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. eμa = ∂μXa with local Lorentz coordinates Xa ia wrong in many sences: it gives the condition ∂μeνa = ∂νeμa, which leads us to the trivial result that the cyclic coefficients vanish identically and to the null Riemannian tensor. Also eμae a ν = gμν is not scalar under the local Lorentz transformation etc. We show how these deficits are remedied by the correct definition, eμa = DμZa with local (Anti) de Sitter coordinates ZA. From the gauge theoretical point of view [1], one of the most important characters of gravity is the soldering of internal space of gauge symmetry of gravity with the external space. The other one is that gravity, at least the leading term at low energy, is linear in the Riemannian tensor unlike the other gauge theories. So the gauge theory of gravity must reflects and explains these peculiarities. In these processes, we can understand that the problem lies in the ambiguous situation of tetrad or metric in the gauge theoretical framework. There seems to exist some prejudice that the local symmetry of gravitation is Lorentz group (as the symmetry before the breaking) and misunderstanding that the tetrad is defined by eaμ = ∂μX (x) ≡ X,μ . (0.1) Here X are the local Lorentz coordinates. The metric tensor is defined by gμν = e a μe b νηab, (0.2) E-mail:[email protected]