Does Special Relativity Follow from Wave Nature?

Correction June 28, 2022: The classical wave-equation should read d/dt d/dt partial W(x,t) = vv d/dx d/dx partial W(x,t) 

Historically two types of  “free” mechanical motions were considered,  that of Newtonian mechanics x=vt (v=constant) and classical waves such as waves on a string which were described in terms of Newton’s laws for a continuous medium. It seems these two were considered as fundamentally different phenomena. Equations of motion were developed for classical mechanics as well as electromagnetic phenomena. Maxwell discovered that electromagnetic equations allowed for a classical wavelike solution which he identified as light. In the early 1900s it was realized that to retain electromagnetic equations invariance as seen from a constant moving frame moving, Lorentz transformations were needed. In 1905 Einstein obtained these transformations in a different manner considering that light has the same vacuum speed when viewed from different constant moving frames. Later in the 1920s quantum mechanics was developed using the idea of a probability wave exp(-iEt+ipx) for a free particle beginning with the work of deBroglie who made use of ideas of special relativity.

   In this note we suggest that there seems to be one fundamental “free” mechanical motion, namely that associated with a wave exp(-iEt+ipx) whether one considers a classical wave or a quantum free particle. Classical mechanics is a mapping into a continuous x-t space, we argue, and describes energy and momentum motion in this continuous space. This same description also occurs for classical waves. The medium undergoes a somewhat complicated nonlocal change, but the “velocity” of the wave maps to a continuous local space time i.e. v=x/t=dx/dt where E= pv. E=pv, however, is only a description of energy momentum propagation given in a simple form. It does not describe the overall picture. We argue the same is true for the simple x=vt (v=constant). It seems that an overall wave structure is associated with energy and momentum and for free motion this is linked with exp(-iEt+ipx). If this wave is fundamental one may argue that its phase may remain constant when viewed from different frames moving at constant v.  Special relativity would then follow from this requirement and so is in a sense a consequence of wave behaviour. We investigate these ideas.