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

This paper employs the covariant coframe/spin-connection formalism in teleparallel $F(T)$ gravity to reconstruct static, spherically symmetric spacetime solutions sourced by Chaplygin and polytropic fluids, revealing diverse geometric branches ranging from stellar interiors and black holes to traversable wormholes while analyzing their horizon structures, energy conditions, and stability within a unified framework.

The Gist

Imagine the universe as a giant, flexible trampoline. For decades, physicists have used Albert Einstein's theory of General Relativity to describe how heavy objects (like stars) bend this trampoline, creating gravity. In Einstein's view, gravity is the curvature of the fabric.

This paper explores a different way to look at the same trampoline. Instead of bending, imagine the fabric is made of tiny, twisting threads. In this alternative theory, called Teleparallel Gravity, gravity isn't about bending; it's about the twisting (or torsion) of these threads. The author, A. Landry, investigates what happens when we twist these threads in a specific, symmetrical way (like a perfect sphere) and fill the space with two very different types of "cosmic fluids."

Here is a breakdown of the paper's findings using everyday analogies:

1. The Two Types of "Cosmic Fluids"

The paper studies how gravity behaves when the universe is filled with two specific kinds of invisible fluids. Think of these as the "ingredients" inside a cosmic balloon.

• The "Anti-Gravity" Fluid (Chaplygin Fluid):
  Imagine a fluid that acts like a spring that wants to push everything apart. It has "negative pressure." In our everyday world, things usually pull together (like gravity), but this fluid pushes out.
  • What it does: The paper finds that this fluid is perfect for creating wormholes (tunnels through space) or exotic dark energy (the force making the universe expand). It naturally creates the "pushy" conditions needed to keep a wormhole open so you could theoretically travel through it.
• The "Star Stuff" Fluid (Polytropic Fluid):
  Imagine a fluid that behaves like the gas inside a star or a giant pressure cooker. It follows standard rules of compression and heat.
  • What it does: This fluid is great for modeling normal stars, neutron stars, and the dense cores of planets. It represents the "regular" matter we are familiar with.

2. The "Recipe" for New Gravity Models

In standard physics, you usually start with a known object (like a star) and calculate the gravity around it. This paper does the reverse. It's like being a chef who decides, "I want to bake a cake that tastes exactly like a wormhole," and then figuring out what ingredients (gravity laws) are needed to make that happen.

The author developed a reconstruction procedure. This is a mathematical recipe that says:

1. Pick a shape (a sphere).
2. Pick a fluid (Chaplygin or Polytropic).
3. Work backward to discover what the "Twisting Gravity" law ($F(T)$) must look like to make that shape and fluid work together.

3. The Results: Different Shapes of Space

By mixing these fluids with the "twisting" gravity, the paper found several distinct types of cosmic structures:

• The "Black Hole" Look: Some solutions look like black holes, with event horizons where nothing can escape.
• The "Wormhole" Look: The Chaplygin fluid (the pushy one) naturally creates shapes that look like wormholes. Interestingly, the paper suggests that the "exotic" push needed to keep the wormhole open doesn't necessarily have to come from the fluid itself. The twisting of space (torsion) can do some of the heavy lifting, acting like a hidden support beam holding the tunnel open.
• The "Star" Look: The Polytropic fluid (the star stuff) creates models that look like the insides of real stars, with dense cores and smooth surfaces.
• The "Constant Radius" Look: Some solutions describe a strange, tube-like universe where the size of the space doesn't change as you move through it, similar to a very specific type of cosmic cylinder.

4. Why This Matters (According to the Paper)

The paper emphasizes that this work is a unified framework. It's like building a single toolbox that can handle both the "weird, exotic" physics of wormholes and the "boring, normal" physics of stars.

• Consistency: The author ensures that if you turn off the "twisting" effects, the math smoothly turns back into Einstein's standard General Relativity. This proves the new theory is a valid extension, not a contradiction.
• Safety Checks: The paper checks if these new models are stable (won't collapse instantly) and if they obey the laws of physics (like not moving faster than light).
• Classification: The author organizes all these new shapes into a "library" based on their mathematical fingerprints (invariants), ensuring that even if two shapes look similar, they are classified correctly if their internal "twisting" is different.

Summary

In simple terms, this paper is a theoretical construction project. It asks: "If gravity is actually about twisting space rather than bending it, what kinds of stars, black holes, and wormholes can we build if we fill the universe with specific types of fluids?"

The answer is: A lot of interesting ones. The "pushy" fluid builds wormholes and dark energy models, while the "squeezable" fluid builds realistic stars. The paper provides the mathematical blueprints to build these objects within this new "twisting" theory of gravity.

Technical Summary

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

Problem Statement
The paper addresses the construction of exact static, spherically symmetric (SS) solutions in covariant teleparallel $F(T)$ gravity sourced by nonlinear fluid equations of state. While previous works have investigated perfect fluids and scalar fields within this framework, there remains a gap in the systematic treatment of nonlinear matter sectors, specifically Chaplygin and polytropic fluids, using a fully covariant formalism. A key challenge in modified teleparallel gravity is the lack of local Lorentz invariance in non-covariant approaches, which necessitates the use of the covariant coframe/spin-connection (CSC) formalism to distinguish between inertial and gravitational effects. Furthermore, the interplay between nonlinear fluid equations of state, torsion reconstruction procedures, and invariant classification schemes has not been fully explored. The authors aim to determine how these specific nonlinear matter sources influence the reconstruction of admissible $F(T)$ functions and the resulting geometric structures, including potential wormhole and compact-object configurations.

Methodology
The study employs the covariant CSC formalism, treating the tetrad $h^a_\mu$ and the flat spin connection $\omega^a_{b\mu}$ as fundamental geometric structures to ensure local Lorentz covariance.

1. Field Equations: The authors derive the symmetric and antisymmetric field equations for $F(T)$ gravity coupled to anisotropic or isotropic nonlinear fluids. The matter sector is defined by an energy-momentum tensor $\Theta^\mu_\nu$ with density $\rho$ and pressures $p_r$ (radial) and $p_t$ (tangential).
2. Conservation Laws: Instead of imposing source-specific constraints, the matter dynamics are governed by the conservation law $\mathring{\nabla}_\mu \Theta^\mu_\nu = 0$. For isotropic fluids, this yields a master differential equation relating the metric function $A_1(r)$ to the equation of state $p(\rho)$.
3. Equations of State: Two specific nonlinear sectors are analyzed:
   • Chaplygin Fluid: $p = -A/\rho^\alpha$, which naturally generates negative pressure and is relevant for dark-energy-like and exotic matter regimes.
   • Polytropic Fluid: $p = K\rho^\Gamma$, representing standard stellar interiors and compact objects.
4. Reconstruction Procedure: A general reconstruction method is developed for two distinct geometric sectors:
   • Constant-Radius Sector ($A_3 = c_0$): Analogous to Nariai or Bertotti–Robinson geometries, suitable for near-horizon limits.
   • Areal-Radius Sector ($A_3 = r$): The standard framework for stellar, black-hole-like, and wormhole-like configurations.
     By assuming power-law coframe ansätze ($A_1 \propto r^a, A_2 \propto r^b$), the torsion scalar $T(r)$ is expressed in terms of radial coordinates. This allows the field equations to be inverted to solve for the function $F(T)$ given a specific fluid source.
5. Classification and Stability: The resulting solution spaces are organized using the Coley–Landry invariant classification framework, which distinguishes inequivalent spacetimes via torsion invariants (e.g., $T$, $T_{\alpha\mu\nu}T^{\alpha\mu\nu}$). Stability is assessed via scalar-torsion conditions ($F_T > 0, F_{TT} > 0$) and the adiabatic sound speed.

Key Contributions and Results

• Derivation of Field Equations: The paper provides the explicit covariant field equations and conservation laws for $F(T)$ gravity coupled to Chaplygin and polytropic fluids, establishing the mathematical foundation for nonlinear fluid reconstruction.
• Reconstructed $F(T)$ Models: The authors derive general reconstruction equations for $F(T)$ functions associated with power-law coframe ansätze. The solutions yield several classes of reconstructed branches, including:
  • Power-law models ($F(T) \sim T^n$).
  • Logarithmic corrections.
  • Exponential-power hybrids.
  • Shifted power laws.
• Geometric Branches: The analysis identifies distinct solution branches based on the fluid source:
  • Chaplygin Sector: Naturally generates effective dark-energy and exotic-matter regimes. These sectors support traversable wormhole geometries and regular-core configurations. The negative pressure of the Chaplygin fluid can violate the Null Energy Condition (NEC), but the authors demonstrate that in $F(T)$ gravity, the required exoticity at a wormhole throat can be shifted to the effective torsion sector, meaning the physical fluid need not violate the NEC entirely.
  • Polytropic Sector: Provides physically relevant models for stellar interiors and compact objects, satisfying standard energy conditions and associated with regular matter sources.
• Invariant Classification: The solution space is categorized within the Coley–Landry framework. The study highlights that different $F(T)$ functions can generate distinct invariant branches even when the metric ansatz possesses the same symmetry structure. This emphasizes the role of nonlinear torsion corrections in shaping the solution space beyond the metric geometry alone.
• TEGR Consistency: The reconstructed models are shown to continuously recover the Teleparallel Equivalent of General Relativity (TEGR) limit ($F(T) = T + \beta$) when nonlinear torsion contributions vanish, serving as a consistency check.

Significance and Claims
The paper claims to provide a unified covariant framework for constructing and interpreting compact objects, effective cosmological sectors, and regular-core strong-field geometries beyond General Relativity.

• Complementarity: By studying Chaplygin and polytropic fluids together, the work bridges the gap between exotic/dark-energy-like sectors and ordinary self-gravitating matter, demonstrating how different matter sectors influence the reconstruction and classification of admissible $F(T)$ models.
• Wormhole Physics: The work addresses the question of NEC violation in wormhole throats, suggesting that in nonlinear teleparallel gravity, the exotic contribution required to sustain a wormhole can be geometrically induced by the torsion sector rather than being carried solely by the physical matter source.
• Systematic Classification: The study contributes to the broader program of understanding how nonlinear matter sectors interact with nonlinear torsion corrections, offering a systematic method to classify teleparallel geometries that is independent of coordinate choices and frame-dependent constraints.

The authors maintain a modest scope, noting that these results provide a starting point for systematic classification and that future work would require numerical integrations, anisotropic extensions, and more detailed stability analyses. The work is presented as a theoretical extension of existing reconstruction programs to nonlinear fluid equations of state within a covariant setting.