University of Virginia Library

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 λgij 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
λgij in the field equations. If we place Tij = 0 in the
field equations (2) they reduce to
Rij = λgij, or Rij - λgij = 0, (1)
and since the term Rij represents the ordinary New-
tonian gravitation of attraction, the term -λgij 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.