V. THE EINSTEIN STATIC UNIVERSE
The first great step in the development of modern
cosmology was taken by
Einstein in his famous 1917
paper, in which he set out to derive the
physical
properties of the universe by applying his field equa-
tions to the kind of distribution of
matter that one
might reasonably expect to find in the universe as a
whole. Here Einstein had to introduce some simpli-
fying assumptions, since we have detailed knowledge
about the distribution of matter only in a relatively
small
region of space (within a few thousand light years
of our own solar system)
and we find that the matter
here is concentrated in lumps (the stars) with
some dust
and gas between the lumps. Einstein therefore intro-
duced the
cosmological
principle, which states that,
except for local irregularities, the
universe has the same
aspect (the same density of matter) as seen from
any
point. This means that what we see in our region of
the universe
is pictured as being repeated everywhere,
like a wall-paper or linoleum
pattern.
Einstein next replaced the lumpiness of the distribu-
tion of matter (as indicated in the existence of
stars
and galaxies) by a smooth, uniform distribution which
we may
obtain by picturing all the matter in the stars
as smeared out to fill
space with a fog of uniform
density (actually a proton gas with a few
protons per
cubic foot of space). Einstein made one other assump-
tion—that the universe is
static; that is, that the density
of matter does not change with time and
that there
are no large scale motions in the universe. At the time
that Einstein did this work, this assumption appeared
to be eminently
justified because the recession of the
distant galaxies had not yet been
discovered and the
stars in our own neighborhood of space were known
to be moving with fairly small random velocities. With
these assumptions,
Einstein still had to make one im-
portant
extrapolation—he had to extend his field equa-
tions to make them applicable to the entire universe
and not just to a small region of empty space around
the sun.
It is useful here (as a guide in our discussion) to
write down Einstein's
field equations in the form in
which Einstein first used them in his study
of cos-
mology:
This equation really
represents ten distinct equations
since the quantities Rij, gij, and Tij are components
of three
different tensors, and there are just ten such
distinct components in each
of these tensors. The tensor
components Rij, which
are constructed in a well-
defined way from
the potentials gij (which are also
called the
components of the metric tensor) determine
the nature of the space-time
geometry. The quantity
R gives the curvature of space-time at any specific
point, and the tensor Tij is the matter-energy-
momentum-pressure tensor. G is the universal gravita-
tional constant and c is the speed of light. This set
of ten equations
thus tells us how the matter and energy
that are present determine the
metric tensor gij at each
point of space-time and
therefore the geometry at each
such point. To determine the potentials gij and hence
the geometry of space time, one must thus solve
the ten field
equations for the known or assumed dis-
tribution of matter and energy as given by the ten-
sor
Tij.
In the case of planetary motion, one simply places
Tij = 0; this leads to Einstein's law of gravity for
empty
space
Rij = 0,
which reduces to
Newton's law for weak gravitational
fields. But for the cosmological
problem, Einstein
placed
Tij equal to a constant
value (the average den-
sity of matter at each
point) and then sought to solve
the field equations (1) under these
conditions. In other
words, he attempted to obtain the potentials
gij from
equations (1) under the assumption that
there is a
constant (but very small) density of matter throughout
the
universe. His idea was that this small density would
introduce a constant
curvature of space-time at each
point so that the universe would be curved
as a whole.
This initial attempt to obtain a static model of the
universe was unsuccessful, however, because the equa-
tions (1) lead to a unique set of potentials
gij only if
one knows the values of these quantities at infinity.
Now
the natural procedure in this kind of analysis is to
assume that
all the values of
gij are zero at infinity,
but this
cannot be done if one keeps the equations (1)
and also retains the
assumption that the density in the
universe is everywhere the same. In
fact, the values
of
gij become infinite at infinity
under these conditions,
so that the equation (1) can give no static model
of
the universe.
This very disturbing development forced Einstein to
alter his field
equations (which he did very reluctantly)
by introducing an additional term
on the left-hand side.
Fortunately, the field equations (1) are such that
this
can be done, for it is clear that the character of these
equations is not changed when one adds to the left
hand side a second order
tensor which obeys the same
conservation principle (it must represent a
quantity
that can neither be destroyed nor created) as the other
two
terms together. Now it can be shown (as Einstein
knew) that the only
physical term that has this impor-
tant
property is λgij, where
λ is a universal constant.
Hence Einstein enlarged his field
equations by the
addition of just this term and replaced (1) by the fol-
lowing most general set of field equations:
Rij
– 1/2
Rgij + λgij = (8π / c4)GTij. (2)
These are now the basic equations of cosmology.
Before discussing the various cosmological models
that can be deduced from
these equations, we should
say a few more words about the famous constant
λ
which has become known in scientific literature as the
“cosmological constant.” It is clear from the
way this
constant was introduced that it has an exceedingly
small
numerical value as compared to the terms in (2)
that give rise to the
ordinary gravitational forces. For
if this were not so, the term
λ
gij would destroy
the
agreement between the observed motions of the planets
(that is,
the motion of Mercury) and those predicted
by (2). It turns out, in fact,
as we shall see, that the
square root of λ (for the static
closed universe that
Einstein first obtained) is the reciprocal of the
radius
of the universe. Finally, we note that the term λ
gij in (2) behaves like a
repulsion—in empty space it has
the opposite sign of the
gravitational term and hence
opposes gravitational attraction. A curious
thing about
it, however, is that the repulsion of an object increases
with its distance from
any observer and is the same
for all objects (regardless of mass) at that distance.
With the inclusion of the cosmical term gij in his
field equations, Einstein was able to derive a static,
finite model of the
universe. In a sense, we can under-
stand
this result in the following way: the small amount
of matter in each until
volume of space introduces the
same curvature everywhere, so that space
bends uni-
formly, ultimately curving back
upon itself to form a
closed spherical universe. If there were no
cosmical
repulsion term, the gravitational force of all the matter
would cause this bubble with a three dimensional sur-
face to collapse. But the cosmical term prevents this;
in fact,
the cosmical repulsion and the gravitational
contraction just balance each
other to give a static
unchanging universe. An interesting property of
this
universe is that it is completely filled; that is, it is as
tightly filled with matter as it can be without changing.
For if we were to
add a bit of matter to it, the gravita-
tional attraction would outweigh the cosmic repulsion
and the
universe would shrink to a smaller size, which
would be just right for the
new amount of matter (again
completely filled). If we remove a bit of
matter, the
universe would expand to a slightly larger size, but
it
would again be completely filled.
Now it may seem that such a completely filled uni-
verse must be jam-packed with matter like a solid, or
like the
nucleus of an atom, but this is not so. In fact,
the density of matter in
such a universe depends on
its radius (that is, its size) and its total
mass. Einstein
found the radius of such a static universe to be about
30 billion light years, with a total mass of about 2 ×
1055 grams.
This would lead to a density of about 10-29
gm/cm3, or about one proton per
hundred thousand
cubic centimeters of space. We see that this is a
quite
empty universe, even though it is as full as it can be!
Before we see why the static Einstein universe had
to be abandoned, we must
try to explain more precisely
the meaning of spherical space. When we speak
of the
universe as we have up to now, we mean the four-
dimensional space-time universe,
but the curvature we
have been referring to is the curvature of the
actual
three-dimensional physical space of our existence. To
understand this, we may picture the physical space of
the universe as the
surface of a rubber balloon and
all the matter (that is, the galaxies) is
to be distributed
over this surface in the form of little specks. Note
that
the physical three-dimensional space of the universe
is the
surface of the balloon, not the whole balloon
itself. Of course, the
surface of a real balloon is two-
dimensional, so that we have lost one dimension in this
picture, but
that does not affect the picture seriously.
The spatial distances of, or
separations among galaxies
are now to be measured along the surface of the
bal-
loon (just as the distance between New
York and
Chicago is measured along the surface of the earth).
With this picture, we thus establish an analogy be-
tween the three-dimensional space of our universe and
the
two-dimensional surface of a sphere like the earth.
The analogy can be made
complete by supposing that
the inhabitants of the earth are capable of only
a
two-dimensional perception (along the surface of the
earth) so that
they know nothing about up or down
and hence cannot perceive that the
earth's surface is
curved in a space of higher dimensions (the three
dimensions of actual space). Even though we, as actual
three-dimensional
creatures, can assign a radius of
curvature to the surface of the earth
(the distance of
the surface of the earth from its center) the two-
dimensional inhabitants of the earth
would find such
a concept difficult to perceive or accept.
To carry this over to the three-dimensional space
of the universe, we must
picture the curvature of this
three-dimensional space as occurring in a
space of
higher dimensions. The radius of the universe is thus
a
distance (actually a number) associated with a direc-
tion at right angles to the three-dimensional curved
surface of
the universe, and hence into a fourth dimen-
sion. In this type of universe, every point is similar
to every
other point and no point of this curved surface
can be taken as the center
of space; in fact, there is
no center, just as there is no center on the
surface of
the earth. The center of the universe, if we can speak
of
it at all, is in the fourth dimension.
