Single-Particle Electrodynamics - Assassination Science

particle in the CACS—and hence is independent of any particular coördinate
frame we choose to use to evaluate components of quantities. On the other
hand, the naïve proper-time derivative of the four-vector Σ α actually furnishes
us with the quantities [ Σ ˙ α ]:
d τ [Σ α ] ≡ [ ˙ Σ α ],
because we generally evaluate the components Σ α in a Lorentz frame, and
then take the derivatives of these components. Thus, the quantity
˙ Σ α + U α ˙U β Σ β
that one frequently sees in the literature should, in the author’s notation, be
surrounded by square brakets—and is indeed simply the right-hand side of
(2.68).
We now turn to the question of finding appropriate names for the quantities
( Σ ˙ ) and [ Σ ˙ ], so that they shall not be confused in the remainder of
this thesis. If one looks, for example, in Jackson’s textbook [113, Sec. 11.11],
one finds that he has, indeed, recognised the fundamental importance of
( ˙ Σ ): he has given it a new symbol F α in his equation (11.166) (to which
we shall return shortly). However, he simply refers to it as “the terms with
coefficient (ge/2mc) in (11.162)”.
Now, it may be acceptable to refer to
this quantity as “the terms with coefficient (ge/2mc) in (11.162)” on one,
or perhaps two, occasions; but if one were to read about “the terms with
coefficient (ge/2mc) in (11.162)” repeatedly, one would no doubt appreciate
the need for a simpler name. Furthermore, we shall shortly have need to consider
proper-time derivatives, similar in nature to “the terms with coefficient
(ge/2mc) in (11.162)”, but for completely different quantities altogether.
We shall therefore refer to ( Σ ˙ ) as the covariant proper-time derivative of
Σ ; and the quantity [ Σ ˙ ] as the partial proper-time derivative of Σ . The justification
for this nomenclature is that it is of course that which is commonly
used in General Relativity (see, e.g., [153]). Although our description above
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