entire universe is expanding at an accelerating rate, driven by another invisible force known as dark energy. While scientists have studied black holes, dark matter, and dark energy separately, a new study asks a more complex question: what happens when a black hole sits at the center of a galaxy, surrounded by both a halo of dark matter and the stretching influence of dark energy?

To understand this, one must first grasp the nature of the forces at play. Dark matter acts like a heavy, invisible blanket, pulling everything inward with its gravity. Dark energy, specifically a form called quintessence, acts in the opposite way, pushing space apart and creating a repulsive force. In the real universe, a black hole is likely caught in the middle of this tug-of-war. The researchers behind this study wanted to map out exactly how these competing forces change the behavior of a charged black hole—a black hole that also carries an electric charge, a scenario that can occur in magnetized environments around real galaxies. They focused on a specific type of galaxy, NGC 4649, which has been well-studied by telescopes, to build a realistic model of the environment surrounding the black hole.

The team constructed a detailed mathematical model of this cosmic neighborhood. They started with the fundamental laws of gravity and added the specific density of the dark matter halo found in NGC 4649, along with the electric charge of the black hole and the repulsive push of the quintessence field. By solving these equations, they created a picture of the space around the black hole. They found that the dark matter halo strengthens the gravitational pull, making the black hole's grip tighter, while the quintessence field pushes back, altering the shape of space far away from the center. This interaction changes the size of the event horizon, the point of no return, showing that the environment around a black hole is not just a passive backdrop but an active participant in shaping the object itself.

Beyond simply describing the shape of space, the researchers investigated the thermodynamics of the system, which is essentially the study of heat, energy, and how the black hole behaves under different conditions. They treated the black hole like a physical system that can exist in different phases, similar to how water can be ice, liquid, or steam. Using a sophisticated method called topological analysis, they mapped out the stability of the black hole. This approach allowed them to count the number of stable states the black hole could occupy and identify critical points where the black hole might undergo a sudden change, much like water boiling into steam. They discovered that the presence of the dark matter halo and the quintessence field significantly shifts these critical points.

The study revealed that the strength of the dark energy field, described by a specific parameter, plays a crucial role in determining the black hole's fate. As the researchers varied this parameter, they observed systematic changes in the black hole's critical temperature and pressure. For instance, as the influence of the quintessence field changed, the critical pressure required for the black hole to shift phases increased dramatically, while the critical radius—the size of the black hole at the moment of change—shrank. These findings suggest that the interplay between dark matter and dark energy is not just a background detail but a fundamental driver of how black holes evolve and transition between different states.

By combining the effects of a realistic dark matter distribution with the repulsive force of dark energy, the authors provided a more complete picture of black hole physics than previous studies that looked at these factors in isolation. Their work shows that the topological structure of the black hole's thermodynamics is robust, meaning the classification of its stable and unstable states remains consistent even as the surrounding conditions change. This research offers a new way to understand the complex environment of supermassive black holes, suggesting that to truly understand these cosmic giants, we must account for the invisible forces that surround them. The study concludes that future observations of black hole shadows or gravitational waves could potentially test these predictions, bridging the gap between theoretical models and the observable universe.

Technical Summary: Topological Analysis of Novel Charged-AdS Black Hole Solutions Embedded in Dark Matter Halos with Quintessence Field

Problem Statement and Motivation
Observational evidence suggests that supermassive black holes reside in complex astrophysical environments dominated by dark matter halos and influenced by dark energy. While the accelerating expansion of the universe is often attributed to a cosmological constant, dynamical dark energy models, specifically quintessence, offer an alternative characterized by an equation of state $p_q = \omega_q \rho_q$ (where $-1 < \omega_q < -1/3$ ). Although previous studies have examined black holes in dark matter backgrounds or in the presence of quintessence separately, a comprehensive analysis simultaneously incorporating electric charge, Anti-de Sitter (AdS) geometry, specific dark matter halo profiles (NGC 4649 and Dehnen-(1,4,5/2)), and a quintessence field remains limited. Furthermore, understanding the thermodynamic phase structure and stability of such systems requires robust analytical tools beyond standard equation-of-state analysis. This work addresses the need to investigate the combined effects of attractive dark matter and repulsive quintessence on the thermodynamic topology of charged AdS black holes.

Methodology
The authors construct static, spherically symmetric black hole solutions by solving the modified Einstein field equations (EFEs). The spacetime metric incorporates four distinct components:

Electric Charge: Represented by the integration constant $Q$ .
Cosmological Constant: $\Lambda$ , defining the AdS geometry.
Dark Matter Halos: Modeled using two observationally constrained profiles:
- The Dehnen-(1,4,5/2) profile, derived from stellar kinematics and globular cluster data.
- The NGC 4649 (M60) halo profile, characterized by a critical tangential velocity $V_c$ and a core radius $a$ .
Quintessence Field: Modeled as an anisotropic fluid with a variable equation of state parameter $\omega_q$ .

The metric function $f(r)$ is derived for both halo models, and the resulting spacetime geometry is analyzed through horizon structures and the Ricci scalar.

To investigate the thermodynamic properties, the study employs extended black hole thermodynamics, treating the cosmological constant as thermodynamic pressure ( $P = -\Lambda/8\pi$ ). The authors derive the Smarr relation and the first law of thermodynamics, treating the halo core radius and quintessence parameters as thermodynamic variables.

The core analytical framework utilizes Duan's $\phi$ -mapping topological theory. This approach involves:

Defining a thermodynamic potential $\phi$ related to the Hawking temperature $T$ .
Constructing a two-dimensional vector field $\Phi = (\Phi_{r_h}, \Phi_\theta)$ in the parameter space of the horizon radius ( $r_h$ ) and an auxiliary angle ( $\theta$ ).
Identifying zero points (defects) of this vector field, which correspond to thermodynamic critical points.
Calculating the winding number ( $W$ ) and topological charge by integrating the topological current density over closed contours enclosing these zero points. This allows for the classification of black hole phases into topological classes (stable vs. unstable) based on the sign and magnitude of the winding number.

The analysis is conducted in both the canonical ensemble (fixed charge) and mixed canonical ensembles (fixed electric potential).

Key Contributions and Results

Metric Construction: The authors successfully derived distinct static spacetime solutions for charged AdS black holes embedded in NGC 4649 and Dehnen halos under the influence of quintessence. The study confirms that the dark matter halo strengthens gravitational attraction, while the quintessence field provides a repulsive contribution that alters the asymptotic behavior of the spacetime.
Thermodynamic Framework: The study establishes the extended first law and Smarr relation for these systems, identifying conjugate variables for the halo core radius ( $a$ ) and the quintessence intensity ( $C_q$ ).
Topological Classification:
- Using the Duan $\phi$ -mapping method, the authors identified critical points (zero points of the vector field) in the thermodynamic parameter space.
- In the canonical ensemble, the analysis revealed a total topological charge of $W = -1$ , indicating a specific topological structure where the system exhibits a non-trivial phase transition.
- In the mixed canonical ensemble, the system also demonstrated a total topological charge of $W = -1$ , though the presence of multiple zero points (e.g., ZP3, ZP4, ZP5) suggested richer equilibrium configurations depending on the parameters.
- The study verified that the topological charge is a robust invariant, remaining unchanged regardless of the specific shape or size of the integration contour, provided the contour encloses the same zero points.
Critical Behavior and Quintessence Influence: Numerical analysis of critical points ( $r_c, T_c, P_c$ ) as a function of the quintessence parameter $\omega_q$ revealed systematic shifts. As $\omega_q$ decreases (becoming more negative), the critical horizon radius decreases, and the critical pressure increases significantly. The fractional shifts relative to a reference value ( $\omega_q = -0.4$ ) highlighted that variations in the quintessence field substantially modify the phase structure and critical behavior of the black hole system.

Significance and Claims
The paper claims that the combined effects of dark matter halos and the quintessence field substantially influence the thermodynamic topology of charged black holes. By applying Duan's topological theory, the authors provide a consistent framework for:

Locating and classifying thermodynamic critical points.
Distinguishing between stable and unstable black hole phases through topological invariants (winding numbers).
Demonstrating that the quintessence field plays a critical role in determining the phase structure, as evidenced by the systematic changes in critical quantities.

The authors conclude that this topological approach offers a powerful, complementary method for analyzing black hole phase transitions in realistic astrophysical environments. They suggest that these results contribute to a deeper understanding of the interplay between dark matter, dark energy, and black hole thermodynamics, potentially offering insights into the microscopic origins of phase transitions and the role of topological invariants in gravitational systems. The study does not propose new experimental tests but suggests that future work could extend this framework to rotating black holes, higher dimensions, and modified gravity theories.

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