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arXiv · 0905.4792

Comments on the Tetrad (Vielbeins)

Takeshi Fukuyama
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Abstract

We want to correct the misunderstandings on the tetrad (or veilbeins in general) appeared in many text books or review articles. The tetrad should be defined without any condition. $e_{μa}=\partial_μX_a$ with local Lorentz coordinates $X_a$ ia wrong in many sences: it gives the condition $\partial_μe_{νa}=\partial_νe_{μa}$, which leads us to the trivial result that the cyclic coefficients vanish identically and to the null Riemannian tensor. Also $e_{μa}e_ν^a=g_{μν}$ is not scalar under the local Lorentz transformation etc. We show how these deficits are remedied by the correct definition, $e_{μa}=D_μZ_a$ with local (Anti) de Sitter coordinates $Z_A$.

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Takeshi Fukuyama. 2009-05-29. Comments on the Tetrad (Vielbeins). https://doi.org/10.1142/s0217732309031739

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