f(T) Gravity and local Lorentz invariance (1010.1041v3)

Abstract: We show that in theories of generalised teleparallel gravity, whose Lagrangians are algebraic functions of the usual teleparallel Lagrangian, the action and the field equations are not invariant under local Lorentz transformations. We also argue that these theories appear to have extra degrees of freedom with respect to general relativity. The usual teleparallel Lagrangian, which has been extensively studied and leads to a theory dynamically equivalent to general relativity, is an exception. Both of these facts appear to have been overlooked in the recent literature on f(T) gravity, but are crucial for assessing the viability of these theories as alternative explanations for the acceleration of the universe.

• The paper demonstrates that f(T) gravity violates local Lorentz invariance by introducing non-divergence terms in the teleparallel Lagrangian.
• The authors derive modified field equations that feature additional dynamical degrees of freedom compared to general relativity.
• These findings challenge the empirical viability of f(T) models for explaining cosmic acceleration without invoking dark energy.

Analysis of the Local Lorentz Invariance in f(T) Gravity

The paper "f(T) gravity and local Lorentz invariance" by Baojiu Li, Thomas P. Sotiriou, and John D. Barrow explores critical aspects of generalized teleparallel gravity theories, specifically focusing on the challenges tied to local Lorentz invariance. This work engages with the landscape of modified gravitational theories that have been increasingly considered as alternative explanations for cosmic acceleration. Analyzing the algebraic functions of the teleparallel Lagrangian, the authors raise important concerns regarding local Lorentz invariance, a topic that has often been neglected in the f(T) gravity literature.

Core Contributions

The authors examine the symmetries of the f(T) gravity, underscoring two pivotal observations:

1. Lorentz Invariance: The paper robustly argues that f(T) gravity theories lack local Lorentz invariance. While standard teleparallel gravity, with action relying purely on the teleparallel Lagrangian $T$, retains local Lorentz invariance due to the Lagrangian's divergence-like behavior, its generalizations do not. The function $f(T)$ introduces terms that are not reducible to a divergence term, hence breaching local Lorentz symmetry.
2. Additional Degrees of Freedom: Beyond the issue of Lorentz invariance, the authors identify the presence of additional degrees of freedom not accounted for in General Relativity (GR). The study indicates that the field equations of f(T) theories entail additional dynamical components compared to the normally constrained dynamics of GR.

These findings are critical for the f(T) gravity model's viability as a physical theory, potentially affecting its empirical adequacy in describing gravitational phenomena, particularly in cosmic settings.

Detailed Insights

Teleparallel gravity is re-analyzed through the lens of its distinctive use of the Weitzenböck connection, which emphasizes torsion rather than curvature as in GR. The bifurcation occurs when a function $f(T)$ replaces $T$ in the Lagrangian density, introducing new dynamic features that lead the authors to question the unconsidered assumptions prevalent in the ongoing scholarly conversation surrounding f(T) gravity.

The paper provides a rigorous derivation of the field equations for f(T) theories, illustrating their deviation from the locally Lorentz-invariant frameworks that GR and standard teleparallel gravity conform to. A significant mathematical exposition reveals that for f(T) gravity, the field equations are not invariant under local Lorentz transformations, unlike GR's equations, which have only ten independent components due to gauge freedom. This insight effectively raises the complexity of reconciling f(T) gravity with empirical observations that traditionally respect Lorentz invariance.

Implications and Future Research

The implications of this study are profound. The lack of local Lorentz invariance in f(T) gravity indicates potential observational discrepancies, particularly in scenarios where Lorentz-breaking effects could manifest. These effects necessitate a re-evaluation of f(T) gravity as a contender in the suite of modified gravitational theories aimed at resolving questions such as cosmic acceleration without introducing dark energy.

Furthermore, the paper suggests the necessity for further investigation into the new degrees of freedom introduced by these theories. This entails a more in-depth theoretical exploration into the stability and physical consequences of these dynamical features, which might pose calculational challenges due to the absence of a standardized tetrad gauge.

Finally, while pointing out the intricacies of breaking Lorentz invariance, the authors also hint at possible modifications to the f(T) framework that could salvage Lorentz symmetry, albeit leading to a theory with different dynamics. Such modifications might yield fertile ground for developing more robust alternative theories still aligned with key gravitational principles.

In summation, this paper offers a compelling critique of f(T) gravity, emphasizing its potential inconsistencies with local Lorentz invariance and the ensuing theoretical ramifications. It serves as a beacon, suggesting a recalibration of theoretical expectations and inspiring subsequent empirical investigations to elucidate the true behavior of our universe under alternative gravitational paradigms.