CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 131

In the following section, we shall employ the material explained in Eqs.
(11-20) to analyze the field produced by a point charge with a relativistic
trajectory.

6.2.3 THE LI É NARD-WIECHERT FIELD

The solution of Eqs. (11f) and (11g) for a particle in arbitrary motion in
Minkowski space was obtained by Liénard and Wiechert; the correspond-
ing potential carries their names and is given by

( )

r b − 1 r

q = charge / 4 πε

A X = qw v

, (retarded potential), (21a)

0

which is fundamental in everything that follows; by the use of Eq. (11a), it
is simple to calculate the associated Faraday tensor :

( )

(21b)

− 2 − 2

F wUkUk wUk = − = ×

rb r b b r r b

means the antisymmetric product. This notation is due to Lowry
[22] and will cause such expressions to be very compact. From Eqs. (11c)
and (21b), it is clear that

where

×

* − 2 ab

, (21c)

FqwUk =− ε

mn mnab

therefore

,

F = 0

(21d)

24 −

Fq =− 2 < 0

2

1

In other words, the electric and magnetic fields generated by the charge
satisfy





EB ⋅ = 0

BE <

, (21e)

),
the invariant F tends to zero, which means that F  is close to the null case
(type C ) far away from the charge.
With Eqs. (21d) and (20c) is valid; therefore, the Stachel theorem [2]
implies that Eq. (21b) can be reduced to Eq. (20b). This is easy to do be-
cause from Eqs. (4) and (5), the following relationships are available:

In consequence F  is type B . Note that in the asymptotic region ( w

→∞