Single-Particle Electrodynamics - Assassination Science

(see equation (A.58) of Section A.8.19). In other words, only three of the
four components of z α (τ) are independent, and hence the parametrisation
(2.84) contains a redundant parameter in each order of τ.
To remove these redundant parameters, it would clearly be sufficient to
eliminate the zero-components c 0 i , in favour of the spatial components c i .
2.8.3 Non-covariant parametrisation
While mathematically quite acceptable, the c i of equation (2.84) do not,
however, have a direct connection with one’s intuitive understanding of the
motion of a pointlike particle. Clearly, any other set of three-vectors, that
are in one-to-one correspondence with the c i , will serve the same purpose,
mathematically.
The author suggests that the most natural parametrisation of the path,
that removes all redundant parameters, is in terms of the lab-frame motion
of the particle. In other words, we consider the three-space position of the
particle, as seen in a particular lab frame, as a function of the time coördinate
in this frame:
z(t) = z| 0
+v| 0
t + 1 2 ˙v| 0 t2 + 6¨v| 1 0 t3 + 1 ...
v|
24 0
t 4 + 1 ....
v|
120 0
t 5 + O(t 6 ), (2.85)
where the overdots on the non-covariant three-vector v denote d t (see Sections
A.3.10, A.3.18, A.8.15 and A.8.20), and our choice of Lorentz frame
sets
z| 0
= 0
and
v| 0
= 0.
The Taylor series (2.85) possesses the dual advantages that it corresponds to
what we would have written down as the trajectory of the particle before the
advent of Special Relativity—and hence is in somewhat more contact with
our intuition than a manifestly-covariant expression,—while still containing
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