MacDowell-Mansouri gravity and Cartan geometry
Derek K. Wise
TL;DR
The paper argues that Cartan geometry provides the natural geometric framework for MacDowell-Mansouri gravity, with the gauge field $A$ splitting as $A = \omega + e$ and curvature decomposing to reveal Einstein gravity with a cosmological constant. The curvature structure and the projection to the Lorentz subalgebra yield the MM action, while a BF reformulation shows gravity as a deformation of a topological theory. The approach unifies Palatini gravity, BF theory, and MM gravity under the umbrella of reductive Cartan connections and the rolling interpretation of model spacetimes, offering a concrete geometric picture of spacetime geometry. It demonstrates explicit links between model spacetimes (de Sitter, anti-de Sitter, Minkowski) and the dynamic geometry of our universe, and it outlines avenues for quantum gravity and matter couplings within this Cartan framework. The work highlights important open questions, including the $\Lambda \to 0$ limit, the role of degenerate coframes, and possible generalizations to deformed relativity and beyond.
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.