VII. THE NONSTATIC MODELS OF A
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 λgij 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
λgij 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
the field equations and which we shall presently discuss.
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 spacetime
interval
between two events for an unaccelerated observer in
empty
space is d2c2t2, 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 spacetime 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
spacetime 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
creases forever or increases to a certain maximum value
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 