D O C . 3 1 I D E A S A N D M E T H O D S 1 4 3
ly established if the measured disk is at rest relative to an inertial system (that is
free of gravitational fields).
If an observer comoving with the previously considered circular disk makes the
same test, then—we claim—the ration of the corresponding two results will be
greater than . This is shown by again viewing the entire process from the non-
comoving coordinate system . Judged from , the measuring rod tangentially
aligned at the periphery of the rotating disk is shortened (Lorentz contraction) due
to its movement along this line; but the radially aligned measuring rod is not. Ac-
cording to the law of Lorentz contraction, one finds
,
where v is the rotation velocity at the edge of the disk.
From this it becomes obvious that on a rotating disk and, therefore, according to
the hypothesis of equivalence also in a field of gravitation, the laws of Euclidean
geometry are not valid for the relative positioning of rigid
rods.[48]
Specifically, it
also becomes impossible to attribute physical meaning to a Cartesian coordinate
system in the theory of general relativity because it will be impossible to construct
a cubic lattice out of identical rods. We are therefore faced with the new difficulty
that spatial and chronological coordinates cannot be physically defined by means
of rigid rods and clocks as in the theory of special relativity. Thus, we face the dif-
ficult question: What is to take the place, in the theory of general relativity, of the
Cartesian coordinate systems and time as defined by means of clocks and light sig-
nals in the theory of special relativity?
20. Gaussian Coordinates. Riemannian Geometry
As shall be shown in the following, the problem arising here has been solved in the
geometric domain by Gauss and
Riemann.[49]
〈Their method can〉
Considered as an axiomatic science, Euclidean geometry has at first nothing to
do with objects of experience. Its theorems are consequences of so-called axioms,
hence, in principle, are already implicitly contained in the latter. The axioms seem
to be related to mere objects of thought that have nothing to do with objects of ex-
perience.
K
U
D
--- -
π
K K
U
D
--- -
π
1
v2-
c2
---- –
------------------ =
[p. 29]