CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 122

FIGURE 6.2(B) Retarded distance from X r to C .

The instantaneous distance (see Figure 6.2(a)) introduced by Dirac [8]
is geometrically simpler than retarded distance w , (see Figure 6.2(b)) pro-
posed by Bhabha [9] and furthered by Synge [7]; nevertheless, has the

big disadvantage of not involving retarded effects (light cone); for this
reason, w has more physical meaning and leads to simpler calculations
because it intrinsically takes in account the fi nite velocity of interaction.
Here, we will work only with w , whose expression is given by the follow-
ing equation:

λ

r

wkv =− ≥ 0

(3a)

r

Bearing in mind that a null vector cannot be orthogonal to a time-like one,
Eq. (3a) points out that

if and only if (3b)

k

r

= 0

w = 0

In other words, the retarded distance is zero only when X r is over C .
When making calculations, we need to know how diverse quantities
change over C when an external event X r varies; for this, it is enough with
having Change's law for τ because x r , v r , a r , etc., are functions of this pa-

rameter: