Metric–Affine Gauge Theory of Gravity I. Fundamental Structure and Field Equations

We give a self–contained introduction into the metric–affine gauge theory of gravity. Starting from the equivalence of reference frames, the prototype of a gauge theory is presented and illustrated by the example of Yang–Mills theory. Along the same lines we perform a gauging of the affine group and establish the geometry of metric–affine gravity. The results are put into the dynamical framework of a classical field theory. We derive subcases of metric– affine gravity by restricting the affine group to some of its subgroups. The important subcase of general relativity as a gauge theory of translations is explained in detail.