Implications of Adler-Finch-Skea solution on charged dark energy star satisfying Karmarkar Condition

Abstract: A possible approach for preventing compact astrophysical objects from gravitational collapse into singularities is the idea of dark energy. Since it is the cause of our universe's accelerated expansion, it has the greatest impact on the cosmos. As a result, it appears that dark energy can interact with any compact astrophysical stellar object [Phys. Rev. D 103, 084042 (2021)]. In this study, our primary objective is to develop a simpler model of a charged strange star coupled with anisotropic dark energy admitting the Adler-Finch-Skea solution [J. Math. Phys. 15, 727 (1974); Class. Quantum Grav. 6, 467 (1989)] within Einstein gravity. To develop this model, the Karmarkar condition was employed to determine the radial metric component, while Adler's methodology was used to choose the time-metric component. For this purpose, we explored a particular strange star, Her X-1, with observed values of mass $(0.85 \pm 0.15)M_{\odot}$ and radius $= 8.1_{-0.41}^{+0.41}$ km. In this context, we proceeded to model dark energy using the equation of state (EoS), such that the density of dark energy is proportional to the density of isotropic perfect fluid matter. The unknown constants in the metric were determined by smooth matching using the Darmois-Israel criterion. We conduct an in-depth examination of the stability and force equilibrium of our suggested star framework, as well as several physical characteristics of the model such as the metric function, pressure, density, mass-radius relation, and dark energy parameters. Thus, the physical consistency and stability of the present model are investigated. Therefore, following a comprehensive theoretical investigation, we discovered that our proposed model is singularity free and meets all the stability requirements to be a stable and physically realistic stellar model.

• The paper presents a singularity-free model embedding dark energy and electric charge into compact stars using the Adler-Finch-Skea metric.
• It rigorously applies the Karmarkar condition and Darmois-Israel junction conditions to achieve smooth interior-exterior matching.
• Numerical analysis on objects like Her X-1 confirms stability and compatibility with observed mass-radius and redshift data.

Implications of Adler-Finch-Skea Solution on Charged Dark Energy Stars Satisfying the Karmarkar Condition

---

Introduction

The paper "Implications of Adler-Finch-Skea solution on charged dark energy star satisfying Karmarkar Condition" (2606.18775) analytically constructs a model for a singularity-free, charged, anisotropic dark energy star based on the Adler-Finch-Skea (AFS) metric, rigorously enforcing the Karmarkar condition within the context of Einstein-Maxwell gravity. The study targets compact objects such as Her X-1, where observational mass and radius data are used to anchor the model parameters; it is motivated by the interplay of dark energy, electric charge, and anisotropy in compact astrophysical bodies, aiming to explore configurations that avoid gravitational collapse into singularities.

---

Mathematical Formalism and Model Construction

The stellar interior is modeled with a spherically symmetric metric employing two key ingredients: (1) the AFS metric, which supplies a physically plausible form for the time component; (2) the Karmarkar condition, which restricts the radial metric component and provides embedding class-I geometry. The energy-momentum tensor includes both normal baryonic matter and a dark energy fluid, with anisotropic pressures and electric field contributions. The dark energy component is specified by an equation of state (EoS) proportional to the baryonic matter density ($\rho^D = \chi \rho$), and satisfies $p_r^D+\rho^D=0$ for radial dark pressure.

Smooth matching to the exterior Reissner-Nordström spacetime is achieved by enforcing the Darmois-Israel junction conditions, ensuring continuity of the metric and its derivatives. The parameters $\mathfrak{B}$, $\mathfrak{C}$, and $\mathfrak{F}$ in the metric, as well as the coupling constant $\chi$, are determined in terms of the stellar mass, radius, and charge.

Key analytic expressions for density, pressure, electric field, and mass profiles are derived in closed form, with the resulting solution class identical to the Finch-Skea model but extended to include anisotropic charge and dark energy.

---

Physical Properties and Numerical Analysis

Central metric potentials are finite and monotonic, indicating regularity and the absence of singularities. Matter density and pressure both decrease outward and peak at the stellar core, accompanied by increasing electric field strength. The dark energy density is always positive, while the radial dark pressure is strictly negative, modeling repulsive effects critical for avoiding collapse.

The mass function grows smoothly and saturates at the surface value, corresponding to the observable stellar mass. Compactness factors remain below the Buchdahl limit ($<4/9$), indicating physical viability. Calculated surface redshifts and gravitational redshifts exhibit contrasting radial profiles; both behave consistently with expectations for compact star atmospheres.

Detailed numerical values are provided for Her X-1 and a suite of other X-ray binary and pulsar candidates, demonstrating the quantitative fit of the model to real stellar objects. Central and surface densities, compactness ratios, pressures, and redshifts are all tabulated to highlight the model's compatibility across multiple systems.

---

Stability, Causality, and Energy Conditions

Stability analysis employs Herrera's cracking method and sound speed constraints. The squared sound speed, including radial and tangential components, remains subluminal ($V_r^2, V_t^2 < 1$), and the stability factor ($|V_t^2 - V_r^2|$) is always less than unity, satisfying Herrera-Abreu criteria for structural stability.

All classical energy conditions—null, weak, strong, and dominant—are explicitly checked and satisfied through the stellar interior, for both the effective and anisotropic-fluid contributions. The model's equilibrium under hydrodynamic forces is demonstrated via a generalized Tolman-Oppenheimer-Volkoff equation, which includes electric and dark energy terms. The total force balance confirms hydrostatic equilibrium, with gravitational and dark energy forces counteracting each other and electric forces contributing stabilizing effects.

Application of the Zel'dovich condition on the EoS parameter ($\Omega = p/\rho$) yields $p_r^D+\rho^D=0$0, further verifying physical acceptability.

---

Implications and Future Directions

This study establishes a robust systematic framework for modeling charged dark energy stars with embedding class-I geometry. The findings underscore the importance of dark energy-induced negative pressure and electric charge in counteracting gravitational collapse, yielding stable, physically realistic models without singularities. Practical implications include improved interpretability of observed compact objects whose behavior diverges from classical neutron or quark star predictions, and the possibility of distinguishing stellar remnants exhibiting dark energy or unusual charge distributions.

Theoretically, the model generalizes prior idealized configurations by enabling double-fluid (baryonic + dark energy) descriptions within realistic astrophysical metrics, paving the way for studies of hybrid stars, QPO dynamics, and more elaborate coupling mechanisms in modified gravity or scalar-tensor frameworks.

Future directions include investigation of dynamic stability under perturbations, extension to rotating or non-spherical geometries, and exploration of observational signatures in gravitational wave and electromagnetic spectra that may differentiate charged dark energy stars from black holes or ordinary compact objects.

---

Conclusion

The constructed charged dark energy star model based on the Adler-Finch-Skea solution and satisfying the Karmarkar condition exhibits regular, stable, and physically plausible behavior for a range of observed compact stars, including Her X-1 (2606.18775). All relevant stability and energy conditions are satisfied, and the model provides consistent mass-radius-redshift relations. The inclusion of anisotropic dark energy and electric charge fundamentally alters the thermodynamic and equilibrium structure, offering a sophisticated alternative to singular endpoints of stellar evolution and contributing to the broader theoretical landscape of compact astrophysical objects influenced by dark energy.