Traversable Wormhole Solutions in massive $F(T)$ gravity

This paper presents exact, horizonless traversable wormhole solutions in massive $F(T)$ gravity by combining a perturbative dRGT graviton-mass term with various redshift profiles, demonstrating that the additional anisotropic pressure from the mass term can sustain the wormhole throat while satisfying or minimally violating energy conditions.

The Gist

Imagine the universe as a giant, stretchy fabric. For decades, physicists have used Albert Einstein's rules (General Relativity) to describe how this fabric bends and twists around heavy objects like stars and black holes. But recently, scientists have started asking: "What if the fabric doesn't just bend, but also has a hidden 'twist' to it?" And, "What if the tiny particles that carry gravity (gravitons) aren't weightless, but have a tiny bit of mass?"

This paper explores a wild idea: Can we build a "wormhole" (a shortcut through space) using these new rules?

Here is a simple breakdown of what the authors did, using everyday analogies.

1. The New Rules of the Game

The authors are working in a modified version of gravity called $F(T)$ gravity.

• The Old Way (Einstein): Gravity is like a heavy bowling ball sitting on a trampoline, causing the fabric to curve.
• The New Way ($F(T)$): Gravity is like a twisted rubber band. Instead of just curving, the fabric has a "torsion" or a twist to it.
• The Extra Ingredient: They added a "massive" component. Think of the graviton (the messenger of gravity) not as a ghost that can fly forever, but as a tiny, heavy ball. This "mass" changes how gravity behaves over long distances.

2. The Goal: A Traversable Wormhole

A wormhole is like a tunnel connecting two distant points in the universe.

• The Problem: In normal physics, these tunnels are unstable. They collapse instantly unless you hold them open with "exotic matter"—a weird, imaginary substance that pushes outward instead of pulling inward (like negative gravity).
• The Question: Can the new "twisted" gravity and the "heavy" graviton do the job of holding the tunnel open, so we don't need that weird exotic matter?

3. The Experiment: Building the Tunnel

The authors tried to build a mathematical model of a wormhole using three different "shapes" for the tunnel's entrance (called the redshift function):

1. Constant: The entrance is a steady, flat tunnel.
2. Logarithmic: The entrance gets wider or narrower in a specific, slow curve.
3. Power-Law: The entrance changes size rapidly, like an exponential curve.

They also tested two different ways the "heavy graviton" behaves:

• General Case: The mass behaves in a complex, standard way.
• Uniform Pressure: The mass pushes out equally in all directions, like a balloon inflating.

4. The Results: It Works!

The authors found that yes, it is possible.

• The "Twist" Holds the Door: The combination of the "twisted" space (torsion) and the "heavy" graviton creates an internal pressure that acts like a structural beam. It holds the throat of the wormhole open.
• No Magic Needed: Crucially, they found that this setup does not require exotic matter. The "twist" and the "mass" of gravity itself provide the necessary force to keep the tunnel from collapsing.
• Safe Passage: The wormholes they found are "traversable," meaning they have no event horizons (the point of no return in a black hole). You could theoretically fly through them without getting stuck or crushed.
• Smooth Landing: As you move away from the wormhole, the strange effects fade away, and the universe looks normal again (asymptotically flat).

5. The "Flare-Out" Condition

To keep a wormhole open, the tunnel must "flare out" at the narrowest point (the throat), like the mouth of a trumpet.

• The authors proved that their new gravity rules naturally create this flare-out shape.
• They checked the "Energy Conditions" (a set of rules that say matter usually behaves nicely, like having positive energy). They found that their wormholes mostly obey these rules, or only break them very slightly and in a controlled way. This makes the solution much more "realistic" than previous theories that required impossible physics.

6. The "What If" Check

The authors also checked what happens if the graviton has zero mass (going back to the old, standard rules).

• The Result: Their new, complex wormhole solutions smoothly turn into the standard wormholes we already know from simpler theories. This proves their math is consistent and doesn't break when you remove the new ingredients.

Summary

Think of this paper as an architect designing a bridge.

• Old Architects said: "We need a magical, invisible material to hold this bridge up."
• These Authors said: "What if we change the laws of physics slightly? What if the ground itself has a twist, and the wind has a little weight?"
• The Conclusion: They showed that with these slight changes, the ground and wind naturally push up against the bridge, holding it steady without needing any magic materials. They built several different blueprints (using different shapes and pressures) and proved they all work mathematically.

This work suggests that if the universe does have these specific "twisted" and "massive" gravity properties, wormholes could exist naturally, held open by the very fabric of spacetime itself.

Technical Summary

Technical Summary: Traversable Wormhole Solutions in Massive F(T) Gravity

Problem Statement
The study of modified gravity theories aims to address the dynamics of the Universe at early and late times, as well as theoretical issues in the ultraviolet regime of General Relativity (GR). While teleparallel gravity (specifically $F(T)$ gravity) attributes gravitation to torsion rather than curvature, and massive gravity theories (specifically the de Rham–Gabadadze–Tolley or dRGT framework) introduce a finite graviton mass to generate self-accelerating solutions, the intersection of these two frameworks remains largely unexplored. Specifically, the potential of combining $F(T)$ gravity with a weak dRGT massive term to generate physically acceptable traversable wormhole solutions has not been fully examined. Standard wormhole solutions often require "exotic matter" that violates energy conditions; this work investigates whether the interplay between torsion and graviton mass can sustain wormhole geometries without such explicit violations.

Methodology
The authors adopt a static, spherically symmetric Morris–Thorne ansatz within a massive $F(T)$ teleparallel framework. The methodology involves:

1. Formalism: Utilizing orthonormal frames (non-proper frames) to ensure well-defined degrees of freedom for the field equations (FEs), solving the antisymmetric parts of the FEs for spin-connection components before addressing the symmetric parts.
2. Action and Equations: The action includes the standard $F(T)$ term, a matter Lagrangian, and a perturbative dRGT massive Lagrangian ($L_{mass}$). The field equations are derived by varying the action, decomposing the effective energy-momentum tensor into cosmological fluid and massive graviton contributions.
3. Redshift Profiles: Three specific redshift functions ($\Phi(r)$) are analyzed to reconstruct exact solutions:
   • Constant ($\Phi(r) = \Phi_0$)
   • Logarithmic ($\Phi(r) = \Phi_0 + a \ln r$)
   • Power-law ($\Phi(r) = \Phi_0 + a_1 r^a$)
4. Massive Sector Realizations: Two cases for the massive sector are considered:
   • The general case with specific diagonal forms for the matrix $K_{ab}$.
   • A special case with uniform pressure ($P_r^{mass} = P_t^{mass}$).
5. Solution Reconstruction: For each redshift profile and massive sector realization, the authors derive the shape function $b(r)$ and the corresponding $F(T)$ model, ensuring the solutions satisfy the flaring-out condition and are asymptotically flat.

Key Contributions and Results

• Exact Solutions: The paper constructs exact, horizonless traversable wormhole solutions for massive $F(T)$ gravity. These solutions are derived for both the general massive gravity case and the uniform-pressure specialization across the three redshift profiles.
• Energy Conditions: The analysis demonstrates that the effective matter sector can either respect standard energy conditions (Weak, Strong, Null, Dominant) or only mildly violate them within controlled parameter ranges. The authors note that while the massive dRGT term provides an additional anisotropic pressure that helps sustain the throat, certain specific forms of the energy conditions (specifically Eq. 31 in the text) may be questionable regarding dark energy existence in specific limits.
• Role of the Massive Term: A central finding is that the dRGT massive term supplies an anisotropic pressure capable of sustaining the wormhole throat without invoking explicitly exotic matter. In the limit where the graviton mass vanishes ($m \to 0$), the configurations smoothly reduce to standard $F(T)$ wormhole solutions, confirming the internal consistency of the framework.
• Torsion-Mass Coupling: The study highlights how the coupling between torsion and the massive term modifies the asymptotic behavior of the solutions. The massive source function $K_3(T)$ is shown to directly influence the form of the $F(T)$ solutions, creating distinct families of wormhole geometries depending on the redshift profile and the relative magnitude of the dRGT parameters ($c_1$ and $c_2$).

Significance and Claims
The paper claims that massive teleparallel gravity admits self-consistent wormhole solutions that remain asymptotically flat and do not require the introduction of additional exotic sources beyond the standard cosmological fluid and the massive graviton term. The work establishes a theoretical bridge between torsion-based modifications of gravity and massive gravity, showing that their combination produces strong-field effects and solution branches absent in either framework alone. The authors conclude that the framework is internally consistent, as the vanishing-mass limit recovers known $F(T)$ results, and the massive contributions act as effective anisotropic sources that can mimic familiar matter at large radii while preserving the essential throat geometry. The study serves as a foundational step for exploring wormhole geometries in theories where graviton mass and torsion are simultaneously non-negligible.