Physical Foundations of Cosmology

52 Propagation of light and horizons

corresponds to a 2-sphere and that the 3-geometries of hypersurfaces along which

the diagrams are glued must match.

To complete the set of the diagrams needed in cosmology, we must also con-

struct the conformal diagrams describing open and flat universes filled by matter

and radiation. As a preliminary step, we first consider the conformal diagram for

Minkowski spacetime, which turns out to be useful in drawing the diagrams of

more complicated infinite spaces.

Minkowski spacetimeIn spherical coordinates the Minkowski metric takes the form

ds^2 =dt^2 −dr^2 −r^2 d 

2. (2.40)

It is trivially conformal but the time and radial coordinates range over infinite

intervals,+∞>t>−∞and+∞>r≥ 0 ,and, therefore, have to be replaced

by coordinates with finite ranges. There exist many such coordinate systems for

Minkowski spacetime. One choice is to introduceηandχcoordinates which are

related to thetandrcoordinates in the same way as closed and open de Sitter

coordinates are related (see (2.39)), namely,

tanht=

sinη

cosχ

, tanhr=

sinχ

cosη

. (2.41)

The Minkowski metric in the new coordinates then becomes

ds^2 =

1

cos^2 χ−sin^2 η

(

dη^2 −dχ^2 −^2 (η,χ)d 

2

)

, (2.42)

wherecan be calculated but is not important for our purposes. Comparing the
Minkowski timetto ̃ηin an open de Sitter universe (2.39), we see thattruns from
−∞to+∞,while ̃ηis restricted to negative values (because the scale factor in
open de Sitter spacetime blows up as ̃η→ 0 −). Therefore, in theη–χplane, the
hypersurfaces of constanttandrspan a large triangle, which can be thought of
as made from two smaller triangles describing the open de Sitter spacetime and its
time-reversed copy (Figure 2.7). Minkowski spacetime possesses two additional
types of infinities compared to an open de Sitter universe: afuture time-like infinity
i+,where all time-like lines end(t→+∞,ris finite),and afuture null infinity
I+(t→+∞,r→+∞witht−rfinite),the region towards which outgoing
radial light geodesics extend. Region I in Figure 2.7 corresponds to a future light
cone which can also be covered by Milne coordinates. The Milne conformal diagram
is geometrically similar to the Minkowski one, though it is four times smaller.

Problem 2.10Draw the conformal diagram for the Milne universe and verify this

last statement.