Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach (1810.12932v2)

Abstract: Teleparallel gravity and its popular generalization $f(T)$ gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure. Various modified teleparallel gravity models are also briefly reviewed.

• The paper introduces a fully invariant formulation of teleparallel gravity by incorporating an inertial spin connection to address Lorentz invariance misconceptions.
• It demonstrates that reinterpretation of gravity through torsion yields predictions equivalent to General Relativity under the TEGR framework.
• The study extends the model to f(T) gravity, showcasing second-order field equations and resolving lingering invariance issues in modified gravity theories.

Teleparallel Theories of Gravity: A Fully Invariant Framework

The paper "Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach" provides a comprehensive exploration of teleparallel gravity and its extensions, emphasizing the critical insight that these theories can be formulated in a manner that is fully invariant under both coordinate transformations and local Lorentz transformations. This approach addresses several misconceptions present in the literature, particularly regarding the local Lorentz invariance of these theories.

The core of teleparallel gravity is the reinterpretation of gravity not as a result of spacetime curvature, as in General Relativity (GR), but as an effect of torsion. In this framework, the gravitational interaction is mediated by a zero curvature, Lorentz connection with non-zero torsion, aptly distinguishing inertial effects from gravitational ones by introducing an "inertial spin connection."

Key Insights and Implications

1. Inertial Spin Connection: A central point of contention in teleparallel gravity is its treatment of the spin connection. Unlike in GR, where the Levi-Civita connection incorporates both inertial and gravitational effects, in teleparallel gravity, the inertial effects are described by a separate spin connection. This form of the connection clarifies the so-called differentiation between good and bad tetrads, showing that any tetrad is essentially valid.
2. Fully Invariant Formulation: By carefully considering the inertial spin connection, the authors construct a version of teleparallel gravity that respects local Lorentz invariance fully. This reconciles past confusions where the lack of apparent Lorentz invariance was mistakenly attributed to the theory's inherent limitations. The paper makes clear that the earlier misunderstandings arose primarily from the neglect of the inertial spin connection.
3. Teleparallel Equivalent to General Relativity (TEGR): The fully invariant formulation leads to a teleparallel model equivalent, in terms of physical predictions, to GR, when specific conditions on the torsion scalar are met, namely the Teleparallel Equivalent of General Relativity (TEGR).
4. Extensions to $f(T)$ Gravity: The paper extends the teleparallel concept to $f(T)$ gravity, where the Lagrangian is an arbitrary function of the torsion scalar $T$. Such models remain second-order in their field equations, distinguishing them from $f(R)$ models that typically generate field equations higher than second-order.
5. Addressing Misconceptions: The invariant approach dispels the myths around the lack of Lorentz invariance in $f(T)$ gravity, demonstrating that proper treatment of non-trivial spin connections yields an invariant theory.
6. Numerical Results and Examples: The paper reviews various teleparallel gravity models and provides examples, such as the spherically symmetric solutions, that illustrate the covariant procedure and the implications of distinct teleparallel models.

Broader Implications and Future Directions

The establishment of a fully invariant teleparallel theory has potential implications both theoretically and practically. It enables a reinterpretation of gravitational phenomena without reliance on spacetime curvature, hence offering alternative insights into dark energy and dark matter. Additionally, by aligning closely with gauge theories, teleparallel models propose a potentially fruitful route toward quantum gravity.

Future developments could explore the implications of teleparallel frameworks in quantum gravity settings, investigations into black hole thermodynamics, or even in the context of cosmological models beyond the standard paradigm. Understanding these models' degrees of freedom and addressing the nature of their solutions in less symmetric cases remains an open field of research.

This paper, therefore, plays a crucial role in clarifying foundational aspects of teleparallel gravity and sets the stage for deeper investigations into its potential as a comprehensive gravitational theory.