Page 46 - Nature Of Space And Time

3 2 3 2
_
action = _ 1 { ( 1 _ L a 2 } ) action = _ 1 { ( 1 _ L a 2 } )
1 +
1
L 3 L 3
+
M
+
M

3-sphere S
of radius a

S
4-sphere of
radius 1 = 3
H L

One can see what the wave function is like from a simple example. Consider a situation
in which there are no matter elds but there is a positive cosmological constant .

Let us take the surface to be a three sphere and the metric h ij to be the round three
sphere metric of radius a. Then the manifold M + bounded by can be taken to be the
four ball. The metric that satis es the eld equations is part of a four sphere of radius 1
H
2
where H = .
3
Z Z
1 1 1 1
4
3
I = (R − 2 )(−g) 2 d x + K(±h) 2 d x
16 8
For a three sphere 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 gure shows the contribution to the wave function that comes from the

action of the metric g 0. When the radius of is less than 1 the wave function increases
H
2
a
exponentially like e . 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 term and maximal symmetry is de Sitter space. This can be embedded
as a hyperboloid in ve dimensional Minkowski space.
One can think of it as a closed universe that shrinks down from in nite 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 coshHt. Putting = it converts the cosh into cos
giving the Euclidean metric on a four sphere of radius 1 .
H

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