VI. THE DE SITTER EMPTY
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
appeared that the universe was indeed static. More-
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
R
ij - 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.