Single-Particle Electrodynamics - Assassination Science

is in terms of the quantities of Special Relativity, the concepts involved are
of course borrowed directly from the General theory.
At this point, we shall make a simplifying definition: for the remainder
of this thesis, we recognise the fact that the quantity ( ˙U) ≡ 0 is both trivial
and useless, and indeed shall not appear again after this paragraph; but the
quantity [ ˙U] is useful, and indeed shall be used quite frequently; hence, we
shall allow [ ˙U] to be denoted by simply the symbol ˙U, on the understanding
that no confusion will result in practice.
We also note that the partial and covariant derivatives are important not
just for the four-spin Σ , but in fact for practically all proper-time derivatives
of manifestly-covariant quantities. For example, we note that ( ˙ Σ ) itself is
a purely spacelike four-vector in the rest frame of the particle; hence, when
we compute its proper-time derivative, we must likewise distinguish between
the partial and covariant derivatives.
And, since this process, for the covariant
derivative, always yields another purely spacelike four-vector in the
rest-frame of the particle, we find that all covariant derivatives of Σ must
be considered in this way; the final term of (2.68) is essentially the Christoffel
symbol term that is “spat out” for each covariant derivative in General
Relativity.
As another example of the wide-ranging need for the covariant derivative,
we note that, since ˙U is also a four-vector that is purely spacelike in the rest
frame, then its derivative, Ü, also has partial and covariant flavours.
particular, we note that
(Ü) ≡ [Ü] + U ˙U 2 , (2.69)
which appears confusing because ˙U itself is both acting as the quantity being
differentiated, as well as appearing in the “Christoffel symbol” of (2.68).
Now, the Abraham term for the radiation reaction force on a point charge is
often written, manifestly-covariantly, as
Γ µ = 2 3
q 2
4π
{Ü µ + U µ ( ˙U α ˙U α )}
.
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