Single-Particle Electrodynamics - Assassination Science

The first way is to simply measure the time-derivatives of the components
of covariant quantities, in the MCLF of the particle: we [65] refer to this as
the partial proper-time derivative of the quantity. Using the notation (A.56),
we have
d τ
{
[C α ] } ≡ [Ċα ].
(A.60)
The second way to define the proper-time derivative of a kinematical
quantity is to compute it as seen in the CACS. In other words, we need to
compute the proper-time derivative of the covariant quantity, in an abstract
way, along the particle’s motion, and then evaluate its components (if required)
in some Lorentz frame: we refer to this as the covariant proper-time
derivative. Using the notation (A.57), we have
{
dτ C } α
≡ ( Ċ) α . (A.61)
It is shown in Chapter 2 that the connection between the partial and
covariant proper-time derivatives, for an arbitrary kinematical four-vector
C, is given by
(Ċ)α = [Ċα ] + U α ( ˙U ·C).
(A.62)
A.8.23
Four-velocity of a particle
The four-velocity of a classical particle is denoted U α , and is defined as
U α ≡ ż α .
(A.63)
A.8.24
Mechanical four-momentum
The mechanical four-momentum of a classical particle of mass m is denoted
p α , and is defined as
p α ≡ mU α .
A massless particle is a particle for which m = 0. It is assumed that
massless particles have finite mechanical momentum components. Thus, τ
and U α are undefined for massless particles.
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