Single-Particle Electrodynamics - Assassination Science

esponding gamma factor γ(v) only appears twice, in the form γ −2 .
The
velocity-dependencies of the results are, in effect, almost entirely “encapsulated”
in the four introduced quantities σ ′ , n ′ , n ′′ and κ. Thus, even if one
goes to the instantaneous rest frame of the particle, one cannot improve on
the simplicity of the expressions much further: all that happens is that the
aforementioned term in E ′d
2 disappears; the factors κ and γ disappear, being
trivially equal to unity; the dashes are dropped on n ′ , n ′′ , σ ′ and ˙σ ′ ; and we
must make the simple substitution
¨σ ′ −→ ¨σ − ( ˙v·σ) ˙v.
(This last subtlety comes about because, in computing two temporal derivatives
of the FitzGerald spin definition (5.74), a term survives in which both
factors of v have been differentiated into ˙v; it has a coëfficient of unity, rather
than the one-half implied by γ/(γ + 1), because of the quadratic dependence
on v.)
It should, at this point, be noted that the expressions (5.75), (5.76),
(5.77), (5.78), (5.79) and (5.80) are only as simple as they are because they
use the FitzGerald three-spin, σ ′ , and its lab-time derivatives. If expressed in
terms of the standard three-spin, σ, and its derivatives, the results are much
more complicated: there are many more terms, most of which contain explicit
factors of v, γ and (γ + 1). Indeed, the author only invented the FitzGerald
three-spin, originally, in an attempt to obtain just such a simplification. (The
quantity σ ′ was actually introduced by the author, previously, as one of the
many “convenient quantities” useful for the expanded-out dipole equations
of motion (see Appendix F, and [62, 63, 64, 65]), but its usefulness as a
general analytical tool was not realised at the time.) The basic reason why
the FitzGerald three-spin yields simpler expressions than the standard threespin
can be gleaned from the expressions listed in Section G.4: in terms of
σ ′ , the components of the four-spin Σ have the simple coëfficients 1 and
γ 2 ; but in terms of σ, there is, instead of γ 2 , a coëfficient of γ 2 /(γ + 1);
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