D O C . 3 1 I D E A S A N D M E T H O D S 1 3 1
motion of the coordinate system. 〈The separate role of the time coordinate vis-à-vis
the space coordinate vanishes.〉 The mutual dependence of the spatial and the chro-
nological necessarily leads to an amalgamation of space and time into a four-di-
mensional 〈total〉 continuum.
In addition, Minkowski also found that this four-dimensional continuum, which
he called “world,” has a deep-rooted formal kinship with the three-dimensional
continuum of Euclidean geometry. We furthermore recognized that this kinship can
be used as a basis for a method of direct formulation of equation systems (mathe-
matically formulized laws of nature) that satisfy the principle of relativity, i.e., are
covariant under Lorentz transformations. For the non-mathematician, it is not easy
to grasp these connections. But still I will try to sketch their essence.
The structures of Euclidean geometry as well as those of theoretical physics
have an existence independent of the spatial orientation of the Cartesian coordinate
system. 〈This corresponds to〉 If I have formulated any relation using orthogonal
coordinates of the coordinate system , then this relation must present
itself in an exactly corresponding form if I use a differently oriented orthogonal co-
ordinate system (with the coordinates ). Example: If are the
radii of two spherical surfaces with and , respectively, the
coordinates of their centers, then these spherical surfaces do not intersect in if
the following equation is
satisfied:[28]
. (8)
With respect to the coordinate system , the condition of non-inter-
section must be the exactly corresponding one, that is,
. (8a)
On the other hand, given the relative position of both coordinate systems, there are
certain relations between the -related coordinates and the
coordinates of the same point such that the can be expressed by
means of the (and vice versa the by means of the ). We want to call these
relations a “Euclidean transformation.” If we express by means of a Euclidean
transformation the coordinates and in (8) through the
and , then we must obtain the condition of intersection of
the two spherical surfaces in reference to i.e., we must get equation (8a). This
x1, x2, x3 K
K′ x1′, x2′x3′ R1, R2
x1, x2, x3
x1∗, x2∗, x3∗
K
R1 R2
x1∗
x1)2
– (
x2∗
x2)2
– (
x3∗
x3)2
– ( + + +
K′( x1′, x2′, x3′)
R1 R2
x1∗′
x1′)2
– (
x2∗′
x2′)2
– (
x3∗′
x3′)2
– ( + + +
K x1, x2, x3 K′-related
x1′, x2′, x3′ P x′
x x x′
[p. 17]
x1, x2, x3
x1∗, x2∗, x3∗
x1′, x2′x3′
x1∗′x2∗′x3∗′
K′,