NonZero Rest Mass Free Particle Action from Electromagnetism?

The Lorentz transformation of (x,t) originally followed from an analysis of Maxwell’s EM equations and included c the speed of light. Later in 1905 Einstein (1) developed an alternative derivation for a particle, but used a photon for measurement and so c again appeared In a previous note (2) we argued that the classical action A for a free particle (nonzero rest mass) (relativistic -mo sqrt(1-vv/cc) or nonrelativistic t.5movv) may be written with v=x/t and x and t varied independently leading to -d/dx partial A = p A and d/dt A = -E. This leads to a solution A = -Et+px. One may note that the relativistic action contains c the speed of light which normally appears in electromagnetic theory. In this note we examine how A=-Et+px arises as a math solution of the electromagnetic continuity equation: d/dt partial energy density + d/dx Poynting vector = 0. Thus A has roots in Maxwell’s electromagnetic equations, naturally includes c and is linked to Lorzentz transformations. It may also be seen that A represents two dispersion relationships: dA=0 = -E dt + p dx which pertains to light and classical waves and dA=0= -t dE + x dp . If this second case is linked with p=Ev which is the dispersion relationship for a particle with rest mass, one may show that E=mocc/ sqrt(1-vv/cc using dp/dE = (dp/dv)/ (dE/dv)). Thus we argue that A=-Etc + pxc follows from electromagnetism hence “c” appears, but is consistent with two dispersion relationships, one for light (or a classical wave) and the other for a particle with nonzero rest mass i.e. p=Ev