Single-Particle Electrodynamics - Assassination Science

Recalling that the naïve differential Ü µ appearing here is in fact the partial
derivative [Ü], we thus see that the Abraham force is in fact simply
Γ = 2 3
q 2
(Ü). (2.70)
4π
The reason for this occurrence of simplicity is that the covariant derivative
is of course just the derivative as seen by the accelerated particle itself , and
is hence the appropriate generalisation of the nonrelativistic result
F = 2 3
q 2
¨v. (2.71)
4π
We likewise find, for the Thomas–Bargmann–Michel–Telegdi equation [214,
25], that the relativistic generalisation of the nonrelativistic torque N is
simply
(Ṡ) ≡ [Ṡ] + U[ ˙U ·S]. (2.72)
Indeed, it is just this quantity that Jackson was referring to in his equation
(11.166). If one uses the expressions listed in Section G.4.8 for the components
of ( ˙ Σ ) in an arbitrary Lorentz frame, one can in fact prove quite
quickly Jackson’s (11.166):
˙σ = 1 γ ( ˙Σ) − 1
γ + 1 ( Σ ˙ )v +
γ2
σ×(v× ˙v). (2.73)
γ + 1
This equation is most important for one to obtain Thomas’s equation for
the ˙σ of a magnetic dipole [113, Sec. 11.11]—indeed, to find the precession
due to any covariant torque N ≡ (Ṡ). It includes—of course!—the Thomas
precession term of the previous section, which automatically emerges through
the use of relativistic kinematics; this effect is independent of the particular
torque N that one may wish to consider; but it is of course (indirectly)
dependent on the force equation of motion, through ˙v.
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