Static Spherically Symmetric Chaplygin and Polytropic Fluid Solutions in Teleparallel $F(T)$ Gravity

Abstract: We investigate static, spherically symmetric (SS) spacetimes in covariant teleparallel $F(T)$ gravity sourced by nonlinear Chaplygin and polytropic fluids. Using the covariant coframe/spin-connection (CSC) formalism, we derive the corresponding field equations and conservation laws governing admissible matter distributions and nonlinear torsion sectors. A general reconstruction procedure is developed, allowing the systematic determination of teleparallel $F(T)$ models for arbitrary coframe ansätze and fluid equations of state. Focusing on power-law configurations, we obtain several classes of reconstructed solution branches, including constant-radius, compact-object-like, black-hole-like (BH-like), and wormhole-like (WH-like) branches. The Chaplygin sector naturally generates effective dark-energy and exotic-matter regimes capable of supporting traversable wormhole geometries, while the polytropic sector provides physically relevant models for stellar interiors and compact objects. We discuss the associated horizon and throat structures, torsion singularities, energy conditions, and stability properties of the reconstructed branches. The resulting geometries are organized within the Coley--Landry invariant classification framework, highlighting the role of nonlinear torsion corrections in shaping the solution space. Overall, this work provides a unified covariant framework for the construction and interpretation of compact objects, effective cosmological sectors, and regular-core strong-field geometries beyond General Relativity.

• The paper introduces static spherically symmetric solutions in covariant teleparallel F(T) gravity by coupling Chaplygin and polytropic fluid models to unify compact, dark-energy, and wormhole scenarios.
• It employs the covariant coframe/spin-connection formalism to derive modified field equations and systematically reconstruct F(T) models under both constant and areal-radius configurations.
• Effective NEC violation emerges through torsion contributions, enabling exotic geometries while ensuring consistency in the TEGR limit with standard General Relativity.

Static Spherically Symmetric Nonlinear Fluid Solutions in Covariant Teleparallel $F(T)$ Gravity

Overview and Context

This work systematically develops static, spherically symmetric (SS) solutions of covariant teleparallel $F(T)$ gravity sourced by nonlinear Chaplygin and polytropic fluids, advancing the covariant coframe/spin-connection (CSC) formalism as the appropriate invariant geometric construction. The analysis establishes a reconstruction program linking matter equations of state with admissible modified-torsion models, thereby unifying the treatment of ordinary compact objects, dark-energy-like sectors, and wormhole-like geometries in a common covariant teleparallel framework.

Teleparallel Gravity Framework and Methodology

The teleparallel approach replaces the Levi-Civita curvature paradigm with one based on the tetrad (coframe) field and a flat spin-connection; the torsion scalar $T$ plays the role of the Lagrangian. In the TEGR limit, $F(T)=T+\beta$ recovers the standard Einstein-Hilbert dynamics with an effective cosmological constant. Promoting $T$ to a general function $F(T)$ leads to modified teleparallel theories, which are known to admit richer solution spaces due to the breakdown of Birkhoff’s theorem and the presence of additional dynamical torsion degrees of freedom.

To address the known local Lorentz invariance pathologies of non-covariant formulations, all field equations are consistently derived using the full coframe and flat spin-connection pair according to the CSC formalism. The analysis proceeds with a diagonal, static, spherically symmetric line element, allowing for both constant and variable areal-radius sectors. The torsion scalar is constructed as an explicit functional of the coframe components.

Fluid Models and Conservation Laws

Two archetypal nonlinear fluid sectors are incorporated:

1. Chaplygin fluid: Equation of state $p=-A/\rho^\alpha$ (with $A>0$, $0\leq\alpha\leq1$) introducing negative-pressure regimes. These sectors have a long tradition in effective dark-energy and exotic-matter modeling, as they permit both NEC violation and transitions between dust-like and cosmological constant-like limits.
2. Polytropic fluid: Equation of state $p=K\rho^\Gamma$ with polytropic exponent $F(T)$0, a canonical choice for realistic modeling of compact stars and relativistic self-gravitating fluids.

In both cases, the nonlinear matter sectors are treated via the general conservation law consistent with the selected equation of state, yielding first-order ODEs for the radial density profile. The interplay of the fluid equation of state and the CSC formalism replaces the standard Tolman-Oppenheimer-Volkoff strategy in GR.

Reconstruction Procedure

Given the non-uniqueness of modified teleparallel gravity models for a specified matter sector, a systematic reconstruction technique is introduced for both constant-radius and areal-radius scenarios:

• Constant-radius sector ($F(T)$1): The torsion scalar $F(T)$2 is a two-term function of the coframe parameters. The field equations in this branch reduce to a second-order ODE for $F(T)$3, enabling reconstruction of explicit models for a given nonlinear equation of state.
• Areal-radius sector ($F(T)$4): The more physically relevant case for stellar and black-hole-like geometries. Here, $F(T)$5 acquires a multi-scale structure with up to three power-law terms dependent on the coframe powers. This leads to reconstructed models of the general form $F(T)$6, encompassing power-law, logarithmic, and exponential corrections.

The conservation law and field equations jointly constrain the admissible density and pressure profiles, the functional form of $F(T)$7, and the geometry.

NEC Violation, Wormhole Sector, and Torsion-exoticity Mechanism

A central investigation concerns the satisfaction or violation of the null energy condition (NEC). In the Chaplygin fluid branch, natural NEC violation arises in low-density regimes, favoring the existence of traversable wormhole-like geometries. A vital result is that, due to nonlinear torsion corrections, the effective (geometric plus matter) NEC can be violated even if the physical fluid satisfies the NEC, allowing the would-be exoticity to be shifted into the torsion sector. This phenomenon is made explicit in the split $F(T)$8, which generalizes the options for wormhole support beyond what is available in GR or non-covariant $F(T)$9 gravity.

Torsion Invariant Classification

All reconstructed solution branches are systematically classified using the Coley–Landry invariant program, which employs torsion scalar invariants (e.g., $T$0, $T$1, and their derivatives) to distinguish inequivalent geometric sectors even when the metric symmetry is unchanged. The classification elucidates the role of nonlinear torsion corrections in constructing regular cores, horizons, and throat structures associated with compact and wormhole-like objects.

Stability and Dynamical Consistency

Scalar-torsion stability is studied via the effective mass $T$2. Absence of tachyonic and gradient instabilities at the linearized level requires $T$3 and $T$4. These conditions are mapped as constraints on the reconstructed $T$5 models and are to be imposed in tandem with the requirement of causal and positive-definite sound speeds in the matter sector.

TEGR Limit and Consistency

The formalism is constructed to recover the Teleparallel Equivalent of General Relativity in the $T$6 limit. TEGR compatibility provides a critical check on the reconstruction scheme: all new branches continuously interpolate with standard GR solution classes for the relevant fluid configurations.

Implications and Future Directions

The results expand the catalog of admissible static SS solutions in teleparallel $T$7 gravity, rigorously incorporating physically and phenomenologically relevant nonlinear fluids. The approach enables:

• Unification of ordinary, dark-energy-like, and exotic matter-supported configurations within one covariant scheme.
• Characterization of wormhole and regular-core geometries supported by Chaplygin fluids, with NEC violation or “effective exoticity” realized via the torsion sector.
• Modeling of compact stars and relativistic objects with polytropic fluids, and tracking deviations from GR as nonlinear torsion corrections.

These findings have direct implications for the structure and viability of modified gravity models, the geometric realization of NEC violation, and the phenomenology of compact astrophysical and cosmological objects.

Conclusion

This work provides an authoritative and rigorous construction and classification of static spherically symmetric Chaplygin and polytropic fluid solutions in covariant teleparallel $T$8 gravity. The methodology exemplifies how nonlinear fluid equations of state interact with geometric torsion corrections, broadening both the physical solution space and the mathematical foundation of teleparallel gravity. Key avenues for further research include numerical treatment of the nonlinear ODEs, investigation of anisotropic and dynamical systems, generalization to more complex torsion-based theories, and detailed stability analyses.

For a comprehensive resource, see "Static Spherically Symmetric Chaplygin and Polytropic Fluid Solutions in Teleparallel $T$9 Gravity" (2606.10100).