permits of our answering it with a moderate degree of certainty, and in
this connection the difficulty mentioned in Section XXX finds its
solution.
XXXII.
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
According to the general theory of relativity, the geometrical
properties of space are not independent, but they are determined by
matter. Thus we can draw conclusions about the geometrical structure of
the universe only if we base our considerations on the state of the
matter as being something that is known. We know from experience that,
for a suitably chosen co-ordinate system, the velocities of the stars
are small as compared with the velocity of transmission of light. We
can thus as a rough approximation arrive at a conclusion as to the
nature of the universe as a whole, if we treat the matter as being at
rest.
We already know from our previous discussion that the behaviour of
measuring-rods and clocks is influenced by gravitational fields, _i.e._
by the distribution of matter. This in itself is sufficient to exclude
the possibility of the exact validity of Euclidean geometry in our
universe. But it is conceivable that our universe differs only slightly
from a Euclidean one, and this notion seems all the more probable,
since calculations show that the metrics of surrounding space is
influenced only to an exceedingly small extent by masses even of the
magnitude of our sun. We might imagine that, as regards geometry, our
universe behaves analogously to a surface which is irregularly curved
in its individual parts, but which nowhere departs appreciably from a
plane: something like the rippled surface of a lake. Such a universe
might fittingly be called a quasi-Euclidean universe. As regards its
space it would be infinite. But calculation shows that in a
quasi-Euclidean universe the average density of matter would
necessarily be _nil_. Thus such a universe could not be inhabited by
matter everywhere; it would present to us that unsatisfactory picture
which we portrayed in Section XXX.
If we are to have in the universe an average density of matter which
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