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67
Relativity: The Special and General Theory
i.e. a smaller value than ?, the difference being the more considerable, the greater is the radius of
the circle in comparison with the radius R of the " world?sphere." By means of this relation the
spherical beings can determine the radius of their universe (" world "), even when only a relatively
small part of their worldsphere is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a spherical " world " and not on
a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane
of the same size.
Thus if the spherical surface beings are living on a planet of which the solar system occupies only a
negligibly small part of the spherical universe, they have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of universe " to which they have
access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that
for our sphere?beings the circumference of a circle first increases with the radius until the "
circumference of the universe " is reached, and that it thenceforward gradually decreases to zero
for still further increasing values of the radius. During this process the area of the circle continues to
increase more and more, until finally it becomes equal to the total area of the whole "
world?sphere."
Perhaps the reader will wonder why we have placed our " beings " on a sphere rather than on
another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the
sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio
of the circumference c of a circle to its radius r depends on r, but for a given value of r it is the same
for all points of the " worldsphere "; in other words, the " world?sphere " is a " surface of constant
curvature."
To this two?dimensional sphere?universe there is a three?dimensional analogy, namely, the
three?dimensional spherical space which was discovered by Riemann. its points are likewise all
equivalent. It possesses a finite volume, which is determined by its "radius" (2?²R³). Is it possible
to imagine a spherical space? To imagine a space means nothing else than that we imagine an
epitome of our " space " experience, i.e. of experience that we can have in the movement of " rigid
" bodies. In this sense we can imagine a spherical space.
Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of
these the distance r with a measuring?rod. All the free end?points of these lengths lie on a
spherical surface. We can specially measure up the area (F) of this surface by means of a square
made up of measuring?rods. If the universe is Euclidean, then F = 4?R² ; if it is spherical, then F is
always less than 4?R². With increasing values of r, F increases from zero up to a maximum value
which is determined by the " world?radius," but for still further increasing values of r, the area
gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge
farther and farther from one another, but later they approach each other, and finally they run
together again at a "counter?point" to the starting point. Under such conditions they have traversed
the whole spherical space. It is easily seen that the three?dimensional spherical space is quite
analogous to the two?dimensional spherical surface. It is finite (i.e. of finite volume), and has no
bounds.
It may be mentioned that there is yet another kind of curved space: " elliptical space." It can be
regarded as a curved space in which the two " counter?points " are identical (indistinguishable from
each other). An elliptical universe can thus be considered to some extent as a curved universe
possessing central symmetry.
It follows from what has been said, that closed spaces without limits are conceivable. From
amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it
are equivalent. As a result of this discussion, a most interesting question arises for astronomers
and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in