IS THE UNIVERSE INFINITE?
_Kant and the number of the stars—Extinct stars and
dark nebulæ—Extent and aspect of the astronomical
universe—Different kinds of universes—Poincaré’s
calculation—Physical definition of the infinite—The
infinite and the unlimited—Stability and
curvature of cosmic space-time—Real and virtual
stars—Diameter of the Einsteinian universe—The
hypothesis of globes of ether._
Is the universe infinite? It is a question that men have asked in all
ages, though they have not defined its meaning very accurately. The
theory of Relativity enables us to approach it from a new and subtle
point of view.
Kant—the genial grumbler who found it so horribly monotonous to see
the same sun shining, and the same spring blossoming, every year—took
his stand on metaphysical considerations when he affirmed that space is
infinite, and is sown with similar stars in all parts.
It is, perhaps, better to confine ourselves in such a matter to the
results of recent observation, and close the doors of our debating-room
against the fog of metaphysics. Indeed, the latter would compel us to
define pure space, about which we know nothing—not even if there is
such a thing.
The proof that we know little about it is the fact that the Newtonians
believe in it, while the Einsteinians regard it merely as an
inseparable attribute of material things. They define space by matter;
and they then have to define the latter. Descartes, on the contrary,
defined matter in terms of extension, which is the same thing as
space. It is a vicious circle. It is therefore better to leave Kant’s
metaphysical arguments out of our discussion, and adhere strictly to
experience, to what is measurable.
To simplify matters, we will admit the reality of this continuum in
which the stars float, which is traversed by their radiations, which
common sense calls space. If there were stars everywhere—if they were
infinite in number—there would also be space and matter everywhere.
Newtonians might find this a triumph equally with Einsteinians. Those
who believe in absolute space and those who deny it—Absolutists and
Relativists—would equally rejoice.
It would be fortunate if astronomical observation were to show that
the number of the stars is infinite, and thus the holders of contrary
opinions could both chant a victory in their writings. But what does
astronomical observation actually report?
There are those who deny _a priori_ that the number of the stars
can be infinite. That number, they said, is capable of increase; it is
therefore not infinite, because nothing can be added to the infinite.
The argument is specious, but unsound; although Voltaire himself was
seduced by it. One need not be a great mathematician to see that it
is always possible to add to an infinite number, and that there are
infinite quantities which are themselves infinitely small in comparison
with others. Let us get on to the facts.
If the stellar universe has no limits, there is no visual line drawn
from the earth to the heavens which will not encounter one of the
stars. The astronomer Olbers has said that the whole nocturnal sky
would in that case shine with the brilliance of the sun. But the total
brilliance of all the stars put together is only three thousand times
greater than that of a star of the first magnitude, or thirty million
times less than the light of the sun.
But that proves nothing, as Olbers’ argument is wrong, for two reasons.
On the one hand, there are necessarily a good many extinct or dark
stars in the heavens. Some of them have been closely studied, even
weighed. They betray their existence by periodically eclipsing brighter
stars, with which they revolve. On the other hand, it was discovered
some time ago that celestial space is occupied over large stretches by
dark gaseous masses and clouds of cosmic dust, which absorb the light
of more distant stars. We thus see that the existence of an infinite
number of stars is quite compatible with the poorness of the light of
the heavens at night.
* * * * *
Now let us put on our spectacles—our telescopes, I mean—and turn from
the province of possibility to that of reality, and we shall see that
recent astronomical observation has yielded a number of remarkable
facts which lead irresistibly to the following conclusions.
The number of the stars is not, as was long supposed, limited by the
range of our telescopes alone. As we get further away from the sun, the
number of stars contained in a unity of space, the frequence of the
stars, the density of the stellar population, do not remain uniform,
but decrease in proportion as we approach the limits of the Milky Way.
The Milky Way is a vast archipelago of stars, our sun lying in its
central region. This mass of stars, to which we belong, has, roughly,
the shape of a watch-case, the thickness being only about half the width
of the structure. Light, which travels from the earth to the moon in
little over a second, from the earth to the sun in eight minutes, and
from the earth to the nearest star in three years, needs at least
30,000 years—three hundred centuries—to pass from end to end of the
Milky Way.
The number of stars in the Milky Way is something between 500 and 1,500
millions. It is a small number: scarcely equal to the human population
of the earth, much smaller than the number of molecules of iron in a
pin’s head.
In addition to these we have discovered dense masses of stars, such
as the Magellanic Clouds, the cluster in Hercules, and so on, which
seem to belong to the fringes of our Milky Way—to be suburbs of it,
so to say. These suburbs seem to stretch a considerable distance,
particularly on one side of the Milky Way. The furthest away is,
perhaps, not less than 200,000 light-years from us.
Beyond these, space seems to be deserted, devoid of stars over expanses
which are enormous in comparison with the dimensions of our galactic
universe as we have described it. What is beyond this?
Well, beyond this we find those strange bodies, the spiral nebulæ,
lying like silver snails in the garden of the stars. We have discovered
several hundred thousand of them. Some astronomers believe that these
spiral masses of stars may be annexes of the Milky Way, reduced models
of it. Most astronomers incline to think, for very good reason, that
the spiral nebulæ are systems like the Milky Way, and comparable to it
in their dimensions. If the former view is correct, the entire system
of stars accessible to our telescopes could be traversed by light in
some hundreds of thousands of years. On the second hypothesis the
dimensions of the stellar universe to which we belong must be
multiplied by ten, and light would take at least millions of years to
traverse it.
On the first view the entire stellar universe, in so far as it is
accessible to us, consists of the Milky Way and its annexes: that is to
say, a local concentration of stars, beyond which we can see nothing.
The stellar universe is, in other words, practically limited, or at
least finite.
On the second view our Milky Way is simply one of the myriads of spiral
universes we see. The spiral nebula (with its hundreds of millions of
stars) plays the same part in this vaster universe that a star has in
the Milky Way. We have the same problem as before, but on a vaster
scale: if the Milky Way consists of a concentration of a finite number
of stars, as observation proves, does the accessible universe consist
of a finite number of spiral nebulæ?
Experience has as yet not pronounced on this point. But in my opinion
it is probable that, when our instruments are powerful enough to tackle
such a problem—in several centuries, perhaps—science will answer
“yes.”
If it were otherwise, if the spiral nebulæ were fairly evenly
distributed as we go outward, we can show by calculation that,
attraction being in inverse proportion to the square of the distance,
gravitation would have an infinite intensity in such a universe, even
in the part in which we live. But this is not the case. It follows
that, either the attraction of two masses decreases at great distances
rather more rapidly than in inverse proportion to the square of the
distance (which is not wholly impossible), or that the number of stellar
systems and stars is finite. Personally I favour the second hypothesis,
but it is incapable of proof. In such matters there is always an
alternative, always a way of escaping in accordance with one’s bias,
and there is really nothing that compels us to say that the stars are
finite in number.
* * * * *
Starting from the mean value, as it has been observed, of the proper
motions of the nearer stars, Henri Poincaré has calculated that the
total number of stars in the Milky Way must be about one thousand
million. The figure agrees fairly well with the results of the
star-gauges effected by astronomers by means of photographic plates.
He has also shown that the proper motions of stars would be greater if
there were many more stars than those which we see. Thus Poincaré’s
calculations are opposed to the hypothesis of an indefinite extension
of the stellar universe, as the number of stars “counted” agrees fairly
closely with the number “calculated.” We should add, however, that
these calculations prove nothing if the law of attraction is not quite
the inverse proportion of the square at enormous distances.
On the other hand, if the universe is finite in space as it is
conceived in classic science, the light of the stars, and isolated
stars themselves, would gradually drift away into the infinite,
and the cosmos would disappear. Our mind resents this consequence,
and astronomical observation discovers no trace whatever of such a
dislocation.
In a word, in the space of the “Absolutists” the stellar universe can
only be infinite if the law of the square of distances is not quite
exact for very remote masses; and it cannot be finite except on the
condition that it is ephemeral in point of time.
For Newton, indeed, the _stellar_ universe might be finite within
an infinite universe, because in his view there can be space without
matter. For Einstein, on the contrary, the universe and the material or
stellar universe are one and the same thing, because there is no space
without matter or energy.
* * * * *
These difficulties and obscurities disappear in great part when we
consider space, or space-time, from the Einsteinian standpoint of
General Relativity.
What is the meaning of the sentence, “The universe is infinite”? From
either the Einsteinian, the Newtonian, or the Pragmatist point of view
it means: If I go straight ahead, going on eternally, I shall never get
back to my starting-point.
Is it possible? Newton is compelled to say yes, because in his view
space stretches out indefinitely, independent of the bodies that occupy
part of it, whether the number of the stars is or is not limited.
But Einstein says no. For the Relativist the universe is not
necessarily infinite. Is it therefore limited, fenced in by some sort
of railings? No. It is not limited.
A thing may be unlimited without being infinite. For instance, a man
who moves on the surface of the earth may travel over it indefinitely
in every direction without ever reaching a limit. The surface of the
earth, thus regarded, or the surface of any sphere whatsoever, is
therefore both finite and unlimited. Well, we have only to apply to
space of three dimensions what we find in two-dimensional space (a
spherical surface), to see how the universe may be at one and the same
time finite and unlimited.
We saw that, in consequence of gravitation, the Einsteinian universe
is not Euclidean, but curved. It is, as we said, difficult, if not
impossible, to visualise a curvature of space. But the difficulty
exists only for our imagination, which is restricted by our life of
sense, not for our reason, which goes farther and higher. It is one of
the commonest of errors to suppose that the wings of the imagination
are more powerful than those of reason. If one wants proof of the
contrary, one has only to compare what the most poetic of ancient
thinkers made of the starry heavens with what modern science tells
about the universe.
Here is the way to approach our problem. Let us not notice for the
moment the rather irregular distribution of stars in our stellar
system, and take it as fairly homogeneous. What is the condition
required for this distribution of the stars under the influence of
gravitation to remain stable? Calculation gives us this reply: The
curvature of space must be constant, and such that space is bent like a
spherical surface.
Rays of light from the stars may travel eternally, indefinitely, round
this unlimited, yet finite, universe. If the cosmos is spherical in
this way, we can even imagine the rays which emanate from a star—the
sun, for instance—crossing the universe and converging at the
diametrically opposite point of it.
In such case we might expect to see stars at opposite points in the
heavens, of which one would be the image, the spectre, the “double” of
the other—in the sense which the ancient Egyptians gave to the word.
Properly speaking, this “double” would represent, not the generating
star as it is, but as it was at the time when it emitted the rays which
form the double, or millions of years earlier.
If we observe the original and the double star, the reality and the
mirage, simultaneously from some remote part of the stellar system,
such as our planet, we shall see a great difference between them, since
the “copy” will show us the original as it was thousands of centuries
before. It may, in fact, happen that the second star is more brilliant
than the first, because in the meantime the first has gradually cooled,
and may even be extinct.
* * * * *
It is improbable that we should find many of these phantom-stars,
or virtual stars, luminous and unreal daughters of heavy suns. The
reason is that the rays in their passage through the universe will
generally be diverted by the stars near which they pass. Concentration
or convergence of them at the antipodes of the real star must be rare.
Moreover, the rays are to some extent absorbed by the cosmic stuff they
meet in space. It is, however, not impossible that the astronomers of
the future may discover such phenomena. It is, in fact, not impossible
that we have already observed such things without knowing it.
In any case, what observers have not done in the past they may very
well do in the future, thanks to the suggestions of the new science.
Possibly it is going to have a great effect on observational astronomy
and induce it to furnish brilliant new verifications of theory. There
may be astonishing results, unforeseen by our folly, of the new
conceptions, surpassing in their fantastic poetry the most romantic
constructions of the imagination. Reality, or at least the possible, is
rising to giddy heights that were far beyond the reach of the golden
wings of fantasy.
I spoke on a previous page of the millions of years which light
takes to travel round our curved universe. Starting from the fairly
well-ascertained value of the quantity of matter comprised in the
Milky Way, it is possible to calculate the curvature of the world and
its radius. We find that the radius has a value equal to at least
150,000,000 light-years.
It therefore takes light at least 900,000,000 years, at a speed of
186,000 miles a second, to travel round the universe, assuming that it
consists only of the Milky Way and its annexes. The figure is quite
consistent with the figures we get from astronomical observation for
the dimensions of the galactic system, and also with the much larger
figures which we find if we regard the spiral nebulæ as Milky Ways.
Thus for the Relativist the universe may be unlimited without being
infinite. As to the Pragmatist, who goes straight ahead—who follows
what he calls a straight line, or the path of light—he will get back
in the end to the body from which he started, provided that he has time
enough at his disposal. He will then say that, if that is the nature of
things, the universe is not infinite.
Hence the question of the infinity or finiteness of the universe can
be controlled by experience, and some day it will be possible to prove
whether the whole cosmos and space are Newtonian or Einsteinian.
Unfortunately, it will have to be a very long experience, with various
little practical difficulties to overcome.
We may therefore prefer not to commit ourselves without further
instructions. We may not feel ourselves obliged to choose between the
two conceptions, and we may leave the benefit of the doubt to whichever
of the two is false.
* * * * *
Moreover, there is perhaps a third issue: if not for the Pragmatist, at
least for the philosopher—I mean, seeing that in England physics comes
under the head of “Natural Philosophy,” for the physicist.
Here it is. If all the heavenly bodies we know belong to the Milky Way,
other and very remote universes may be inaccessible to us because they
are optically isolated from us; possibly by the phenomena of the cosmic
absorption of light, to which we have already referred.
But this might also be due to something else which will, perhaps, shock
Relativists, but will seem to Newtonians quite possible. The ether, the
medium that transmits the luminous waves, and which Einstein has ended
by admitting once more (refusing, however, to give it its familiar
kinematic properties), and matter seem more and more to be merely
modalities. We explained this, on the strength of the most recent
physical discoveries, in a previous chapter. There is nothing to prove
that these two forms of substance are not always associated.
Does this not give me the right to think that perhaps our whole visible
universe, our local concentration of matter, is only an isolated clump
or sphere of ether? If there is such a thing as absolute space (which
does not mean that it is accessible to us), it is independent of ether
as well as matter. In that case there would be vast empty spaces,
devoid of ether, all round our universe. Possibly other universes
palpitate beyond these; and for us such worlds would be for ever as if
they did not exist. No ray of knowledge would ever reach us from them.
Nothing could cross the black, dumb abysses which environ our stellar
island. Our glances are confined for ever within this giant—yet too
small—monad.
“Are there, then,” some will cry in astonishment, “things which exist,
yet we will never know them?” Naive pretension—to want to embrace
everything in a few cubic centimetres of grey brain-stuff!