Space-Time--Time

Abstract

Space-time--time is a natural hybrid of Kaluza's five-dimensional geometry and Weyl's conformal space-time geometry. Translations along the secondary time dimension produce the electromagnetic gauge transformations of Kaluza--Klein theory and the metric gauge transformations of Weyl theory, related as Weyl postulated. Geometrically, this phenomenon resides in an exponential-expansion producing ``conformality constraint'', which replaces Kaluza's ``cylinder condition''. The curvature tensors exhibit a wealth of ``interactions'' among geometrical entities with physical interpretations. Unique to the conformally constrained geometry is a sectionally isotropic, ultralocally determined ``residual curvature'', useful in construction of an action density for field equations. A space-time--time geodesic describes a test particle whose rest mass m and electric charge q evolve according to definite laws. The particls's motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a gradient d(ln phi), where the scalar field phi is essentially the space-time--time residual radius of curvature. The particle appears suddenly at an event E1 with q = -phi(E1) and vanishes at an event E2 with q = phi(E2). At E1 and E2 the phi-force infinitely dominates the others, causing E1 and E2 to tend to occur near where phi has an extreme value; application to the modeling of orbital transitions of atomic electrons suggests itself. The equivalence of a test particle's inertial mass and its passive gravitational mass follows from the gravitational force's proportionality to m. No connection is apparent between m and active gravitational mass or between q and active electric charge, nor does the theory seem to require any.