Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

2 |

3 |

9 |

2 | VI. |

V. |

VI. |

3 | I. |

VI. |

2 | V. |

2 | III. |

3 | III. |

2 | VI. |

1 | VI. |

6 | V. |

5 |

3 | V. |

1 | III. |

2 | VII. |

VI. |

1 | VI. |

1 | III. |

III. |

8 | II. |

3 | I. |

2 | I. |

1 | I. |

2 | V. |

1 | VII. |

2 | VI. |

4 | V. |

9 | III. |

4 | III. |

5 | III. |

16 | II. |

2 | I. |

9 | I. |

1 | I. |

1 | VI. |

VII. |

2 | III. |

1 | VII. |

3 | VII. |

2 | VII. |

2 | V. |

VI. |

1 | VI. |

1 | VI. |

2 | VI. |

2 | VI. |

1 | VII. |

III. |

IV. |

10 | VI. |

VI. |

1 | VI. |

1 | V. |

3 | V. |

4 | V. |

10 | III. |

6 | III. |

2 | VII. |

4 | III. |

I. |

7 | V. |

2 | V. |

2 | VII. |

1 | VI. |

5 | I. |

4 | I. |

7 | I. |

8 | I. | COSMOLOGY SINCE 1850 |

1 | VI. |

12 | III. |

4 | IV. |

4 | III. |

2 | IV. |

1 | IV. |

1 | IV. |

VI. |

1 | VI. |

3 | VI. |

1 | V. |

2 | III. |

1 | VI. |

Dictionary of the History of Ideas | ||

#### COSMOLOGY SINCE 1850

####
*I. INTRODUCTION*

The last forty years of the nineteenth century
were

among the most remarkable in the history of science,

for this was
a period of amazing scientific achievements

and contradictions; on the one
hand classical physics

and astronomy were enjoying some of their
greatest

successes during this period, but at the same time

observational and experimental data, which were ulti-

mately to overthrow the classical laws of physics,
were

slowly being collected. Until the year 1860 physics and

astronomy
were dominated by Newton's concepts of

space and time and by his laws of
mechanics and

gravitation; these seemed sufficient to explain observa

tions ranging all the way from the motion of the planets

to the
behavior of the tides on the earth. The great

eighteenth- and
nineteenth-century mathematicians

such as Euler, Laplace, Lagrange,
Hamilton, and

Gauss had cast the Newtonian laws into beautiful and

magnificent mathematical forms which had their

greatest applications to
celestial mechanics. Astrono-

mers happily
used these techniques to show how excel-

lent
was the agreement between observation and the-

ory. The two domains of physics that still lay outside

the Newtonian
laws—electromagnetism and optics—

were also soon to
be incorporated into a satisfying

theoretical structure. In the year 1865,
James Clerk

Maxwell published his famous papers on his electro-

magnetic theory of light, which
gave a precise and

beautiful mathematical formulation of Faraday's ex-

perimental discoveries, unified
electricity, magnetism,

and optics, and opened up the whole field of electro-

magnetic technology.

Thus, at the end of the first decade of the last forty

years of the
nineteenth century, everything seemed to

fall neatly into place in the
world of science. To the

scientists of that period, the universe appeared
to be

a well ordered arrangement of celestial bodies moving

about in
an infinite expanse of absolute space, and with

all the events in the
universe occurring in a unique

and absolute sequence in time. There was no
question

at that time as to the correctness of this Newtonian

universe
based on the concepts of absolute space and

time; only the observational
and experimental details

were lacking to make the picture complete,
and

everyone was confident that, with improved technol-

ogy, these details would be obtained in time.

This absolute concept of the universe and of the laws

of nature was very
satisfying to the late nine-

teenth-century man, who saw in the orderly and abso-

lute scheme of things the demonstration of the
Divine

Omnipotence which he worshipped and which gave

him the reason
for his existence; moreover, the infini-

tude
of space and time required by the Newtonian

universe was also required by
the concept of an in-

finitely powerful
deity, as described by Alexander

Pope:

A hero perish or a sparrow fall;

Atoms or systems into ruin hurl'd;

And now a bubble burst, and now a world.

(*An Essay on Man* III. 87-90)

####
*II. DISCREPANCIES IN THE*

NEWTONIAN UNIVERSE

NEWTONIAN UNIVERSE

But even while this neat, orderly scheme of the

universe was being eagerly
incorporated into Victorian

philosophical and social concepts, its very
basis was

data, and by logical analysis in four different realms

of physics and astronomy: in the realm of optics, the

experiments of Michelson and Morley on the speed of

light were to destroy the Newtonian concepts of abso-

lute space and time and to replace them by the Ein-

steinian space-time concept (the special theory of rela-

tivity); in the realm of radiation, the discoveries of the

properties of the radiation emitted by hot bodies were

to upset the Maxwell wave-theory of light and to

introduce the quantum theory (the photon) with its

wave-particle dualism; in the realm of observational

astronomy, the discrepancy between the deductions

from Newtonian gravitational theory and the observed

motion of Mercury (the advance of its perihelion)

indicated the need for a new gravitational theory

which Einstein produced in 1914 (the general theory

of relativity); finally, in the realm of cosmology, var-

ious theoretical analyses showed that the nine-

teenth-century models of the universe, constructed

with Newtonian gravitational theory and space-time

concepts, were in serious contradiction with stellar

observations.

Although the investigation of each of these depar-

tures from classical physics is of extreme importance

and each
one has an important bearing on the most

recent cosmological theories, we
limit ourselves here

to the cosmological realm and, where necessary in
our

discussion, use the results of modern physics without

concern
about how they were obtained. However,

before we discuss the difficulties
inherent in Newtonian

cosmology, we must consider one other important

nineteenth-century discovery which, at the time,

seemed to have no bearing
on the structure of the

universe but which ultimately played a most
important

role in the development of cosmology. This was the

discovery
of the non-Euclidean geometries by Gauss,

Bolyai, Lobachevsky, Riemann, and
Klein. At the time

that these non-Euclidean geometries were
discovered,

and for many years following, scientists in general

considered them to be no more than mathematical

curiosities, with no
relevance to the structure of the

universe or to the nature of actual
space. Most mathe-

maticians and
scientists simply took it for granted

that the geometry of physical space
is Euclidean and

that the laws of physics must conform to Euclidean

geometry.

This attitude, however, was not universal and Gauss

himself, the spiritual
father of non-Euclidean geometry,

proposed a possible (but in practice,
unrealizable) test

of the flatness of space by measuring the interior
angles

of a large spatial triangle constructed in the neigh-

borhood of the earth. Also, the
mathematician W. K.

Clifford, in *The Common Sense of the
Exact Sciences*

(1870; reprint, New York, 1946), speculated that the

geometry of actual space might not be Euclidean. He

proposed
the following ideas: (1) that small portions

of space are, in fact, of a
nature analogous to little

hills on a surface which is, on the average,
flat—

namely, that the ordinary laws of geometry are not

valid in them; (2) that this property of being curved

or distorted is
continually being passed on from one

portion of space to another after the
manner of a wave;

(3) that this variation of the curvature of space is
what

really happens in that phenomenon which we call

motion of matter,
whether ponderable or ethereal; (4)

that in the physical world nothing else
takes place but

this variation, subject (possibly) to the laws of con-

tinuity.

Clifford summarized his opinion as follows:

The hypothesis that space is not homaloidal and, again, that

its
geometrical character may change with time may or may

not be destined
to play a great part in the physics of the

future; yet we cannot refuse
to consider them as possible

explanations of physical phenomena because
they may be

opposed to the popular dogmatic belief in the
universality

of certain geometrical axioms—belief which has
arisen from

centuries of indiscriminating worship of the genius of

Euclid.

These were, indeed, prophetic words, for, as we shall

see, in the hands of
Einstein the non-Euclidean geome-

tries
became the very foundation of modern cosmo-

logical theory. But let us first examine the flaws and

difficulties inherent in the Newtonian cosmology of the

nineteenth century.

####
*III. CONTRADICTIONS IN THE*

NEWTONIAN COSMOLOGY

NEWTONIAN COSMOLOGY

We first consider what is now called the Olbers

paradox, a remarkable
conclusion about the appear-

ance of the
night sky deduced by Heinrich Olbers in

1826. Olbers was greatly puzzled by
the fact that the

night sky (when no moon is present) appears as dark

as it does instead of as bright as the sun, which, he

reasoned, is how it
should appear if the basic New-

tonian
concepts of space and time were correct. In

deducing this paradox, Olbers
assumed the universe to

be infinite in extent, with the average density and
the

average luminosity of the stars to be the same every-

where and at all times. He assumed, further, that
space

is Euclidean and that there are no large systematic

movements of
the stars. With these assumptions we

can see, as Olbers did, that each
point of the night

sky should appear as bright as each point of the
surface

of the sun (or any other average star). The reason for

this is
that if the stars were distributed as assumed,

a line directed from our eye
to any point in space

would ultimately hit a star so that the whole sky
should

appear to be covered with stars.

Until quite recently this apparent paradox was taken

as a very strong
argument against an infinite Newtonian

universe (or at least against
Olbers' assumptions) but

E . R. Harrison (1965) has shown that Olbers' conclu-

sions are contrary to the principle of
conservation of

energy. To understand this, we first note that a star

(like the sun) can radiate energy at its present rate for

only a finite
time because only a finite amount of

nuclear fuel is available for this
release of energy. Now

if we assume that stars (or galaxies) are
distributed

everywhere the way we observe them to be in our part

of
the universe, it would take about 1023 years before

the radiation from
these stars would fill the universe

to give the effect deduced by Olbers.
But all stars

would have used up their nuclear fuel long before this

time and their luminosities would have changed drasti-

cally. Thus Olbers' assumption that the luminosities
of

the stars do not change during their lifetimes is not

tenable.
Harrison has shown that the radiation emitted

by stars in a period of about
1010 years (which, on the

basis of modern theories we may take as a
reasonable

estimate of the age of the universe) should give just

about
the kind of night sky we observe.

Although Harrison's analysis of the Olbers paradox

removes this flaw in a
static infinite Newtonian uni-

verse, another
difficulty, first pointed out by Seeliger

in 1895 and also by C. G.
Neumann, still remains. In

a static Newtonian universe (one which is not
expand-

ing), with stars (or galaxies)
extending uniformly out

to infinity, the gravitational force at each point
must

be infinitely large, which is contrary to what we actu-

ally observe. This difficulty with a Newtonian
universe

can be expressed somewhat differently by considering

the
behavior of the elements of matter in it. These

elements could not remain
fixed but would move to-

wards each other so
that the universe could not be

static. In fact, a Newtonian universe can
remain static

only if the density of matter in it is everywhere zero.

To overcome this difficulty Neumann (1895) and

Seeliger (1895) altered
Newton's law of gravity by the

addition of a repulsive term which is very
small for

small distances but becomes very large at large dis-

tances from the observer. In this way a
static, but

modified, Newtonian universe can be constructed.

We may also exclude a Newtonian universe of in-

finite extent in space but containing only a finite

amount of
matter. The principal difficulty with such

a universe is that, in time,
matter would become in-

finitely dispersed or
it would all coalesce into a single

globule—contrary to
observation.

####
*IV. COSMOLOGY AND THE THEORY*

OF RELATIVITY

OF RELATIVITY

When it became apparent at the end of the nine-

teenth century that pure Newtonian theory (that is,

without the addition of a repulsive term to Newton's

law of
force) could not lead to a static model of the

universe, most scientists
lost interest in the cosmologi-

cal problem
and very little work was done in this field

until the whole subject was
dramatically reopened by

Einstein in 1917, when he published his famous
paper

on relativistic cosmology. New life was suddenly given

to
cosmology by the appearance of this paper, since

it now appeared that the
flaws in Newtonian cosmol-

ogy would be
eliminated with the introduction of the

Einsteinian space-time concept. As
we shall presently

see, this is indeed true, but difficulties still arise
because

a number of different model universes can be obtained

from
general relativity theory, and we are then left

with the problem of
deciding which of these is the

correct model. This is a somewhat
unsatisfactory situa-

tion since one of the
purposes of a theory is to restrict

the theoretical models that can be
deduced from it to

just those that we actually observe in nature; but
in

spite of this drawback, we must turn to the general

theory of
relativity for an understanding of cosmology,

since it is the best theory
of space and time that we

now have and Newtonian theory has certainly
been

disproved. However, before we can discuss relativistic

cosmology
meaningfully, we must understand the basic

concepts of the theory of
relativity itself.

This theory was developed in two stages: the first

(1905) is called the
special or restricted theory of

relativity and the second (1915) is called
the general

theory. The basic feature of the special theory is that

all observers moving with uniform speed in straight

lines relative to the
distant background stars (such

observers are said to be moving in inertial
frames of

reference) are equivalent in the eyes of nature, in the

sense that the laws of nature are the same for all of

them. Put
differently, the special theory states that an

observer in an inertial
frame cannot determine his state

of motion by any kind of experiment (or
observation)

performed entirely in his frame of reference (that is,

without referring to the background stars). Before the

time of Einstein,
this formulation of the special theory

was accepted by physicists only
insofar as it applied

to the laws of Newtonian mechanics. They
believed

that an observer in an inertial frame could not detect

his
uniform motion by means of any mechanical exper-

iment, but they assumed that the principle did not

apply to
optical phenomena and that an inertial ob-

server
*could* detect his motion through the ether (whose

existence had been postulated to account for the prop-

agation of light) by observing the way light moves
(that

is, by measuring the speed of light) in various directions

in
his frame of reference. Physicists believed this to

be so because the
Newtonian concepts of absolute

space and absolute time lead precisely to
this very

conclusion.

One can deduce from these concepts that the speed

of light is not the same
in all directions, as measured

by a moving observer—the measured
value of the

speed of light should be a maximum for a beam of

light
moving against the motion of the observer and

a minimum for a beam moving
in the same direction

as the observer. This deduction, however, is
contrary

to the experimental evidence. In 1887 Michelson and

Morley
demonstrated experimentally that the speed of

light is the same in all
directions for all inertial ob-

servers. Thus
the constancy of the speed of light for

all such observers must be accepted
as a law of nature.

This means, as emphasized by Einstein, that the
special

theory of relativity must apply to optical phenomena

just as
it does to mechanical phenomena, so that an

observer in an inertial frame
cannot deduce his state

of motion from optical phenomena. Since this is
con-

trary to the deductions from the
Newtonian concepts

of absolute space and absolute time, Einstein
rejected

these absolute Newtonian notions and replaced them

by
relative time and relative space.

To illustrate the essential difference between the two

concepts (absolute
and relative) we may consider two

events separated in space by a certain
distance *d* and

in time by the interval *t as measured by some particular observer in an inertial
frame.* Now, according to the

absolute concepts of Newton,

*all other*inertial ob-

servers recording these two events would find the same

distance

*d*between them and the same interval

*t.*This

is what Einstein denies, for, as we have noted, this

contradicts the observed fact that the speed of light

is the same in all directions for all observers. This

means that the distance

*d*and the time interval

*t*are

different for observers moving with different speeds,

so that space and time separately vary as we pass from

one inertial frame to another. The special theory of

relativity replaces the separate absolute Newtonian

concepts of space and time with a single absolute

space-time concept for any two events, which is con-

structed as follows by any observer: Let this observer

measure the distance between these two events and

square this number to obtain

*d*2. Next, let him measure

the time interval between the two events and square

this to obtain

*t*2. He now constructs the numerical

quantity

*d*2-

*c*2

*t*2, where

*c*is the speed of light.

*This*

quantity, which is called the square of the space-time

interval between the two events, is the same for all

observers moving in different inertial frames of refer-

ence.

quantity, which is called the square of the space-time

interval between the two events, is the same for all

observers moving in different inertial frames of refer-

ence.

We see from this that the absolute three-dimensional

Newtonian spatial
universe, with its events unfolding

in a unique (absolute) temporal
sequence, is replaced

by a four-dimensional space-time universe in which
the

spatial separation and the time interval between any

two events
vary from observer to observer, but in

which all observers measure the same space-time in-

terval. We may state this somewhat differently by

saying that the universe of the special theory of rela-

tivity is a four-dimensional space-time universe *gov- erned by Euclidean
geometry.* The last part of this

statement is important since it is equivalent to saying

that the square of the space-time interval in a universe

governed by special relativity is exactly

*d*2-

*c*2

*t*2. In such

a universe, free bodies (bodies that are not pulled or

pushed by ropes, or rods, or by some other force) move

in straight lines in space-time.

We must now see how this theory, which is restricted

to observers in
inertial frames of reference, is to be

extended when we introduce
gravitational fields and

observers undergoing any arbitrary kind of
motion

(rotation, linear acceleration, etc.). That the theory as

it
stands (that is, the special theory of relativity) is not

equipped to treat
observers in accelerated frames of

reference or to deal with gravitational
fields can be

seen easily enough if we keep in mind that the special

theory is based on the premiss that all inertial observers

are equal in the
eyes of nature and that there is no

observation, mechanical or optical,
that an inertial

observer can make to indicate how he is moving.

Now it appears at first sight that such a statement

cannot be made about
observers in accelerated frames

of reference since the acceleration causes
objects to

depart from straight line motion. If one is in a train

which is moving at constant speed in a straight line,

objects in the train
behave just as they would if the

train were standing still; thus one can as
easily pour

coffee into a cup when the train is moving with con-

stant speed as when it is at rest. But any
departure

from constant motion (that is, any kind of acceleration)

can
at once be detected, because such things as pouring

liquids from one vessel
into another become extremely

difficult. We should therefore be able to
detect that

we are in an accelerated frame by observing just such

phenomena. It thus appears that inertial frames of

reference and
accelerated frames are not equivalent.

This, then, at first blush, would
seem to eliminate the

possibility of generalizing the theory of relativity.
But

we shall presently see just how Einstein overcame this

difficulty.

That the law of gravity, as stated by Newton, is not

in conformity with the
special theory of relativity, is

evident from the fact that, according to
this theory,

clocks, measuring rods, and masses change when

viewed
from different inertial frames of reference. But,

according to Newton's law
of gravity, the gravitational

force between two bodies is expressed in
terms of the

masses of the bodies and the distance between them

at a
definite instant of time. Hence this force can have

no absolute
meaning—in fact, there is no meaningful

way for an inertial
observer to calculate this force since

masses of the two bodies and the distance between

them. This breakdown of the Newtonian law of gravity,

and the impossibility of incorporating accelerated

frames of reference in the framework of special rela-

tivity, convinced Einstein that a generalization of the

theory of relativity was not only necessary, but possi-

ble. For if it were not possible to generalize the theory,

a whole range of observers and of physical phenomena

related to gravity would not be expressible in terms

of a space-time formulation.

To see how Einstein set about generalizing his the-

ory, we may first note that two apparently unrelated

classes of
phenomena—those arising from accelerations

and those arising
from gravitational fields—are ex-

cluded from the special theory. Einstein therefore

proceeded on the
assumption that these two groups

of phenomena must be treated together and
that a

generalization of the theory of relativity must stem

from some
basic relationship between gravitational

fields and accelerated frames of
reference. This basic

relationship is contained in Einstein's famous
principle

of equivalence, a principle which permits one to state

that
all frames of reference (in a small enough region

of space) are equivalent
and that in such a region there

is no way for an observer to tell whether
he is in an

inertial frame of reference, in an accelerated frame,

or
in a gravitational field. Another way of putting this

is that the principle
of equivalence permits one to use

any kind of coordinate system (frame of
reference) to

express the laws of physics. This means, further, that

no law of physics can contain any reference to any

special coordinate
system, for if a law did contain such

a reference, this in itself could be
used by an observer

to determine the nature of his frame of reference.
Thus

all laws must have the same form in all coordinate

systems.

To understand how the principle of equivalence

leads to the general theory,
we must first see just what

the basis of this principle is and what it
states. The

principle itself stems from Galileo's observation that

all
bodies allowed to fall freely (that is, in a vacuum

with nothing impeding
them) fall with the same speed.

This can be stated somewhat differently if
we consider

the mass of a body (the amount of matter the body

contains). This quantity appears in two places in the

laws of Newtonian
physics. On the one hand, it is the

quantity that determines the inertia of
a body (that

is, the resistance a body offers to a force that tries to

move it). For this reason, the quantity is referred to

as the *inertial mass* of the body. But the concept of

mass
also appears in Newton's formula that expresses

the gravitational force
that one body exerts on another;

this mass is then referred to as the *gravitational mass*

of the body. The fact that all
bodies fall with the same

speed in a gravitational field means that the inertial

mass and
the gravitational mass of a body must be

equal.

This remarkable fact had been considered as no more

than a numerical
coincidence before Einstein devel-

oped his
general theory of relativity. Einstein started

out on the assumption that
the equality of the inertial

and gravitational masses of a body is not a
coincidence

but, instead, must have a deep significance. To see what

this significance is, consider the way bodies behave in

an accelerated
frame of reference somewhere in empty

space (far away from any masses) and
the way they

behave in a gravitational field (for example, on the

surface of the earth). Owing to their inertial masses,

all the bodies in
the accelerated system behave as

though they were being pulled opposite to
the direc-

tion of the acceleration and they
all respond in exactly

the same way (that is, they all
“fall” with the same

speed). To Einstein, this meant
that there is no way

to differentiate between an accelerated frame of refer-

ence and a frame that is at rest (or
moving with con-

stant speed) in a
gravitational field. This is called the

principle of equivalence. Another
way of stating it is

to say that the apparent force that a body
experiences

when it is in an accelerated frame of reference is

identical with the force this body would experience

in an appropriate
gravitational field; thus inertial and

gravitational forces are
indistinguishable.

Since the principle of equivalence makes it impossi-

ble to assign any special quality or physical significance

to
inertial frames of reference, the special theory

(which is based on the
assumption that inertial frames

are *special* in the
sense that only in such frames do

the laws of physics have their correct
and simplest

form) must be discarded for a more general theory

which
puts all frames of reference and all coordinate

systems on the same
footing. In such a theory, the laws

of physics must have the same form in
all coordinate

systems. With this in mind, we can now see how

Einstein
constructed his general theory of relativity.

We begin by noting that the
special theory replaces

the concepts of absolute distance *d* and absolute time

*t* between events by a single absolute space-time inter-

val whose square is *d*2-*c*2*t*2. Consider now a
freely

moving particle as viewed by an observer in an inertial

frame
of reference in a region of space where no

gravitational fields are
present. If this particle moves

a distance *d* in a
time *t,* the quantity *d*2-*c*2*t*2 must be

the same for all
observers in inertial frames. This simply

means that the natural space-time
path of a free parti-

cle for inertial frames
of reference is a straight line

and that the space-time geometry of the
special theory

of relativity is Euclidean. We may take this
formulation

then as the law of motion (and hence a law of nature)

of a
free particle.

Now if we are to carry out our program of extending

the principle of
relativity to cover observers in gravi-

tational fields and in accelerated frames of reference,

we must say
that this same law of motion (straight line

motion) applies to a body
moving freely in a gravita-

tional field
or in an accelerated frame of reference.

But we know that the space-time
path of a free particle

in a gravitational field (or in a rotating system)
appears

to be anything but straight. How, then, are we to

reconcile
this apparent contradiction? We must re-

define
the concept of a straight line! We are ordinarily

accustomed to think of a
straight line in the Euclidean

sense of straightness, because the geometry
of our

world is very nearly Euclidean and we have been

brought up on
Euclidean geometry. In a sense, we

suffer from the same kind of geometrical
bias concern-

ing space-time as does the man
who thinks the earth

is flat because he cannot detect its sphericity in
his

small patch of ground.

To overcome this parochial attitude, we note that

we can replace the
“straightness” concept by the con-

cept of the shortest distance between two points. We

can now state the law of motion of a free particle as

follows:

A free particle moving between two space-time

points always moves in such a
way that its space-time

path between these two points is shorter than any
other

space-time path that can be drawn between the two

points.

This statement of the law of motion makes no refer-

ence to the way the space-time path of the particle

looks, but
refers only to an absolute property of the

path which has the same meaning
for all observers.

If no gravitational fields or accelerated observers
are

present, the shortest space-time path is *d*2-*c*2*t*2 and the

geometry is
Euclidean. But if gravitational fields are

present, the shortest space-time
path of the particle

(that is, its geodesic) is not given by *d*2-*c*2*t*2,
but by

a different combination of *d* and *t* because the space-

time geometry is non-Euclidean. The essence of Ein-

stein's general theory is, then, that a gravitational
field

distorts space-time (it introduces a curvature into

space-time)
and the behavior of a free particle (that

is, the departure from Euclidean
straight-line motion)

is not due to a “gravitational
force” acting on the

particle, but rather to the natural
inclination of the

free particle to move along a geodesic. In a sense,
this

is similar to what happens whan a ball is allowed to

roll freely
on a perfectly smooth piece of ground. The

ball appears to us to move in a
“straight line,” but

we know that this cannot be so
because it is following

the contour of the earth, which is spherical.
Actually

the ball is moving along the shortest path on the

smooth
surface, which is the arc of a great circle.

From this discussion we see that in the general

theory of relativity, the space-time path of a freely

moving
particle is not *d*2-*c*2*t*2, but some variation of

this, which depends on the
kind of gravitational fields

that are present, and on the acceleration of
our coordi-

nate system. We can therefore go
from the special

theory to the general theory of relativity by
replacing

the space-time interval (*d*2-*c*2*t*2) by the quantity *gd*2-*qc*2*t*2, where *g* and *q* are
quantities that vary from point

to point. The value of the quantities *g* and *q* at any

point for a
given observer will depend on the intensity

of the gravitational field at
that point and on the

acceleration of the frame of reference of the
observer.

Just as the special theory of relativity is based on the

statement that the quantity *d*2-*c*2*t*2 is the same for all

observers in
inertial frames of reference, the general

theory of relativity is based on
the statement that the

quantity *gd*2-*qc*2*t*2 must be the same for all
observers,

regardless of their frames of reference.

Now the use of the latter expression as the absolute

space-time interval
instead of the former means that

we pass from Euclidean to non-Euclidean
geometry

in going from the special to the general theory, and

the
quantities *g* and *q* (they are also
referred to as the

Einstein gravitational potentials) determine by how

much the geometry at any point of space-time departs

from Euclidean
geometry—in other words, these

quantities determine the
curvature of space time at

each point. If, then, we know how to find *g* and *q,*

we can determine the
nature of the geometry in any

region of space-time and hence the path of a
free

particle in that region. The curvature of space-time

thus becomes
equivalent to the intensity of the gravi-

tational field, so that the gravitational problem is re-

duced to a problem in non-Euclidean geometry.
The

next step, then, in this development was to set down

the law that
determines the quantities *g* and *q,*
and

this Einstein did in his famous field equations—a set

of ten partial differential equations that show just how

the quantities *g* and *q* (there are actually ten
such

quantities, but in the gravitational field arising from

a body
like the sun, only two of these ten quantities

are different from zero)
depend on the distribution of

matter. These gravitational field equations
are the basis

of all modern cosmological theories which we shall

now
discuss.

####
*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.

####
*VI. THE DE SITTER EMPTY*

EXPANDING UNIVERSE

EXPANDING UNIVERSE

When Einstein first obtained his static universe the-

ory, it seemed to be just what was wanted, for it agreed

with the
astronomical observations as they were known

in 1917. The measured
velocities of the stars were

small, and the large scale speed of recession
of the

distant galaxies had not yet been detected. It thus

over, it appeared to Einstein at the time that the

solution of the field equations he had obtained with

the introduction of the cosmical constant λ

*gij*was a

logical necessity which intimately linked up space and

matter, so that one could not exist without the other.

He was led to this opinion because he thought that

the field equations (2) with a positive value of λ have

no solution for

*Tij*= 0 (that is, in the absence of mat-

ter). But, as de Sitter (1917) later showed, this con-

clusion was wrong. He found a solution for empty

space; that is, for

*Tij*= 0 everywhere. Now such a

universe is an expanding one in the sense that if a test

particle (a particle of negligible mass) is placed at any

point in the universe, it recedes from the observer with

ever increasing speed. In other words, the speed of

recession increases with distance from the observer. In

fact, if the de Sitter universe had test particles distrib-

uted throughout, they would all recede from each

other. The reason for this is found in the cosmical term

λ

*gij*in the field equations. If we place

*Tij*= 0 in the

field equations (2) they reduce to

*Rij*= λ

*gij*, or Rij - λ

*gij*= 0, (1)

and since the term

*Rij*represents the ordinary New-

tonian gravitation of attraction, the term -λ

*gij*repre-

sents repulsion, owing to the minus sign.

The de Sitter universe aroused interest initially be-

cause it showed that the cosmological field equations

(2) do not
have a unique solution, and that more than

one model of a universe based on
these equations can

be constructed. Beyond this, however, the de
Sitter

model of the universe was not taken seriously, since

it seemed
to contradict the observations in two re-

spects: it is an empty universe, whereas the actual

universe
contains matter; it is an expanding universe,

whereas the observations
seemed to indicate that the

actual universe was static. But then, in the
early 1920's,

the recession of the distant nebulae was discovered by

Hubble, Slipher, Shapley, and others. The work of

these investigators on
the Doppler displacement (to-

wards the red) of
the spectral lines of the extragalactic

nebulae (or galaxies) indicates
that the universe is, in-

deed, expanding.
Moreover, the rate of recession of

the galaxies increases with distance
(the famous Hubble

law, 1927) in line with what one would expect from

the de Sitter universe. These discoveries demonstrated

the inadequacies of
the Einstein universe and brought

the de Sitter model into prominence.

Another difficulty associated with the Einstein static

universe is that it
is not a stable model but must un-

dergo either
expansion or contraction if there is the

slightest departure from the
precise balance between

the gravitational attraction and the cosmic repulsion.

Thus, if by some process or other some of the mass

were to be
changed into energy, or if condensations

were to occur, the universe would
have to begin to

expand or collapse. This point, taken together with
de

Sitter's work and the observed recession of the distant

galaxies,
led cosmologists to the idea that the actual

model of the universe might be
an expanding one, that

is, intermediate between the empty de Sitter
model

and the Einstein static model. One must therefore look

for
solutions of the field equations which give models

that are expanding, but
not empty. Such models were

first obtained by the Russian mathematician
Friedmann

in 1922 when he dropped Einstein's assumption that

the
density of matter in the universe must remain

constant. By dropping this
assumption, Friedmann was

able to obtain nonstatic solutions of the field
equations

which are the basis of most cosmological models. This

same
problem was independently investigated later by

Weyl (1923),
Lemaître (1931), Eddington (1933),

Robertson (1935), and Walker
(1936). Since the treat-

ment of this problem
as given by Robertson, and,

independently, by Walker, is the most general
one, we

shall use their analysis as a guide in our discussion of

the
current models of the expanding universe.

####
*VII. THE NONSTATIC MODELS OF A*

NONEMPTY UNIVERSE

NONEMPTY UNIVERSE

In the previous section we saw that an expanding

model of the universe can
be obtained without altering

Einstein's original assumptions if we remove
all the

matter from the universe and, at the same time, intro-

duce into the field equations a cosmical
repulsion term.

Friedmann escaped this unrealistic situation by re-

moving Einstein's assumption that there are
no large

scale motions in the universe. He assumed immediately

that
the average distance between bodies in the uni-

verse does not remain constant but changes steadily

with time. This
means that the right hand side of the

field equations (2) does not remain
constant, so that

the density of matter in the universe changes with
time.

Owing to this variation of density it is not necessary

to keep
the cosmical term λ*gij* in
the left hand side

of (2) to obtain nonstatic solutions; in fact,
Friedmann

discarded this term in his work and obtained two

nonstatic
models of the universe—one which represents

a universe that
expands forever, and the other a pul-

sating
universe. In the investigations that followed the

work of Friedmann, the
general field equations (2) with

λ*gij* present, and with the right hand side
changing

with time, were used. This introduces a whole range

of
expanding and pulsating models whose properties

depend on whether
λ is negative, positive, or zero, and

on the value of still
another constant (the curvature

constant) which also enters into the final
solution of

To see how these two constants determine the vari-

ous models of the universe, we first consider briefly

the manner in
which Robertson and Walker repre-

sented the
solution of the field equations for a nonstatic

universe. We first recall,
according to what we said

in Section IV, that the square of the space-time
interval

between two events for an unaccelerated observer in

empty
space is *d*2-*c*2*t*2, and we have Euclidean space.

The presence of matter alters this
by distorting space

and changing the geometry from Euclidean to non-

Euclidean. Suppose now that the two
events we are

talking about are close together (so that *d* and *t* are

small) and that they are both at
about the same distance

*r* from us. We then find (following Robertson and

Walker) that the space-time interval between these

events for an expanding
universe with matter in it can

always be written as

* R2d2 / (1 +
kr2/4)2 - c2t2
*
, (2)

where *R* is a
quantity that changes with time and *k*

is the
curvature constant referred to above; it can have

one of the three values:
-1, 0, +1. If *k* = -1, the

curvature of the universe
is negative (like a saddle

surface) and the geometry is hyperbolic. The
universe

is then open and infinite. If *k* = 0, the
curvature is

zero and space is flat (Euclidean); the universe is open

and infinite. If *k* = +1, the curvature is positive
and

the universe is finite and closed. The quantity *R* is the

scale factor of the universe; it measures the expansion

(or contraction) and is often referred to as the radius

of the universe.
However, it is not in itself a physical

distance that can be observed or
measured directly,

but rather the quantity that shows how the
distances

between objects in the real universe change; if, in a

given
time, *R*(*t*) doubles, all distances
and dimensions

in the universe double.

To obtain a model of the universe, one must find

the law that tells us how
*R* varies with time, and this

is done by using
the field equations (2) in conjunction

with the above expression for the
space-time interval.

When we do this, we obtain the equations that
tell

us exactly how *R* changes with time, but we find
that

these equations also contain the cosmic constant λ and

the curvature constant *k* so that many different
models

of the universe are possible, depending on the choice

of these
constants. Before Friedmann and those follow-

ing him did their work, it was thought that λ neces-

sarily had to be positive, but the
equations for *R* show

that we can obtain models of
the universe for which

λ can be negative, zero, or positive. If
we combine

these three possibilities for λ with the three possible

values (-1, 0, +1) for *k,* we obtain a large
variety

of model universes, and there is no way for us, at the

present
time, to say with certainty which of these

models give the correct
description of the universe.

Owing to this uncertainty we shall give a brief dis-

cussion of these models as a group and then see which

of these
is most favored by the observational evidence.

We designate a model
universe as either expanding or

oscillating (pulsating) depending on
whether *R*
in-

and then decreases. In the expanding models, two cases

are possible, depending on the choice of λ and

*k.*In

the first case (expanding I),

*R*increases from a zero

value, at a certain initial time, to an infinitely large

value, after an infinite time. In the second case (ex-

panding II),

*R*increases from some finite value, at a

certain initial time, to an infinite value, after an infinite

time. In all the oscillating models,

*R*expands from

zero to a maximum value and then decreases to zero

again. This fluctuation is then repeated over and over

again. In Figure 1 graphs are shown giving the varia-

tion of

*R*with time for the expanding and oscillating

cases.

We summarize the various model universes in

Table I.

TABLE I

λ | k (or curvature) | ||

-1 | 0 | +1 | |

negative | oscillating | oscillating | oscillating |

zero | expanding I | expanding I | oscillating |

positive | expanding I | expanding I | oscillating |

expanding I | |||

expanding II |

####
*VIII. MODEL UNIVERSES WITH THE*

COSMICAL CONSTANT
EQUAL TO ZERO

COSMICAL CONSTANT EQUAL TO ZERO

We have seen that the Einstein field equations lead

to both expanding and
oscillating models of the uni-

verse, but these
field equations do not permit us to

determine which one of the eleven
models listed in

Table I corresponds to the actual universe. The
reason

for this is that three unknowns, viz., the cosmical con-

stant λ, the sign of the
curvature *k,* and the scale of

the universe (the
units in which *R* and the time are

to be expressed)
appear in the final solutions, whereas

direct observations of the galaxies
can give us only the

rate of expansion of the universe (Hubble's law)
and

its average density. Another possible observation is the

deceleration of the expansion of the universe, and some

work has been done
on that possibility which we shall

discuss later. If the deceleration could
be measured

accurately, we could decide among the various models,

but
until we have reliable observational evidence on

this point, we must
proceed by making some assump-

tion about
either λ or *k.*

For the time being, we proceed as Einstein did after

Friedmann's work and
place λ = 0. Einstein was very

unhappy about the introduction of
λ in the first place

since he considered it to be an ad hoc modification

of the
theory which spoiled “its logical simplicity”;

he
therefore felt that the models with λ = 0 were the

ones to be
favored. From Table I we see that λ = 0

leads to two expanding
models of type I for *k* = -1

and *k* = 0, and to a single oscillating model for *k*
> 0.

To decide between the expanding and oscillating

models, we must have the equation that tells us just

how *k* depends on the density of the universe and its

rate of
expansion when λ = 0. This relationship, which

is obtained from
the solution of the field equations,

is the following:

* k = R2/c2
(8/3πGρ - H2
*
, (3)

where *G* is the gravitational constant, *c* is the speed

of light, ρ is the average
density of the universe, and

*H* is Hubble's constant—that is, the rate of
expansion

of the universe.

The important quantity in equation (3) is that con-

tained in the parenthesis on the right hand side; for

it
determines whether *k* is negative, zero, or positive,

and hence whether the universe is expanding or oscil-

lating. If we express distance in centimeters, mass
in

grams, and time in seconds, the quantity (8/3)π*G* equals

5.58 × 10-7
and the parenthesis in (3) becomes (5.58 ×

10-7 ρ - *H*2). If we knew ρ and
*H* accurately, we could

see at once from this
expression whether our universe

(with λ = 0) is expanding or
oscillating, but neither

ρ nor *H* is well
known. Hubble was the first to measure

*H* by analyzing the recession of the galaxies and
placed

it equal to 550 km per sec per million parsecs; but

we now know
that this is too large. According to A.

Sandage (1961), observations on the
recession of the

galaxies indicate that *H* is about
100 km per sec per

million parsecs. If we use this value, *H*2 becomes (in

cm-gm-sec units) 9 × 10-36 and the
quantity in the

critical parenthesis becomes (5.58 × 10-7 ρ - 9 ×

10-36) or 5.58
× 10-7 (ρ - 1.61 × 10-29).

This is a most remarkable result, for it tells us that

the model of the
universe (for a given value of the

recession) is determined by the density
of matter in

the universe. In our particular case (the cosmical con-

stant zero) the density ρ must be
larger than 1.61 ×

10-29 gms per cc (one
proton per 100,000 cubic cm.

of space) for the universe to be an
oscillating one. If

the density just equals this value, the universe is
ex-

panding and Euclidean (no curvature),
and if the den-

sity is less than this value,
the universe is expanding

but it has negative curvature. It is precisely
here that

we run into difficulty in drawing a definite conclusion

because the density ρ is not accurately known.

In terms of our present data, the density appears

to be about 7 ×
10-31, which would make *k* = -1,

tive curvature). But there may be great quantities of

undetected matter that can increase ρ considerably.

One must therefore try to get other observational

evidence which can permit us to decide between ex-

panding and oscillating models. This can be done if

one determines (from observational evidence) whether

the Hubble constant

*H*is changing with time, and, if

so, how rapidly. If the value of

*H,*as determined from

the recession of nearby galaxies, is sufficiently smaller

than the value as determined from the recession data

of the distant galaxies, we must conclude that

*H*was

considerably larger when the universe was younger (the

distant galaxies show us a younger universe) than it

is now. This would mean that the rate of expansion

had decreased and that ultimately the universe must

stop expanding and begin to collapse. This means that

the universe is oscillating. This sort of analysis has been

carried out jointly by Humason, Mayall, and Sandage

(1956) and the evidence favors an oscillating universe.

This means either that the value of the density ρ has

been greatly understimated or that the correct model

of the universe is one in which λ is different from zero.

Of course, it may be that

*H*is even smaller than 100

km per sec per M pc, but it cannot be much smaller

than this value, and reducing

*H*by a small amount

does not help.

Before leaving these Friedmann models with λ = 0,

we briefly
consider the principal properties of the

models associated with the three
different values of

*k.* For *k* = 0 there is no curvature
and space is infinite.

The age of the universe (as measured from some
initial

moment *t* = 0 when the expansion began) is
then equal

to 2/3(1/*H*), and we obtain about 8
× 109 years, which

appears to be too small to account for the
evolution

of the stars and galaxies. For this kind of universe, the

expansion parameter *R* increases as the 2/3 power of

the time.

For *k* = -1, space is negatively curved and infinite;

the expansion is continuous and endless, so that the

universe finally
becomes completely empty and Eu-

clidean. At
some initial moment, *t* = 0, the universe

was in an
infinitely condensed state and then began

to expand. According to this
model, the age of the

universe is 1/*H* or 1.2
× 1010 years, which gives ample

time for stellar evolution.

For *k* = +1, we obtain the oscillating universe

which
began from an infinitely condensed state at

*t* = 0. This is a positively curved, closed universe,

whose radius *R* will reach a maximum value and then

decrease down to zero again. A similar expansion will

then begin again and
this will be repeated ad infinitum.

The age of this model of the universe
is smaller than

that of the other two.

####
*IX. MODELS OF THE UNIVERSE*

WITH THE COSMICAL
CONSTANT

DIFFERENT FROM ZERO

WITH THE COSMICAL CONSTANT

DIFFERENT FROM ZERO

We saw in the last section that placing λ = 0 se-

verely restricts the number of models, and that these

models represent ages that are somewhat too small for

stellar evolutionary
comfort. For this reason, a group

of investigators, particularly
Lemaître, Eddington,

Robertson, Tolman, and McVittie, in the early
days (all

independently of each other and without knowledge

of
Friedmann's work) and Gamow (1946) later, con-

structed various models with λ different from zero.

There are many more such models than one can obtain

with λ = 0,
and among them are both the expanding

and oscillating types, as we have
already noted. The

most popular of these models during the earlier
period

of this work is the one first proposed by Lemaître in

1927 and strongly supported by Eddington. This is the

expanding II model
listed in Table I, for which both

λ and *k*
are positive. In this model the universe is

always closed and finite and
began its expansion from

some finite nonzero value of *R.* But the moment of

the beginning of the expansion was not the
moment

of zero time (that is, the moment of the origin of the

universe) because in this model the universe could have

remained in a
nonexpanding, static state for as long

as one might desire—in
fact, for an infinite time in

the past.

Since this model starts expanding from a static

model, both
Lemaître and Eddington assumed this

initial static model to be the
original Einstein static

model. In this model the value of λ and
the radius *R*

are chosen (in relationship to the mass
*M* of the uni-

verse)
in such a way as to give a closed spherical

universe in which the cosmical
repulsion is just bal-

anced by the
gravitational attraction. However, this

Einstein universe is unstable, as
we have already noted,

so that any initial expansion reduces the density
and

causes this model to expand still more, with further

reduction in
density, and so on. The expansion thus

proceeds faster and faster until the
universe is infinitely

expanded and the density is everywhere zero. On
the

other hand, a slight compression could have caused the

Einstein
model to have contracted indefinitely, finally

ending up as an infinitely
condensed point of matter.

If, then, we accept this Lemaître-Eddington picture,

the universe
was in a static Einstein state for an infinite

time in the past and then at
some finite time in the

past, for some unknown reason, began to expand,
at-

taining its present rate of expansion
after a few billion

years. Although Eddington never abandoned this con-

cept and fought for it vigorously to the end
of his life,

Lemaître revised his thinking in 1931 and replaced
this

type II expanding model by a type I expanding model.

the universe can be constructed with λ positive and

*k*= 1: an oscillating type, an expanding I type, and

an expanding II type. If we reject the last of these

(which corresponds to the original Lemaître-Eddington

model, which we have just discussed) we still have the

oscillating and the expanding I models.

The reason Lemaître replaced the expanding II

model by the
expanding I model is that he had no

reasonable explanation for the start of
the initial ex-

pansion of the actual universe
from an Einstein static

state. Although his own theoretical investigations
and

those of McCrea and McVittie (1931) strongly sug-

gested that any local condensation of the matter in

the Einstein static universe (for example, the formation

of a single galaxy
or star) would cause it to start ex-

panding,
these investigations left unanswered the

question as to why other galaxies
were formed. If

expansion began after the formation of a single
galaxy,

the density of the universe would immediately begin

to
decrease and other condensations into galaxies would

be precluded. This
would mean, of course, that the

cosmological principle defined in Section V
would be

untenable, since the distribution of matter in the

neighborhood of this initial condensation would be

different from that
elsewhere in the universe. More-

over, it is
difficult to see how the heavy elements such

as iron, lead, and uranium
could have originated in

an Einstein static-state universe, since we know
from

nuclear theory that the formation of such elements

from hydrogen
in great abundance requires extremely

high temperatures and pressures. This
means that the

entire universe, or at least parts of it, must have
passed

through a high temperature-high pressure phase. Thus

the very
existence of the stars and heavy elements

argues against the Einstein
static state as the initial

phase of our present universe.

Owing to these difficulties, inherent in the assump-

tion that our present universe evolved from an Einstein

static
universe of finite radius, Lemaître introduced the

assumption that
we live in an expanding universe of

type I, which began its expansion from
a highly con-

densed state. He referred to
this initial condensation

as the primordial atom or nucleus and assumed
that

a vast, radioactive explosion occurred in this atom and

that what
we now see in the recession of the galaxies

all about is the result of this
explosion. In this picture,

the expanding universe is always finite in
size, but

closed like a sphere. The initial condensed state (that

is,
the Lemaître primordial atom) may be pictured as

having been
present for an infinite time in the past

or we may suppose that the
universe began its life in

the Einstein static state and then collapsed
violently

into a primordial atom from which it began to expand.

According to Lemaître, this expansion carried the uni-

verse back to its initial Einstein state,
but it did not

stop there. Its velocity of expansion carried it beyond

this static phase, and after that its expansion proceeded

with ever
increasing speed.

Whether we are discussing an Einstein-Friedmann

expanding model, with
λ = 0; or an oscillating model,

with λ = 0; or a
Lemaître model, with λ > 0 and

*k* = +1 (expanding II or oscillating), we are dealing

with a group of models that are referred to as the “big

bang” models of the universe, since all of them picture

the
universe as having originated explosively from a

point. The term
“big bang” was first introduced by

Gamow (1948) who,
together with Alpher and Herman

(1950), sought to account for the origin of
the heavy

elements by supposing that they were formed from the

original protons and neutrons in the very early and

very hot stage of the
explosion. According to this

picture of the origin of the universe,
neutrons were

the principal components of the original material

ejected from the primordial atom or point source. Some

of these neutrons
quickly decayed into protons and

electrons, and these protons then captured
other neu-

trons to build up the heavy
elements. This whole

buildup of heavy elements must have occurred
during

the first thirty minutes after the initial explosion, for

the
temperature of this primordial material dropped

very rapidly after that and
everything then remained

frozen.

Gamow's theory was very appealing at first since

no other theory of the
elements was available then;

the theory of stellar structure and evolution
had not

yet reached a point of development where it could

be shown
that heavy elements can be and are built

up inside stars, as they evolve
from structures like the

sun into red giants like Antares and Betelgeuse,
with

their internal temperatures rising to billions of degrees.

Gamow's theory of the buildup of the heavy elements

during the first thirty
minutes of the life of the universe

had to be discarded, however, since
there are no stable

nuclei of atomic masses 5 and 8, so that neutron cap-

ture alone could not have bridged the nuclear
gap

between the light and heavy nuclei. Even if some heavy

nuclei were
formed by neutron capture in this early

fireball stage of the universe (and
all nuclei capture

neutrons very readily) a half hour would hardly
have

been long enough for the heavy elements to have been

formed in
their present abundances. Since we now

know that the heavy elements can all
be baked in the

stellar furnaces at various stages of evolution, this
phase

of the Gamow “big bang” theory is not essential
and

one can discard it without invalidating the overall

concept.

If we then accept this Lemaître-Gamow hot “big

through a very high temperature phase (about 1010 to

1011 degrees K) soon after the initial explosion, and

some observable evidence of this may still be around.

That this should be so was first pointed out by Gamow

himself, who argued that there must have been a con-

siderable amount of very hot black body radiation

present in this initial phase of the universe and most

of it must still be around, but in a very much red-shifted

form. He estimated that its temperatures would now

be 6°K. Without knowing about Gamow's suggestion,

Dicke proposed the same idea in 1964 (he called it

the “primordial fireball radiation”) and later, in collab-

oration with Peebles, Roll, and Wilkinson (1965), dem-

onstrated that the initial hot black body radiation (at

a temperature of 1010 degrees K) must now be black

body radiation (at a temperature of 3.5°K). The general

idea behind this deduction is the following: if the

universe was initially filled with very hot black body

radiation (that is, of very short wavelength), this radia-

tion would remain black body radiation during the

expansion of the universe, but it would become redder

and redder owing to the Doppler shift imparted to it

by the expansion. This is similar to radiation that is

reflected back and forth from the walls of an expanding

container. This 3.5°K black body radiation was de-

tected by Penzias and Wilson in 1965 and has since

been verified by other observers. It is present in the

form of isotropic, unpolarized microwave background

radiation in the wavelength range from 1/10 to 10 cm.

One other residual feature of the “big bang” should

still be visible, or at least amenable to verification—the

present helium abundance. During the initial fireball

period when the
temperature was considerably larger

than 1010 degrees K, the thermal
electrons and neu-

trinos that were present
would have resulted in very

nearly equal abundances of neutrons and
protons.

When the temperature of the fireball dropped to 1010

degrees
K these neutrons and protons would have

combined to form deuterium, which,
in turn, would

have been transformed into He4, and no heavier ele-

ments would have been formed. Two questions
then

arise. (1) Is the helium that we now observe all about

us, in our
own galaxy and in others, still this primordial

helium? (2) If so, what can
this tell us about the models

of our universe?

The evidence relating to the first question is some-

what ambiguous because we know that helium burning

occurs during
the giant stage of a star's evolution, so

that some of the original helium
must certainly have

been transformed into heavy elements in stellar inte-

riors, and thus disappeared. But we may
assume that

the helium that is found in stellar atmospheres is pri-

mordial and the evidence here is that
although there

is an overall helium abundance of about 25%, some

stars have
been observed with very weak helium lines.

In spite of these, however, the
overall evidence favors

the 25% abundance, which is in agreement with
the

“big bang” hypothesis.

Taking all of the observed data into account (the

3°K black body
radiation and the helium abundance)

the preponderance of the evidence
favors the “big

bang” theory and points to an age of
at least 1010,

i.e., ten billion years for our universe. The observed

helium abundance (if we accept 25% as the primeval

abundance) also
indicates that the density of matter

in the universe must be at least 4
× 10-31 grams per

cc. But if the density of matter in the universe
is no

larger than this, we run into difficulty with the obser-

vations on the rate at which the
expansion of the

universe is decelerating. We have already noted that

Humason, Mayall, and Sandage have given a value for

this deceleration which
indicates that the universe must

ultimately stop expanding and begin to
collapse. This

means that the correct model of the universe is an

oscillating one, rather than expanding, but, as we have

seen, this requires
the density of matter to be about

10-29 gms/cc, as compared to the observed
density of

7 × 10-31

In spite of this, the evidence for an oscillating uni-

verse has been greatly strengthened recently by the

analysis of
the distribution of quasars and of quasi-

stellar radio sources in general. Since these objects

(according to
their red shifts) are at enormous distances

from us, they give us the rate
of expansion of the

universe in its earliest stages. By comparing this
with

the present rate of expansion, we obtain a very reliable

value
for the deceleration, which shows the universe

to be oscillating. To
account for the discrepancy be-

tween the
observed and required density of matter for

such a model of the universe,
we must suppose that

there are large quantities of dark matter in inter-

galactic space—in the form
of hydrogen clouds, black

dwarf stars, and streams of neutrinos. But until
we have

direct evidence of this, we cannot be sure about the

validity
of the oscillating model.

####
*X. THE STEADY-STATE THEORY AND*

OTHER COSMOLOGIES

OTHER COSMOLOGIES

We shall conclude our discussion of modern cos-

mologies with brief descriptions of theories that are

related
to, but do not spring directly from, Einstein's

field equations, whether or
not we place λ = 0. Of

these, the most popular, and one which,
has been

strongly supported by outstanding cosmologists and

physicists, is the steady state or continuous creation

theory of Bondi and
Gold (1948) and Hoyle (1948).

On the basis of what they call the *perfect cosmological
*

*568*

*principle,*which is an extension of Einstein's cosmo-

logical principle, they assert that not only must the

universe present the same appearance to all observers,

regardless of where they are, but it must appear the

same at all times—it must present an unchanging as-

pect on a large scale. The immediate consequence of

this theory is that mass and energy cannot be conserved

in such a universe. Since the universe is expanding,

new matter must be created spontaneously and contin-

uously everywhere so as to prevent the density from

decreasing.

It can be shown from this theory that matter would

have to be created at a
rate equal to three times the

product of the Hubble constant and the
present density

of the universe, in order to keep things as they are.

One nucleon must be created per thousand cubic cen-

timeters, per 500 billion years to maintain the status

quo.
Hoyle arrived at the same result by altering

Einstein's field equations.

Although the steady-state theory was very popular

because it eliminated
entirely the question of the origin

of the universe, it was rejected by
most cosmologists

because of its continuous creation and the
consequent

denial of the conservation of mass energy. But the

strongest argument against the steady state theory is

the existence of the
3°K radiation, which shows clearly

that our universe has evolved
from a highly condensed

state. In addition, the observed distribution of
quasars,

radio sources, and other distant celestial bodies shows

that
the density of matter in the universe was much

higher a few billion years
ago than it is now. The

observational evidence seems weighted against
the

steady-state theory.

Other general principles have been invoked to derive

cosmological theories.
Perhaps the most ambitious of

these theories is that of Eddington (1946),
who at-

tempted, in his later years, to deduce
the basic con-

stants of nature by combining
quantum theory and

general relativity. Starting from the idea that the
re-

ciprocal of the square root of the
cosmical constant

represents a natural unit of length in the universe,
and

that the number of particles in the universe must de-

termine its curvature, he derived numerical values for

such constants as the ratio of the mass of the proton

to that of the
electron, Planck's constant of action, etc.

But very few physicists have
accepted Eddington's

numerology since his analysis is often obscure,
difficult

to follow, and rather artificial. In any case, the exist-

ence of nuclear forces and new particles
which Ed-

dington was unaware of when he did
his work, and

which therefore are not accounted for in his theory,

destroys the universality which he claims for his theory.

During the period that Eddington was developing

his quantum cosmology, three
other cosmological sys

tems were introduced: the kinematic cosmology of

Milne (1935)
and the cosmologies of Dirac (1937) and

Jordan (1947). Although these
theories are extremely

interesting and beautifully constructed, we can
only

discuss them briefly here. Of all the cosmological theo-

ries that we shall have discussed in this
essay, Milne's

is the most deductive, for instead of starting with the

laws of nature as we know them locally, and then

constructing a model of
the universe based upon these

laws, he introduces only the cosmological
principle and

attempts to deduce, by pure reasoning, not only a

unique
model of the universe, but also the laws of

nature themselves. To do this,
Milne had to assume

the existence of a class of ideal observers attached
to

each particle of an ideal homogeneous universal sub-

stratum, which is expanding according to Hubble's law.

To carry out his analysis consistently, Milne had to

introduce two different
times; a kinematic time which

applies to the ideal observer and which also
governs

electromagnetic and atomic phenomena, and according

to which
the universe is expanding; and a dynamic

time, so that a good deal of
arbitrariness is inherent

in this theory, particularly at the boundary
region

where we pass from one kind of time to another. But

the major
objection to this theory arises from its basic

assumption that an absolute
substratum exists in the

universe, and that a privileged class of observers
is

associated with this substratum.

Although a cosmological principle of one sort or

another is at the basis of
the cosmologies which we

have discussed here, other types of principles
have also

been used. The most notable of these is that proposed

by
Dirac in 1937 (and later in a slightly different form

by Jordan), according
to which certain basic numbers

associated with matter and the universe are
not con-

stant, as had been assumed in all
previous cosmologies,

but vary with time. The numbers Dirac had in
mind

are certain dimensionless quantities which are obtained

by taking
the ratio of atomic quantities to cosmological

quantities of the same kind.
Dirac expressed this prin-

ciple as follows:
“All very large dimensionless numbers

which can be constructed
from the important constants

of cosmology and atomic theory are simple
powers of

the epoch.”

One consequence of this principle is that the univer-

sal gravitational constant would have to decrease with

time. But
one can show, as E . Teller did (1948), that

this would lead to a sun that
was much too hot during

the Cambrian period; the temperature of the
earth

would then have been so high that its oceans would

have been
boiling. Owing to this discrepancy, Dirac's

theory has generally been
discarded, although, more

recently, C. Brans and R. H. Dicke have
introduced

a variation of it, starting from a different point of view.

####
*SUMMARY*

At this point in our narrative, the reader may well

feel that modern
cosmology is a welter of conflicting

theories, all of which contain some
elements of truth,

but none of which gives a complete picture of the

actual universe. This, however, would be a wrong

conclusion to draw from
the present state of affairs.

It is true that a few years ago this would
have been

a fair assessment, since the observational evidence then

was
far too meager to permit us to choose from among

the various cosmologies
that stem from the basic field

equations. But even then, the common
heritage of all

of these theories (the general theory of relativity) indi-

cated that the basic differences among
them are more

apparent than real.

The situation in the early 1970's was quite different,

for a threshold had
been reached for a cosmological

breakthrough; as we have seen, enough
observational

evidence was available to show us that our universe

originated explosively, about ten to twenty billion years

ago, from a
highly condensed state. Even though we

still could not decide unequivocally
between an ex-

panding and an oscillating
universe on the basis of the

observational evidence, the major problem of
the origin

of the universe had been solved and we had a self-

consistent picture. It accounted not
only for the reces-

sion and distribution of
the distant galaxies but also

for many diverse phenomena, ranging from the
back-

ground radiation all around us in
space (the 3° K. iso-

tropic
radiation which we have already discussed) to the

formation of the stars
and the heavy elements.

The most remarkable thing about the state of matter,

whether in the form of
stars or interstellar dust and

gas all around us, is that it points to some
momentous

event that must have occurred some billions of years

ago and
which led to the pronounced differentiation

that we see now. Starting from
the “big bang,” to

which all these observations
point, we can now arrange

the succession of events that led to the present
state

of the universe into a well-ordered, meaningful, and

understandable sequence. After the original explosion,

when the temperature
was still very high, about 30%

of the primordial neutrons and protons were
fused into

He4, but the expanding gas cooled off much too rapidly

for
elements above He4 to be built up in any appreci-

able quantities, and these had to wait for the stellar

ovens
that were to be formed when the rapidly ex-

panding gas of hydrogen and helium was fragmented

into stars by
turbulence and the gravitational forces.

The fragmentation of the original hydrogen-helium

gaseous mixture into
galaxies and stars occurred when

the exploding universe had cooled off to
very nearly

its present value—about two hundred million
years

after the initial explosion. The density of matter and

radiation was then favorable for gravitational contrac-

tion to take over in local regions
and to compress the

gas into huge clouds. This, however, could occur
only

after another process had come into operation—the

natural and unavoidable fragmentation of the expand-

ing gas into local eddies. One can show that a stream

of gas
becomes unstable against such a fragmentation

when the length of the stream
exceeds a certain num-

ber whose value can be
derived from hydrodynamical

theory. In an expanding universe this is bound
to hap-

pen after the expansion has progressed
beyond a given

point. The average size of the turbulent eddies that

are formed during this kind of fragmentation is deter-

mined by the speed and density of the expanding gas.

The details of this fragmentation process were

worked out many years ago by
J. H. Jeans. According

to his calculation, we know that the expanding gas
must

have broken up into fragments having an average size

equal to
that of a typical galaxy. These galaxies in turn

also suffered
fragmentation (on a smaller scale) by the

same process and the oldest stars
were thus formed.

These oldest stars (about 8 billion years old) were

formed at the center of the galaxies; and that is where

we find them now,
although they also constitute the

globular clusters that surround the core
of a galaxy.

Since the very oldest stars were formed almost ex-

clusively from the primordial hydrogen and helium,

at least some
of the heavy elements that we now ob-

serve all
about us in the universe must have been

synthesized in the interiors of
these stars as they

evolved. This, indeed, is the case, for we now
know,

from the theory of stellar interiors, that thermonuclear

processes occur near the center of a star, resulting in

the transmutation
of the light to the heavy elements.

When the oldest stars were first
formed, they con-

tracted very rapidly until
their central temperatures

reached about 10 million degrees, at which point
ther-

monuclear energy was released
with the transformation

of hydrogen to helium; this process kept the stars
in

equilibrium and supplied them with their energy for

the first few
billion years of their lives—in fact, until

about 12% of their
hydrogen had been transformed into

helium.

The core of each star, consisting entirely of helium,

then began to contract
quite rapidly under its own

weight, and the central temperature rose (in a
few

hundred million years) to about 100,000,000 degrees.

At this high
temperature, the helium nuclei in the core

were transformed to
carbon—the first step in the

buildup of the heavy elements. This
led to the forma-

tion of a carbon core which
contracted still further,

resulting in still higher core temperatures. In
fact, the

temperature in the core continued to increase until the

billion degree mark was reached, and the heavy ele-

that point a drastic change occurred in the evolution

of the star, for very little of its nuclear fuel was left

to supply the energy required to support its own

weight. The star, which by this time had evolved into

a very large and luminous object, collapsed violently

and became a supernova, ejecting great quantities of

material from its outer regions.

Following the supernova explosion, the hot residual

core (consisting of such
nuclei as iron, calcium, magne-

sium, and free
electrons) continued to contract, finally

becoming a white dwarf of
enormous density. It re-

mains in this stage
when the outward pressure of the

free electrons just balances the
gravitational contrac-

tion. But this is not
so in all cases, and the star must

continue to contract beyond the white
dwarf stage if

it is massive enough—ultimately becoming a very
hot

neutron star, about ten miles in diameter. Although

such stars
have not yet been observed directly, astron-

omers believe that they constitute some of the X-ray

sources now
being observed and are the recently dis-

covered “pulsars.” But even neutron stars are not
the

final stage of stellar evolution, for the theory of relativ-

ity tells us that such stars must
continue to contract

until they disappear from sight.

But what of the material that was ejected from each

star that became a
supernova? This was swirled into

the outer regions of the galaxy, where it
became the

gas and dust that formed the spiral arms that we now

see.
From this gas and dust—consisting not only of the

primordial
hydrogen and helium, but also of such heavy

elements as carbon, oxygen,
sodium, calcium, and

iron—the second generation, and hence
younger stars

such as our sun, were formed. But something else

happened at the same time—planets were also formed.

It can be
shown, as has been done by C. F. von

Weizsäcker, G. F. Kuiper, H.
Urey, H. Alfvén, and

others, that the turbulences that must occur
when a

star like the sun is formed by gravitational contraction,

from
dust and gas, must lead to the formation of planets

at fairly definite
distances from the star. This is in

agreement with the arrangement of the
planets in our

solar system.

We thus see that the cosmological theories that stem

from Einstein's
gravitational field equations agree with

the overall architectural and
dynamical features of the

universe as we now observe them. At the same
time,

these theories show us how the present state of the

universe has
evolved from a highly condensed initial

state, and tell us what to expect
in the future evolution

of the universe. Although many of the details are
still

missing from this forecast, the dominant features are

clearly
indicated, and we have every reason to believe

that we shall soon be able
to answer most of the ques

tions about the universe that seemed so unanswerable

just a few
years ago, for never before in the history

of science have so many capable
scientists been work-

ing on this exciting
problem.

##
*BIBLIOGRAPHY*

R. Alpher and R. Herman, *Reviews of Modern Physics,*

22 (1950), 153. H. Bondi and T. Gold, *Monthly Notices, Royal Astronomical Society,*
108 (1948), 252. R. H. Dicke,

P. J. E . Peebles, P. G. Roll, and D. T. Wilkinson,

*The*

Astrophysical Journal,142 (1965), 414. P. A. M. Dirac,

Astrophysical Journal,

*Proceedings of the Royal Astronomical Society, A,*165 (1938),

199. A. S. Eddington,

*The Expanding Universe*(Cambridge,

1933); idem,

*Fundamental Theory*(Cambridge, 1946). A.

Einstein,

*Sitzungsberichte der Preussische Akademische*

Gesellschaft,142 (1917). A. Friedmann,

Gesellschaft,

*Zeitschrift für*

Physik,10 (1922), 377. G. Gamow,

Physik,

*Physical Review,*70

(1946), 572; 74 (1948), 505. E . R. Harrison,

*Monthly Notices,*

Royal Astronomical Society,131 (1965), 1. F. Hoyle,

Royal Astronomical Society,

*Monthly*

Notices, Royal Astronomical Society,108 (1948), 372. M. L.

Notices, Royal Astronomical Society,

Humason, N. U. Mayall, and A. Sandage,

*The Astronomical*

Journal,61 (1956), 97. J. Jeans,

Journal,

*Astronomy and Cosmology*

(Cambridge, 1928, reprint 1961). P. Jordan,

*Die Herkunft*

der Sterne(Stuttgart, 1947). G. Lemaître,

der Sterne

*Monthly Notices,*

Royal Astronomical Society,91 (1931), 490. W. H. McCrea

Royal Astronomical Society,

and G. C. McVittie,

*Monthly Notices, Royal Astronomical*

Society,92 (1931), 7. A. A. Michelson and E . M. Morley,

Society,

*Philosophical Magazine,*190 (1887), 449. E . A. Milne,

*Rela-*

tivity, Gravitation, and World Structure(Oxford, 1935). C. G.

tivity, Gravitation, and World Structure

Neumann,

*Über das Newtonische Prinzip der Fernwirkung*

(Leipzig, 1895). A. A. Penzias and R. W. Wilson,

*The Astro-*

physical Journal,142 (1965), 419. H. P. Robertson,

physical Journal,

*The*

Astrophysical Journal,82 (1935), 284; 83 (1936), 187, 257.

Astrophysical Journal,

A. Sandage,

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Seeliger,

*Astronomische Nachrichtungen,*137 (1895), 129. W.

de Sitter,

*Monthly Notices, Royal Astronomical Society,*78

(1917), 3. R. C. Tolman,

*Relativity, Thermodynamics, and*

Cosmology(Oxford, 1932). A. G. Walker,

Cosmology

*Proceedings of the*

London Mathematical Society(

London Mathematical Society

*2*), 42 (1936), 90. H. Weyl,

*Physikalische Zeitschrift,*24 (1923), 230.

*GENERAL BIBLIOGRAPHY*

H. Bondi, *Cosmology* (Cambridge, 1961); *Rival Theories of Cosmology* (Oxford, 1960). P.
Couderc,

*The Expansion of the*

Universe(London, 1952). G. Gamow,

Universe

*The Creation of the*

Universe(New York, 1952). E . Hubble,

Universe

*Realm of the Nebulae*

(Oxford, 1961). G. C. McVittie,

*Fact and Theory in Cosmology*

(New York, 1961). M. K. Munitz, ed.,

*Theories of the Universe*

(New York, 1957). D. Sciama,

*The Unity of the Universe*

(Garden City, N.Y., 1961). J. Singh,

*Great Ideas and Theories*

of Modern Cosmology(London, 1961). W. de Sitter,

of Modern Cosmology

*Kosmos*

(Cambridge, Mass., 1932). E . Teller,

*Physical Review,*73

(1948), 801.

LLOYD MOTZ

[See also Cosmic Images; Cosmic Voyages; Cosmology fromAntiquity to 1850; Infinity; Relativity; Space; Time.]

Dictionary of the History of Ideas | ||