Single-Particle Electrodynamics - Assassination Science

B.5 Four-vector cross-product
The four-cross-product A×B×C has explicit components
A×B×C ≡ ( A·B×C, A 0 B×C + B 0 C ×A + C 0 A×B ) .
(B.45)
Relation (B.45) illustrates explicitly the invariance of the four-cross-product
under a cyclic permutation of its three four-vector entries:
A×B×C ≡ B×C ×A ≡ C ×A×B.
(B.46)
B.6 Radiation reaction gradients
The following identities are of use when computing the gradient terms in
the radiation reaction calculations of Chapter 6. (The three-vectors A and
B are “external” quantities—such as ˙v—that are independent of r.) Note
that, since the resulting quantities are often to be integrated over all of r d –r s
space, symmetry may be used to eliminate many terms from the calculations.
(σ·∇)rd
m = mrd
m−1 (n d·σ),
{ }
(σ·∇)rd m n d = rd
m−1 σ + (m − 1)(nd·σ)n d ,
{
(σ·∇)rd m (n d·A) p = rd
m−1 p(A·σ)(nd·A) p−1
+ (m − p)(n d·A) p (n } d·σ) ,
{
(σ·∇)rd m (n d·A) p n d = rd
m−1 p(A·σ)(nd·A) p−1 n d + (n d·A) p σ
{
(σ·∇)rd m (n d·A)(n d·B)n d = rd
m−1 (nd·A)(n d·B)σ
+ (m − p − 1)(n d·A) p (n d·σ)n d
}
,
+ (A·σ)(n d·B)n d
+ (B·σ)(n d·A)n d
+ (m − 3)(n d·A)(n d·B)(n d·σ)n d
}
,
(σ·∇)rd m r s (n s·A) = rd m (A·σ) + mrd
m−1 r s (n d·σ)(n s·A),
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