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b
r
−
4
F p
=
qw Wv k
b
Fp
σ
=
0
−
,
(24)
()
cb
,
cr
r
,
b
of great importance in the next section on the deduction of the radiative
superpotential.
Teitelboim started an era in electrodynamics by employing only re-
tarded fi elds and studying Faraday and Maxwell tensors near to and away
from a point charge. This type of analysis is generated by substituting Eqs.
(5) and (7) in Eq. (21b) to obtain the following decomposition:
rb
FFF
−
=+
,
(25a)
()
rb
()
rb
1
−
2
where
(
)
−
2
−
1
F wwWvak
=
+
×
(25b)
r
r
b
()
rb
−
1
(
)
−
1
c
=
qw
a p v
× +×+×
p
a
v
a
p
(25c)
c
r
b
r
b
r
b
(
)
F wvk wvp
=
−
3
×
=
−
2
×
(25d)
r
b
r
b
rb
()
−
2
w
−
is clear be-
cause the parenthesis in Eqs. (25c) and (25d) are independent of the re-
tarded distance; their terms are functions of
x
, which remains stationary
when we move away over the light cone. Thus,
i
w
−
; the dependence on
i
Meaning that
()
Fi
−
,
=
, 2
varies like
rb
i
F
−
are dominant
away from (
w
> > 1) and near to (
w
< < 1), respectively, then
and
F
−
rb
()
rb
2
()
1
F
−
being re-
sponsible for the Larmor formula that provides the radiation speed toward
infinitum. Note that Eqs. (25b) and (25c) depend on the particle accelera-
tion, which is an expected result because of the Schild theorem [26]:
1
ij
()
“Radiation exists if and only if
a
r
≠
0
”
(26)
Schild was the first author to give a covariant definition of radiation even
though some of his ideas were already implicit in Synge [27], Appendix
A, whose first edition was made in 1955. We call
F
−
the radiative part of
1
ij
()
F
because it is a null field in classification Eq. (16):
ab
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