# Disformal Transformations in Modified Teleparallel Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 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

## References

- Aldrovandi, R.; Pereira, J.G. Teleparallel Gravity: An Introduction; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Golovnev, A. Introduction to teleparallel gravities. arXiv
**2018**, arXiv:1801.06929. [Google Scholar] - Adak, M.; Sert, O. A Solution to symmetric teleparallel gravity. Turk. J. Phys.
**2005**, 29, 1–7. [Google Scholar] - Adak, M.; Kalay, M.; Sert, O. Lagrange formulation of the symmetric teleparallel gravity. Int. J. Mod. Phys. D
**2006**, 15, 619–634. [Google Scholar] [CrossRef][Green Version] - Ferraro, R.; Fiorini, F. Modified teleparallel gravity: Inflation without inflaton. Phys. Rev. D
**2007**, 75, 084031. [Google Scholar] [CrossRef][Green Version] - Beltrán-Jiménez, J.; Heisenberg, L.; Koivisto, T.S.; Pekar, S. Cosmology in f(Q) geometry. arXiv
**2019**, arXiv:1906.10027. [Google Scholar] - Bekenstein, J.D. Sixth Marcel Grossmann Meeting on General Relativity; Sato, H., Nakamura, T., Eds.; World Scientific: Singapore, 1992; p. 905. [Google Scholar]
- Bekenstein, J.D. The Relation between physical and gravitational geometry. Phys. Rev. D
**1993**, 48, 3641. [Google Scholar] [CrossRef][Green Version] - Horndeski, G.W. Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys.
**1974**, 10, 363. [Google Scholar] [CrossRef] - Bittencourt, E.; Lobo, I.P.; Carvalho, G.G. On the disformal invariance of the Dirac equation. Class. Quant. Grav.
**2015**, 32, 185016. [Google Scholar] [CrossRef] - Hohmann, M. Disformal Transformations in Scalar-Torsion Gravity. Universe
**2019**, 5, 167. [Google Scholar] [CrossRef][Green Version] - Krššák, M.; Pereira, J.G. Spin Connection and Renormalization of Teleparallel Action. Eur. Phys. J. C
**2015**, 75, 519. [Google Scholar] [CrossRef][Green Version] - Bengochea, G.R.; Ferraro, R. Dark torsion as the cosmic speed-up. Phys. Rev. D
**2009**, 79, 124019. [Google Scholar] [CrossRef][Green Version] - Yang, R.J. Conformal transformation in f(T) theories. EPL
**2011**, 93, 60001. [Google Scholar] [CrossRef][Green Version] - Wright, M. Conformal transformations in modified teleparallel theories of gravity revisited. Phys. Rev. D
**2016**, 93, 103002. [Google Scholar] [CrossRef][Green Version] - Eling, C.; Jacobson, T.; Mattingly, D. Einstein-Aether Theory. arXiv
**2004**, arXiv:Gr-qc/0410001. [Google Scholar] - Ferraro, R.; Guzmán, M.J. Hamiltonian formalism for f(T) gravity. Phys. Rev. D
**2018**, 97, 104028. [Google Scholar] [CrossRef][Green Version] - Beltrán-Jiménez, J.; Dialektopoulos, K.F. Non-Linear Obstructions for Consistent New General Relativity. arXiv
**2019**, arXiv:1907.10038. [Google Scholar] - Beltrán-Jiménez, J.; Heisenberg, L.; Iosifidis, D.; Jiménez-Cano, A.; Koivisto, T.S. General Teleparallel Quadratic Gravity. arXiv
**2019**, arXiv:1909.09045. [Google Scholar] - Koivisto, T.; Tsimperis, G. The spectrum of teleparallel gravity. Universe
**2019**, 5, 80. [Google Scholar] [CrossRef][Green Version] - Raatikainen, S.; Rasanen, R. Higgs inflation and teleparallel gravity. arXiv
**2019**, arXiv:1910.03488. [Google Scholar] [CrossRef][Green Version] - Krššák, M.; Saridakis, E.N. The covariant formulation of f(T) gravity. Class. Quant. Grav.
**2016**, 33, 115009. [Google Scholar] [CrossRef][Green Version] - Golovnev, A.; Koivisto, T.; Sandstad, M. On the covariance of teleparallel gravity theories. Class. Quant. Grav.
**2017**, 34, 145013. [Google Scholar] [CrossRef] - Hohmann, M.; Järv, L.; Ualikhanova, U. Covariant formulation of scalar-torsion gravity. Phys. Rev. D
**2018**, 97, 104011. [Google Scholar] [CrossRef][Green Version] - Krššák, M.; Hoogen, R.J.v.; Pereira, J.G.; Boehmer, C.G.; Coley, A.A. Teleparallel theories of gravity: Illuminating a fully invariant approach. Class. Quant. Grav.
**2019**, 36, 183001. [Google Scholar] [CrossRef][Green Version] - Bejarano, C.; Ferraro, R.; Fiorini, F.; Guzmán, M.J. Reflections on the covariance of modified teleparallel theories of gravity. Universe
**2019**, 5, 158. [Google Scholar] [CrossRef][Green Version] - Bamba, K.; Capozziello, S.; Laurentis, M.D.; Nojiri, S.; Sáez-Gómez, D. No further gravitational wave modes in F(T) gravity. Phys. Lett. B
**2013**, 727, 194. [Google Scholar] [CrossRef][Green Version] - Cai, Y.F.; Li, C.; Saridakis, E.N.; Xue, L. f(T) gravity after GW170817 and GRB170817A. Phys. Rev. D
**2018**, 97, 103513. [Google Scholar] [CrossRef][Green Version] - Hohmann, M.; Krššák, M.; Pfeifer, C.; Ualikhanova, U. Propagation of gravitational waves in teleparallel gravity theories. Phys. Rev. D
**2018**, 98, 124004. [Google Scholar] [CrossRef][Green Version] - Chen, S.H.; Dent, J.B.; Dutta, S.; Saridakis, E.N. Cosmological perturbations in f(T) gravity. Phys. Rev. D
**2011**, 83, 023508. [Google Scholar] [CrossRef][Green Version] - Li, B.; Sotiriou, T.P.; Barrow, J.D. Large-scale Structure in f(T) Gravity. Phys. Rev. D
**2011**, 83, 104017. [Google Scholar] [CrossRef][Green Version] - Izumi, K.; Ong, Y.C. Cosmological Perturbation in f(T) Gravity Revisited. JCAP
**2013**, 1306, 29. [Google Scholar] [CrossRef][Green Version] - Golovnev, A.; Koivisto, T. Cosmological perturbations in modified teleparallel gravity models. JCAP
**2018**, 1811, 12. [Google Scholar] [CrossRef][Green Version] - Toporensky, A.; Tretyakov, P. Spin connection and cosmological perturbations in f(T) gravity. arXiv
**2019**, arXiv:1911.06064. [Google Scholar] - Li, M.; Miao, R.X.; Miao, Y.G. Degrees of freedom of f(T) gravity. JHEP
**2011**, 1107, 108. [Google Scholar] [CrossRef][Green Version] - Ferraro, R.; Guzmán, M.J. Quest for the extra degree of freedom in f(T) gravity. Phys. Rev. D
**2018**, 98, 124037. [Google Scholar] [CrossRef][Green Version] - Blixt, D.; Hohmann, M.; Pfeifer, C. Hamiltonian and primary constraints of new general relativity. Phys. Rev. D
**2019**, 99, 084025. [Google Scholar] [CrossRef][Green Version] - Ong, Y.C.; Izumi, K.; Nester, J.M.; Chen, P. Problems with Propagation and Time Evolution in f(T) Gravity. Phys. Rev. D
**2013**, 88, 024019. [Google Scholar] [CrossRef][Green Version] - Finch, A.; Said, J.L. Galactic Rotation Dynamics in f(T) gravity. Eur. Phys. J. C
**2018**, 78, 560. [Google Scholar] [CrossRef][Green Version] - Ferraro, R.; Fiorini, F. Remnant group of local Lorentz transformations in f(T) theories. Phys. Rev. D
**2015**, 91, 064019. [Google Scholar] [CrossRef][Green Version] - Bejarano, C.; Ferraro, R.; Guzmán, M.J. Kerr geometry in f(T) gravity. Eur. Phys. J. C
**2015**, 75, 77. [Google Scholar] [CrossRef][Green Version] - Nashed, G.G.L. Exact Axially Symmetric Solution in f(T) Gravity Theory. Adv. High Energy Phys.
**2014**, 2014, 857936. [Google Scholar] [CrossRef][Green Version] - Bejarano, C.; Ferraro, R.; Guzmán, M.J. McVittie solution in f(T) gravity. Eur. Phys. J. C
**2017**, 77, 825. [Google Scholar] [CrossRef] - Guzmán, M.J.; Ferraro, R. Degrees of freedom and Hamiltonian formalism for f(T) gravity. arXiv
**2019**, arXiv:1910.03100. [Google Scholar] - Chamseddine, A.H.; Mukhanov, V. Mimetic Dark Matter. JHEP
**2013**, 1311, 135. [Google Scholar] [CrossRef][Green Version] - Golovnev, A. On the recently proposed Mimetic Dark Matter. Phys. Lett. B
**2014**, 728, 39. [Google Scholar] [CrossRef][Green Version] - Deruelle, N.; Rua, J. Disformal Transformations, Veiled General Relativity and Mimetic Gravity. JCAP
**2014**, 2, 1409. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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