Single-Particle Electrodynamics - Assassination Science

G.4.12
Radiation reaction torque
We now compute just one consequence of the radiation reaction results of
Chapter 6, namely, the finite radiation reaction torque on a charged magnetic
dipole, due to its Thomas–BMT motion. It was found in Chapter 6 that the
finite terms in the torque expression, for v=0, are
N self = 1 3 µ2 η 0 σ× { 2 ... σ − σ×( ˙vרv) } .
(G.10)
From the expressions in Section G.4.8, one finds that (G.10) can be written
in covariant form as
(Ṡ) = 2 3 µ2 η 0 U ×Σ × { ( ...
Σ ) + ˙U 2 ( ˙ Σ ) } .
(G.11)
From (G.11), one can find the rate of change of the three-spin σ, in any lab
frame:
˙σ = σ×
{ 1
γ C − 1 }
γ + 1 C0 v + ˙σ T ≡ σ×Ω RR + ˙σ T , (G.12)
where ˙σ T is the Thomas precession contribution to ˙σ, and
C ≡ 2 µ 2
3 s η { ...
0 ( Σ ) + ˙U 2 ( Σ ˙ ) } .
We remove inconvenient constants in Ω RR by defining a related vector Ω ′ RR:
Ω RR ≡ µ2
6πs Ω′ RR.
(G.13)
Apart from the Thomas precession contribution, we use (G.11) and (G.13),
and the expressions in Section G.4.8, to find the general rate of change of
spin in any lab frame:
Ω ′ RR = γ 2 ...
σ + 3γ 4 (v· ˙v)¨σ + γ 4 (v·¨v) ˙σ + 3γ 6 (v· ˙v) 2 ˙σ
+ 3γ 4 (γ + 1) −3 ( ˙v· ˙σ) ˙v + γ 4 (γ + 1) −3 (¨v·σ) ˙v
+ 2γ 4 (γ + 1) −2 ( ˙v·σ)¨v + γ 4 (γ + 1) −1 (v·σ) v
...
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