Teleparallel $F(T)$ electromagnetic static spherically symmetric spacetime solutions

This paper investigates static, spherically symmetric spacetimes in covariant teleparallel $F(T)$ gravity with electromagnetic sources, deriving field equations and conservation laws to establish a general reconstruction procedure that yields exact charged solutions—including black-hole-like and wormhole-like branches—which generalize Reissner–Nordström spacetimes and offer new insights into strong-field physics beyond General Relativity.

The Gist

Imagine gravity not as a smooth, curved sheet (like the classic image of a bowling ball on a trampoline), but as a twisting, turning force called torsion. This paper explores a specific version of gravity theory called Teleparallel $F(T)$ Gravity, where this "twist" is the main character, rather than the curvature we are used to in Einstein's General Relativity.

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

1. The New Rulebook: The "CSC" Pair

In the past, scientists trying to use this "twisting" gravity theory ran into a problem: the rules changed depending on how you looked at them (like a magic trick that only works from one angle). This paper uses a new, more robust rulebook called the Coframe/Spin-Connection (CSC) pair.

• The Analogy: Think of the "Coframe" as the map you draw, and the "Spin-Connection" as the compass that tells you which way is "straight" without getting confused by the map's distortion. By using both together, the authors ensure their math works no matter how you rotate or move your viewpoint. This prevents "fake" solutions that only exist because of a bad choice of map.

2. The Players: Gravity and Electricity

The authors are studying what happens when you have a heavy, round object (like a star or black hole) that also has an electric charge. They are mixing the "twisting" gravity with Maxwell's equations (the rules for electricity and magnetism).

• The Constraint: In this twisting gravity, you can't just have any old electric field. The "twist" of space acts like a strict bouncer at a club. It only lets in radial electric or magnetic fields (fields pointing straight out from the center, like spokes on a wheel). It kicks out any "sideways" or "transverse" fields.
• The Result: The electric charge behaves somewhat like it does in standard physics (getting weaker as you move away), but the gravity around it gets weird and modified by the "twist."

3. The Three Types of Cosmic Objects

The paper finds three main types of solutions (shapes of space) that can exist with this twisted gravity and electric charge:

A. The "Constant Radius" Zone (The Nariai/Bertotti-Robinson Branch)

• The Analogy: Imagine a cylinder that goes on forever in both directions, or a box where the size of the "room" doesn't change as you move around.
• What happens: Here, the "twist" of space is constant. It acts like a background vacuum energy (similar to a cosmological constant). The electric field is also constant everywhere. This isn't a black hole; it's more like a special, uniform state of the universe.

B. The "Black Hole-Like" Zone (The $A_3 = r$ Branch)

• The Analogy: This is the familiar black hole, but with a twist. Imagine a funnel that gets narrower and narrower until it hits a point.
• The Twist: In standard physics, these funnels always end in a sharp, infinite point (a singularity) where the math breaks. In this paper, the authors show that by changing the "twist" rules (using different mathematical functions for $F(T)$), you can:
  • Keep the sharp point: Just like a normal black hole.
  • Smooth it out: The "twist" can act like a cushion, making the center of the black hole finite and smooth, avoiding the infinite breakdown.
  • Change the horizon: The "event horizon" (the point of no return) can shift, appear, or disappear depending on how strong the "twist" is.

C. The "Wormhole-Like" Zone

• The Analogy: Instead of a funnel that ends in a point, imagine a tunnel that goes through a mountain and comes out the other side. The narrowest part is the "throat."
• The Twist: In standard physics, building a wormhole requires "exotic matter" (stuff with negative energy) to hold the throat open. Here, the authors suggest that the twist of space itself can do the heavy lifting. The "torsion" acts as the glue holding the tunnel open, potentially allowing a wormhole without needing weird, unphysical matter.
• Caveat: The paper is careful to say these are possible local solutions. It doesn't guarantee they are stable or that you could actually travel through them, but it shows the math allows for them.

4. The "Reconstruction" Tool

One of the paper's main tools is a "reconstruction" method.

• The Analogy: Imagine you see a shadow on the wall (the shape of space and the electric field). The authors work backward to figure out what object cast that shadow.
• How it works: They start with a guess for how the space looks (the "ansatz"), calculate the "twist," and then ask: "What specific rule for gravity ($F(T)$) would create exactly this twist?" This allows them to build a library of different gravity theories that produce specific, interesting shapes of space.

5. Stability: Is it Safe?

Just because a shape exists mathematically doesn't mean it's stable.

• The Analogy: Think of a pencil balanced on its tip. It's a valid position, but the slightest breeze knocks it over.
• The Finding: The authors check if these solutions are "ghost-free" (no weird negative energy) and "tachyon-free" (no runaway instability). They find that some of the "smoothed out" black holes and wormholes are stable, while others are prone to collapsing or exploding. The stability depends heavily on the specific "twist" parameters chosen.

Summary

This paper is a blueprint for building new types of cosmic objects using a "twisting" version of gravity. It shows that:

1. Electricity is picky: It only plays nicely with radial fields in this theory.
2. Black holes can be fixed: The "twist" can potentially smooth out the infinite center of a black hole.
3. Wormholes are possible: The "twist" of space might hold a wormhole open without needing exotic matter.
4. Not all shapes are safe: Only specific combinations of "twist" and charge create stable, physical objects.

The authors provide a unified way to classify these shapes using "invariants" (mathematical fingerprints that don't change no matter how you look at them), ensuring that the solutions they find are real physical possibilities, not just mathematical artifacts.

Technical Summary

Technical Summary: Teleparallel F(T) Electromagnetic Static Spherically Symmetric Spacetime Solutions

Problem Statement
The paper addresses the construction and classification of static, spherically symmetric (SS) spacetime solutions in covariant teleparallel $F(T)$ gravity coupled to electromagnetic sources. While teleparallel gravity offers an alternative geometric description of gravitation where torsion, rather than curvature, encodes the interaction, early formulations suffered from a lack of local Lorentz invariance. This issue was resolved through the covariant coframe/spin-connection (CSC) formalism, which separates inertial and gravitational contributions. However, a systematic and covariant treatment of Einstein–Maxwell systems within this framework remains incomplete. Specifically, the interplay between torsion invariants, Maxwell fields, and admissible nonlinear $F(T)$ sectors is not fully understood. The presence of electromagnetic sources introduces constraints via antisymmetric field equations that may artificially restrict solution spaces in non-covariant formulations. The paper aims to develop a unified, covariant analysis of SS solutions with electromagnetic sources, bridging formal developments with physically relevant models like charged black holes and wormholes.

Methodology
The investigation is grounded in the covariant CSC pair formalism, utilizing a tetrad (coframe) $h^a_\mu$ and a flat spin-connection $\omega^a_{b\mu}$ to ensure local Lorentz invariance. The action includes the $F(T)$ gravitational term and the standard Maxwell term.

1. Field Equations: The authors derive the covariant Einstein–Maxwell field equations by varying the action. These equations are decomposed into symmetric (dynamical) and antisymmetric (constraint) sectors. The antisymmetric sector imposes strict conditions on the admissible CSC pairs and electromagnetic configurations.
2. Ansätze: Two primary geometric sectors are analyzed:
   • Constant-Radius Sector ($A_3 = c_0$): Analogous to generalized Nariai or Bertotti–Robinson geometries.
   • Areal-Radius Sector ($A_3 = r$): Corresponding to standard static SS geometries, including black hole (BH) and wormhole (WH) configurations.
3. Reconstruction Procedure: A systematic reconstruction method is employed. By assuming power-law (PL) ansätze for the coframe functions ($A_1(r) \sim r^a, A_2(r) \sim r^b$), the reduced field equations are transformed into ordinary differential equations for the unknown function $F(T)$. This allows for the determination of admissible nonlinear $F(T)$ models compatible with specific symmetry structures.
4. Invariant Classification: The Coley–Landry invariant classification program is applied. Solutions are categorized using scalar torsion invariants ($T, \nabla T, \dots$) to distinguish inequivalent geometries independent of coordinate or gauge choices. Classes include TEGR, Power-Law (PL), Logarithmic (LOG), Exponential (EXP), and Composite (COMP) models.
5. Analysis: The study examines horizon structures, torsion singularities, energy conditions (specifically the Null Energy Condition, NEC), and stability properties (via the effective mass parameter $m^2_{eff} \sim F_T / F_{TT}$).

Key Contributions

• Covariant Field Equations: Derivation of the full covariant Einstein–Maxwell field equations for static SS teleparallel geometries, explicitly separating symmetric and antisymmetric sectors.
• Conservation Laws: Identification of explicit conservation law solutions for radial electric, magnetic, and mixed electromagnetic sectors. The analysis confirms that in the covariant SS sector, transverse electromagnetic modes are strongly constrained or eliminated, favoring Coulomb-like radial configurations.
• Reconstruction Framework: Establishment of a closed-form reconstruction procedure that determines admissible nonlinear $F(T)$ models for arbitrary coframe ansätze. This yields explicit classes of solutions (PL, LOG, EXP, COMP) derived from power-law coframe configurations.
• Invariant Classification Extension: Extension of the Coley–Landry invariant classification to electromagnetic sectors, providing a coordinate-independent method to distinguish teleparallel branches.
• Solution Branches:
  • Constant-Radius ($A_3=c_0$): Identification of vacuum branches behaving as effective cosmological sectors and charged branches with constant electromagnetic density.
  • Areal-Radius ($A_3=r$): Construction of BH-like solutions that generalize the Reissner–Nordström (RN) spacetime, revealing modified horizon structures and near-core behaviors.
  • Wormhole-Like (WH-like): Demonstration that nonlinear torsion contributions can effectively support WH-like geometries by shifting the NEC-violating requirement from the physical matter sector to the effective torsion sector.

Results

• Electromagnetic Constraints: The antisymmetric field equations impose severe restrictions on admissible CSC pairs. In the vacuum sector, only radial electric and magnetic fields are physically admissible; transverse modes are excluded unless symmetry-breaking sectors are introduced.
• Horizon and Singularity Structure:
  • In the $A_3=r$ sector, the parameter $b$ (from the coframe ansatz) governs the ultraviolet behavior of torsion invariants.
  • $b > -1$: Leads to divergent torsion invariants and central singularities (BH-like).
  • $b = -1$: Corresponds to a critical, softer singular branch.
  • $b < -1$: May regularize the torsion sector, generating regular BH-like cores or WH-like geometries.
  • The parameter $a$ controls the redshift structure and horizon formation.
• Stability: Stability is analyzed via the ratio $F_T / F_{TT}$. Stable configurations require $F_T > 0$ and $F_{TT} > 0$ to avoid ghost and tachyonic instabilities. PL models with $n > 1$ (where $n = 2a/(b+1)$) are conditionally stable, while EXP models are noted to be more sensitive to perturbations.
• Wormhole Viability: WH-like solutions require a delicate balance. While torsion can mimic exotic matter to satisfy the flaring-out condition, full viability demands finite torsion invariants at the throat, compatibility with antisymmetric equations, and dynamical stability of the throat radius.
• Asymptotics: The solutions include branches that asymptotically approach RN–AdS geometries modified by nonlinear torsion corrections, acting as effective radial-dependent deformations of the cosmological term.

Significance and Claims
The paper claims to provide a unified and covariant approach to constructing and interpreting physically relevant compact objects, effective cosmological sectors, and regularized strong-field sectors in nonlinear teleparallel gravity.

• It resolves the issue of spurious constraints in non-covariant formulations by strictly adhering to the CSC formalism.
• It demonstrates that nonlinear teleparallel gravity enlarges the space of admissible charged solutions compared to General Relativity (GR), allowing for multiple inequivalent branches for the same symmetry class (violating Birkhoff's theorem in the strict GR sense).
• The work highlights the role of torsion in shaping the solution space, showing that torsion corrections can deform horizon structures, regularize singularities, and support wormhole geometries without necessarily requiring exotic matter in the physical sector.
• The authors suggest that these reconstructed models may lead to observable deviations from GR, such as modifications to BH shadows, gravitational lensing, and quasi-normal mode spectra, offering potential avenues for strong-field tests beyond GR.
• The study emphasizes that while torsion can regularize geometries, this is not automatic but depends on specific regions of the parameter space where torsion invariants remain finite.

The paper concludes that the interplay between torsion dynamics, electromagnetic contributions, and symmetry constraints creates a rich landscape of teleparallel geometries, providing a framework to investigate modified compact objects and effective cosmological sectors within $F(T)$ gravity.