Single-Particle Electrodynamics - Assassination Science

More important is the fact that they have used a (+, +, +, −) metric,
whereas we use a (+, −, −, −) metric. The fact that they call the temporal
component X 4 , rather than X 0 , is irrelevant: they (in contradistinction to
Kolsrud and Leer [124]) have not used “ict” at all, and so their metric is
equally well considered to be (−, +, +, +).
Thus, we only need to worry
about converting from signature +2 to signature −2. To do this, one need
only note that the only time that the metric enters into the derivation is
whenever one forms a contraction over two indices:
(A·B) ≡ g αβ A α B β .
Thus, we can convert all of the Cohn and Wiebe expressions by performing
the transformation
(A·B) −→ −(A·B)
(D.5)
for any Lorentz dot-product that appears. (It will be noted that their definition
of the proper-time τ with a minus sign, with their metric, means that
τ needs no conversion.) It is convenient to first convert the quantities (D.4):
a µ −→ a µ ,
ρ −→ −ρ,
U µ ≡ − Rµ
ρ − V µ ,
a U −→ −a U ,
ȧ U −→ −ȧ U ,
(D.6)
where it will be noted that the change in ρ has necessitated a change in the
definition of U µ , rather than a transformation of it.
Written in terms of
these converted quantities (D.6), and performing the conversion (D.5) where
necessary, the fields (D.3) become
F µν
(−1) = − e {
− 3 4πmρ 2 2 a2 UM [να U α R µ] + 1 2 aUṀ[να U α R µ] − a U
ρ Ṁ [να R α R µ]
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