Abstract
In general relativity, the only dynamical field describing the gravitational interaction of matter, is the metric. It induces the causal structure of spacetime, governs the motion of physical bodies through its Levi-Civita connection, and mediates gravity via the curvature of this connection. While numerous modified theories of gravity retain these principles, it is also possible to introduce another affine connection as a fundamental field, and consider its properties—curvature, torsion, nonmetricity—as the mediators of gravity. In the most general case, this gives rise to the class of metric-affine gravity theories, while restricting to metric-compatible connections, for which nonmetricity vanishes, comprises the class of Poincaré gauge theories. Alternatively, one may also consider connections with vanishing curvature. This assumption yields the class of teleparallel gravity theories. This chapter gives a simplified introduction to teleparallel gravity, with a focus on performing practical calculations, as well as an overview of the most commonly studied classes of teleparallel gravity theories.
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Notes
- 1.
This equation takes the same role as \(\stackrel {\circ }{\nabla }_{\nu }G_{\mu }{ }^{\nu } = 0\) for the Einstein tensor, which is satisfied identically as a consequence of the Bianchi identities.
- 2.
In the literature, the abbreviation TEGR is more common, since it was developed prior to the other equivalent theories. Another proposed nomenclature is “antisymmetric teleparallel equivalent of general relativity” (ATEGR) [27], since the distortion tensor becomes antisymmetric in its first two indices. However, the term “metric” or “metric-compatible” is more abundant in the contemporary literature on teleparallel gravity to denote the case of vanishing nonmetricity.
- 3.
They can be reduced to second order by introducing an auxiliary scalar field.
- 4.
This term is also, more commonly, used for a particular subclass of theories, in which \(2a_1 + a_2 = 0\) and \(a_3 = -1\), so that there is only one free parameter besides the gravitational constant \(\kappa \) [34].
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Acknowledgements
The author thanks Claus Lämmerzahl and Christian Pfeifer for the kind invitation to contribute this book chapter. He acknowledges the full financial support of the Estonian Ministry for Education and Science through the Personal Research Funding Grant PRG356, as well as the European Regional Development Fund through the Center of Excellence TK133 “The Dark Side of the Universe”.
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Hohmann, M. (2023). Teleparallel Gravity. In: Pfeifer, C., Lämmerzahl, C. (eds) Modified and Quantum Gravity. Lecture Notes in Physics, vol 1017. Springer, Cham. https://doi.org/10.1007/978-3-031-31520-6_4
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