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.
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