three-dimensional space
x
,
y
,
z
{\displaystyle x,y,z}
, the three first components of vector
V
I
4
{\displaystyle V_{I}^{4}}
constitute a three-dimensional vector
V
3
{\displaystyle V^{3}}
,
r
{\displaystyle {\mathfrak {r}}}
, the fourth (
u
{\displaystyle u}
) a three-dimensional scalar (
S
3
{\displaystyle S^{3}}
).
A four-dimensional vector of the second kind
(
V
I
I
4
)
{\displaystyle \left(V_{II}^{4}\right)}
denotes a system of six magnitudes, which are transformed like the following expressions, formed by the components
x
1
,
y
1
,
z
1
,
u
1
{\displaystyle x_{1},y_{1},z_{1},u_{1}}
and
x
2
,
y
2
,
z
2
,
u
2
{\displaystyle x_{2},y_{2},z_{2},u_{2}}
of two
V
I
4
{\displaystyle V_{I}^{4}}
:
(1)
{
a
x
=
|
y
1
z
1
y
2
z
2
|
,
a
y
=
|
z
1
x
1
z
2
x
2
|
,
a
z
=
|
x
1
y
1
x
2
y
2
|
;
b
x
=
|
x
1
u
1
x
2
u
2
|
,
b
y
=
|
y
1
u
1
y
2
u
2
|
,
b
z
=
|
z
1
u
1
z
2
u
2
|
.
{\displaystyle \left\{{\begin{array}{ccccc}{\mathfrak {a}}_{x}=\left|{\begin{array}{cc}y_{1}&z_{1}\\y_{2}&z_{2}\end{array}}\right|,&&{\mathfrak {a}}_{y}=\left|{\begin{array}{cc}z_{1}&x_{1}\\z_{2}&x_{2}\end{array}}\right|,&&{\mathfrak {a}}_{z}=\left|{\begin{array}{cc}x_{1}&y_{1}\\x_{2}&y_{2}\end{array}}\right|;\\\\{\mathfrak {b}}_{x}=\left|{\begin{array}{cc}x_{1}&u_{1}\\x_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{y}=\left|{\begin{array}{cc}y_{1}&u_{1}\\y_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{z}=\left|{\begin{array}{cc}z_{1}&u_{1}\\z_{2}&u_{2}\end{array}}\right|.\end{array}}\right.}
Obviously, when projecting into three-dimensional space,
V
I
I
4
{\displaystyle V_{II}^{4}}
is composed of two
V
3
{\displaystyle V^{3}}
, which, in the symbolism of ordinary vector analysis, we can write:
(1a)
a
=
[
r
1
r
2
]
,
b
=
r
1
u
2
−
r
2
u
1
{\displaystyle {\mathfrak {a}}=\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right],\ {\mathfrak {b}}={\mathfrak {r_{1}}}u_{2}-{\mathfrak {r}}_{2}u_{1}}
From two
V
I
4
{\displaystyle V_{I}^{4}}
:
r
,
u
{\displaystyle {\mathfrak {r}},\ u}
and
r
1
,
u
1
{\displaystyle {\mathfrak {r}}_{1},\ u_{1}}
we can compose a four-dimensional scalar (
S
4
{\displaystyle S^{4}}
) as follows:
(2)
S
=
x
x
1
+
y
y
1
+
z
z
1
+
u
u
1
=
r
r
1
+
u
u
1
{\displaystyle S=xx_{1}+yy_{1}+zz_{1}+uu_{1}={\mathfrak {rr}}_{1}+uu_{1}}
Conversely, from any four-dimensional scalar
φ
(
x
,
y
,
z
,
u
)
{\displaystyle \varphi (x,y,z,u)}
, we obtain (derived with respect to their coordinates) a
V
I
4
{\displaystyle V_{I}^{4}}
:
(3)
X
=
∂
φ
∂
x
,
Y
=
∂
φ
∂
y
,
Z
=
∂
φ
∂
z
,
U
=
∂
φ
∂
u
.
{\displaystyle X={\frac {\partial \varphi }{\partial x}},\ Y={\frac {\partial \varphi }{\partial y}},\ Z={\frac {\partial \varphi }{\partial z}},\ U={\frac {\partial \varphi }{\partial u}}.}
So the operators
∂
∂
x
,
∂
∂
y
,
∂
∂
z
,
∂
∂
u
{\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}},\ {\frac {\partial }{\partial u}}}
transform as the components of a
V
I
4
{\displaystyle V_{I}^{4}}
, and these operators were denoted by Minkowski as the components of the operator "lor".
We can compose a
S
4
{\displaystyle S^{4}}
from four
V
I
4
{\displaystyle V_{I}^{4}}
, which determines the space of the parallelepiped of the four vectors:
(4)
φ
=
|
x
y
z
u
x
1
y
1
z
1
u
1
x
2
y
2
z
2
u
2
x
3
y
3
z
3
u
3
|
{\displaystyle \varphi =\left|{\begin{array}{ccccccc}x&&y&&z&&u\\x_{1}&&y_{1}&&z_{1}&&u_{1}\\x_{2}&&y_{2}&&z_{2}&&u_{2}\\x_{3}&&y_{3}&&z_{3}&&u_{3}\end{array}}\right|}
If we apply scheme (3) to
S
4
{\displaystyle S^{4}}
, we obtain a
V
I
4
{\displaystyle V_{I}^{4}}
, which is composed of three other
V
I
4
,
r
1
u
1
,
r
2
u
2
,
r
3
u
3
{\displaystyle V_{I}^{4},\ {\mathfrak {r}}_{1}u_{1},\ {\mathfrak {r}}_{2}u_{2},\ {\mathfrak {r}}_{3}u_{3}}
, whose components are:
(5)
{
X
=
∂
φ
∂
x
=
|
y
1
z
1
u
1
y
2
z
2
u
2
y
3
z
3
u
3
|
,
Y
=
∂
φ
∂
y
=
|
z
1
x
1
u
1
z
2
x
2
u
2
z
3
x
3
u
3
|
,
Z
=
∂
φ
∂
z
=
|
x
1
y
1
u
1
x
2
y
2
u
2
x
3
y
3
u
3
|
,
U
=
∂
φ
∂
u
=
−
|
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
|
;
{\displaystyle \left\{{\begin{array}{ccc}X={\frac {\partial \varphi }{\partial x}}=\left|{\begin{array}{ccccc}y_{1}&&z_{1}&&u_{1}\\y_{2}&&z_{2}&&u_{2}\\y_{3}&&z_{3}&&u_{3}\end{array}}\right|,&&Y={\frac {\partial \varphi }{\partial y}}=\left|{\begin{array}{ccccc}z_{1}&&x_{1}&&u_{1}\\z_{2}&&x_{2}&&u_{2}\\z_{3}&&x_{3}&&u_{3}\end{array}}\right|,\\\\Z={\frac {\partial \varphi }{\partial z}}=\left|{\begin{array}{ccccc}x_{1}&&y_{1}&&u_{1}\\x_{2}&&y_{2}&&u_{2}\\x_{3}&&y_{3}&&u_{3}\end{array}}\right|,&&U={\frac {\partial \varphi }{\partial u}}=-\left|{\begin{array}{ccccc}x_{1}&&y_{1}&&z_{1}\\x_{2}&&y_{2}&&z_{2}\\x_{3}&&y_{3}&&z_{3}\end{array}}\right|;\end{array}}\right.}