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as a scalar, but not as an invariant; the
electromagnetic fi eld only possesses two Lorentz invariants, namely
We are placing emphasis on
φ
(
)
ab
2
2
,
FFF
≡
*
ab
=
4
EB
⋅
(15)
FFF
≡
=
2
B E
−
1
ab
2
ab
and
BB
With
EE
=
,
=
Just like Weyl tensor invariants allow to establish the Petrov classifi -
cation [13] for the gravitational fi eld, the quantities Eq. (15) lead to the
Synge [1]-Piña [14] classifi cation for the Faraday tensor:
F
≠
0
Type A:
2
F
<
0
F
=
0
Type B:
and
(16)
1
2
F
=
0
Type C:
F
=
0
and
null fi eld
2
1
F
>
0
Type D:
and
F
=
0
1
2
and
EB
. A
nonnull field implies a different type of
C
. Classification Eq. (16) is alge-
braic, but the type of electromagnetic field may change from one point to
another.
Further, we will see that the fi eld that produces a relativistic charge is
B
type, which tends to type
C
(plane wave) toward infi nity.
There are very important identities for the Maxwell fi eld:
=
A point with a null field means that in such an event,
EB
⊥
(
)
ar
*
ar
*
a
FF
−
F F
=
F
1
/2
δ
(17a)
br
br
b
(
)
*
ar
a
(17b)
FF
=
F
2
/4
δ
br
b
which do not have a specific name, are well known and can be found in
Rainich [15], Plebañski [16], Wheeler [17] pp. 239, Penney [18], and Piña
[14]—Expressions (17) correspond to Lanczos identities [18] between the
Riemann tensor and its double dual. If Eq. (17a) is multiplied by
F
or
*
i
F
and Eq. (17b) is employed, valuable identities result in the calculation of
an antisymmetric matrix's exponential function [14, 20]:
ia
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