Electricity as a Natural Property of Riemannian Geometry

Field equations of a Riemannian geometry which can be deduced from a Hamiltonian principle have the following general property: Due to the conservation law of Einstein's curvature tensor $R_{\mathrm{ik}}$ there appears in the integration of the field equations a free vectorial function ${\phi }_i$, determined, however, by the conservation law. This has been shown by applying a mathematical formulation of Mach's principle to the extremely weak deformation of a given arbitrary metric. Specifying the Hamiltonian function by the evident condition of gauge invariance this free vectorial function ${\phi }_i$ has all the fundamental properties of the electro-magnetic vector potential: the law of continuity is strictly fulfilled everywhere, the potential equation is to be deduced in first approximation and also the Lorentz' ponderomotive force of a particle. In this theory the material particle is to be considered as a proper solution of the field equations.