# Disformal Transformations in Modified Teleparallel Gravity

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## Abstract

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## 1. Introduction

## 2. Teleparallel Gravity

## 3. Modified Teleparallel Gravity and Degrees of Freedom

#### 3.1. Generalized Teleparallel Gravity

#### 3.2. On the Extra Degree(s) of Freedom of $f\left(\mathbb{T}\right)$ Gravity

- Lorentz invariance: It is believed by some authors [1,22,23,24,25] that the loss of local Lorentz invariance could be restored by dropping the assumption of the Weitzenböck connection and resorting to a more general one, inertial connection. However, see the Ref. [26] for another viewpoint, especially when it comes to modified teleparallel gravities.
- Weak gravity: It has been shown that there are no new propagating gravitational modes in the linearized approximation around Minkowski spacetime with the trivial choice of the tetrad [27,28]. This follows from a simple observation that $f\left(\mathbb{T}\right)$ gravity reduces to TEGR in this limit [29], and in particular, the local Lorentz invariance is restored.
- Cosmological perturbations: Extra propagating modes also do not appear in the linear cosmological perturbations around a spatially flat FRW universe with the standard choice of the background tetrad [30,31,32,33,34]. When a dynamical scalar mode is seen in the perturbations [32], it actually comes from an additional scalar field added as a matter field to the model.
- Hamiltonian formalism: Probably, the first attempt to count the number of degrees of freedom in the Hamiltonian formalism of $f\left(\mathbb{T}\right)$ gravity can be found in the Ref. [35], where the authors claim that the theory has $n-1$ extra degrees of freedom in n spacetime dimensions. Later work finds only one extra degree of freedom in any dimension [17,36]. It is suggested that the extra mode could be related with the proper parallelization of spacetime. For other kinds of modified teleparallel gravities, the consideration of this mode is still pending [37].
- Conformal transformations: We proved above that a clean interpretation of the extra degree(s) of freedom of $f\left(\mathbb{T}\right)$ gravity cannot be achieved with the use of conformal transformations. This result was first obtained in [14], and some more details and refinements, and also extensions have been given in [15].
- Claims of acausality: Assuming the presence of extra modes, the cosmological results indicate a strong coupling regime for them, which is bad news for predictability. Also, acausality claims for $f\left(\mathbb{T}\right)$ gravity can be found in the Ref. [38] through the method of characteristics; however, they use the number of degrees of freedom calculated in the Ref. [35] to draw some of their conclusions, which are currently in doubt.
- Galaxy rotation curves: An indirect indication for extra degree(s) of freedom can be found in [39], where they use a model with $f\left(\mathbb{T}\right)=\mathbb{T}+\alpha {\mathbb{T}}^{n}$ to reproduce galactic rotation curves. When the function approaches GR/TEGR ($n=1$), the curves are different to those obtained in GR, indicating the presence of some extra mode.
- Remnant symmetry: Even though the full Lorentz group is no longer a symmetry of $f\left(\mathbb{T}\right)$ gravity, there still exist subgroups of remnant symmetries which have been studied in the Ref. [40]. Obviously, a good understanding of symmetries is an important key to understanding the degrees of freedom.
- Null tetrads: Last but not least, null tetrads can be used to find solutions with $\mathbb{T}=0$ (or any other constant instead of zero) [41,42,43], which could be useful for exhibiting the extra d.o.f. [44], even though, at the background level, these solutions are not different from general relativistic ones, up to rescaling of fundamental constants.

## 4. Disformal Transformations in the Teleparallel Framework

#### 4.1. Generalities on Disformal Transformation of the Tetrad Field

#### 4.2. Disformal Transformation of the Torsion Tensor

#### 4.3. Contortion and Torsion Vector

## 5. Disformal Transformations in $\mathbf{f}\left(\mathbb{T}\right)$ Gravity and the Search for the Einstein Frame

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Golovnev, A.; Guzmán, M.J.
Disformal Transformations in Modified Teleparallel Gravity. *Symmetry* **2020**, *12*, 152.
https://doi.org/10.3390/sym12010152

**AMA Style**

Golovnev A, Guzmán MJ.
Disformal Transformations in Modified Teleparallel Gravity. *Symmetry*. 2020; 12(1):152.
https://doi.org/10.3390/sym12010152

**Chicago/Turabian Style**

Golovnev, Alexey, and María José Guzmán.
2020. "Disformal Transformations in Modified Teleparallel Gravity" *Symmetry* 12, no. 1: 152.
https://doi.org/10.3390/sym12010152