constant is small compared to the Planck value, the curvature of the Euclidean four sphere
should be small. This will mean that the saddle point approximation to the path integral
should be good, and that the calculation of the wave function of the universe won’t be
affected by our ignorance of what happens in very high curvatures.
One can also solve the field equations for boundary metrics that aren’t exactly the
round three sphere metric. If the radius of the three sphere is less than 1 , the solution is a H
real Euclidean metric. The action will be real and the wave function will be exponentially
damped compared to the round three sphere of the same volume. If the radius of the three
sphere is greater than this critical radius there will be two complex conjugate solutions
and the wave function will oscillate rapidly with small changes in hij.
Any measurement made in cosmology can be formulated in terms of the wave function.
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Thus the no boundary proposal makes cosmology into a science because one can predict the
result of any observation. The case we have just been considering of no matter fields and
just a cosmological constant does not correspond to the universe we live in. Nevertheless,
it is a useful example, both because it is a simple model that can be solved fairly explicitly and because, as we shall see, it seems to correspond to the early stages of the universe.
Although it is not obvious from the wave function, a de Sitter universe has thermal
properties rather like a black hole. One can see this by writing the de Sitter metric in a
static form rather like the Schwarzschild solution.
Static form of the de Sitter metric
ds 2 = −(1 − H 2 r 2) dt 2 + (1 − H 2 r 2) − 1 dr 2 + r 2( dθ 2 + sin2 θdφ 2) observers
event horizon
future infinity
r = ∞
4
r = 1/H
0
3
1
0
1/H
r=
=
r=
r
2
r = ∞
observers
past infinity
world line
There is an apparent singularity at r = 1 . However, as in the Schwarzschild solution, one H
can remove it by a coordinate transformation and it corresponds to an event horizon. This
can be seen from the Carter-Penrose diagram which is a square. The dotted vertical line on
the left represents the center of spherical symmetry where the radius r of the two spheres goes to zero. There is another center of spherical symmetry represented by the dotted
vertical line on the right. The horizontal lines at the top and bottom represent past and
future infinity which are space like in this case. The diagonal line from top left to bottom
right is the boundary of the past of an observer at the left hand center of symmetry. Thus
it can be called his event horizon. However, an observer whose world line ends up at a
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different place on future infinity will have a different event horizon. Thus event horizons
are a personal matter in de Sitter space.
If one returns to the static form of the de Sitter metric and put τ = it one gets a Euclidean metric. There is an apparent singularity on the horizon. However, by defining
a new radial coordinate and identifying τ with period 2 π , one gets a regular Euclidean H
metric which is just the four sphere. Because the imaginary time coordinate is periodic, de
Sitter space and all quantum fields in it will behave as if they were at a temperature H .
2 π
As we shall see, we can observe the consequences of this temperature in the fluctuations in
the microwave background. One can also apply arguments similar to the black hole case
to the action of the Euclidean-de Sitter solution. One finds that it has an intrinsic entropy of π , which is a quarter of the area of the event horizon. Again this entropy arises for H 2
a topological reason: the Euler number of the four sphere is two. This means that there
can not be a global time coordinate on Euclidean-de Sitter space. One can interpret this