Comments on the Tetrad (Vielbeins)

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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