Signal Propagation in a Positive Definite Riemannian Space

Abstract

It is commonly accepted that the propagation of signals requires a space which is hyperbolic in time. The indefiniteness of the metric thus established contradicts the natural requirements of a rational metric. The proof is given that a genuinely Riemannian (positive definite) space of fourfold lattice structure is well suited to the propagation of signals, if the $g_{\mathrm{ik}}$ assume very large values along some narrow ridge surfaces. The resulting signal propagation is strictly translational and has the nature of a particle which moves with light velocity (photon). According to this theory the discrepancy between classical and quantum phenomena is caused by the misinterpretation of a Riemannian metric in Minkowskian terms. The Minkowskian metric comes about (in high approximation) macroscopically, in dimensions which are large in comparison to the fundamental lattice constant. Since this constant is of the order ${10}^{-32\mathrm{}}$ cm, this condition is physically always fulfilled.