1 3 4 D O C . 3 1 I D E A S A N D M E T H O D S
(12)
is measurable by means of measuring rods and clocks and is independent of the
choice of coordinates (inasfar as this choice is free). is the fundamental invari-
ant of the system and is called the elementary distance of neighboring points of the
space-time continuum because of its formal analogy with the fundamental invariant
of Euclidean geometry.
But, since is imaginary and therefore negative, there is, aside from the
number of dimensions, a deep-rooted formal difference between the invariant
of Euclidean geometry and the invariant of the theory of special relativity.
Namely, is always positive as long as the adjacent space points and —to
which is associated—do not coincide. However, when and are neighbor-
ing space-time point-events (“world points” in Minkowski’s terminology), then
one must distinguish three possibilities as to their relative position, all of them in-
dependent of the choice of the coordinate system:
The coordinate system for the first case can be chosen such that only differs
from zero while , , vanish; the distance of events is then called
“timelike” and the distance
[32]
of can be directly measured by a clock that
is at rest relative to the coordinate system chosen. For the second case, the coordi-
nate system can be chosen such that the pointlike events and occur simulta-
neously relative to it. is then called “spacelike”; the associated can be
measured by a measuring rod that is at rest relative to the coordinate system chosen.
The third case, which is a limiting case, is physically characterized by the fact
that the two pointlike events (world points) can be connected by a vacuum light sig-
nal.
Therefore, the analogy existing between the properties of the four-dimensional
“world” of the theory of special relativity and the “space” of Euclidean geometry
is only a mathematically formal one, not a physical one.
dσ2
dx1
2
dx2
2
dx3
2
dx4
2
+ + + =
dσ
ds
x4 dx4
2
ds
dσ
ds2 P P′
ds P P′
dσ2 0
dσ2 0
dσ2 0. =
dx4
dx1 dx2 dx3 PP′
dσ PP′
P P′
PP′ –dσ2