Comments on the Tetrad (Vielbeins)

TL;DR

This paper clarifies common misconceptions about tetrads in general relativity, emphasizing the correct definition using covariant derivatives of (Anti) de Sitter coordinates to avoid incorrect conditions and transformations.

Contribution

It provides a corrected definition of tetrads using covariant derivatives of (Anti) de Sitter coordinates, rectifying errors in standard textbook treatments.

Findings

Incorrect definitions lead to trivial or null results.

Proper definition avoids these issues and maintains correct transformation properties.

Clarifies the mathematical structure of tetrads in gravitational theories.

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. $e_{\mu a}=\partial_\mu X_a$ 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 $Z_A$.