Sidelights on Relativity
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Albert Einstein >> Sidelights on Relativity
After having gained a vivid mental image of the geometrical
behaviour of our _L'_ spheres, let us assume that in our space there
are no rigid bodies at all in the sense of Euclidean geometry, but
only bodies having the behaviour of our _L'_ spheres. Then we shall
have a vivid representation of three-dimensional spherical space,
or, rather of three-dimensional spherical geometry. Here our spheres
must be called "rigid" spheres. Their increase in size as they
depart from _S_ is not to be detected by measuring with
measuring-rods, any more than in the case of the disc-shadows on
_E_, because the standards of measurement behave in the same way as
the spheres. Space is homogeneous, that is to say, the same
spherical configurations are possible in the environment of all
points.* Our space is finite, because, in consequence of the
"growth" of the spheres, only a finite number of them can find room
in space.
* This is intelligible without calculation--but only for the
two-dimensional case--if we revert once more to the case of the disc
on the surface of the sphere.
In this way, by using as stepping-stones the practice in thinking
and visualisation which Euclidean geometry gives us, we have acquired
a mental picture of spherical geometry. We may without difficulty
impart more depth and vigour to these ideas by carrying out special
imaginary constructions. Nor would it be difficult to represent the
case of what is called elliptical geometry in an analogous manner.
My only aim to-day has been to show that the human faculty of
visualisation is by no means bound to capitulate to non-Euclidean
geometry.