CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 139

2. To construct an “electromagnetic Weyl tensor” through prescrip-
tion Eq. (40a) (in this case rj

KT =

):

rj

B B

C KKKKgTgTgTgT =−+−+ − + −

(40c)

jm ij ri rm

jrim jri m , jrm i , imj r , imr j , ir mr mj ij

B B B B B B B B B

C , see Refs. [39, 41],
resulting in Type II in the Penrose diagram, which is:

The Petrov classification [13] can be applied to jrim
B

“The Liénard-Wiechert field is type II”; (40d)

This strengthens the analogies found by Newman [50] between Robin-
sonTrautman metrics (Einstein's equations solution type II) [51], and the
electromagnetic fi eld of a point charge. The physical meaning of the Weert
superpotential was elucidated in [40].
The idea Eq. (40b) motivates the following question:
Can sar
B

K be written as the sum of two or more Minkowskian spinten-

sors?
The answer is affi rmative because the terms in Eq. (38b) can be grouped
in form [33]:



KKK =+

(41a) sar sar sar

B B B

with



( )

− 2

⎡

− 3

⎤

Kqwqwvk Fk = × −

, (41b)

⎣ ⎦

s c sc r

scr

B

( )

( )

Kq w wvkkgkgk =

2

/4

− 4

⎡ 3

− 1

× + − ⎤

(41c)

⎣ ⎦

a s r ra s rs a

sar

B

K satisfy Eq. (39); therefore, they are spintensors. By sub-
stituting Eq. (41a) in Eq. (38a), we obtain in a natural manner the splitting
of López [52]:

Both parts of sar
B



TTT =+ (42a)

ra ra ra

B B B

  ,

TK =

a

a

TK =

where (42b)

rs sr a ,

rs sr a ,

B B

B B