Test particles in Kaluza-Klein models
Joao Baptista
TL;DR
The paper develops a generalized Kaluza–Klein framework in which test-particle geodesics on a higher-dimensional space $P = M_4 \times K$ encode 4D mass and electromagnetic charge through internal momentum. By modeling the higher-dimensional metric $g_P$ as a Riemannian submersion data $(g_M, A, g_K)$, it derives coupled geodesic equations that reveal how rest mass $m(s)$ and charges $q_\xi(s)$ vary along trajectories in regions with nonconstant internal geometry or massive gauge fields, and identifies constants of motion in massless sectors. It also analyzes the space of null geodesics, showing that a photon-like dispersion in higher dimensions can reproduce 4D dynamics while implying a natural “unique speed” in the higher-dimensional theory; it discusses how such a framework may alleviate traditional KK obstacles and argues for higher-dimensional null paths as a unifying classical picture. The work provides concrete mass- and charge-variation formulas, explicit constants of motion linked to the Killing algebra of $g_K$, and a detailed treatment of how to choose backgrounds and actions to yield physically reasonable 4D reductions, with notes on the role of Higgs-like scalars and massive gauge fields in shaping 4D particle properties.
Abstract
Geodesics in general relativity describe the behaviour of test particles in a gravitational field. In 5D Kaluza-Klein, geodesics reproduce the Lorentz force motion of particles in an electromagnetic field. This paper studies geodesic motion on a higher-dimensional $M_4 \times K$ with background metrics encoding general 4D gauge fields and Higgs-like scalars. It shows that the classical mass and charge of a test particle become variable quantities when the geodesic traverses regions of spacetime with massive gauge fields, such as the weak force field, or with non-constant Higgs scalars. This agrees with the physical fact that interactions mediated by massive bosons can change the mass and charge of particles. The variation rates of mass and charge along a geodesic are given by natural geometric formulae. In regions where mass is preserved, there are additional constants of motion, one for every abelian or simple summand in the Killing algebra of $K$. The last part of the paper discusses traditional difficulties of Kaluza-Klein models, such as the low $q / m$ ratios in the 5D model. It suggests possible ways to circumvent them. It also remarks the naturalness of a model in which elementary particles always travel at the speed of light in higher dimensions.