CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 126

()

(8b)

γ r γ

σ

ee = δ

()

σ r

Vector p r does not necessarily have to be orthogonal to four-acceleration
a r .

The triad e (σ) r is arbitrary except for the orthonormality conditions Eq.
(6a); nevertheless, some triads may be more convenient than others in
some set calculations. For our theoretical purposes, the Fermi triad [10] is
very important; it satisfi es over C the transport law (which we use in this
work):

r b r r

de d τ = e a v = a v

(9a)

() () ()

σ σ b σ

This type of transport has been very fundamental in gravitation, for exam-
ple, Pirani [11] and Synge [12]; but in electrodynamics, we shall show its
participation in the deduction of superpotential for the radiative part of the
Maxwell tensor (see Section (6.4). In Eq. (9a)) we have used the notation:

r

aae
σ

= (9b)

() ()

σ r

because a r is space-like type; remember that the triad is only defined over
C .

To end this section, we give some useful expressions:

, , , ,

b
wk

= w

b
wv

= W

, b

b
wa

ww =− 12 W

=− WB

,

,

, b

,

r a
kv

, ,

, c r

,

= 0

,

()

Ww =+ WB ks

r σ

W =− pa =− pa

c
Wk

= W

, c r ,

()

,

r σ

c
Bk

w −

1

=−

( )

,

c
wp

= wB , ,

a
a ww W

, − 1

, , (10)

w −

1, a

= 212 −

= 0

kp p

r

=

,

,

, a ,

, ar a

( )

,

⎡ ⎤

,

r − 1 r − 1 r − 1 r r − 1 r − 1 r

Up =− wW p = w δ + wvk + a + wv − wBk k

⎣ ⎦

r , a a a a

pk

r r

=− wWk

− 1

, a a

( )

r − 22

r pw W

r

=

− 1

2 −

pw = wWk

r
pk

r a
pk

,

= 0

,

= 0

, ,

,

, ar , a

()

,

σ ,

p = p e

a

()

ar ,

()

σ , r σ