MacDowell–Mansouri gravity and Cartan geometry

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 connections.