Single-Particle Electrodynamics - Assassination Science

where we set
u(τ) = r (3.7)
because we do not wish the body to rotate. This is, indeed, the transformation
that the author had used, until after the computer algebra programs of
Appendix G had been completed and the final equations of motion obtained
(see Section G.2.1). However, (3.7) is, in fact, incorrect, for the following
subtle reason: The non-covariant three-vector u is, according to (3.7), constant
in time:
˙u = 0. (3.8)
But we know, from Section 2.6.8, that the three-vector u should Thomas
precess, as seen from the (fixed) lab frame, as the velocity and acceleration
of the body increase from zero, so that the corresponding u measured in the
CACS does not precess. In other words, by using the transformation (3.6)
with (3.7), we would, in fact, be unwittingly specifying that the body should
start to rotate—as seen by the particle itself, in its CACS,—in such a way
so that, when this rest-frame rotation is added to the Thomas precession in
the lab frame, the net result would be zero.
This rest-frame rotation would violate the very assumptions underlying
the construction of the relativistically rigid body (see Section 3.2.2), and
hence the conclusions originally drawn by the author from the use of (3.7)
were rendered invalid. Somewhat surprisingly, the final equations of motion
obtained by the author (to be described in Chapter 6) were unchanged by
his correction of this oversight (to be described below)—even though all
expressions up to the penultimate step were affected. The author has no
clear understanding of this phenomenon as yet; it is discussed further in
Section 6.9.
Returning to the derivation under consideration, it is clear that the correct
way to compute the trajectory of the constituent r is to add the required
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