Analysis of Charged Compact Stars with Bardeen Black Hole in $f(\mathfrak{Q}, \mathcal{T})$ Gravity

Abstract: This study investigates the behavior of charged compact stars within the $f(\mathfrak{Q}, \mathcal{T})$ gravitational framework, where $\mathfrak{Q}$ denotes the non-metricity scalar and $\mathcal{T}$ represents the trace of the energy-momentum tensor. Recognized as a promising model for explaining the accelerated expansion of the universe, this approach offers a strong theoretical foundation. A central focus of this research is the application of Bardeen's model to describe the exterior spacetime. Additionally, the study examines the internal structure of compact stars using a solution based on the Finch-Skea metric potential. Various physical properties, including energy density, pressure components, anisotropy, energy conditions and equation of state parameters are analyzed through graphical representations. Equilibrium conditions are explored via the Tolman-Oppenheimer-Volkoff equation while key characteristics such as the mass-radius relationship, compactness, surface redshift and stability criteria based on the causality condition and adiabatic index are thoroughly evaluated. The analysis concludes that the proposed solutions for charged compact stars in this framework are both theoretically consistent and physically valid.

• The paper demonstrates that charged compact stars modeled in f(Q, T) gravity with a Bardeen exterior satisfy all classical energy and stability conditions.
• The study employs the Finch-Skea metric for the anisotropic charged interior and uses analytical matching to derive key physical parameters.
• The paper highlights that the second-order field equations in f(Q, T) gravity avert singularities and provide a realistic framework for high-density astrophysical objects.

Comprehensive Analysis of Charged Compact Stars in $f(\mathfrak{Q}, \mathcal{T})$ Gravity with Bardeen Exterior

Context and Motivation

The investigation addresses the structural and dynamical properties of charged compact stars within the framework of $f(\mathfrak{Q}, \mathcal{T})$ gravity, utilizing the Bardeen black hole solution for the exterior region and the Finch-Skea metric for the interior. $f(\mathfrak{Q}, \mathcal{T})$ gravity—where $\mathfrak{Q}$ is the non-metricity scalar and $\mathcal{T}$ is the trace of the energy-momentum tensor—provides a second-order field equation alternative to GR, offering increased flexibility in the modeling of astrophysical systems and potential explanations for accelerated cosmic expansion without dark energy. The Bardeen solution mitigates central singularities via a nonlinear electromagnetic source and enables regular interior-exterior matching, while the Finch-Skea metric robustly describes the anisotropic charged interior.

Mathematical Formulation and Methodology

The action for $f(\mathfrak{Q}, \mathcal{T})$ gravity is constructed by extending GR to incorporate both non-metricity and matter trace contributions. A linear functional form $$f(\mathfrak{Q}, \mathcal{T}) = \zeta \mathfrak{Q} + \eta \mathcal{T}$$ is adopted for tractable analytic treatment, where $\zeta$ and $\eta$ modulate geometric and matter effects, respectively, and simplify the modified field equations.

Charged anisotropic matter is modeled in a spherically symmetric spacetime, with radial and tangential pressures, electric field contributions from Maxwell’s equations, and charge density self-consistently integrated. The interior metric potentials are chosen per the Finch-Skea ansatz, ensuring a regular core and compatibility with observed compact stars. Exterior matching employs the Bardeen black hole metric, achieving seamless transition at $r=R$ via continuity in the metric and its derivatives. Analytical expressions for the unknown interior constants $f(\mathfrak{Q}, \mathcal{T})$0, $f(\mathfrak{Q}, \mathcal{T})$1, and $f(\mathfrak{Q}, \mathcal{T})$2 are derived from boundary conditions.

Physical Analysis and Numerical Results

The paper provides a thorough evaluation of key physical properties of the solution:

Energy Density and Pressure Profiles

Numerical analysis reveals that central density ($f(\mathfrak{Q}, \mathcal{T})$3), radial ($f(\mathfrak{Q}, \mathcal{T})$4), and tangential ($f(\mathfrak{Q}, \mathcal{T})$5) pressures peak at the core and decrease outward. The tangential pressure transitions to negative values in the envelope, signifying physical anisotropy—a typical trait in high-density compact stars. The pressure vanishing at the boundary $f(\mathfrak{Q}, \mathcal{T})$6 ensures physical viability.

Anisotropy and Charge Effects

Anisotropy ($f(\mathfrak{Q}, \mathcal{T})$7) is positive throughout most of the interior, corresponding to an outward-directed force that counteracts collapse and is accentuated by electric charge. The anisotropy decreases with radius, supporting a stable configuration.

Energy Conditions

The model satisfies all classical energy conditions (NEC, DEC, WEC, SEC) for all tested $f(\mathfrak{Q}, \mathcal{T})$8 values. This implies the constructed solution describes ordinary (non-exotic), physically acceptable matter.

Equation of State

The EoS parameters $f(\mathfrak{Q}, \mathcal{T})$9 and $f(\mathfrak{Q}, \mathcal{T})$0 remain in the interval $f(\mathfrak{Q}, \mathcal{T})$1, confirming that the matter is neither ultrarelativistic nor unphysical, consistent with stellar interiors.

Tolman-Oppenheimer-Volkoff (TOV) Equilibrium

The generalized TOV equation, adjusted for anisotropy and charge, confirms the interplay of hydrostatic, gravitational, electric, and anisotropic forces yields a net zero force, guaranteeing equilibrium.

Mass-Radius, Compactness, and Redshift

The mass function $f(\mathfrak{Q}, \mathcal{T})$2 increases monotonically with radius, with $f(\mathfrak{Q}, \mathcal{T})$3, consistent with Buchdahl’s bound. Compactness $f(\mathfrak{Q}, \mathcal{T})$4 remains below the critical threshold, and gravitational redshift $f(\mathfrak{Q}, \mathcal{T})$5 stays finite and below the maximum bound for anisotropic stars, confirming no formation of event horizons or forbidden regimes.

Stability Analysis

The causality condition ($f(\mathfrak{Q}, \mathcal{T})$6) is met, ensuring physical signal propagation. Herrera's cracking criterion ($f(\mathfrak{Q}, \mathcal{T})$7) is satisfied, and the adiabatic indices $f(\mathfrak{Q}, \mathcal{T})$8, $f(\mathfrak{Q}, \mathcal{T})$9 throughout, indicating robust dynamical stability against radial perturbations and gravitational collapse.

Comparative Discussion

Compared to previous works in $\mathfrak{Q}$0 gravity, notably [9b], the inclusion of $\mathfrak{Q}$1 enhances the degrees of freedom, allowing consistent balance between hydrostatic and gravitational forces and eliminating core singularities. The second-order nature of $\mathfrak{Q}$2 theory offers computational efficiency and avoids the complexities often encountered in fourth-order formalisms like $\mathfrak{Q}$3, while the Finch-Skea metric remains compatible across several MGTs. The Bardeen exterior is preferable over Reissner-Nordström in eliminating singularities and achieving regularity at the stellar interface.

Implications and Prospective Developments

The findings substantiate that $\mathfrak{Q}$4 gravity, combined with the Bardeen regular black hole exterior, delivers stable, physically realistic charged compact star solutions, directly addressing historical pathological issues (singular cores, unstable equilibrium, violation of energy conditions) in standard GR and other MGTs. These results pave the way for detailed modeling of exotic compact objects (e.g., magnetars or quark stars) within second-order MGTs, and support further exploration into the effects of non-metricity and matter coupling on stellar evolution and gravitational wave phenomenology.

Future directions might include: incorporating rotation, extending to non-spherical geometries, statistical analysis of parameter space ($\mathfrak{Q}$5, $\mathfrak{Q}$6), investigating the impact on neutrino emission and gravitational wave signatures, and systematic comparison with observational constraints from pulsar timing and X-ray measurements.

Conclusion

This paper demonstrates that charged compact star configurations derived from $\mathfrak{Q}$7 gravity with a Bardeen black hole exterior and Finch-Skea interior obey all physical, energy, and stability criteria for a wide range of model parameters. The non-singular nature, equilibrium, ordinary matter composition, and dynamical stability make this framework a promising candidate for realistic modeling of high-density astrophysical objects and broader exploration in modified gravitational theories (2604.03332).