Single-Particle Electrodynamics - Assassination Science

A.8.19
Proper time of a particle
The proper time for a classical particle is denoted τ, and is defined as the
cumulative time measured in the particle’s CACS.
Referred to an arbitrary lab frame with coördinates (t, x), the differential
of τ is given by
dτ ≡ ±dt √ 1 − v 2 ,
(A.58)
where the sign is + (−) if the particle’s mechanical energy in the lab frame
is positive (negative).
A.8.20
Overdots
An overdot on a Lorentz-covariant quantity always denotes d τ . Refer to
Sections A.8.21 and A.8.22 for a description of the various ways the propertime
derivative may be computed.
An overdot on a non-covariant quantity always denotes d t .
A.8.21
Convective derivative
The proper-time derivative of a quantity that is “external” to the particle in
question (i.e., is not an intrinsic quality of the particle itself, but is due to
an outside agent; e.g., an externally applied field) is given by the relativistic
convective derivative,
d τ E ≡ (U ·∂)E,
(A.59)
where E is the external quantity in question. Overdots are not to be used
to denote such a derivative.
A.8.22
Kinematical proper-time derivatives
The proper-time derivative of an intrinsic kinematical property of a particle
can be computed in two different ways.
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