Einstein, Weyl and Asymmetric Geometries

TL;DR

This paper explores the relationship between asymmetric geometries, Weyl's conformal geometry, and Einstein's spacetime, revealing implications for light-like intervals, spacetime oscillations, and quantum models like hydrogen atom behavior.

Contribution

It demonstrates how projective changes relate asymmetric connections to Weyl's geometry and analyzes the effects on spacetime intervals and quantum phenomena.

Findings

Weyl's geometry is conformal and linked to asymmetric geometry via projective change.

Integrability conditions imply microscopic spacetime oscillations.

Qualitative agreement with Bohr's hydrogen model and Schrödinger's mechanics.

Abstract

In a previous paper, we presented new results on non-Riemannian geometry. For an asymmetric connection, we showed that a projective change in the symmetric part generates a vector field that is not arbitrary, but is the gradient of a non-arbitrary scalar field. As a consequence, Weyl's geometry is a conformal differential geometry and is associated with asymmetric geometry by this projective change. In the present paper, important differences between light-like and non-light-like intervals are analysed. We show that integrability condition in Weyl's geometry implies microscopic spacetime oscillations of Weyl's and Einstein's geometries. We show that the integrability condition in Weyl's geometry, together with the condition that null and massive particles interact locally in an Einstein spacetime. We construct an equation for linear transverse waves and make a qualitative application in a hydrogen atom, which, from qualitative viewpoint, agrees with Bohr's hydrogen model and Schrodinger's quantum mechanics.