Wormhole geometries in modified teleparralel gravity and the energy conditions

Abstract: In this work, we explore the possibility that static and spherically symmetric traversable wormhole geometries are supported by modified teleparallel gravity or f(T) gravity, where T is the torsion scalar. Considering the field equations with an off-diagonal tetrad, a plethora of asymptotically flat exact solutions are found, that satisfy the weak and the null energy conditions throughout the spacetime. More specifically, considering T=0, we find the general conditions for a wormhole satisfying the energy conditions at the throat and present specific examples. Secondly, considering specific choices for the f(T) form and for the redshift and shape functions, several solutions of wormhole geometries are found that satisfy the energy conditions throughout the spacetime. As in their general relativistic counterparts, these f(T) wormhole geometries present far-reaching physical and cosmological implications, such as being theoretically useful as shortcuts in spacetime and for inducing closed timelike curves, possibly violating causality.

Wormhole Geometries in Modified Teleparallel Gravity and Energy Conditions

The paper by Böhmer, Harko, and Lobo explores the complex topic of wormhole geometries within the framework of modified teleparallel gravity, specifically $f(T)$ gravity. This exploration is significant as it seeks alternative gravitational theories capable of supporting wormhole solutions that comply with known energy conditions.

Key Contributions and Findings

1. Framework of Teleparallel Gravity: The study begins by contextualizing teleparallel gravity, an alternative to General Relativity (GR) that uses the tetrad as a fundamental variable rather than the metric tensor. Teleparallel gravity can be seen as equivalent to GR but described through torsion instead of curvature. This equivalence permits the introduction of the $f(T)$ models where $T$ is the torsion scalar.
2. Wormhole Solutions and Energy Conditions: The paper finds exact solutions for asymptotically flat, static, and spherically symmetric traversable wormholes in $f(T)$ gravity. These solutions satisfy the weak and null energy conditions, which are notoriously violated in GR-based wormhole solutions. This violation typically requires "exotic" matter to sustain a wormhole, an aspect the authors aim to address within the scope of $f(T)$ gravity.
3. Specific Cases Explored:
   • $T \equiv 0$ Solutions: By considering the torsion scalar $T$ to be identically zero, the authors derive general conditions for a wormhole that satisfies energy conditions at the throat. The study discusses shape functions and redshift functions that ensure energy density is non-negative throughout spacetime.
   • Teleparallel Gravity: By setting $f(T) = T$, which corresponds to equivalent teleparallel GR, the authors verify that the standard GR field equations for wormhole physics are retrieved, reinforcing the conceptual consistency of the theory.
   • Modified Gravity Functions: Specific functional forms of $f(T)$, including $f(T) = T + T_0T^2$, are investigated. Within these frameworks, wormhole solutions satisfying energy conditions at and near the throat are discovered, differing significantly from their GR counterparts.
4. Physical and Cosmological Implications: The paper speculates on the implications of these $f(T)$ wormhole geometries. Theoretically, such wormholes could serve as shortcuts through spacetime and potentially induce closed timelike curves, challenging classical notions of causality. While primarily theoretical, the implications broaden the discussion of spacetime structures in modified gravity theories.

Theoretical and Practical Implications

The work expands the theoretical landscape of viable wormhole solutions within modified gravity, specifically demonstrating scenarios where the stress-energy tensor does not require exotic matter. This brings $f(T)$ gravity into broader consideration for researchers exploring singularity-free models and the completion of spacetime topology. Practically, these insights could inform future investigations into the nature of gravity and the potential for observational evidence supporting alternative gravitational theories.

Future Directions

While this paper provides a foundational step toward understanding wormholes in $f(T)$ gravity, several avenues for future research exist. These include:

• Stability Analysis: Investigation into the stability of these wormhole solutions under perturbations or dynamic conditions is essential for assessing their physical plausibility.
• Cosmological Scalars and Observations: Further study involving cosmological parameters like the cosmic microwave background or gravitational wave signatures could offer insights into the detectability of such wormhole solutions.
• Comparative Studies: More comparative analysis with other modified gravity theories, such as $f(R)$ gravity, could illuminate the unique or advantageous properties of $f(T)$ models.

In conclusion, Böhmer, Harko, and Lobo's paper innovatively explores the possibility of traversable wormholes in teleparallel gravity, potentially circumventing the thorny issue of exotic matter. This work contributes a new dimension to gravitational theory, inviting deeper exploration of spacetime's fundamental nature and its compatibility with observed cosmological phenomena.