Lorentz Transformation of Photon EM Field Energy and Poynting Momentum and Phase Invariance Physically Defining a Wave

 In general, given a charge at rest, its EM field energy  .5eo Integral E” dot E” dv, where E” is electric field in the rest frame, does not transform as the fourth component of a 4-vector. In the moving frame, the field energy is:   .5eo Integral E dot E + .5/uo B dot B and the Poynting momentum 1/uo Integral dv ExB. 

   In the moving frame, one also has the equations:  D’Alembertian phi = 1/(4*3.14 eo) charge density and D’Alembertian A = uoeo /(4*3.14 eo) v charge density for a point charge. If one sets the charge density to 0, then exp(-iwt+ikx) is a solution for phi, where w/k=c, the speed of light in a vacuum. 

   In a previous note, we argued that one may set phi”= exp(-iwt+ikx) and A”= exp(-iwt+ikc) (1,1,1) and obtain E” electric and B” magnetic fields perpendicular to each other and to the direction of motion, i.e. x direction and of equal magnitude.  For these solutions, one may show that:  Integral dx  .5eo Integral E dot E + .5/uo B dot B / Integral dx is  w  (we set c=1), and that the average Poynting momentum is k. Thus, there is a physical link for the parameters w,k. For a rest charge, however, these energies and momentum do not transform like a 4-vector. We show, however, that for the phi” and A” chosen above, the em field energy and Poynting momentum do transform as 4-vectors. 

   It is already known that exp(-iwt+ikx) is a solution in one frame, and exp(-iw1t’+ik1x) in another, such that c= w/k=w1/k1, but that does not prove that one has a Lorentz invariant phase, By directly transforming phi” and A”, however, one may show explicitly that em field energy and poynting momentum, equal to w and k do transform as a 4 -vector. This means that -iwt+ikx is a Lorentz invariant, i.e. the phase is a Lorentz invariant. We argue there is physical relevance to the phase of exp(-iwt+ikx), namely that one has physical implications to the wave solution because the top of a crest in one frame is the top of a crest in another.