Single-Particle Electrodynamics - Assassination Science

The component ∆L 0 is our warning flag: it tells us that, to first order in v,
the two charges are not only separated in space, they are also separated in
time—even though they are “simultaneous” in their rest frame. This failure
of simultaneity is an elementary result in any text on Special Relativity, but
due attention is not always brought to the fact this effect is first order in the
velocity v.
How does Galilean kinematics warn of this complication? It doesn’t. Newton’s
“universal time” admits of no such manipulation.
It is at about this point that one’s faith in nonrelativistic physics generally
takes a sharp turn for the worse. Galilean kinematics is not even correct to
first order in v, in general. Those for whom group theory is second nature
recognise this simply by virtue of the fact that the transformations of Lorentz
group, and those of the Galileo group, are of a completely different inherent
structure (see, e.g., [237, 24, 111, 97, 98, 133]).
How, then, is one to generalise a valid analysis of nonrelativistic physics
into the relativistic domain? The answer is, in one sense, obvious, but on
deeper reflection, somewhat subtle: Galilean kinematics is correct to zeroth
order in particle velocities. One need therefore only obtain the physically
correct equations of motion for a physical system to zeroth order, convert
them to relativistic form, and, voila!, one has the answers one seeks. That
this is indeed a valid procedure may be recognised by recalling that, if one’s
equations are valid in some chosen Lorentz frame, and they are written in
manifestly covariant form, then they are automatically valid in all Lorentz
frames.
There is, however, a catch: The equations of Galilean physics are formulated
in terms of three-vectors and scalars. To convert these equations to
relativistic form, we need to place these three-vectors and scalars into appropriate
Lorentz quantities: four-scalars, four-vectors, four-tensors, six-vectors,
etc. (See Section A.8 for a description of the nomenclature used here.) The
catch is that, a priori, there is no way of fundamentally knowing just which
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