Connecting connections

Abstract

As true as it is that a bricklayer needs a plumb line and a T-square, so it is that a physicist using general relativity needs how to draw geodesics and use fields of congruent vector frames of reference. While the first part of the preceding statement depends on the Christoffel connection and related metric and curvature concepts, the second part depends on the Weitzenb\"{o}ck connection and the concept of torsion. This dual structure has been considered before as a possibility of using either one of them to describe General relativity. We claim here that both structures have to be correlated to produce useful interpretations of any space-time model.