Single-Particle Electrodynamics - Assassination Science

and employing the FitzGerald spin three-vector σ ′ of Section G.4.5,
σ ′ ≡ σ −
γ (v·σ)v, (5.74)
γ + 1
one finds (after an excruciating amount of algebra) that
{ [n−v]× ¨σ ′
E ′d
+ ˙v× ˙σ ′
1 = n×
+ (σ′·n)[n−v]רv
[1−v·n] 3 [1−v·n] 4
( ( ˙σ′·n)
+ 3
[1−v·n] + (σ′·n)( )
}
˙v·n)
[n−v]× ˙v ,
4 [1−v·n] 5
E ′d
2 = 3 ( ˙σ′·{[n−v] − v×(n×v)}) [n−v] ˙σ ′
−
[1−v·n] 4
γ 2 [1−v·n] 3
+ 3 (σ′·{[n−v] − v×(n×v)}) n×([n−v]× ˙v)
[1−v·n] 5
+ 3 (σ′·n) ( ˙v·{[n−v] − v×(n×v)}) [n−v]
[1−v·n] 5
− σ′ ×([n−v]× ˙v) + ( ˙v·n)σ ′
[1−v·n] 3
+ ( ˙σ′·v)[n−v]
[1−v·n] 3 ,
E ′d
3 = 3 (σ′·{[n−v] − v×(n×v)}) [n−v] σ ′
−
γ 2 [1−v·n] 5
γ 2 [1−v·n] . 3
In order to obtain these results, it is necessary to use the relations listed in
Section G.4 extensively; many of the hundreds of terms that result cancel
among themselves; the remainder simplify into the expressions above. The
results are further simplified—notationally, at least—by using the convenient
quantities n ′ , n ′′ and κ of (5.38), (5.39) and (5.40). One then finds
{
E ′d
1 = κ 3 n× n ′ × ¨σ ′ + ˙v× ˙σ ′ + κ(σ ′·n)n′ רv
+ 3κ [ ( ˙σ ′·n) + κ(σ ′·n)( ˙v·n) ] }
n ′ × ˙v , (5.75)
{
E ′d
2 = 3κ 4 ( ˙σ ′·n′′ )n ′ + κ(σ ′·n′′ )n×(n ′ × ˙v) + κ(σ ′·n)( }
˙v·n ′′ )n ′
{
− κ 3 γ −2 ˙σ ′ + σ ′ ×(n ′ × ˙v) + ( ˙v·n)σ ′ − ( ˙σ ′·v)n }
′ , (5.76)
E ′d
3
{
= γ −2 κ 3 3κ 2 (σ ′·n }
′′ )n ′ − σ ′ . (5.77)
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