Single-Particle Electrodynamics - Assassination Science

when the particle is in arbitrary motion—i.e., when ˙v, ˙σ, etc., are not zero?
It is tempting to think that one may proceed simply from the static case,
by formulating a Lorentz-covariant expression, and then deeming that this
holds in all frames. But this brings us dangerously close to the error that
Cohn made in his unfortunate 1969 paper [53]: one cannot simply analyse
a static system, boost the results, and hope for the best, since the effects
of acceleration will then automatically be lost—as a static system knows
nothing of acceleration, and hence cannot possibly have terms involving it as
a factor.
Instead, we must proceed a little more carefully. Our plan of attack is
as follows: Firstly, we shall evaluate the full expressions for the arbitrarymotion
retarded fields, obtained in Section 5.4.6, for the extended model
considered in the previous sections—namely, an infinitesimally small sphere
(in its instantaneous rest frame) which has a uniform density of electromagnetic
moment. We shall then (for technical reasons) shrink these expressions
to a point, but then “regularise” them again using a mathematical trick used
in standard electrodynamics textbooks. We shall then compute the threedivergence
of the regularised point particle electric and magnetic fields thus
computed: this will, from Maxwell’s equations, tell us what source densities
are, assuming the expressions of Section 5.4.6 only. Finally, we can then add
extra fields—such as the extra Maxwell field of Section 5.5.3—so that the
Maxwell equations indicate the physically correct source current densities.
To perform this procedure, however, we face a problem of a logistical
nature, which is somewhat of an embarassment to the author: namely, in
carrying out the steps above, we would in fact need to compute almost all
of the aspects of the radiation reaction calculations, that is actually going
to take up the remaining chapter of this thesis. It may therefore seem that
it would have been better if the author had postponed this section to the
end of Chapter 6, by which time all of those prerequisite results had in fact
been presented. But the problem is that the worldline fields are, themselves,
218