Test particles in Kaluza–Klein models

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> 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.