Comments on the Tetrad (Vielbeins)

Details Comments on the Tetrad (Vielbeins) Comments on the Tetrad (Vielbeins)

2009-05-29

arxiv.org/abs/0905.4792v1

Mod.Phys.Lett.A24:2459-2466,2009

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

8 pages, no Figure

Scientific paper

10.1142/S0217732309031739

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. $e_{\mu a}=\partial_\mu X_a$ with local Lorentz coordinates $X_a$ ia wrong in many sences: it gives the condition $\partial_\mu e_{\nu a}=\partial_\nu e_{\mu a}$, which leads us to the trivial result that the cyclic coefficients vanish identically and to the null Riemannian tensor. Also $e_{\mu a}e_\nu^a=g_{\mu\nu}$ is not scalar under the local Lorentz transformation etc. We show how these deficits are remedied by the correct definition, $e_{\mu a}=D_\mu Z_a$ with local (Anti) de Sitter coordinates $Z_A$.

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