Single-Particle Electrodynamics - Assassination Science

where we are taking as understood the fact that the equation is referring to
the instant t = 0.
Now let us turn our attention to the description of the particle, at time
t = dt, in the MCLF . The three-velocity of the particle, as seen in the MCLF,
is at this time given by
v(dt) = ˙v dt + O(dt 2 ), (2.64)
where ˙v is its three-acceleration in the MCLF at t = 0. We may thus apply
the Lorentz transformation (G.1) to the four-velocity U, by the three-velocity
˙v dt:
[U(dt)] = (1, ˙v dt) + O(dt 2 ),
and hence
[ ˙U] = (0, ˙v). (2.65)
Now, let us pretend, for the moment, that we wished to have a way to connect
the rate of change ( ˙U) in the CACS, equation (2.62), with the rate of change
[ ˙U] in the MCLF, equation (2.65). This is of course a hypothetical scenario,
since the rate of change of U in the CACS is trivially zero, and hence ( ˙U)
is not a quantity that can be in any way useful to us; but nevertheless the
procedure is intructive: We could write
( ˙U) = [ ˙U] − (0, ˙v). (2.66)
The connection (2.66) reminds us that, while the CACS and MCLF coïncide
at t = 0, they fail to coïncide, in general, for t > 0, and hence time
derivatives of the components (U) in the CACS are not equivalent to the
time derivatives of the components [U] in the MCLF.
We now apply the same considerations to the four-spin Σ . In the MCLF,
at t = dt, we may again apply the Lorentz transformation (G.1), by the
velocity ˙v dt of (2.64), to the components of the four-spin. Before doing so,
however, we must again recall that, in the time interval dt, the three-spin σ
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