The experimental fact of the constancy of the speed of light has already in pre- relativistic physics led to the conclusion that certain physical quantities have to be formulated in a velocity dependent way in order for the equations of electrodynamics to remain invariant in reference frames moving relatively to each other. In this sense, Lorentz postulated for instance an actual length contraction of moving objects, whereas Larmor predicted moving clocks to actually run slower. As if these theories were not already unphysical enough, Einstein later elevated them to a purely metaphysical concept by postulating that time and space itself are the quantities to be transformed if objects are moving. As explained under lightspeed.htm
, these conclusions were reached because the usual concept of speed (and with it the inherent frame dependence of the latter) was incorrectly applied to light signals. As a consequence, Newton's second law
(1) ma = F0
is not valid anymore in Special Relativity, but instead one has
(2) ma = [F0 -(F0.β)β]/γ ,
(3) β = v/c
(4) γ = 1/√(1-β2) .
(bold letters shall indicate vectors in all equations above and in the following).
Because of the above mentioned inconsistency of Einstein's interpretation of the constancy of the speed of light, Eq.(2) is in principle without any theoretical foundation, but there is experimental evidence (from particle accelerators etc.) that the dynamics of particles with a speed approaching c can indeed be closely described by this function. However, on a purely empirical basis, the only possible interpretation of Eq.(2) would be to assume that the interaction force is velocity dependent, and if one sets
(5) F = [F0 -(F0.β)β]/γ ,
the Newtonian form
(6) ma = F
is indeed recovered. There is consequently no need to give up the usual notion of mass and acceleration for relativistic velocities if one transforms the electric field by
(7) E = [E0 -(E0.β)β]/γ ,
and the magnetic field by
(8) v×B = [v×B0 -((v×B0).β)β]/γ = v×B0/γ ,
are the field values experienced by a particle stationary to the objects creating the electric or magnetic field (as given by the usual Coulomb law and Biot- Savart law).
Although Eqs.(7) and (8) yield exactly the same particle dynamics as Special Relativity would predict, this is now fully interpreted as a property of the physical interaction rather than metaphysically as a consequence of the nature of space and time. It is therefore in principle not justified to extend these transformations to areas other than the movement of charged particles in electric and magnetic fields (because the speed of light c is only associated with the latter). In any case, c is only a limiting velocity as far as the relative speed of the interacting particles is concerned. With regard to a given observer, a particle can obviously exceed c if accelerated successively by co-moving field sources.
With regard to stationary particle accelerators, it is of interest to note that a velocity dependent Coulomb and Lorentz force in the suggested form limits the maximum energy to just 0.25 MeV for electrons and 450 Mev for protons. This would be consistent with my alternative theory for Synchrotron Radiation
(although the theory would have to be modified somewhat such that the relativistic γ-factor does in fact depend on the ratio (velocity relative to the center of mass)/(speed of light)).