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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
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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-
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

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:
Rij1/2 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 λgij 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 λgij
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.