Charged Letelier--Alencar Black Strings with a Quintessence and the Cosmolgical Constant

Technical Summary: Charged Letelier–Alencar Black Strings with Quintessence and the Cosmological Constant

Problem Statement
This study addresses the gravitational configuration of charged black strings within Einstein–Maxwell–$\Lambda$ theory, incorporating two specific matter sources: a generalized cloud of strings (the Letelier–Alencar model) and a quintessence field. While standard black strings (cylindrical solutions with negative cosmological constants) and black holes surrounded by string clouds or quintessence have been studied individually, this work seeks to unify these elements. The primary objective is to derive exact solutions for a charged black string coupled to a Letelier–Alencar cloud of strings and quintessence, and to systematically analyze the resulting spacetime geometry, thermodynamic properties, and null geodesic structure.

Methodology
The authors begin by formulating the field equations for a four-dimensional static spacetime with cylindrical topology. The system includes:

1. Einstein–Maxwell–$\Lambda$ equations: Incorporating the cosmological constant ($\Lambda$) and an electromagnetic field.
2. Matter Sources:
   • A Letelier–Alencar cloud of strings, characterized by parameters $g_s$ (effective coupling) and $l_s$ (string length), which introduces a fully anisotropic energy-momentum tensor with a magnetic-like component.
   • A quintessence field described by a Kiselev anisotropic fluid with a barotropic equation of state parameter $\omega$ ($-1 < \omega < -1/3$) and normalization constant $\sigma$.
3. Metric Ansatz: A static line element $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\phi^2 + r^2\alpha^2dz^2$ is assumed, with a radial electric field derived from a gauge potential $A_\mu$.

The authors solve the resulting differential equations to obtain the metric function $f(r)$, which involves hypergeometric functions due to the string cloud contribution. Subsequent analyses include:

• Curvature Analysis: Calculation of Ricci and Kretschmann scalars to identify singularities and asymptotic behavior.
• Thermodynamics: Derivation of Hawking temperature, entropy (via the area law), electric charge, electric potential, and mass (using the Ashtekar-Magnon-Das approach). Stability is assessed via heat capacity (local stability) and Gibbs free energy (global stability).
• Extended Phase Space: The cosmological constant is treated as thermodynamic pressure ($P = -\Lambda/8\pi$), allowing for the derivation of an extended first law and a modified Smarr relation.
• Geodesic Structure: Investigation of null geodesics to derive the equation for the photon cylinder radius, analyzing the effective potential for massless particles.

Key Contributions and Results

• Exact Solution: The paper derives a new metric function $f(r)$ that generalizes known charged black string solutions. The solution reduces to the standard charged black string when the string cloud and quintessence parameters vanish. The metric function includes terms dependent on the mass $m_0$, charge $q$, cosmological constant $\Lambda$, and the specific parameters of the matter fields ($g_s, l_s, \sigma, \omega$).
• Geometric Structure: Curvature invariants confirm an essential curvature singularity at the origin ($r=0$). The metric function $f(r)$ generally possesses an event horizon enclosing this singularity. Numerical analysis indicates that increasing the mass $m_0$, coupling $g_s$, or state parameter $\omega$ enlarges the event horizon, while increasing the string length $l_s$, charge $q$, or quintessence normalization $\sigma$ reduces it.
• Thermodynamic Properties:
  • Temperature: The Hawking temperature is shown to be negative for small horizon radii (indicating non-physical branches) and positive for large radii (provided $\Lambda < 0$). The transition point depends on the model parameters.
  • Stability: Local thermodynamic stability (determined by heat capacity) is found only for large black strings ($r_+ > r_{+, C=0}$). The domain of stability shrinks with increasing $g_s$, $l_s$, and $q$, but expands with increasing $\sigma$.
  • Global Stability: Analysis of the Gibbs potential reveals that global stability is sensitive to the parameters. For sufficiently large values of $g_s$, $l_s$, or $q$, small black strings become globally unstable, while large ones remain stable. Conversely, the quintessence parameter $\sigma$ tends to preserve global stability across the entire domain.
• Extended Thermodynamics: By treating $\Lambda$ as pressure, the authors establish the validity of the extended first law of thermodynamics, $dM = TdS + \Phi dQ + VdP + Gdg_s + Ldl_s + \Sigma d\sigma + \Omega d\omega$. They derive a corresponding Smarr relation that includes work terms associated with the string cloud and quintessence parameters.
• Photon Cylinder: The study derives a transcendental equation for the radius of the photon cylinder ($r_{pc}$). Unlike neutral black strings without matter fields, the presence of the Letelier–Alencar cloud and quintessence allows for the existence of photon cylinders even in neutral configurations. Numerical results show that the first photon cylinder radius increases with the string length $l_s$ and coupling $g_s$, while the second radius behaves inversely with $l_s$ but increases with $g_s$.

Significance
The paper claims to provide a physically well-motivated framework for investigating the interplay between electromagnetic effects, the cosmological constant, and generalized string-like matter distributions. By generalizing known charged black string spacetimes to include the Letelier–Alencar cloud and quintessence, the work elucidates how anisotropic matter distributions modify horizon structures, thermodynamic stability regimes, and null geodesic properties. The derivation of the extended first law and Smarr relation for this specific system offers a rigorous thermodynamic description that accounts for the additional degrees of freedom introduced by the string cloud and quintessence. The results suggest that these matter sources significantly alter the stability criteria and the existence of photon cylinders, providing a broader context for understanding black string thermodynamics in modified matter environments.