CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 124

This tetrad forms a base for each vector associated to the world line, in
particular for null vector Eq. (2b):

r σ r 4 r

kbe = + be ; (6b)

() ()

σ 4

From now on the Greek indexes shall only take values 1, 2, and 3.
Equation (6b) can be written as follows:

with

Mbe

r

=

σ r

,

r

, (6c)

Mv = 0

r r r
kMbv

4

=+

()

σ r

M r is space-like type because it is a lineal combination of the three space-
like vectors of the tetrad (see Figure 6.4):
If ( )

1 2

MMM =

r

is the magnitude of M r , then

r

pp

r

= 1

r r
MMp

with (6d)

=

r

and by Eq. (6c):

, (6e)

r
pv

= 0

FIGURE 6.4 Spatial triad.

Therefore, p r is a space-like unitary vector. From Eqs. (2c) and (3a), it
is apparent that

4
Mb w

; and as a consequence, Eqs. (6b) through (6e)
imply the important Synge [7]-Teitelboim [4] decomposition for k r :

==