Single-Particle Electrodynamics - Assassination Science

whence the reverse transformation is
r s ≡ r + r ′ , (6.11)
r ≡ 1 2(
rs + r d
)
,
r ′ ≡ 1 2(
rs − r d
)
. (6.12)
Then the integrals, over r d - and r s -space, of the terms in (6.10) are much
simpler: the terms all involve r d , and are independent of r s .
Of course, when we perform the full calculations, we will find that not
all terms will possess these simplifying qualities—recall, we only substituted
the static retarded time expression above; nevertheless, most terms will, in
fact, be of this form, and the remainder can be treated as special cases.
The change of variable (6.11) may seem obvious; indeed, it is usually
glossed over in most accounts of the Lorentz method of derivation. However,
it is, in reality, far from trivial. In the following sections we examine the
subtleties accompanying any use of the transformation (6.11), without due
regard for which one cannot validly proceed any further whatsoever.
6.4.1 The six-dimensional hypervolume
The problem with the tranformation (6.11) from r–r ′ space to r d –r s space
is that our original integrals over r and r ′ were each over finite volumes—
namely, two independent three-spheres—rather than over all space.
The
regions of integration for r s and r d are thus dependent, rather than independent.
The essence of the complication can be seen by considering the simplified
case in which r and r ′ are one-dimensional, rather than three-dimensional:
the double-integral
∫ ε
−ε
∫ ε
dx
−ε
dx ′
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