Section thirty cosmological difficulties of Newton's theory. A far from the difficulty discussed in
section twenty one, there is a second fundamental difficulty attending classical celestial mechanics,
which, to the best of my knowledge, was first discussed in detail by
the astronomers Selucre. If we ponder over the question as to how the universe,
considered as a whole, is to be regarded, the first answer that
suggestsself to us is surely this, as reguard space and time, universe is
infinite. There are stars everywhere, so that the density of matter, although
very variable in detail, is nevertheless, on the average everywhere the same.
In other words, however far we might travel through space, we should find
everywhere in attenuated swarm of fixed stars of approximately the same kind and density.
This view is not in harmony with the theory of Newton, but the latter
theory rather requires that the universe should have a kind of center in which the
density of the stars is a maximum, and that as we proceed outwards from
the center, the group density of the stars should diminish, until finally,
at great distances it is succeeded by an infinite region of emptiness. The stellar
universe ought to be a finite island in the infinite ocean of space. Begin
fine proof. According to the theory of Newton, the number of lines of
force which come from infinity and terminate in a mass M is proportional to the
mass M. If on the average, the mass density rose sum zero is
constant throughout the universe, then a sphere of volume V will enclose the average
man rose zero V. Thus, the number of lines of force passing through
the surface F of the sphere into its interior is proportional to row subzero V
or unit area of the surface of the sphere. The number of lines of
force which enters the sphere is thus proportional to row subzero V over F or
two row subzero R. Hence, the intensity of the field at the surface
would ultimately become infinite with increasing radius R of the sphere, which is impossible
n footnote this conception is in itself not very satisfactory. It is still less
satisfactory because it leads to the result that the light emitted by the stars and
also individual stars of the stellar system are perpetually passing out into infinite space,
never to return and without ever again coming into interaction with other objects of nature,
Such a finite material universe would be destined to become gradually but systematically impoverished.
In order to escape this dilemma, Seeliger suggested a modification of Newton's law,
in which he assumes that for great distances, the force of attraction between
two masses diminishes more rapidly than would result from the inverse square law. In
this way, it is possible for the mean density of matter to be constant
everywhere, even to infinity, without infinitely large gravitational fields being produced. We
thus free ourselves from the distasteful conception that the material universe ought to possess something
of the nature of a center. Of course, we purchase our emancipation from
the fundamental difficulties mentioned at the cost of a modification and complication of Newton's law,
which is neither empirical nor theoretical foundation. We can imagine innumerable laws which
would serve the same purpose without our being able to state a reason why one
of them is to be preferred to the others. For any one of these
laws would be founded just as little on more general theoretical principles as is the
law of Newton end a Section thirty Section thirty one the possibility of a finite
and yet unbounded universe. But speculations on the structure of the universe also move
in quite another direction. The development of non Euclidean geometry led to the recognition
of the fact that we can cast out on the infiniteness of our space without
coming into conflict with the laws of thought or with experience Raymond Home whole.
These questions have already been treated in detail and with unsurpassedbu lucidity by Helmwoltz and
Poincare, whereas I can only touch on them briefly. Here. In the
first place, we imagine in existence in two dimensional space flat beings with flat
implements, and in particular flat rigid measuring rods, are free to move in
a plane. For them, nothing exists outside of this plane. That which
they observe to happen to themselves and their flat things is the all inclusive reality
of their plane. In particular, the constructions of plain Euclidean geometry can be
carried out by means of the rods. For example, the lattice construction considered
in Section twenty four. In contrast to ours. The universe of these beings
is two dimensional, but like ours, it extends to infinity. In their
universe there is room for an infinite number of identical squares made up of rods
ie its volume surface is infinite. If these beings say their universe is quote
plain, there is sense in the statement, because they mean that they can
perform the constructions of plain Euclidean geometry with their rods. In this connection,
the individual rods always represent the same distance, independently of their position. Let
us consider now a second two dimensional existence, but this time on a spherical
surface. Instead of auto play. The flat beings, with their measuring rods
and other objects, fit exactly on the surface. They are unable to leave
it. Their holy universe of observation extends exclusively over the surface of the sphere.
Are these beings able to regard the geometry of their universe as being plain
geometry and the rods with all as a realization of distance. They cannot do
this, for if they attempt to realize a straight line, they will obtain
a curve, which we three dimensional beings designate as a great circle, i
e. A self contained line of definite phenite length which can be measured up
by means of a measuring rod. Similarly, this universe has a phenite area
that can be compared with the area of a square constructed with rods. Great
charm resulting from this consideration lies in the recognition of the fact that the universe
of these beings is phenite and yet has no limits. But the spherical surface
beings do not need to go on a world tour in order to perceive that
they are not living any Euclidean universe. They can convince themselves of this on
every part of their world, provided they do not use too small a piece
of it. Starting from a point, they draw straight lines. Arcs of
circles is judged in three dimensional space of equal length in all directions. They
will call the line joining the free ends of these lines a circle. For
a plane surface, the ratio of the circumference of a circle to its diameter,
both lengths being measured with the same round is, according to Euclidean geometry
of the plane, equal to a constant value pie, which is independent of
the diameter of the circle. On their spherical surface. Our flat beings would
find for this ratio the value equation twenty seven Pie times sign parentheses little R
over big r unparentheses divided by parentheses little r over big r unparentheses i e.
Smaller value than pie, the difference being the more considerable greater is the
radius of the circle in comparison with the radius R of the world's sphere.
By means of this relation, spherical beings can determine the radius of their universe
quote world unquote, even when only relatively small part of their world's sphere it's
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,
of 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 am no
means of determining whether they are living in a finite or an infinite universe,
because a piece of universe to which they have access is in both cases practically
play more Euclidean. It follows directly from this discussion that for our sphere beings,
the circumference of a circle first increases with the radius till the circumference of
the universe is reached, and that it thenceforward gradually decreases to cyril for still
further increasing values of the radius. During this process, the area of the
circle continues to increase more and more, until finally becomes equal to the total
area of the whole world's 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 justifications in the fact that of all closed surfaces, sphere is
unique in possessing the property that all points on it are equivalent. I admit
that the ratio of the circumference 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 world's sphere. In other words, the world's 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 Ramond. Its points are likewise all equivalent. It possesses a finite volume,
which is determined by its radius two pie squared are cubed. 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
are the measuring ron. All the free ends of these lengths lie on a
spherical surface. We can specially measure up the area F of the surface by
means of a square made up of measuring rods. If the universe is Euclidean,
then F equals four pi are squared. If it is spherical, then
F is always less than four pi are squared. 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 quote counterpoint unquote 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 phinite i e. A phenite volume,
and has no bounds. It may be mentioned that there is yet another kind
of curved space quote elliptical space. It can be regarded as a curved space
in which the two counterpoints 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, 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 the manner of
the spherical universe. Our experience is far from being sufficient to enable us to
answer this question, But the general theory of relativity permits of our answering it
with a moderate degree of certainty, and in this connection the difficulty mentioned in
section thirty finds solution. End of section thirty one. Section thirty two structure
a space according to the general theory of relativity. According to the general theory
of relativity, the geometrical properties of space are not independent, but they are
determined by matter. Thus, we can draw conclusions about the geometrical structure of
the universe only if we base our considerations on the state of the matter as
being something that is known. We know from experience that for a suitably chosen
coordinates 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 behavior of measuring rods in 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.
As regards its space, it would be infinite. But calculation shows that in
a quasi Euclidian universe the average density of matter would necessarily be nil. Thus
such a universe could not be inhabited by matter everywhere. It would present to
us that unsatisfactory picture which we portrayed in section thirty. If we are to
have in the universe an average density of matter which differs from zero, however
small may be, that difference, then the universe cannot be quasi euclidian.
On the contrary, the results of calculation indicate that if matter be distributed uniformly,
the universe would necessarily be spherical or elliptical. Since in reality, the
detailed distribution of matter is not uniform, the real universe will deviate in individual
parts from the spherical ie. The universe will be quasi spherical, but it
will be necessarily finite. In fact, the theory supplies us with a simple
connection. Begin footnote. For the radius R of the universe, we obtain
the equation R squared equals two over a capa row. The use of the
C G S system in this equation gives two over a K equals one to
the eighth power times ten to the twenty seventh power. P is the average
density of the matter, and K is a constant connected with the Newtonian constant
of gravitation. End footnote between the space expanse of the universe and the average
density of matter in it. End Section thirty two, end of Part three.
End of relativity special in general theory by Albert Einstein.