Page 49 - Nature Of Space And Time

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 elds 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

2
2 2
2
2
2
2
2 2 −1
2
ds = −(1 − H r )dt +(1 − H r ) dr 2 + r (d +sin d )
observers
event horizon
future infinity
r= ¥
4

3 r= 1/H 1
0 0
r= 1/H 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 in nity 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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