Since I've given up trying to put out a separate paper quickly on the superstrong magnetic force between highly accelerating ultrarelativistic charges, as I said in the previous post, and have gone back to trying to finish my more general paper on the relationship between electrodynamics and relativistic kinematics, I should report out a little on what I found and didn't find.
The strong attractiveness of the magnetic force in the retarded magnetic acceleration field is already shown in the version posted on arxiv. What I was trying to determine was whether there's an obvious way there can be net attraction in the time symmetric case, as considered by Schild, where the magnetic force due to the advanced field tends to cancel the force due to the retarded field. My idea was that in the ultrarelativistic case the delay and advance angles approach 90 degrees, so maybe it might be possible to change the phase relationship so that the net force is strongly attractive on average.
It turned out that although I was able show tentatively that the retarded and advanced forces don't have to exactly cancel and can easily exceed in magnitude Coulomb repulsion, I wasn't able to generate a net attractive force. I may try further later but for now I have gone back to trying to finish the more general argument.
What I did was to assume two charges were circularly orbiting each other in an approximately circular orbit with an orbit diameter smaller than one one-hundredth the size of a proton, and at a velocity very close to the speed of light. I wrote a matlab program to calculate the full retarded and advanced, and non-radiative and radiative, electric and magnetic fields at the position of one particle due to the other, and accounting for delay and advancement, where the motion was assumed to be circular and periodic, but allowing the accelerations to depart from the strict centripetal acceleration of a pure circular orbit. That is, I let the non-centripetal acceleration affect the fields but not the orbit. Then I looked at the induced acceleration of one particle due to the other, and attempted to construct a configuration where the motions of each particle induced by the other would be consistent. I totally ignored radiation damping, as did Schild, although it's enormous in this configuration.
It turned out to be pretty simple to build a configuration where the motions seem approximately consistent in the time-symmetric electrodynamic sense. A lot more work would be needed to determine if this is a real or meaningful result, and I don't mean to assert that it is. If I had more confidence I could build something convincingly meaningful in a reasonable amount of time, I'd continue to work on it, but for now I think my time is better spent elsewhere.
To illustrate what I'm trying to describe, I captured a plot from my matlab program, see below. Clicking on the figure should expand it. The top two strip plots are what is used to calculate the full em field at the position of the second (test) particle, and then the bottom two plots are the acceleration induced on the test particle. The scales are not very meaningful because the magnitudes depend on how close the velocity is to the speed of light, and the orbital radius, and the invariant particle masses, and in a complicated way.
The top two strip plots show the motion of the charge the generates the field that the test particle moves in. That's the field source particle. The field that the test particle generates isn't allowed to affect the source particle here. The top plot shows that I arbitrarily imposed a strong radial oscillating (at the orbital period) acceleration on top of the constant radial centripetal acceleration in it's circular circular orbit. The second plot is showing that there's no comparatively significant motion in the tangential or axial directions. That's just the noise level when some large positive and negative numbers got added together, at matlab default precision.
Then the time retarded and advanced fields at the test particle position, moving oppositely in a nominally circular orbit, are calculated and the corresponding acceleration of the test particle due to their sum is plotted on the two lower strips. What's interesting to me is that the oscillating radial acceleration of the source particle has induced a similar radial acceleration in the test particle, out of phase such that if it were allowed to act back on the source particle has a hope of leading to a consistent periodic motion, perhaps. There is also a tangential acceleration induced, but it's much smaller in magnitude. The smaller magnitude is in part at least due to the difference in relativistic mass along track versus cross-track.
This would be a rabbit hole to pursue seriously, that one might never emerge from. But it would be fun.
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