Charged Bardeen black hole with a cosmological constant and surrounded by quintessence and a cloud of strings

Technical Summary: Charged Bardeen Black Hole with a Cosmological Constant and Surrounded by Quintessence and a Cloud of Strings

Problem Statement
The paper addresses the construction and analysis of exact solutions to Einstein's field equations that generalize the Bardeen regular black hole metric. While the original Bardeen solution provides a singularity-free black hole model by introducing a mass function $M(r)$ dependent on the radial coordinate, this work seeks to extend that framework to include a more complex physical environment. Specifically, the authors investigate a charged Bardeen black hole situated within a cosmological constant ($\Lambda$), surrounded by a cloud of strings, and immersed in a quintessence field. The primary objective is to derive the metric for this generalized system and analyze its regularity and geodesic structure to determine how these additional sources affect the singularity-free nature of the original Bardeen solution.

Methodology
The authors employ a theoretical framework based on General Relativity coupled with nonlinear electrodynamics. The methodology proceeds as follows:

1. Action Formulation: The total action is constructed by combining the Einstein-Hilbert action (including the cosmological constant $\Lambda$) with contributions from a nonlinear electromagnetic field coupled to quintessence, a standard electromagnetic field (representing electric charge $Q$), and a cloud of strings (modeled via the Nambu-Goto action).
2. Field Equations: By varying the total action with respect to the metric, the authors derive the Einstein field equations. The energy-momentum tensors for the nonlinear electrodynamics, the electromagnetic field, and the string cloud are explicitly calculated.
3. Metric Derivation: Assuming a static, spherically symmetric spacetime, the authors solve the resulting system of differential equations to obtain the metric function $f(r)$. This yields a generalized metric containing parameters for mass ($m$), magnetic charge ($q$), electric charge ($Q$), the string cloud parameter ($a$), the quintessence parameter ($\alpha$ and equation of state $\omega_q$), and the cosmological constant ($\Lambda$).
4. Regularity Analysis: To test for singularities, the Kretschmann scalar ($K = R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$) is calculated for the general solution and for specific limiting cases (e.g., removing specific sources). The behavior of $K$ is analyzed in the limits $r \to 0$ and $r \to \infty$.
5. Geodesic Analysis: The motion of test particles is studied using the Lagrangian formalism. The authors derive the geodesic equations and the effective potential ($V_{eff}$) for both massive (time-like) and massless (null) particles, examining radial and non-radial trajectories under various parameter configurations.

Key Contributions and Results
The paper derives a new exact metric representing a charged Bardeen black hole with a cosmological constant, surrounded by quintessence and a cloud of strings. The analysis yields the following specific results:

• Metric Generalization: The derived metric (Eq. 33) successfully recovers known solutions as special cases, including the standard Bardeen black hole, Bardeen-Reissner-Nordström, Bardeen-Letelier (string cloud), Bardeen-Kiselev (quintessence), and Bardeen-AdS spacetimes.
• Regularity and Singularities:
  • The pure Bardeen black hole remains regular (finite Kretschmann scalar at $r=0$) and asymptotically flat.
  • The addition of a cosmological constant preserves the regularity of the Bardeen metric at the origin.
  • The inclusion of a cloud of strings (parameter $a$) or an electric charge ($Q$) destroys the regularity, introducing a curvature singularity at $r=0$.
  • The presence of quintessence has a dual effect depending on the equation of state parameter $\omega_q$:
    • For the quintessence regime ($\omega_q = -2/3$), the solution becomes singular at the origin.
    • For the phantom energy regime ($\omega_q = -4/3$), the solution is singular at the origin (the Kretschmann scalar diverges as $r \to 0$) and also exhibits a singularity at infinity (the curvature diverges as $r \to \infty$).
• Geodesic Motion and Effective Potential:
  • Non-radial time-like geodesics: For the pure Bardeen case ($q=0$), no stable circular geodesics exist. However, when the magnetic charge $q > 0$, stable circular orbits become possible, depending on the values of $Q$, $\omega_q$, $\Lambda$, and $a$. The effective potential diverges both near the horizon and at infinity.
  • Non-radial null-like geodesics: Similar to time-like particles, stable circular orbits for massless particles only exist when $q > 0$. The effective potential behavior at infinity varies based on the cosmological constant and quintessence parameters.
  • Radial geodesics: Radial time-like geodesics exhibit stability only when $q > 0$. Radial null-like geodesics result in a vanishing effective potential.

Significance
The paper claims significance in extending the study of regular black holes to more diverse and realistic astrophysical environments. By systematically analyzing the interplay between the Bardeen core and external sources (strings, quintessence, cosmological constant, and charge), the authors demonstrate that the "regularity" of a black hole is not an intrinsic property of the core alone but is highly sensitive to the surrounding matter and energy fields. The work highlights that while the Bardeen metric provides a singularity-free core, the introduction of certain physical sources (specifically electric charge, string clouds, and phantom energy) reintroduces singularities, whereas others (like the cosmological constant) may preserve regularity. This provides a nuanced understanding of how different energy components in the universe might influence the structural integrity of regular black hole models.