Directional Derivatives

So the tetrad at each point x in whatever space we are dealing with is a local coordinate system expressed in coordinates relative to the Minkowski metric. This allows us to pull back from flat space the concept of local directions and thus local directional derivatives. This assignment of basis has the very important property that it leaves the metric (not the basis) unchanged if the coordinate system is changed by a local Lorentz tranfrorm.

If $$\text{ee}_{\text{ }\mu }^I(x)=\Lambda _{\text{  }J}^I(x)e_{\text{  }\mu }^J(x)$$ and $$\text{gg}_{\mu \nu }(x)=\text{ee}_{\text{  }\mu }^I(x)\text{ee}_{\text{  }\mu }^J(x)\eta _{IJ}$$ then $$gg_{\mu \nu }(x)= g_{\mu \nu }(x)$$.