2+1 gravity on the conformal sphere

TL;DR

This paper demonstrates two equivalent first-order descriptions of 2+1 gravity with a cosmological constant, linking spacetime and conformal geometry via Cartan geometry, and explores their implications for phase space reductions.

Contribution

It introduces a novel perspective by using Cartan geometry to connect spacetime and conformal descriptions of 2+1 gravity, offering insights into their equivalence and phase space structures.

Findings

Two equivalent first-order formulations of 2+1 gravity with cosmological constant.

Cartan geometry as a unifying tool linking spacetime and conformal descriptions.

Different phase space reductions lead to GR and Shape Dynamics interpretations.

Abstract

We show that there are 2 equivalent first order descriptions of 2+1 gravity with non-zero cosmological constant. One is the well-known spacetime description and the other is in terms of evolving conformal geometry. The key tool that links these pictures is Cartan geometry, a generalization of Riemannian geometry that allows for geometries locally modelled off arbitrary homogeneous spaces. The two different interpretations suggest two distinct phase space reductions. The spacetime picture leads to the 2+1 formulation of General Relativity due to Arnowitt, Deser, and Misner while the conformal picture leads to Shape Dynamics. Cartan geometry thus provides an alternative to symmetry trading for explaining the equivalence of General Relativity and Shape Dynamics.