4-Dimensional optics, an alternative to relativity

Abstract

The starting point of this work is the principle that all movement of particles and photons in the observable Universe must follow geodesics of a 4-dimensional space where time intervals are always a measure of geodesic arc lengths, i.e. $c^2(\mathrm{d}t)^2 = g_{\alpha \beta} \mathrm{d} x^\alpha \mathrm{d} x^\beta$, with $c$ is the speed of light in vacuum, $t$ time, $g_{\alpha \beta}$ and the metric tensor; $x^\alpha$ represents any of 4 space coordinates. The last 3 coordinates $(\alpha = 1,2,3)$ are immediately associated with the usual physical space coordinates, while the first coordinate $(\alpha=0)$ is later found to be related to \emph{proper time}. The work shows that this principle is applicable in several important situations and suggests that the underlying principle can, in fact, be used universally. Starting with special relativity it is shown that there is perfect mapping between the geodesics on Minkowski space-time and on this alternative space. The discussion than follows through light propagation in a refractive medium, and some cases of gravitation, including Schwartzschild's outer metric. The last part of the presentation is dedicated to electromagnetic interaction and Maxwell's equations, showing that there is a particular solution where one of the space dimensions is eliminated and the geodesics become equivalent to light rays in geometrical optics. A very brief discussion is made of the implications for wave-particle duality and quantization.