A better demonstration of the similarity of the anticentrifugal force to the strong magnetic force

The current version of my paper ( 1108.4343v5 ), like previous versions 2-4, has a section that attempts to characterize the anticentrifugal force of the Thomas precession in the highly relativistic limit, and to show that it can, like the strong force, overcome Coulomb repulsion as needed to bind quarks into nucleons.  At the time it was written, over a year ago, and until just a couple of weeks ago, I was thinking that the anticentrifugal force was not present in Maxwell-Lorentz electrodynamics, and said or implied as much in the earlier versions.  I didn't elaborate on this much, but I was thinking that the Maxwell fields didn't contain the strong force, and so although I stated or implied the Lorentz force was incomplete, I didn't expect that strong force could be added to electrodynamics by a modification of the Lorentz force law without an accompanying modification of the electromagnetic field.  Now of course I think this was a mistaken belief, and that the strong force is already apparently present in Maxwell-Lorentz electrodynamics as the magnetic force between highly relativistic mutually-Coulomb-accelerating charges.  This has resulted in an explicit form (if only approximate so far, due to my neglect so far of retardation effects, which cannot be considered insignificant here) of the strong magnetic force, which can be compared with my earlier characterization.  This comparison has forced me to realize the previous characterization is at best confusing and less than clear.

The problem of my initial characterization of the anticentrifugal force, which is in section V of the version at the link above, is that it obtains a force law that is inversely proportional to only the first power of the interparticle separation.  The Coulomb repulsion of course is inversely proportional to the square of the separation, so in order to overcome it, the anticentrifugal force should be inverse to a higher power of separation than two.  The strong magnetic force obtained in section VI is inverse to the third power of separtion, and so meets this expectation.  On the other hand, when I equated the anticentrifugal force magnitude with that of Coulomb repulsion, I got essentially the same formula that I got by equating the Coulomb repulsion with the strong magnetic force (and immediately declared success).  Naturally when I examined this situation further I was perplexed and at least a little disturbed by it.  I'm still in the process of sorting this out, but I think there's probably a straightforward explanation, that there's a hidden dependence on separation in the assumption of near light-like particle velocities, that can contribute additional inverse dependence on separation.  However, while looking into this, I realized there's an easier and I think more straightforward way to see the direct correspondence between the anticentrifugal force and the strong magnetic force.  I put this into a new draft version of my paper, but I don't want to do another update on arxiv just yet, pending addressing the issue of retardation, so I think I will copy it in here instead, for now.

 

The above equation  (303), derived as the anticentrifugal force, is essentially the same as Eq. (34) of my version 5 at the link above, that is derived from the Lienard-Wiechert potentials, if the test and source particles are of equal mass, apart from some gamma factors that still need to be sorted out carefully.   This shows more explicitly than the current arxiv version how the strong magnetic force is the embodiment of the anticentrifugal force of the Thomas precession.