Single-Particle Electrodynamics - Assassination Science

( Σ ...
) 0∣ ∣ ∣v=0 = 0,
( Σ)
...
∣ = σ ... + ˙v 2 ˙σ + 1
v=0 2 ( ˙v·σ)¨v − 1 (¨v·σ) ˙v,
2
G.4.9
FitzGerald spin derivatives
The spin derivatives computed in Sections G.4.7 and G.4.8 were given in
terms of the three-spin σ; this vector is intuitively understandable, and
has been universally used historically. However, as noted in Section G.4.5,
some theoretical results are considerably simplified algbraically if rewritten
in terms of the FitzGerald three-spin σ ′ . We now recompute the partial and
covariant derivatives of Σ in terms of σ ′ .
It may be shown that
σ = σ ′ + γ 2 (γ + 1) −1 (v·σ ′ )v;
to verify this, simply substitute σ ′ into the right-hand side; one finds
σ = σ.
Using σ ′ in the definition of Σ , and then differentiating the resulting expressions
anew, both partially and covariantly, we find
Σ 0 = γ 2 (v·σ ′ ),
Σ = σ ′ + γ 2 (v·σ ′ )v,
[ Σ ˙ 0 ] = γ 3 (v· ˙σ ′ ) + γ 3 ( ˙v·σ ′ ) + 2γ 5 (v· ˙v)(v·σ ′ ),
[ ˙Σ] = γ ˙σ ′ + γ 3 (v·σ ′ ) ˙v + γ 3 (v· ˙σ ′ )v + γ 3 ( ˙v·σ ′ )v + 2γ 5 (v· ˙v)(v·σ ′ )v,
[ ¨Σ 0 ] = γ 4 (v· ¨σ ′ ) + 2γ 4 ( ˙v· ˙σ ′ ) + γ 4 (¨v·σ ′ ) + 2γ 6 ˙v 2 (v·σ ′ )
+ 5γ 6 (v· ˙v)(v· ˙σ ′ ) + 5γ 6 (v· ˙v)( ˙v·σ ′ ) + 2γ 6 (v·¨v)(v·σ ′ )
+ 10γ 8 (v· ˙v) 2 (v·σ ′ ),
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