3 - Cartan’s Tetrad Formulation

Summary

This is the first central Chapter of the book that describes Riemannian geometry using Cartan's notion of soldering. Gravity first appears in this Chapter as a dynamical theory of a collection of differential forms rather than a metric. We describe thegeneral notion of geometric structures and then specialise to the case of a geometric structure corresponding to a metric. We describe the notion of a spin connection, its torsion, and then present examples of caclulations of Riemann curvature in the tetrad formalism. We then describe the Einstein-Cartan formulation of GR in terms of differential forms, and present its teleparallel version. We introduce the idea of the pure connection formulation, and compute the corresponding actino perturbatively. We then describe theso-called MacDowell-Mansouri formulation. We briefly describe the computations necessary to carry out the dimensional reduction from 5D to 4D. We then describe the so-called BF formulation of 4D GR, which in particular allows to determine the pure connection action in a closed form. We then describe the field redefinitions that are available when one works in BF formalism, and the associated formulation of BF-type plus potential for the B field.