Relationship Between Zero and NonZero Rest Mass in Special Relativity

The Lorentz transformation was discovered by Lorentz by considering the invariance of Maxwell’s electromagnetic equations as seen by a moving frame. These same equations yield a solution for a photon which shows its wavelike behaviour.. Shortly after, Einstein obtained the same Lorentz transformations as well as energy-momentum ones by considering motion of a particle with rest mass together with motion by a photon. This lead to the well-known nonzero rest mass equations E=mcc, E=mo/sqrt(1-(v/c)(v/c)) and p=Ev for a particle with a rest mass, but the transformation also applied to a photon although with a different dispersion relationship i.e. E=pc (a classical wave relationship). About two decades later, de Broglie linked particle behaviour to photons and argued for a matter wave. In previous notes (1) we argued that special relativity i.e. the Lorentz transformation may be derived strictly for a particle with rest mass without the need to include electromagnetic equations or photons. The result, however, includes a “special” constant velocity to allow for proper units. Furthermore this velocity is the same in all constant moving frames. In such an approach, this special velocity is not defined. Furthermore, p=Ev and E=mo/sqrt(1-(vv/cc)) so as mo→0 energy and momentum also tend to zero. There is, however, electrostatic energy and force associated with a charge which seems to have nothing to do with rest mass. Thus p=Ev and nonzero rest mass cannot be the full picture it seems. In other words, for any (p,E) one wishes to use the same transformation, so there may be p,E pairs which do not correspond to a particle with nonzero rest mass. One might postulate a particle with momentum p which does not have a rest mass, but may still create impulse reactions within the electrostatic potential just as a particle with rest mass may create impulses. In such a case, the “special” single velocity might apply to this particle and it must be the same as viewed in any frame. Given that the special velocity is already part of the special relativistic transformation developed for a particle with rest mass, one may argue that this transformation should contain two solutions, one for a nonzero rest mass particle and one for a zero rest mass one. In particular, one may find these using dA=0 where A= -Et+px a Lorentz invariant. We have already argued this in previous notes. Given that special relativity contains solutions for both zero and nonzero rest mass particles and that zero rest mass particles exhibit wave properties, one may argue one should use this theory to argue for wave properties for a particle with nonzero rest mass we suggest. The difference is that for a particle with rest mass there is a special frame with mo, x=0 and t=to such that x and to (time) are decoupled. (x=0, and t=to) have already been shown to follow from any general (x,t) in -Et+px in (1). A Lorentz transformation yields x/t=v and one may show that dA=0 and that -dE/dv x + dp/dv t =0 and -E dt/dv + p dx/dv = 0 separately. For the photon solution one may use dA=0 = -Edt + pdx. This couples t and x, but is also linked with a scaling time and length proportional to 1/E and 1/p. For a photon these two are coupled to give the velocity. For a particle with rest mass, 1/E and 1/p are decoupled because the velocity follows from a Lorentz boost. Nevertheless 1/E and 1/p physically exist and delineate space and time, they just do not “create” velocity as in the photon case. Thus we argue that special relativity is really a dual theory describing both particles with and one special case without rest mass and that one has to consider the full set of similarities between the two including scaling of x and t in -Et+px which gives rise to “wave” type motion for both particles with rest mass and without