Conformal Gravity as a Deformed Topological Field Theory

TL;DR

This paper reformulates general relativity as a deformed topological field theory using the MacDowell-Mansouri approach, and shows that making the length scale a dynamical scalar leads to conformal Einstein equations, highlighting geometric differences via Cartan geometry.

Contribution

It introduces a novel perspective by treating the fundamental length scale as a dynamical scalar, connecting topological field theory formulations to conformal Einstein equations.

Findings

Field equations are equivalent to conformal Einstein equations when the length scale is dynamical.

Points in spacetime are approximated by homogeneous spaces with radii determined by the scalar field.

The approach reveals geometric differences between general relativity and conformal transformations.

Abstract

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, $\omega$. From the curvature form $F$ of $\omega$ 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.