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50
Relativity: The Special and General Theory
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 I, since, according to Section 12, moving bodies
suffer a shortening in the direction of the motion. On the other hand, the measaring?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 A = 3.14 . . ., but a larger number,²
)
whereas of course, for a disc which is at rest
with respect to K, this operation would yield A 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 I 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.
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.
Next: Euclidean and Non?Euclidean Continuum
Footnotes
1
)
The field disappears at the centre of the disc and increases proportionally to the distance
from the centre as we proceed outwards.
2
)
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 K¹ a gravitational field prevails).
Relativity: The Special and General Theory