Comparison of basic equations of the Kaluza–Klein theory with the nonholonomic model of space–time of the sub-Lorentzian geometry

Abstract

The field equations of gravity coupled to electromagnetism and equations of motion of a charged particle are part of the Kaluza–Klein theory where general relativity is extended to five dimensions. These equations can also be obtained if nonholonomic constrains are imposed on the 5-vector of particle’s velocity. Hence, further development of the general relativity theory can be sub-Riemannian (or sub-Lorentzian) geometry. Gauge transformations become a special case of coordinate transformations in both the Kaluza–Klein theory and the nonholonomic model. Sub-Riemannian geodesics are proved to be equations of motion of a charged particle. The Dirac operator can be extended for a 5-dimensional manifold as a first-order differential operator. Since the base manifold in physics contains the electromagnetic gauge group [Formula: see text], the eigenvalues of the charge operator are always an integer multiplied by the fundamental electric charge.