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. $e_{\mu a}={\mathrm{\partial }}_{\mu }{X}_a$ 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 $Z_A$.