50 Propagation of light and horizons
spacetime, a triangle whose lower boundary coincides with the particle horizon. On
the particle horizon, ̄η→−∞, ̄χ→+∞, and hence, the flat coordinates become
singular.
Problem 2.8To understand the shape of the constant ̄ηand constant ̄χhypersur-
faces near the corners of the triangular conformal diagram,χ= 0 ,η=−πand
χ=π, η= 0 ,calculate the derivativesdη/dχalong these hypersurfaces.
Viewing the flat de Sitter solution as describing an infinite space, we can cat-
egorize the types of infinities that arise. For instance,space-like infinity, where
χ ̄→+∞along a hypersurface of constant ̄η, is represented on the conformal di-
agram by a point which is denoted asi^0 .Thepast time-like infinity, from where
all time-like lines emanate, occurs at ̄η→−∞for finite ̄χand is denoted byi−.
The lower diagonal boundary of the flat de Sitter diagram corresponds to the region
from which incoming light-like geodesics originate. It is easy to verify that as we
approach this boundary, ̄χ→∞and ̄η→−∞but the sum ̄χ+η ̄remains finite.
This infinity is calledpast null infinityand denoted byI−.
In anopen de Sitter universe, the relation between physical and conformal times
is
sinh(
H ̃t)
=− 1 /sinh ̃η (2.37)and the metric becomes
ds^2 =1
H^2 sinh^2 η ̃(
dη ̃^2 −dχ ̃^2 −sinh^2 χ ̃d
2)
. (2.38)
The coordinates run over the same range as in a flat de Sitter universe, 0>η> ̃ −∞
and+∞>χ> ̃ 0, therefore the conformal diagrams of these two spaces will look
similar. We can again use the closed coordinates to determine which part of the
de Sitter spacetime is covered by the open coordinates. The corresponding relation
between coordinate systems follows from (1.99):
tanh ̃η=sinη
cosχ, tanh ̃χ=sinχ
cosη. (2.39)
In this case, the coordinates ̃η,χ ̃cover only one eighth of the whole de Sitter
spacetime (Figure 2.6), and thus, cover an even smaller part of the de Sitter manifold
than the flat coordinates. Of course, it only makes sense to compare the sizes of
different diagrams when they describe the same spacetime, as in the case of the de
Sitter manifold. Otherwise, as noted before, the size of the diagram has no invariant
meaning.
Problem 2.9Calculate the derivativedη/dχalong the hypersurfaces ̃η=const
and ̃χ=const,neari−andi^0 respectively.