Single-Particle Electrodynamics - Assassination Science

Lorentz performed this calculation quantitatively,—using, however, Galilean
rigidity,—and found, in the rest frame of the body,
F self = − 4 3 mq e.m. ˙v + 2 3
q 2
¨v + O(ε), (6.1)
4π
where m q e.m. is the static field mechanical self-energy (5.97) of Section 5.5.8,
and where the terms of order ε and higher in (6.1) disappear in the point
limit ε → 0.
The properties of (6.1) are, arguably, quite astounding. Firstly, we note
that (6.1) has been derived directly from the Lorentz force law; thus, while
it is has not been obtained explicitly from a Lagrangian, it is nevertheless
implicitly contained in the Lagrangian derivation of the Lorentz force law
itself, when the fields E and B appearing therein are taken to be the total
fields, including the self-fields.
Secondly, it was shown by Abraham [3] that the second term in (6.1)
completely describes the force of radiation reaction. (For example, for circular
motion, ˙v is radially inwards, and hence ¨v is antiparallel to the velocity v;
the coëfficient (2/3)(q 2 /4π) gives the correct balancing of the loss of energy.)
Thirdly—and, perhaps, most remarkably—the first term of (6.1) provides
a completely dynamical explanation of how the “inertial mass” of the
charge’s self-field works: it provides a real, quantifiable force opposite to the
acceleration of the charge; one does not have to insert this inertial property
by hand. Of course, the factor of 4/3 in this term of Lorentz’s result is incorrect;
the coëfficient should be unity. It has been shown by a number of
authors, in numerous instructive ways, that this is due to an inappropriate
use of nonrelativistic concepts; we shall discuss this in greater detail shortly.
It may seem that the author is devoting an unjustifiable amount of effort
to simply re-tell a story that is already told in any good electrodynamical
textbook (see, e.g., [113]). Perhaps so. But the author wishes to emphasise
most strongly that the Lorentz method of derivation of the radiation reaction
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