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Teleparallelism

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Geometrodynamics of Gauge Fields

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

According to Feynman (1962/63) (Feynman, Lectures on gravitation, 1962/63), “gravity is that field which corresponds to a gauge invariance with respect to displacement transformations.” Taking this literally would favor Einstein’s teleparallelism equivalent of GR, which has been recast Mielke (1992) (Mielke, Ann Phys 219(1), 78–108, 1992); Mielke, Baekler, Hehl, Macías & Morales-Técotl (1996) (Mielke et al. Gravity particles and space-time, 1996) into a Yang–Mills-type gauge theory of translations.

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Notes

  1. 1.

    The self-dual formulation of gravity was anticipated already by Plebański (1977), whereas Hojman et al. (1980) discussed the pseudoscalar curvature as a parity-violating Lagrangian for gravity and noted already 1980 its relationship to a complete divergence, before Nieh & Yan (1982), Nieh (2007).

  2. 2.

    In the presence of matter, the first Noether identity enters the game. For a resolution of (6.2.24) in terms of the momenta \({\buildrel {(\pm )}\over \varPi }{}_{\alpha }\), it would be convenient to have a relocalized energy–momentum current \({\buildrel {(\pm )}\over \varSigma }{}_{\alpha }\) for which \({\buildrel {(\pm )}\over D}{\buildrel {(\pm )}\over \varSigma }{}_{\alpha } \cong 0\) holds.

  3. 3.

    The transposed connection, which has the property (Mielke et al. 1989) that \({\overset{\frown }{D}}\eta _{\alpha }:=\) \( D\eta _{\alpha } - (e_{\alpha }\rfloor T^{\beta })\wedge \eta _{\beta } \equiv 0\, \), may be regarded as a special real version of our Sen-type connection, for which \({\buildrel {(\pm )}\over D}\eta _{\alpha }= \pm (i/2) \ell ^{2}\eta _{\alpha \beta }\wedge {\buildrel {(\pm )}\over \varPi }{}^{\beta }\) holds.

  4. 4.

    Interestingly, in the gauge \(\underline{\vartheta }{}^{\hat{0}}=3\kappa d\theta _\mathrm{L}=h dr =\pm df/2\), such instantons are solutions to the topological Eq. (6.7.12), due to \(T^{\hat{0}}=0\).

  5. 5.

    The translational angle \(\theta _\mathrm{T}=2/\gamma \) is at times identified (Freidel et al. 2005) with the inverse Barbero–Immirzi parameter \(\gamma \). Such \(\theta \)-terms and the canonical transformation induced by the translational Chern–Simons term \(dC_\mathrm{TT}\) were considered earlier by Mielke (1992).

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  1. Departamento de Física, CBI Iztapalapa, Universidad Autónoma Metropolitana, Mexico City, México

    Eckehard W. Mielke

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Mielke, E.W. (2017). Teleparallelism. In: Geometrodynamics of Gauge Fields. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-29734-7_6

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