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.
Notes
*) The field disappears at the centre of the disc and increases proportionally to
the distance from the centre as we proceed outwards.
**) 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).
EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
The surface of a marble table is spread out in front of me. I can get from any one
point on this table to any other point by passing continuously from one point to a
" neighbouring " one, and repeating this process a (large) number of times, or, in
other words, by going from point to point without executing "jumps." I am sure
the reader will appreciate with sufficient clearness what I mean here by "
neighbouring " and by " jumps " (if he is not too pedantic). We express this
property of the surface by describing the latter as a continuum.
Let us now imagine that a large number of little rods of equal length have been
made, their lengths being small compared with the dimensions of the marble