Geometrical 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 $\omega$ and the soldering form $\theta$ 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 $A=\omega+\theta$. 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 $(P_{H}\rightarrow M, A)$ can be obtained from a $G$-principal bundle $P_{G}\rightarrow M$ 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 $P_{G}$, 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 $M$.