Time and dark matter from the conformal symmetries of Euclidean space

TL;DR

This paper explores the geometric structure of a conformal space derived from Euclidean 4-space, revealing properties akin to relativistic phase space, including Lorentzian submanifolds and new tensor fields, with implications for dark matter and cosmology.

Contribution

It introduces a novel conformal geometric framework with phase space properties, identifying new tensor fields and linking the geometry to dark matter and cosmological constant phenomena.

Findings

Existence of Lorentzian submanifolds within Euclidean conformal space.

Identification of two new tensor fields influencing timelike directions.

Geometric terms equivalent to dark matter and cosmological constant in absence of curvature.

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.