42 Propagation of light and horizons
where the indicesa,bandcrun over only two values, 0 and 1, corresponding to
the time and radial coordinates respectively. The angular part of the metric is rather
simple. It is proportional to
d
2 ≡dθ^2 +sin^2 θdφ^2 (2.16)and describes a 2-sphere of radiusR(xc). The only nontrivial piece of the metric is
the temporal–radial part, which can describe spaces with different causal structure.
The causal structure can be represented by a two-dimensionalconformal diagram,
in which every point corresponds to a 2-sphere.
The global properties of the spacetime can be completely explored by considering
the radial geodesics of light. As we showed in Section 2.1, in a coordinate system
where metric (2.15) takes the form
ds^2 =a^2 (η,χ)[
dη^2 −dχ^2 − 2 (η,χ)d
2]
, (2.17)
the radial propagation of light is described by the equation
χ(η)=±η+const, (2.18)or in other words, by straight lines at±45 degree angles in theη–χplane.
In principle, it is always possible to find a coordinate system that allows us to
write (2.15) as (2.17). In the coordinate transformation
xa→x ̃a≡(
η(
xa)
,χ(
xa))
,
the freedom to choose the two functionsηandχmeans we can impose the two
conditions
g ̃ 01 = 0 , g ̃ 00 =g ̃ 11 ≡a^2 (η,χ).Solving the equations forηandχcan be difficult in general, but in cosmologically
interesting cases the metric is already in the required form.
Typically,ηandχ may extend over infinite or semi-infinite intervals. Since
our goal is to visualize the causal structure of the full spacetime, in these cases
we perform a further coordinate transformation that preserves the form of metric
(2.17) but maps unbounded coordinates into coordinates which vary over a finite
interval. We shall see that it is always possible to find such transformations. In this
section, we reserve the symbolsηandχto refer only to bounded coordinates.
A conformal diagram is a picture of a spacetime plotted in terms ofηandχ.
Hence,a conformal diagram always has finite size and light geodesics (null lines)
arealwaysrepresentedbystraightlinesat± 45 degreeangles.These are the defining
features of a conformal diagram. Although the finite ranges spanned by the coordi-
nates and the size of the diagram can be altered, its shape is uniquely determined.