# Tetrads (vierbeins) basis vectors and basis vectors of the local inertial coordinates in general relativity

In local inertial coordinates $g_{\mu \nu}$ becomes $\eta_{\mu \nu}$. Also, tetrads are defined as the basis vectors ${\hat{e}}_{a}, {\hat{e}}_{b}$... such that $g( {\hat{e}}_{a},{\hat{e}}_{b} ) = \eta_{ab}$. So, this means that in tetrad basis $g_{\mu \nu} = \eta_{\mu \nu}$. Hence the bases of local inertial coordinates should be the same as the tetrad bases (apart from a Lorentz transformation). Is that correct?

• In inertial coordinates (in general) the metric is $\eta_{\mu\nu}$ only at one point.
– MBN
Nov 12, 2017 at 10:28

Yes, that is correct. If you Lorentz transform a tetrad, $\eta_{\mu\nu}$ is preserved by the transformation.