Single-Particle Electrodynamics - Assassination Science

6.5.11 Final redshift expression
Finally, from the definition (3.21), we have
λ = 1 + 1 2 r d(n d· ˙v) + 1 2 r s(n s· ˙v). (6.66)
6.5.12 Retarded field expressions
We can now use the preceding expressions to compute the various retarded
fields of the particle, generated by the sending constituent r ′ and received
by the receiving constituent r.
It will be noted that, in all cases, the “accelerative redshift” factor λ(r)—
that relates the constituent proper-time derivatives to body proper-time
derivatives—will be multiplied in as the final step of the self-interaction calculations.
Most of these calculations will involve the self-generated electric
and magnetic fields as a simple factor; for these calculations, it is convenient
to compute λ times the electric or magnetic field from the outset.
On the other hand, the self-interaction terms involving spatial gradients
of the electromagnetic fields must be treated more carefully. Here, we must
note that the operations of multiplying by λ(r), and that of taking the spatial
derivative ∇ r in r-space, do not commute, because λ(r) of course has an
explicit dependence on r. Thus, we may not compute (for example)
(σ·∇)λ(r)E(r);
we must, rather, compute
λ(r)(σ·∇)E(r).
The explicit expressions for the retarded self-fields of the particle are
lengthy; we list them explicitly in Section G.6.19 of Appendix G.
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