All Physics Faculty Publications
Document Type
Article
Journal/Book Title/Conference
Classical and Quantum Gravity
Volume
31
Issue
21
Publisher
IOP Publishing
Publication Date
10-16-2014
Arxiv Identifier
1305.6972v2
Abstract
The quotient of the conformal group of Euclidean 4-space by its Weyl subgroup results in a geometry possessing many of the properties of relativistic phase space, including both a natural symplectic form and non-degenerate Killing metric. We show that the general solution posesses orthogonal Lagrangian submanifolds, with the induced metric and the spin connection on the submanifolds necessarily Lorentzian, despite the Euclidean starting pont. By examining the structure equations of the biconformal space in an orthonormal frame adapted to its phase space properties, we also find that two new tensor fields exist in this geometry, not present in Riemannian geometry. The first is a combination of the Weyl vector with the scale factor on the metric, and determines the timelike directions on the submanifolds. The second comes from the components of the spin connection, symmetric with respect to the new metric. Though this field comes from the spin connection it transforms homogeneously. Finally, we show that in the absence of conformal curvature or sources, the configuration space has geometric terms equivalent to a perfect fluid and a cosmological constant.
Recommended Citation
Hazboun, Jeffrey S. and Wheeler, James Thomas, "Time and Dark Matter from the Conformal Symmetries of Euclidean Space" (2014). All Physics Faculty Publications. Paper 2030.
https://digitalcommons.usu.edu/physics_facpub/2030
Comments
Originally published by IOP Publishing in Classical and Quantum Gravity.