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Dictionary of the History of Ideas | ||

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*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

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 λ*gij,* 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* + λ*gij* = (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

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.

Dictionary of the History of Ideas | ||