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Page 8
It appears less problematical to extend the ideas of practical
geometry to spaces of cosmic order of magnitude. It might, of course,
be objected that a construction composed of solid rods departs more
and more from ideal rigidity in proportion as its spatial extent
becomes greater. But it will hardly be possible, I think, to assign
fundamental significance to this objection. Therefore the question
whether the universe is spatially finite or not seems to me
decidedly a pregnant question in the sense of practical geometry.
I do not even consider it impossible that this question will be
answered before long by astronomy. Let us call to mind what the
general theory of relativity teaches in this respect. It offers
two possibilities:--
1. The universe is spatially infinite. This can be so only if the
average spatial density of the matter in universal space, concentrated
in the stars, vanishes, i.e. if the ratio of the total mass of the
stars to the magnitude of the space through which they are scattered
approximates indefinitely to the value zero when the spaces taken
into consideration are constantly greater and greater.
2. The universe is spatially finite. This must be so, if there is
a mean density of the ponderable matter in universal space differing
from zero. The smaller that mean density, the greater is the volume
of universal space.
I must not fail to mention that a theoretical argument can be adduced in
favour of the hypothesis of a finite universe. The general theory
of relativity teaches that the inertia of a given body is greater as
there are more ponderable masses in proximity to it; thus it seems
very natural to reduce the total effect of inertia of a body to
action and reaction between it and the other bodies in the universe,
as indeed, ever since Newton's time, gravity has been completely
reduced to action and reaction between bodies. From the equations
of the general theory of relativity it can be deduced that this
total reduction of inertia to reciprocal action between masses--as
required by E. Mach, for example--is possible only if the universe
is spatially finite.
On many physicists and astronomers this argument makes no impression.
Experience alone can finally decide which of the two possibilities
is realised in nature. How can experience furnish an answer? At first
it might seem possible to determine the mean density of matter by
observation of that part of the universe which is accessible to our
perception. This hope is illusory. The distribution of the visible
stars is extremely irregular, so that we on no account may venture
to set down the mean density of star-matter in the universe as
equal, let us say, to the mean density in the Milky Way. In any
case, however great the space examined may be, we could not feel
convinced that there were no more stars beyond that space. So it
seems impossible to estimate the mean density. But there is another
road, which seems to me more practicable, although it also presents
great difficulties. For if we inquire into the deviations shown
by the consequences of the general theory of relativity which are
accessible to experience, when these are compared with the consequences
of the Newtonian theory, we first of all find a deviation which
shows itself in close proximity to gravitating mass, and has been
confirmed in the case of the planet Mercury. But if the universe
is spatially finite there is a second deviation from the Newtonian
theory, which, in the language of the Newtonian theory, may be
expressed thus:--The gravitational field is in its nature such as
if it were produced, not only by the ponderable masses, but also by
a mass-density of negative sign, distributed uniformly throughout
space. Since this factitious mass-density would have to be enormously
small, it could make its presence felt only in gravitating systems
of very great extent.
Assuming that we know, let us say, the statistical distribution
of the stars in the Milky Way, as well as their masses, then by
Newton's law we can calculate the gravitational field and the mean
velocities which the stars must have, so that the Milky Way should
not collapse under the mutual attraction of its stars, but should
maintain its actual extent. Now if the actual velocities of the stars,
which can, of course, be measured, were smaller than the calculated
velocities, we should have a proof that the actual attractions
at great distances are smaller than by Newton's law. From such a
deviation it could be proved indirectly that the universe is finite.
It would even be possible to estimate its spatial magnitude.
Can we picture to ourselves a three-dimensional universe which is
finite, yet unbounded?
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