On the embedding of spacetime in five-dimensional Weyl spaces

TL;DR

This paper explores how Weyl geometry can be used to embed four-dimensional spacetime into five-dimensional spaces, analyzing stability, confinement, and the influence of Weyl fields on particle motion.

Contribution

It extends Riemannian theorems to Weyl geometries and investigates local embeddings, confinement, and stability in five-dimensional Weyl spaces with warped geometries.

Findings

Riemannian spacetime can be locally embedded in Weyl bulk

Weyl fields influence confinement and geodesic stability

Weyl scalar fields can model quantum confinement effects

Abstract

We revisit Weyl geometry in the context of recent higher-dimensional theories of spacetime. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian spacetime may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical confinement and the stability of motion of particles and photons in the neighbourhood of branes for the case when the Weyl bulk has the geometry of a warped product space. We show how the confinement and stability properties of geodesics near the brane may be affected by the Weyl field. We construct a classical analogue of quantum confinement inspired in theoretical-field models by considering a Weyl scalar field which depends only on the extra coordinate.