CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 138

a

, (38a)

TTK ==

rn nr nr a ,

B B B

( )

( )( ) ( )

2 − 4

⎡

− 1

⎤

Kq =− /4 ww 3 − 4 Wv × k k + 4 a × k k + g k − g k , (38b)

⎣ ⎦

s a r s a r rs a ra s

sar

B

K . This idea of the superpo-
tential is not Weert's original; it is actually quite old and was introduced
by Freud [36] constructing the superpotential for the canonical energy-
momentum pseudotensor of Einstein [37, 38].
Weert did not study deeply the algebraic and differential properties of

T

which means that ij
B

is the divergence of sar
B

K , which was remedied in [33, 39-41] obtaining a better comprehension
of such superpotential structures:

sar

B

KK =−

Antisymmetry

sar sar

B B

r

K = 0

Null trace (39)

sr

B

r

K = 0

Null divergence sa r ,

B

KKK ++ = 0

Cyclic

sar ars rsa

B B B

Surprisingly, Eq. (39) is also satisfied in curved spaces (replacing par-
tial derivatives with covariant ones) for the Lanczos spintensor sa K [42],
which generates the Weyl conformal tensor in four-dimensions [43-49]:

(40a)

CK K KK g = − + − + Kg Kg Kg − + − K

jrim jri m ; jrm i ; imj r ; imr j ; jm ir ij mr ri mj rm ij

a

KK K ==

so that rj jr rj a ;

This fact suggests at least two things:
1.

The introduction in electrodynamics of the definition:

“A Minkowski spintensor is that which satisfies Eq. (39),”
(40b)

therefore, the Weert superpotential is a Minkowskian spintensor.