The Equations of Motion of a Charged Particle in the Five-Dimensional Model of the General Relativity Theory with the Four-Dimensional Nonholonomic Velocity Space

TL;DR

This paper explores the motion of charged particles in a five-dimensional general relativity model using nonholonomic geometry, revealing that their equations of motion align with classical physics and analyzing the properties of geodesic wavefronts.

Contribution

It demonstrates that the equations of motion for charged particles in this model match those in general relativity, employing sub-Lorentzian geometry and variational principles.

Findings

Equations of motion coincide with classical charged particle dynamics.

Geodesic spheres in a magnetic field have singular points.

Long geodesics are non-optimal, defining a nonholonomic wavefront.

Abstract

We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's maximum principle we proved that the equations of the horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general relativity theory. This is a Kaluza -- Klein problem of classical and quantum physics solved by methods of sub-Lorentzian geometry. We study the geodesics sphere which appears in a constant magnetic field and its singular points. Sufficiently long geodesics are not optimal solutions of the variational problem and define the nonholonomic wavefront. This wavefront is limited by a convex elliptic cone. We also study variational principle approach to the problem. The Euler -- Lagrange equations are the same as those obtained by the Pontryagin's maximum principle if the restriction of the metric tensor on the distribution is the same.