Single-Particle Electrodynamics - Assassination Science

frame arbitrary.
2.8.1 Covariant Taylor series expansion
We now make the following assumption: the worldline of the particle, z(τ),
is an analytical function of τ.
Without this assumption, it is practically
impossible to proceed; we shall, however, leave philosophical questions on
this topic aside, for the purposes of this thesis.
Expanding z α (τ) as a Taylor series about τ = 0, we have
z α (τ) ≡ c α 0 + c α 1 τ + c α 2 τ 2 + c α 3 τ 3 + c α 4 τ 4 + c α 5 τ 5 + c α 6 τ 6 + O(τ 7 ), (2.84)
where the c α i are constants, dependent on the physical motion around τ = 0.
(See Section A.3.17 for a description of the +O( ) notation.)
The keeping of terms up to sixth order in τ in (2.84) is not an arbitrary
choice: the considerations of this thesis require precisely this many orders be
retained, and no more.
2.8.2 Redundancies in the covariant parametrisation
By setting the origin of spacetime to be at the zero event, we have set
c α 0 = 0
in (2.84); the choice of zero velocity at τ = 0 likewise sets
c α 1 = (1, 0).
However, it is clear that the manifestly covariant parametrisation (2.84) still
contains a greater number of parameters than is required to fully specify the
path of the particle. To see this, one need only recall that, once the threeposition
z(τ) of the particle is specified for all proper time, the lab-time of
the particle, t(τ), is automatically specified, by virtue of the identity
dt ≡
dτ
√
1 − v 2 (τ)
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