The covariant formulation of f(T) gravity (1510.08432v2)

Abstract: We show that the well-known problem of frame dependence and violation of local Lorentz invariance in the usual formulation of $f(T)$ gravity is a consequence of neglecting the role of spin connection. We re-formulate $f(T)$ gravity starting, instead of the "pure-tetrad" teleparallel gravity, from the covariant teleparallel gravity, using both the tetrad and the spin connection as dynamical variables, resulting in the fully covariant, consistent, and frame-independent, version of $f(T)$ gravity, which does not suffer from the notorious problems of the usual, pure-tetrad, $f(T)$ theory. We present the method to extract solutions for the most physically important cases, such as the Minkowski, the FRW and the spherically-symmetric ones. We show that in the covariant $f(T)$ gravity we are allowed to use an arbitrary tetrad in an arbitrary coordinate system along with the corresponding spin connection, resulting always to the same physically relevant field equations.

• The paper introduces a covariant reformulation of f(T) gravity by including both the tetrad and spin connection to restore Lorentz invariance.
• The authors derive explicit field equations that remain consistent across frames, overcoming the limitations of the pure-tetrad approach.
• Computations in Minkowski, FRW, and spherically symmetric spacetimes demonstrate the model's capability to explain cosmic acceleration without coordinate artifacts.

The Covariant Formulation of f(T) Gravity

The paper "The covariant formulation of f(T) gravity" by Martin Krssak and Emmanuel N. Saridakis tackles the long-standing issue of local Lorentz invariance within the context of modified teleparallel gravity theories. Extending the teleparallel equivalent of General Relativity (TEGR), the f(T) gravity seeks to introduce non-linear modifications in the form of a function of the torsion scalar $T$, analogous to the transition from General Relativity to f(R) gravity. However, the naive implementation of f(T) gravity compromises Lorentz invariance, a critical concept in modern theoretical physics.

Key Contributions

1. Covariant Reformulation: The authors propose a covariant reformulation of f(T) gravity by incorporating both the tetrad and the spin connection as dynamic variables. This approach resolves the issues of frame-dependence and the violation of Lorentz invariance that plague the traditional pure-tetrad formulation. By retaining the spin connection, the reformed theory remains locally Lorentz invariant without renouncing the second-order field equations that are the hallmark advantage of f(T) over f(R) gravity.
2. Derivation and Consistency: Theoretical developments within the paper extend beyond a conceptual overhaul. Explicit field equations for the new covariant formulation are derived, showcasing how these equations maintain consistency across different frames. This is a clear advantage over the earlier formulations, where solutions were incorrectly perceived to depend on the chosen coordinate frame.
3. Practical Illustrations: The paper includes computations for notable cases, such as the Minkowski spacetime, the Friedmann-Robertson-Walker (FRW) universe, and spherically symmetric spacetimes. For instance, within the FRW universe context, the derived Friedmann equations justify the ability of f(T) theories to account for the accelerated expansion of the universe without generating artifacts from coordinate system choices.

Implications

The covariant formulation has significant implications in the theoretical exploration of gravitational modifications. By restoring Lorentz invariance, the theory aligns with the fundamental symmetries respected across physics, providing a firmer foundation for exploring ramifications such as the ultraviolet behavior of gravity and possible alternatives to cosmological acceleration. Additionally, this work enhances the viability of f(T) theories as contenders or complements to f(R) theories, offering a distinctive avenue that engages with the torsional rather than the curvature paradigm of spacetime geometry.

Future Directions

The covariant reformulation invites further research focused on:

• Cosmological Applications: Investigating complex cosmological phenomena and probing the role of f(T) modifications in early universe models, dark energy, and inflation.
• Theoretical Extensions: Exploring extensions or synergistic theories such as hybrid models that incorporate both curvature and torsion approaches, possibly enhancing our understanding of quantum gravity.
• Numerical Simulations: Implementing robust computational models to simulate scenarios under this bifactorial approach (tetrad and spin connection), especially where analytic solutions remain elusive.

This work reopens foundational discussions regarding formulation choices in gravitational theory modifications by promising consistency without sacrificing foundational symmetries. With a covariant approach, f(T) gravity may present more than a mathematical curiosity but a thematic resonance with physics' ongoing narrative to describe nature's laws comprehensively.