2012/03/26 by Hans Westman, H. F. Westman, Westman, H. F. +3
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #gr-qc
paper · pdf · doi:10.48550/arxiv.1203.5709
Some typos corrected and presentation cleaned up in places. Much of the content of this paper may also be found in the larger and more recent paper arXiv:1411.1679
openalex publication_date 2012/03/26 · arxiv created 2015/02/16 · arxiv updated 2015/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \VA(x),AAB(x)\, each of which admit simple interpretations: VA is the point of contact between the waywiser's idealized wheel and the manifold whose geometry one wishes to characterize, and AAB=Aμ\phantomμ ABdxμ 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 gμν, affine connection Γρμν, co-tetrad eI, torsion TI, spin-connection ωIJ, Riemannian curvature RIJ) can be seen as compound objects made out of the waywiser variables \VA,AAB\. 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.