|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.|| 1|
| 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
| If we are to have in the universe an average density of matter
which differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of calculation
indicate that if matter be distributed uniformly, the universe would necessarily
be spherical (or elliptical). Since in reality the detailed distribution
of matter is not uniform, the real universe will deviate in individual parts
from the spherical, i.e. the universe will be quasi-spherical. But
it will be
necessarily finite. In fact, the theory supplies us with a simple connection 1
between the space-expanse of the universe and the average density of matter