Gravity, Cartan geometry, and idealized waywisers

Abstract

The primary aim of this paper is to provide a simple and concrete interpretation of Cartan geometry in terms of the mathematics of idealized waywisers. Waywisers, also called hodometers, are instruments traditionally used to measure distances. The mathematical representation of an idealized waywiser consists of a choice of symmetric space called a {\em model space} and represents the `wheel' of the idealized waywiser. The geometry of a manifold is then completely characterized by a pair of variables \(\{V^A(x),A^{AB}(x)\}\), each of which admit simple interpretations: \(V^A\) is the point of contact between the waywiser's idealized wheel and the manifold whose geometry one wishes to characterize, and \(A^{AB}=A_\mu^{\phantom{\mu} AB}dx^\mu\) is a connection one-form dictating how much the idealized wheel of the waywiser has rotated when rolled along the manifold. The familiar objects from differential geometry (e.g. metric \(g_{\mu\nu}\), affine connection \(\Gamma^\rho_{\mu\nu}\), co-tetrad \(e^I\), torsion \(T^I\), spin-connection \(\omega^{IJ}\), Riemannian curvature \(R^{IJ}\)) can be seen as compound objects made out of the waywiser variables \(\{V^A,A^{AB}\}\). We then generalize this waywiser approach to relativistic spacetimes and exhibit action principles for General Relativity in terms of the waywiser variables for two choices of model spacetimes: De Sitter and anti-De Sitter spacetimes.