# Reflections on the Covariance of Modified Teleparallel Theories of Gravity

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Teleparallel Gravity

#### 2.1. Geometric Foundations of General Relativity

#### 2.2. The Teleparallel Equivalent of General Relativity

## 3. Modified Teleparallel Gravities

## 4. Covariance in Modified Teleparallel Gravity

## 5. Remnant Symmetries and The Lorentz Group

- Why, for a given spacetime $({\mathcal{T}}^{\star}\mathcal{M},{\mathbf{E}}^{a}\left(x\right))$, there are certain proper tetrads ${\mathbf{E}}^{a}$, and others that definitely do not lead to a consistent set of equations of motion for a function $f\left(T\right)$ other than the one corresponding to GR?
- Once the above point was established, is there any way of counting the number of proper tetrads, and some systematic procedure in order to obtain them?
- What is the physical meaning of the proper tetrads?

#### The Remnant Group

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Sundermeyer, K. Symmetries in Fundamental Physics; Springer International Publishing: Berlin, Germany, 2014. [Google Scholar]
- Aldrovandi, R.; Pereira, J.G. Teleparallel Gravity; Springer: Dordrecht, The Netherlands, 2013; Volume 173. [Google Scholar]
- Ferraro, R.; Guzmán, M.J. Hamiltonian formulation of teleparallel gravity. Phys. Rev. D
**2016**, 94, 104095. [Google Scholar] [CrossRef] - Ferraro, R.; Guzmán, M.J. Hamiltonian formalism for f(T) gravity. Phys. Rev. D
**2018**, 97, 104028. [Google Scholar] [CrossRef] - 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] - Beltrán Jiménez, J.; Heisenberg, L.; Koivisto, T. Coincident general relativity. Phys. Rev. D
**2018**, 98, 044048. [Google Scholar] [CrossRef] [Green Version] - Beltrán Jiménez, J.; Heisenberg, L.; Koivisto, T. The geometrical trinity of gravity. arXiv
**2019**, arXiv:1903.06830. [Google Scholar] - Hayashi, K.; Shirafuji, T. New general relativity. Phys. Rev. D
**1979**, 19, 3524. [Google Scholar] [CrossRef] - Ferraro, R.; Fiorini, F. Modified teleparallel gravity: Inflation without inflaton. Phys. Rev. D
**2007**, 75, 084031. [Google Scholar] [CrossRef] - Bengochea, G.; Ferraro, R. Dark torsion as the cosmic speed up. Phys. Rev. D
**2009**, 79, 124019. [Google Scholar] [CrossRef] - Cai, Y.F.; Capozziello, S.; De Laurentis, M.; Saridakis, E.N. f(T) teleparallel gravity and cosmology. Rep. Prog. Phys.
**2016**, 79, 106901. [Google Scholar] [CrossRef] [PubMed] - Boehmer, C.G.; Fiorini, F. The regular black hole in four dimensional Born Infeld gravity. Class. Quant. Grav.
**2019**, 36, 12LT01. [Google Scholar] [CrossRef] - Ferraro, R.; Guzmán, M.J. Quest for the extra degree of freedom in f(T) gravity. Phys. Rev. D
**2019**, 98, 124037. [Google Scholar] [CrossRef] - Yang, R.J. Conformal transformation in f(T) theories. Eur. Phys. Lett.
**2011**, 93, 60001. [Google Scholar] [CrossRef] - Wright, M. Conformal transformations in modified teleparallel theories of gravity revisited. Phys. Rev. D
**2016**, 93, 103002. [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] - Li, B.; Sotiriou, T.P.; Barrow, J.D. Large-scale structure in f(T) gravity. Phys. Rev. D
**2011**, 83, 104017. [Google Scholar] [CrossRef] - Izumi, K.; Ong, Y.C. Cosmological perturbation in f(T) gravity revisited. J. Cosmol. Astropart. Phys.
**2013**, 1306, 029. [Google Scholar] [CrossRef] - Golovnev, A.; Koivisto, T. Cosmological perturbations in modified teleparallel gravity models. J. Cosmol. Astropart. Phys.
**2018**, 1811, 012. [Google Scholar] [CrossRef] - Ferraro, R.; Fiorini, F. Remnant group of local Lorentz transformations in f(T) theories. Phys. Rev. D
**2015**, 91, 064019. [Google Scholar] [CrossRef] - Krššák, M.; Pereira, J.G. Spin Connection and Renormalization of Teleparallel Action. Eur. Phys. J. C
**2015**, 75, 519. [Google Scholar] [CrossRef] - Krššák, M. Holographic renormalization in teleparallel gravity. Eur. Phys. J. C
**2017**, 77, 44. [Google Scholar] [CrossRef] [Green Version] - Sotiriou, T.P.; Li, B.; Barrow, J.D. Generalizations of teleparallel gravity and local Lorentz symmetry. Phys. Rev. D
**2011**, 83, 104030. [Google Scholar] [CrossRef] - Ong, Y.C.; Nester, J.M. Counting components in the lagrange multiplier formulation of teleparallel theories. Eur. Phys. J. C
**2018**, 78, 568. [Google Scholar] [CrossRef] - Krššák, M.; Saridakis, E.N. The covariant formulation of f(T) gravity. Class. Quant. Grav.
**2016**, 33, 115009. [Google Scholar] [CrossRef] - 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.; Krššák, M.; Pfeifer, C. Teleparallel theories of gravity as analogue of nonlinear electrodynamics. Phys. Rev. D
**2018**, 97, 104042. [Google Scholar] [CrossRef] [Green Version] - 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.; Van Den Hoogen, R.J.; Pereira, J.G.; Boehmer, C.G.; Coley, A.A. Teleparallel theories of gravity: Illuminating a fully invariant approach. arXiv
**2018**, arXiv:1810.12932. [Google Scholar] - Maluf, J.W.; Ulhoa, S.C.; da Rocha-Neto, J.F. Difficulties of Teleparallel Theories of gravity with local Lorentz symmetry. arXiv
**2018**, arXiv:1811.06876. [Google Scholar] - 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] - Hohmann, M.; Järv, L.; Krššák, M.; Pfeifer, C. Modified teleparallel theories of gravity in symmetric spacetimes. arXiv
**2019**, arXiv:1901.05472. [Google Scholar]

1 | Other authors define the affine connection as ${\nabla}_{{\mathbf{e}}_{c}}{\mathbf{e}}_{b}={\mathsf{\Gamma}}_{cb}^{a}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{e}}_{a}\phantom{\rule{3.33333pt}{0ex}}$. |

2 | The anti-symmetry of the spin connection implies that $D{\eta}_{ab}=d{\eta}_{ab}-{\mathit{\omega}}_{\phantom{\rule{4pt}{0ex}}a}^{c}{\eta}_{cb}-{\mathit{\omega}}_{\phantom{\rule{4pt}{0ex}}b}^{c}{\eta}_{ac}=-{\mathit{\omega}}_{ba}-{\mathit{\omega}}_{ab}=0$, and $D{\epsilon}_{abcd}=0$. |

3 | Notice that the coefficients ${E}_{\mu}^{a}\phantom{\rule{3.33333pt}{0ex}}$, ${e}_{a}^{\mu}$ are the links between anholonomous and coordinate basis. |

© 2019 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**

Bejarano, C.; Ferraro, R.; Fiorini, F.; Guzmán, M.J.
Reflections on the Covariance of Modified Teleparallel Theories of Gravity. *Universe* **2019**, *5*, 158.
https://doi.org/10.3390/universe5060158

**AMA Style**

Bejarano C, Ferraro R, Fiorini F, Guzmán MJ.
Reflections on the Covariance of Modified Teleparallel Theories of Gravity. *Universe*. 2019; 5(6):158.
https://doi.org/10.3390/universe5060158

**Chicago/Turabian Style**

Bejarano, Cecilia, Rafael Ferraro, Franco Fiorini, and María José Guzmán.
2019. "Reflections on the Covariance of Modified Teleparallel Theories of Gravity" *Universe* 5, no. 6: 158.
https://doi.org/10.3390/universe5060158