Planar Lorentz invariant velocities with a wave equation application

In this paper we determine the functional form of those planar velocity fields for which the associated system of two ordinary differential equations are automatically invariant under a Lorentz transformation. For planar motion we determine first order partial differential equations for the velocity components $u(x,y,t)$ and $w(x,y,t)$ in the $x-$ and $y-$directions respectively and their general solutions in terms of two arbitrary functions. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz invariant energy–momentum relations and appear not to have been given previously in the literature. For a particular special relativistic model, one example is given involving similarity solutions of the wave equation. An interesting special case gives rise to a family of particle paths which are characterized by a single arbitrary function, and for which the magnitude of their velocities is the speed of light. This is indicative of the abundant possibilities existing in the “fast-lane”.