Single-Particle Electrodynamics - Assassination Science

(often counter-intuitive) complications involved when the system is moving
with an arbitrarily relativistic speed. One therefore often decides to analyse
the system in question in the nonrelativistic limit; and then, after the correct
physics has been ascertained, kinematically “extrapolate” the results to the
relativistic domain.
Such a procedure sounds simple enough. It is, however, a potential minefield.
In this section, we will describe the steps that need to be taken to
ensure that one is not led down an incorrect path; on the other hand, it is
shown that, with suitable care, the process is not as difficult as sometimes
imagined.
Firstly, it must be noted that Galilean kinematics and Lorentzian kinematics
do not fit together as well as one might at first think. On the surface
of it, it may appear that the former is the “first approximation” of the latter:
Galilean expressions appear to be correct up to first order in a particle’s velocity,
v, but not to second order. For example, the mechanical momentum
of a point particle is given, in Galilean kinematics, by
p = mv,
whereas in relativistic kinematics it is given by
p = √ mv = mv + 1 1 − v
2 2 mv2 v + · · · .
But the first-order-in-v rule is often incorrect. Consider a dumbbell model
of an electric dipole: two charges on the ends of a stick. Call their relative
displacement ∆l. Relativistically, this relative displacement is actually the
three-vector part of a four-vector, ∆L α , which has vanishing zero-component
in the object’s rest frame. If this object is boosted by a velocity v, then the
Lorentz transformation (G.1) shows that, to first order in v, the components
of the four-vector ∆L are transformed into
∆L 0 = (v·∆l) + O(v 2 ),
∆L = ∆l + O(v 2 ). (2.81)
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