Special Relativity Energy-Momentum Relation in terms of Projections

Addendum: Sept. 2, 2022 In the note below we argue that v/c acts as a projection i.e. as cos(theta), but give no direct evidence for this. In a second note posted Sept. 2, 2022 and called Part II we address this issue and show that a photon bouncing back and forth perpendicular to the direction of motion has both a momentum and a rest mass type of term and v/c is cos(theta). Special relativity is often described in terms of a Lorentz transformation. Historically Lorentz considered Maxwell’s electromagnetic equations viewed from a moving frame. Einstein considered motion in a moving frame using light as a measuring device with the assumption that the speed of light is the same in all frames and cannot be surpassed. In this note we try to think of a single equation describing E and p, but with the ability to describe two different objects, namely a particle with rest mass and a photon. The photon is known to have the dispersion relationship E=pc while one may postulate p=E(v) v for a particle with rest mass i.e. a usual flux equation. At first these seem very different, but if one considers Ev/c with c being the largest velocity attainable (as Einstein assumed) then v/c maps into sin(theta) i.e. it is a projection. If v→c then E→E which is the photon result so there can be no extra component. What is the extra component for v