Geometric Implications of Maxwell's Equations

Maxwell's synthesis of the varied results of the accumulated knowledge of electricity and magnetism, based largely on the searching insights of Faraday, still provide new issues to explore. A case in point is a well recognized anomaly in the Maxwell equations: The laws of electricity and magnetism require two 3-vector and two scalar equations, but only six dependent variables are available to be their solutions, the 3-vectors $\bf E$ and $\bf B$. This leaves an apparent redundancy of two degrees of freedom (J. Rosen, AJP $\bf 48$, 1071 (1980); Jiang, Wu, Povinelli, J. Comp. Phys. $\bf 125 $, 104 (1996)). The observed self-consistency of the eight equations suggests that they contain additional information. This can be sought as a previously unnoticed constraint connecting the space and time variables,$ \bf r$ and $t$. This constraint can be identified. It distorts the otherwise Euclidean 3-space of $\bf r$ with the extremely slight, time dependent curvature $k(t)=R_{\rm curv}^{-2}(t)$ of the 3-space of a hypersphere whose radius has the time dependence $dR_{\rm curv}/dt=\pm c$ nonrelativistically, or $dR_{\rm curv}^{\rm Lor}/dt=\pm ic$ relativistically. The time dependence is exactly that of the Hubble expansion. Implications of this identification will be explored.