Single-Particle Electrodynamics - Assassination Science

Equation (5.28) is, in the above notation, the manifestly covariant expression
for the retarded fields generated by the point charge. It is, however,
useful to further split it into two parts: F q 1 , which has a dependence on ζ of
ζ −1 ; and F q 2 , which has a dependence on ζ of ζ −2 ; while we are at it, we shall
also deal with the ever-inconvenient factor 4π, and the value of the electric
charge, q:
F q ≡ q {
F
q
1 + F q }
2 .
4π
An examination of (5.28), and the definitions (5.24), (5.25) and (5.26), allows
us to slot the terms of (5.28) into F q 1 and F q 2 :
F q 2
= ϕ 3 ζ ∧U,
F q 1 = ϕ 2 ζ ∧ ˙U − ϕ 3 ˙χζ ∧U. (5.29)
5.3.5 Explicit form for the retarded fields
While being mathematically elegant, it is somewhat difficult to appreciate the
true physical content of the expressions (5.29), as they stand. One therefore
usually reëxpresses them in terms of explicit, non-covariant quantities: the
fields E and B in the case of F αβ , and the kinematical quantities v and ˙v in
the case of U α and ˙U α . The lightlike four-separation ζ α ≡ (x − z) α between
the observation point x α and the retarded lightcone four-position z α may be
parametrised as
(x − z) 0 ≡ R,
(x − z) ≡ Rn, (5.30)
where n 2 = 1. Clearly, n can be interpreted as the unit normal in the direction
of the observation point, from the position of the charge as it was at the
retarded time τ ret ; R is the simple three-distance between these temporallyseparated
events. From the relations listed in Section G.4.7, and the defini-
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