Modified teleparallel theories of gravity (1508.05120v2)

Abstract: We investigate modified theories of gravity in the context of teleparallel geometries. It is well known that modified gravity models based on the torsion scalar are not invariant under local Lorentz transformations while modifications based on the Ricci scalar are. This motivates the study of a model depending on the torsion scalar and the divergence of the torsion vector. We derive the teleparallel equivalent of $f(R)$ gravity as a particular subset of these models and also show that this is the unique theory in this class that is invariant under local Lorentz transformation. Furthermore one can show that $f(T)$ gravity is the unique theory admitting second order field equations.

• The paper demonstrates that f(T) gravity maintains second-order field equations but compromises local Lorentz invariance.
• It shows that combining the torsion scalar with a boundary term recovers f(R) gravity while preserving local Lorentz symmetry.
• The study generalizes teleparallel models, offering insights to reconcile gravitational theory with dark matter and dark energy observations.

Teleparallel Gravity: An Examination of Modified Theories

The paper "Modified Teleparallel Theories of Gravity" by Sebastian Bahamonde, Christian G. Böhmer, and Matthew Wright offers an in-depth analysis of teleparallel geometries to modify general relativity (GR). The core exploration is dedicated to understanding how modifications based on the torsion scalar and the divergence of the torsion vector can yield theories with distinct properties concerning local Lorentz invariance and the order of field equations.

Teleparallel Gravity, also known as the Teleparallel Equivalent of General Relativity (TEGR), reformulates GR in a spacetime devoid of curvature; instead, torsion describes gravitational interactions. In this framework, the gravitational fields are represented by the torsion tensor instead of the metric-induced curvature. This reformulation maintains equivalence with GR quantitatively but offers a distinct qualitative perspective, using the Weitzenböck connection instead of the Levi-Civita connection.

The significance of this investigation lies in addressing known issues within GR, particularly the phenomena of dark matter and dark energy, which are not directly elucidated by the classical GR framework. By analyzing $f(T)$ and $f(R)$ gravity models, where $T$ and $R$ are the torsion scalar and the Ricci scalar respectively, the paper seeks to uncover formulation nuances that provide second-order field equations while discussing the implications of Lorentz invariance.

Key Findings and Methodologies

The paper expounds the conditions under which certain gravity models can maintain desirable mathematical properties:

1. $f(T)$ Gravity: This theory retains second-order field equations, a notable advantage in mathematical tractability over $f(R)$ models, which inherently introduce fourth-order derivatives. However, the formulation is not invariant under local Lorentz transformations due to the dependence of the torsion scalar $T$ on the frame choice.
2. Teleparallel Equivalent of $f(R)$ Gravity: A modified approach that involves the combination of the torsion scalar and a boundary term, translating to a total derivative, thus recovering $f(R)$ gravity's standard formulation, which inherently respects local Lorentz invariance. This equivalence emphasizes the necessity of higher-order derivatives in maintaining such invariances.
3. General $f(T, B)$ Gravity: By introducing a boundary term $B$, the paper synthesizes a more generalized action $f(T, B)$, capturing both $f(T)$ and $f(R)$ theories as specific cases. It shows that only specific formulations can fulfill second-order equations or Lorentz invariance exclusively when formulated correctly.

Practical and Theoretical Implications

This exploration into modified teleparallel theories of gravity opens pathways to potentially reconcile GR with cosmological observations that necessitate dark components. The reformulation offers theoretical insights that challenge and refine our understanding of the gravitational and inertial interplay wielded by an underlying geometrical perspective.

Practically, understanding these modifications is crucial for interpreting observational data about the universe’s expansion history and the behavior of cosmic large-scale structures. The paper suggests further research paths, including scalar field couplings to boundary terms, which could yield viable cosmological scenarios providing an alternative to scalar-tensor theories like Brans-Dicke.

Conclusion and Future Speculation

The paper highlights essential advances in understanding modified gravity theories within the teleparallel paradigm. The distinctions between local Lorentz invariance and the order of derivatives in field equations offer fertile ground for future research, particularly in cosmology's intersection with fundamental gravitational theories.

Anticipating future developments in theoretical and computational models may direct teleparallel gravity toward more comprehensive models that efficiently incorporate quantum mechanics and predict hitherto unexplained phenomena within general relativity's standard framework. The paper is a critical stepping stone in a broader quest for a unified theory of gravity.