46 Propagation of light and horizons
to clarify what happens to light after passing through the pole, one has to use another
coordinate system which is regular nearχ=π.
Problem 2.5Show that light propagating away from an observer atχ=0inthe
direction (θ, φ) begins to propagate back towards the observer along the direction
(θ ̃=π−θ,φ ̃=φ+π) after it passes through the pole atχ=π.
Thus, a light geodesic is “reflected” from the boundary atχ=πand its angular
coordinatesθandφchange. This change of the angular coordinates is not apparent
from the conformal diagram because they are suppressed there.
Let us use a conformal diagram to infer how a galaxy located atχ=χg=
const appears to an observer atχ=0 in a dust-dominated universe. As is clear
from Figure 2.2, atη> 2 π−χg,when the universe is contracting, there are two
geodesics along which light emitted by the galaxy can reach the observer. Hence, the
observer simultaneously sees two images of the same galaxy in opposite directions
in the sky. One image is older than the other byη= 2
(
π−χg)
. In a radiation-
dominated universe, only one image of the galaxy can be seen because light does
not have enough time to travel around the pole atχ=πand reach an observer
before the universe recollapses.
Problem 2.6Using (1.83), draw the conformal diagram for a closed universe filled
with a mixture of dust and radiation.
De Sitter universeDe Sitter spacetime is an example of how different coordinate
systems used for the same spacetime can lead to different conformal diagrams. We
begin by rewriting metric (1.106) in terms of conformal time instead of physical
timet. For aclosed universe, the relation is
η=∫t∞dt
H−^1 cosh(Ht)=arcsin[tanh(Ht)]−
π
2. (2.26)
The conformal timeηis always negative and ranges from−πto0astvaries from
−∞to+∞.It follows from (2.26) that
cosh(Ht)=−(sinη)−^1 , (2.27)which allows us to write the metric of theclosedde Sitter universe as
ds^2 =1
H^2 sin^2 η(
dη^2 −dχ^2 −sin^2 χd
2)
. (2.28)
Since the spatial coordinateχvaries from 0 toπand the temporal coordinateη
changes from−πto 0, the conformal diagram for a closed de Sitter universe is