MacDowell-Mansouri gravity and Cartan geometry

TL;DR

This paper explains how Cartan geometry provides a clear geometric interpretation of the MacDowell-Mansouri formulation of gravity, unifying various models of spacetime and connecting gauge theory with geometric structures.

Contribution

It offers a comprehensive geometric perspective on MacDowell-Mansouri gravity using Cartan geometry, including its reformulation as BF theory and the interpretation of connections.

Findings

Cartan geometry clarifies the geometric meaning of the MacDowell-Mansouri trick.

Spacetime can be modeled as a tangent space approximated by various homogeneous spaces.

The MacDowell-Mansouri connection can be viewed as rolling a model spacetime along physical spacetime.

Abstract

The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan geometry.