Gravity, Cartan geometry, and idealized waywisers

Details Gravity, Cartan geometry, and idealized waywisers Gravity, Cartan geometry, and idealized waywisers

2012-03-26

arxiv.org/abs/1203.5709v3

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Citations added, several typos corrected

Scientific paper

The primary aim of this paper is to provide a simple and concrete interpretation of Cartan geometry by pointing out that it is nothing but 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 $A^{AB}=A_\mu^{\ AB}dx^\mu$ 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-connecion $\omega^{IJ}$, Riemannian curvature $R^{IJ}$) can be seen as merely different characterizations of the change of contact point. 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 {\em spacetimes}: De Sitter and anti-De Sitter spacetimes. In one approach we treat the contact vector $V^A$ as a non-dynamical {\em \`a priori} postulated object, and in another $V^A$ is treated as a dynamical field subject to field equations of its own. We do so without the use Lagrange multipliers and the resulting equations are shown to reproduce Einstein's General Relativity in the case of vacuum.

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