g
{\displaystyle {\mathfrak {g}}}
:
(18b)
f
=
c
g
=
1
c
S
;
{\displaystyle {\mathfrak {f}}=c{\mathfrak {g}}={\frac {1}{c}}{\mathfrak {S}};}
finally, the tenth part of
T
4
{\displaystyle T^{4}}
which is derived from (18),
ψ
{\displaystyle \psi }
, determines the density of electromagnetic energy.
To arrive at a suitable four-dimensional scalar, which is a second-order homogeneous function of coordinates
x
,
y
,
z
,
u
{\displaystyle x,y,z,u}
, with bilinear coefficients in the components of the electromagnetic vectors, we form the first one according to scheme (6), the radius vector
{
r
,
u
}
{\displaystyle \{{\mathfrak {r,u}}\}}
in a space of four dimensions, and
V
I
4
{\displaystyle V_{I}^{4}}
from
V
I
I
4
{
a
,
b
}
{\displaystyle V_{II}^{4}\{{\mathfrak {a,b}}\}}
:
R
=
u
a
+
[
r
b
]
,
U
=
−
(
r
a
)
{\displaystyle {\mathfrak {R}}=u{\mathfrak {a}}+[{\mathfrak {rb}}],\ U=-({\mathfrak {ra}})}
Similarly, from another
V
I
I
4
{
a
′
,
b
′
}
{\displaystyle V_{II}^{4}\{{\mathfrak {a',b'}}\}}
and from
V
I
4
{
r
,
a
}
{\displaystyle V_{I}^{4}\{{\mathfrak {r,a}}\}}
which is comprised of
V
I
4
{\displaystyle V_{I}^{4}}
:
R
′
=
u
a
′
+
[
r
b
′
]
,
U
′
=
−
(
r
a
′
)
{\displaystyle {\mathfrak {R}}'=u{\mathfrak {a}}'+[{\mathfrak {rb}}'],\ U'=-({\mathfrak {ra}}')}
Now, according to the scheme (2), we obtain the
S
4
{\displaystyle S^{4}}
:
S
=
R
R
′
+
U
U
′
=
u
2
a
a
′
+
u
a
[
r
b
′
]
+
u
a
′
[
r
b
]
+
[
r
b
]
[
r
b
′
]
+
(
r
a
)
(
r
a
′
)
{\displaystyle S={\mathfrak {RR}}'+UU'=u^{2}{\mathfrak {aa}}'+u{\mathfrak {a}}[{\mathfrak {rb}}']+u{\mathfrak {a}}'[{\mathfrak {rb}}]+[{\mathfrak {rb}}][{\mathfrak {rb}}']+({\mathfrak {ra}})({\mathfrak {ra}}')}
that can be written:
(19)
S
=
(
r
a
)
(
r
a
′
)
−
(
r
b
)
(
r
b
′
)
+
r
2
(
b
b
′
)
+
u
r
[
b
a
′
]
+
u
r
[
b
′
a
]
+
u
2
(
a
a
′
)
{\displaystyle S=({\mathfrak {ra}})({\mathfrak {ra}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\mathfrak {r}}^{2}({\mathfrak {bb}}')+u{\mathfrak {r}}[{\mathfrak {ba}}']+u{\mathfrak {r}}[{\mathfrak {b'a}}]+u^{2}({\mathfrak {aa}}')}
As it follows from (6a), we can permute
a
{\displaystyle {\mathfrak {a}}}
with
b
{\displaystyle {\mathfrak {b}}}
and
a
′
{\displaystyle {\mathfrak {a'}}}
with
b
′
{\displaystyle {\mathfrak {b'}}}
, and obtain in a corresponding way another
S
4
{\displaystyle S^{4}}
:
(19a)
S
∗
=
(
r
b
)
(
r
b
′
)
−
(
r
a
)
(
r
a
′
)
+
r
2
[
a
a
′
]
+
u
r
[
a
b
′
]
+
u
r
[
a
′
b
]
+
u
2
(
b
b
′
)
{\displaystyle S^{*}=({\mathfrak {rb}})({\mathfrak {rb}}')-({\mathfrak {ra}})({\mathfrak {ra}}')+{\mathfrak {r}}^{2}[{\mathfrak {aa}}']+u{\mathfrak {r}}[{\mathfrak {ab}}']+u{\mathfrak {r}}[{\mathfrak {a'b}}]+u^{2}({\mathfrak {bb}}')}
Putting
4
φ
=
S
−
S
∗
{\displaystyle 4\varphi =S-S^{*}}
it is given:
(20)
{
2
φ
=
(
r
a
)
(
r
a
′
)
−
1
2
r
2
[
a
a
′
]
−
(
r
b
)
(
r
b
′
)
+
1
2
r
2
[
b
b
′
]
+
u
r
[
b
′
a
]
+
u
r
[
b
′
a
]
+
1
2
u
2
{
(
a
a
′
)
−
(
b
b
′
)
}
.
{\displaystyle \left\{{\begin{array}{c}2\varphi =({\mathfrak {ra}})({\mathfrak {ra}}')-{\frac {1}{2}}{\mathfrak {r}}^{2}[{\mathfrak {aa}}']-({\mathfrak {rb}})({\mathfrak {rb}}')+{\frac {1}{2}}{\mathfrak {r}}^{2}[{\mathfrak {bb}}']\\\\+u{\mathfrak {r}}[{\mathfrak {b'a}}]+u{\mathfrak {r}}[{\mathfrak {b'a}}]+{\frac {1}{2}}u^{2}\left\{({\mathfrak {aa}}')-({\mathfrak {bb}}')\right\}.\end{array}}\right.}
Now, by identifying the homogeneous second-order function
φ
{\displaystyle \varphi }
of
x
,
y
,
z
,
u
{\displaystyle x,y,z,u}
which is invariant under the Lorentz transformation, with
S
4
{\displaystyle S^{4}}
as given by (18), we find the expressions:
(20a)
2
Φ
=
(
r
a
)
(
r
a
′
)
−
1
2
r
2
(
a
a
′
)
−
(
r
b
)
(
r
b
′
)
+
1
2
r
2
(
b
b
′
)
{\displaystyle 2\Phi =({\mathfrak {ra}})({\mathfrak {ra}}')-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {aa}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {bb}}')}
(20b)
2
f
=
i
[
b
′
a
]
+
i
[
b
′
a
]
{\displaystyle 2{\mathfrak {f}}=i[{\mathfrak {b'a}}]+i[{\mathfrak {b'a}}]}
(20c)
2
ψ
=
(
a
a
′
)
−
(
b
b
′
)
{\displaystyle 2\psi =({\mathfrak {aa}}')-({\mathfrak {bb}}')}
We introduce the electrodynamic
V
I
I
4
{\displaystyle V_{II}^{4}}
of Minkowski , by setting
(21)
{
a
=
H
,
b
=
−
i
D
;
a
′
=
B
,
b
′
=
−
i
E
.
{\displaystyle {\begin{cases}{\mathfrak {a}}={\mathfrak {H}},&{\mathfrak {b}}=-i{\mathfrak {D}};\\{\mathfrak {a}}'={\mathfrak {B}},&{\mathfrak {b}}'=-i{\mathfrak {E}}.\end{cases}}}
By taking into account (18a), the following expressions are resulting now:
(21a)
{
X
x
x
2
+
Y
y
y
2
+
Z
z
z
2
+
2
Y
z
y
z
+
2
Z
x
z
x
+
2
X
y
x
y
=
(
r
E
)
(
r
D
)
−
1
2
r
2
(
E
D
)
+
(
r
H
)
(
r
B
)
−
1
2
r
2
(
H
B
)
,
{\displaystyle {\begin{cases}X_{x}x^{2}+Y_{y}y^{2}+Z_{z}z^{2}+2Y_{z}yz+2Z_{x}zx+2X_{y}xy\\=({\mathfrak {rE}})({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {ED}})+({\mathfrak {rH}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {HB}}),\end{cases}}}
(21b)
2
f
=
[
E
H
]
+
[
D
B
]
{\displaystyle 2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]}
(21c)
2
ψ
=
E
D
+
H
B
.
{\displaystyle 2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}.}