2025/08/05 by James A. Reid, Reid, James A.
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)
paper · pdf · doi:10.48550/arxiv.2508.05683
openalex publication_date 2025/08/05 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
In the MacDowell-Mansouri formulation of general relativity, the spin connection and coframe variables are incorporated into a single Lie algebra-valued connection called the MacDowell-Mansouri connection, ω. From the curvature form F of ω and an auxiliary field, B, one may formulate general relativity as a deformed topological field theory by constructing an action functional whose variation yields a set of field equations that are equivalent to the Einstein equations on shell. In this article, we show that when the fundamental length scale of the MacDowell-Mansouri connection is regarded as a dynamical variable -- a cosmological scalar field -- the field equations obtained from the variation of the resulting action are equivalent to the conformal Einstein equations on shell. Through the lens of Cartan geometry, we then discuss a notable geometrical difference between general relativity and its conformally transformed counterpart. Specifically, for the latter, we show that points in spacetime are infinitesimally approximated by homogeneous spaces (restricted to a point) whose radii are parameterised by the value of the cosmological scalar field.