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ra
*
ra
FF
=
FF
=
0
,
(27a)
()
()
()
()
ra
ra
−
1
−
1
−
1
−
1
F
−
This does not happens with
()
ij
2
ra
qw
−
24
FF
=
2
<
0
*
ra
FF
−
=
0
,
,
(27b)
()
()
ra
−
2
()
()
ra
2
−
2
−
2
which belongs to type
B
—this portion will be designated as the bounded
part of
F
, besides:
mni
*
FF
ab
=
F F
*
ab
=
0
FF
−
ab
=
0
,
(27c)
()
()
()
()
()
()
ab
ab
ab
−
2
−
2
2
−
1
−
1
−
1
These Eq. (27) relations can be found in Weert [28], page 465.
It is possible to write Eq. (25a) in the following form:
ab
NwF
−
=
ab
MwF
−
=
2
−
1
−
2
FwNwM
=
+
so that
,
(28)
()
1
ab
()
ab
2
ab
ab
ab
with the following properties:
,
,
,
b
r
−
1
N
ξ
=
0
b
ξξ =
0
ξ
=
wk
= −
τ
Mq
ξ
=
ξ
r
r
,
r
ab
ab
a
r
Therefore, we see that Eq. (28) is coherent with Eqs. (1), (2), and (3) of
the Goldberg-Kerr theorem [29] for the asymptotic behavior of electro-
magnetic fields.
Teitelboim's decomposition Eq. (25a) is fundamental in everything
that follows, and it is interesting that Eq. (7) generates such splitting in a
natural manner:
r
−
1
r
r
vwkp
=
−
which substituting in Eq. (21a) gives
r r r
AAA
=+
r r
A wp
−
1
=−
A wk
r
=
−
2
r
with
,
(29a)
1
2
1
2
This partition of the Liénard-Wiechert four-potential is found in the Teitel-
boim [4] well-known article; however, it was also published by Migglietta
[5] not-knowing Ref. [4]. Expressions (29b) are simpler than Migglietta's
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