1 Kaluza-Klein Theory and Lorentz Force Geodesics

We examine a Kaluza-Klein-type theory of classical electrodynamics and gravitation in a five-dimensional Riemannian geometry. Based solely on the condition that the electrodynamic Lorentz force law must describe geodesic motion in this five-dimensional geometry, it appears possible to place all of Maxwell’s electrodynamics on a solid geometrodynamic footing. We make no choice as between the fifth dimension being timelike or spacelike, but simply point out the impact in those places where this choice makes a difference. In the end, we deduce the Maxwell stress energy tensor from a four-dimensional variation applied to the five-dimensional geometry, and in the process, learn that this fifth dimension must be spacelike