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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
1 occurrence of Tonelli, Giorgio
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1 occurrence of Tonelli, Giorgio
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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


563

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 space-time interval
between two events for an unaccelerated observer in
empty space is d2-c2t2, 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-


564

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  +1 
negative  oscillating  oscillating  oscillating 
zero  expanding I  expanding I  oscillating 
positive  expanding I  expanding I  oscillating 
expanding I 
expanding II