Geodesics in a Toroidal space-time

TL;DR

This paper constructs a Weyl space-time based on a toroidal Euclidean metric, analyzes its properties, and investigates geodesic motion within this unique geometric setting.

Contribution

It introduces a novel Weyl space-time derived from toroidal coordinates and examines geodesic behavior in this context, combining analytical and computational methods.

Findings

Weyl equations are satisfied in cylindrical coordinates

Geodesic motion along the symmetric axis is characterized

Geodesic motion along radii of the singularity is analyzed

Abstract

We take a three dimensional Euclidian metric in toroidal coordinates and consider the corresponding Laplace equation. The simplest solution of this equation is taken. Based on this we build a Weyl space-time. This space-time is transformed to cylindrical coordinates. It is shown by using Mathematica that Weyl equations in cylindrical coordinates are satisfied. Geodesic motion is considered along the symmetric axis as well as along the radii of the singularity, which is the cause of the space time.