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

VI. |

V. |

VI. |

I. |

VI. |

V. |

III. |

III. |

VI. |

VI. |

V. |

V. |

III. |

VII. |

VI. |

VI. |

III. |

III. |

II. |

I. |

I. |

I. |

V. |

VII. |

VI. |

V. |

III. |

III. |

III. |

II. |

I. |

I. |

I. |

VI. |

VII. |

III. |

VII. |

VII. |

VII. |

V. |

VI. |

VI. |

VI. |

VI. |

VI. |

VII. |

III. |

IV. |

VI. |

VI. |

VI. |

V. |

V. |

V. |

III. |

III. |

VII. |

III. |

I. |

V. |

V. |

VII. |

VI. |

I. |

I. |

I. |

I. |

VI. |

III. |

IV. |

III. |

IV. |

IV. |

IV. |

VI. |

VI. |

VI. |

V. |

III. |

VI. |

Dictionary of the History of Ideas | ||

####
*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 |

Dictionary of the History of Ideas | ||