Geometric foundations of Cartan gauge gravity

Abstract

We use the theory of Cartan connections to analyze the geometrical structures underpinning the gauge-theoretical descriptions of the gravitational interaction. According to the theory of Cartan connections, the spin connection ω and the soldering form θ that define the fundamental variables of the Palatini formulation of general relativity can be understood as different components of a single field, namely a Cartan connection A = ω + θ. In order to stress both the similarities and the differences between the notions of Ehresmann connection and Cartan connection, we explain in detail how a Cartan geometry (PH→ M, A) can be obtained from a G-principal bundle PG→ M endowed with an Ehresmann connection (being the Lorentz group H a subgroup of G) by means of a bundle reduction mechanism. We claim that this reduction must be understood as a partial gauge fixing of the local gauge symmetries of PG, i.e. as a gauge fixing that leaves "unbroken" the local Lorentz invariance. We then argue that the "broken" part of the symmetry — that is the internal local translational invariance — is implicitly preserved by the invariance under the external diffeomorphisms of M.