84 Cosmological Models
Einstein Universe. Consider now the static universe cherished by Einstein. This is
defined by푎(푡)being constant,푎(푡 0 )=1, so that푎̇=0and푎̈=0 and the age of the
Universe is infinite. Equations (5.4) and (5.5) then reduce to
푘푐^2 =8 휋
3
퐺휌 0 =−
8 휋
푐^2
퐺푝 0. (5.15)
In order that the mass density휌 0 be positive today,푘must be+1. Note that this leads
to the surprising result that the pressure of matter푝 0 becomes negative!
Einstein corrected for this in 1917 by introducing a constant Lorentz-invariant
term휆푔휇휈into Equation (3.28), where thecosmological constant훺휆corresponds to
a tiny correction to the geometry of the Universe. Equation (3.28) then becomes
퐺휇휈=푅휇휈−^1
2푔휇휈푅−휆푔휇휈. (5.16)
In contrast to the first two terms on the right-hand side, the휆푔휇휈term does not
vanish in the limit of flat space-time. With this addition, Friedmann’s equations take
the form
푎̇^2 +푘푐^2
푎^2−
휆
3
=
8 휋퐺
3
휌, (5.17)
2 푎̈
푎
+푎̇
(^2) +푘푐 2
푎^2
−휆=−^8 휋퐺
푐^2
푝. (5.18)
A positive value of휆curves space-time so as to counteract the attractive gravitation
of matter. Einstein adjusted휆to give a static solution, which is called theEinstein
universe.
The pressure of matter is certainly very small, otherwise one would observe the
galaxies having random motion similar to that of molecules in a gas under pressure.
Thus one can set푝=0 to a good approximation. In the static case when푎=1,푎̇ 0 = 0
and푎̈ 0 =0, Equation (5.17) becomes
푘푐^2 −휆
3=^8 휋퐺
3
휌 0.
It follows from this that in a spatially flat Universe휌휆= 휆
8 휋퐺=−휌 0. (5.19)
But Einstein did not notice that the static solution is unstable: the smallest imbal-
ance between휆and휌would make푎̈nonzero, causing the Universe to accelerate into
expansion or decelerate into contraction. This flaw was only noticed by Eddington in
1930, soon after Hubble’s discovery, in 1929, of the expansion that caused Einstein
to abandon his belief in a static universe and to withdraw the cosmological constant.
This he called ‘the greatest blunder of my lifetime’.
The Friedmann–Lemaitre Universe. If the physics of the vacuum looks the same
to any inertial observer, its contribution to the stress–energy tensor is the same as
Einstein’s cosmological constant휆,aswasnotedbyLemaitre.The휆term in Equa-
tion (5.16) is a correction to the geometrical terms in퐺휇휈, but the mathematical con-
tent of Equations (5.17) and (5.18) are not changed if the휆terms are moved to the