When projecting into the space of three-dimensional (
x
,
y
,
z
{\displaystyle x,y,z}
), then the first six components form a three-dimensional tensor (
T
3
{\displaystyle T^{3}}
), which are transforming as the squares and products of (
x
,
y
,
z
{\displaystyle x,y,z}
); the following three components of
T
4
{\displaystyle T^{4}}
are forming a
V
3
{\displaystyle V^{3}}
, the tenth a scalar
S
3
{\displaystyle S^{3}}
.
The four components of the operator "lor" transform as components of
V
I
4
{\displaystyle V_{I}^{4}}
, we can deduce – from a four-dimensional scalar
φ
{\displaystyle \varphi }
, given as a function of
x
,
y
,
z
,
u
{\displaystyle x,y,z,u}
– a
T
4
{\displaystyle T^{4}}
which is twice differentiable with respect to
x
,
y
,
z
,
u
{\displaystyle x,y,z,u}
:
∂
2
φ
∂
x
2
,
∂
2
φ
∂
y
2
,
∂
2
φ
∂
z
2
,
∂
2
φ
∂
y
∂
z
,
∂
2
φ
∂
z
∂
x
,
∂
2
φ
∂
x
∂
y
;
∂
2
φ
∂
x
∂
u
,
∂
2
φ
∂
y
∂
u
,
∂
2
φ
∂
z
∂
u
;
∂
2
φ
∂
u
2
.
{\displaystyle {\frac {\partial ^{2}\varphi }{\partial x^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial z^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial z}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial x}},\ {\frac {\partial ^{2}\varphi }{\partial x\ \partial y}};\quad {\frac {\partial ^{2}\varphi }{\partial x\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial u}};\quad {\frac {\partial ^{2}\varphi }{\partial u^{2}}}.}
A
S
4
{\displaystyle S^{4}}
is given, being a quadratic homogeneous function of
x
,
y
,
z
,
u
{\displaystyle x,y,z,u}
:
(13)
{
φ
(
x
,
y
,
z
,
u
)
=
1
2
c
11
x
2
+
1
2
c
22
y
2
+
1
2
c
33
z
2
+
c
23
y
z
+
c
31
z
x
+
c
12
x
y
+
c
14
x
u
+
x
24
y
u
+
c
34
z
u
+
1
2
c
44
u
2
,
{\displaystyle {\begin{cases}\varphi (x,y,z,u)&={\frac {1}{2}}c_{11}x^{2}+{\frac {1}{2}}c_{22}y^{2}+{\frac {1}{2}}c_{33}z^{2}\\\\&+c_{23}yz+c_{31}zx+c_{12}xy\\\\&+c_{14}xu+x_{24}yu+c_{34}zu+{\frac {1}{2}}c_{44}u^{2},\end{cases}}}
,
the 10 coefficients:
c
11
,
c
22
,
c
33
,
c
23
,
c
31
,
c
12
;
c
14
,
c
24
,
c
34
;
c
44
,
{\displaystyle c_{11},\ c_{22},\ c_{33},\ c_{23},\ c_{31},\ c_{12};\ c_{14},\ c_{24},\ c_{34};\ c_{44},}
form a four-dimensional tensor.
In the electrodynamics of moving bodies, the equations of the momentum and energy apply:[ 1]
(14)
{
K
x
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
−
∂
g
x
∂
t
,
K
y
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
−
∂
g
y
∂
t
,
K
z
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
−
∂
g
z
∂
t
,
c
q
K
+
Q
=
−
∂
S
x
∂
x
−
∂
S
y
∂
y
−
∂
S
z
∂
z
−
∂
ψ
∂
t
.
{\displaystyle {\begin{cases}&{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\&{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\&{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}},\\\\\\c{\mathfrak {qK}}+&Q=-{\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}-{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}-{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}}-{\frac {\partial \psi }{\partial t}}.\end{cases}}}
In order to give to these four equations a more symmetrical from, we put:
(15)
u
=
i
c
t
,
K
u
=
i
q
K
+
i
Q
c
,
U
u
=
ψ
;
{\displaystyle u=ict,\ {\mathfrak {K}}_{u}=i{\mathfrak {qK}}+i{\frac {Q}{c}},\ U_{u}=\psi ;}
(15a)
X
u
=
−
i
c
g
x
,
Y
u
=
−
i
c
g
y
,
Z
u
=
−
i
c
g
z
;
{\displaystyle X_{u}=-ic{\mathfrak {g}}_{x},\ Y_{u}=-ic{\mathfrak {g}}_{y},\ Z_{u}=-ic{\mathfrak {g}}_{z};}
(15b)
U
x
=
−
i
c
S
x
,
U
y
=
−
i
c
S
y
,
U
z
=
−
i
c
S
z
.
{\displaystyle U_{x}=-{\frac {i}{c}}{\mathfrak {S}}_{x},\ U_{y}={\frac {-i}{c}}{\mathfrak {S}}_{y},\ U_{z}={\frac {-i}{c}}{\mathfrak {S}}_{z}.}
↑ M. Abraham , l. c. equations (6) and (7).