Single-Particle Electrodynamics - Assassination Science

6.3 Infinitesimal rigid bodies
We now consider afresh the problem of deriving the radiation reaction equations
of motion for a point particle carrying electric charge and electric and
magnetic dipole moments, based on the results of the previous chapters.
From the discussion of Section 6.1, we know that it is necessary to “regularise”
a pointlike particle, before we compute self-fields or self-interactions,
if one is to obtain meaningful answers. Since we have chosen to follow the
Lorentz self-interaction method of derivation of radiation reaction, it is only
natural that we employ his technique of expanding the point particle into a
small rigid sphere, of radius ε.
Now, in Chapter 3, we considered the various problems associated with
defining rigidity in a relativistically meaningful way. It was found that, indeed,
it is possible to define rigidity in a meaningful way, but that the constituents
of such a rigid system may well end up “crossing the accelerative
horizon” if the body is subject to a sufficiently large acceleration.
If we consider an infinitesimal relativistically rigid body, however, such a
problem disappears, in the transition to the point limit. For the acceleration
˙v of the system is a finite, “external” quantity, that does not vary as we take
the point limit; hence, the “accelerative horizon” is, at any time of a particle’s
motion, at some particular fixed distance from the centre of the system; and
thus, as we shrink the body smaller and smaller, all of its constituents must
eventually, at some point of this transition, be contained completely within
this horizon.
6.3.1 Orders of expansions
If we are solely concerned with interactions, propagated at the speed of light,
within the infinitesimal sphere—as we are for the self-interaction calculations,—then
clearly the maximum temporal period that will be of relevance
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