2
As before the term linear in the perturbations vanishes. The quadratic term can be re-
garded as giving the contribution of gravitons on the background and the higher order
terms as interactions between the gravitons. These can be ignored when the radius of
curvature of the background is large compared to the Planck scale. Therefore
Ψ ≈
1
e−I[ go]
1
(det I 2) 2
45
Λ
3
3
2
Λ
2
action = _ 1
Λ 1 _ 1 _
a2
{ (
) } action = _ 1
a2
{ (
) }
3
Λ 1 + 1 _ 3
M +
M +
3-sphere
Σ
of radius a
Σ
4−sphere of
1
3
radius
=
H
Λ
One can see what the wave function is like from a simple example. Consider a situation
in which there are no matter fields but there is a positive cosmological constant Λ.
Let us take the surface Σ to be a three sphere and the metric hij to be the round three sphere metric of radius a. Then the manifold M+ bounded by Σ can be taken to be the four ball. The metric that satisfies the field equations is part of a four sphere of radius 1
H
where H 2 = Λ .
3
Z
Z
1
1
1
1
I =
( R − 2Λ)( −g) 2 d 4 x +
K( ±h) 2 d 3 x
16 π
8 π
For a three sphere Σ of radius less than 1 there are two possible Euclidean solutions:
H
either M+ can be less than a hemisphere or it can be more. However there are arguments that show that one should pick the solution corresponding to less than a hemisphere.
The next figure shows the contribution to the wave function that comes from the
action of the metric g 0. When the radius of Σ is less than 1 the wave function increases H
exponentially like ea 2 . However, when a is greater than 1 one can analytically continue H
the result for smaller a and obtain a wave function that oscillates very rapidly.
One can interpret this wave function as follows. The real time solution of the Einstein
equations with a Λ term and maximal symmetry is de Sitter space. This can be embedded
as a hyperboloid in five dimensional Minkowski space.
One can think of it as a closed universe that shrinks down from infinite size to a minimum
radius and then expands again exponentially. The metric can be written in the form of a
Friedmann universe with scale factor cosh Ht. Putting τ = it converts the cosh into cos giving the Euclidean metric on a four sphere of radius 1 .
H
46
maximum
minimum
3
3
=
=
radius
radius
Λ
Λ
Euclidean
Lorentzian
4-sphere
de Sitter space
Ψ
a
1
3
=
H
Λ
Lorentzian – de Sitter Metric
1
ds 2 = −dt 2 +
cosh Ht( dr 2 + sin2 r( dθ 2 + sin2 θdφ 2))
H 2
Thus one gets the idea that a wave function which varies exponentially with the three
metric hij corresponds to an imaginary time Euclidean metric. On the other hand, a wave function which oscillates rapidly corresponds to a real time Lorentzian metric.
Like in the case of the pair creation of black holes, one can describe the spontaneous
creation of an exponentially expanding universe. One joins the lower half of the Euclidean
four sphere to the upper half of the Lorentzian hyperboloid.
47
Euclidean Metric
1
ds 2 = dτ 2 +
cos Hτ ( dr 2 + sin2 r( dθ 2 + sin2 θdφ 2))
H 2
Lorentzian
de Sitter solution
Euclidean
4-sphere
Unlike the black hole pair creation, one couldn’t say that the de Sitter universe was created out of field energy in a pre-existing space. Instead, it would quite literally be created out of nothing: not just out of the vacuum but out of absolutely nothing at all because there
is nothing outside the universe. In the Euclidean regime, the de Sitter universe is just a
closed space like the surface of the Earth but with two more dimensions. If the cosmological