Tetrads and Fermions

The basic point of this section is that relativity in the absence of matter can be formulated with the metric alone (i.e. $$g_{\mu \nu }(x)$$) whereas the physics of matter requires Dirac's equation which depends on being able to take derivatives ($$i\gamma ^I \partial _I\psi - m \psi  = 0 $$). Thus we need not only the metric but an underlying set of local coordinate systems: the tetrads.

I think of the tetrads as a kind of "square root" of the metric, the Pythagorean theorem generalized.

The Minkowski metric $$\eta =\eta _{IJ}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right)$$is considered to be the "flat spacetime metric" in the sense that it describes the canvas on which we do all of our mathematics. Then an arbitrary metric (i.e. quadratic form, symmetric 4x4 matrix) is related to a local coordinate system $$

☀e_{\text{  }\mu }^I(x) $$ by the relationship $$g_{\mu \nu }(x)=e_{\text{  }\mu }^I(x)e_{\text{  }\nu }^J(x)\eta _{IJ}$$.