with respect to the body of reference. A similar difficulty presents
itself when we attempt to apply our earlier definition of simultaneity
in such a case, but I do not wish to go any farther into this question.
Moreover, at this stage the definition of the space co-ordinates also
presents insurmountable difficulties. If the observer applies his
standard measuring-rod (a rod which is short as compared with the
radius of the disc) tangentially to the edge of the disc, then, as
judged from the Galileian system, the length of this rod will be less
than 1, since, according to Section XII, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measuring-rod will not experience a shortening in length, as judged
from _K_, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
the one by the other, he will not obtain as quotient the familiar
number = 3.14 . . ., but a larger number,[19] whereas of course, for
a disc which is at rest with respect to _K_, this operation would yield
exactly. This proves that the propositions of Euclidean geometry
cannot hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length 1 to the rod
in all positions and in every orientation. Hence the idea of a straight
line also loses its meaning. We are therefore not in a position to
define exactly the co-ordinates _x, y, z_ relative to the disc by means
of the method used in discussing the special theory, and as long as the
co-ordinates and times of events have not been defined, we cannot
assign an exact meaning to the natural laws in which these occur.
[19] Throughout this consideration we have to use the Galileian
(non-rotating) system _K_ as reference-body, since we may only assume
the validity of the results of the special theory of relativity
relative to _K_ (relative to _K _ a gravitational field prevails).
Thus all our previous conclusions based on general relativity would
appear to be called in question. In reality we must make a subtle
detour in order to be able to apply the postulate of general relativity
exactly. I shall prepare the reader for this in the following
paragraphs.
57