The Robertson-Walker metric is used in cosmology to describe space-times that are compatible with the cosmological principle of spatial homogeneity and isotropy. It defines the proper distance and intervals between events in an expanding universe using comoving coordinates and the scale factor a(t) that describes the expansion over time. Key relationships defined include the luminosity distance and angular diameter distance, relating observed properties of distant objects to the scale factor at emission and observation times.

1. The Robertson-Walker metric
In cosmology it is most useful to use spherical polar coordinates
rather than a Cartesian system. In this system the only metric
that describes space-times compatible with the cosmological
principle is the Robertson-Walker metric:
𝑑𝑠2 = 𝑐2 𝑑𝑡2 − 𝑎2(𝑡)
𝑑𝑟2
1 − 𝑘𝑟2
+ 𝑟2 𝑑𝜃2 + 𝑠𝑖𝑛2 𝜃𝑑𝜙2
Here 𝑘 is the spatial curvature and 𝑟, 𝜃, 𝜙 are comoving
coordinates; the cosmic scale factor is 𝑎 𝑡 . The time coordinate is
cosmological proper time.
𝑑𝑠2 = 𝑐2 𝑑𝑡2 − 𝑎2(𝑡)
𝑑𝑟2
1 − 𝑘𝑟2
+ 𝑟2 𝑑Ω2

2. Intervals
The time coordinate 𝑡 is chosen so that hypersurfaces of
simultaneity (spatial sections) are chosen to have the same
density everywhere on them.
There are different types of interval
𝑑𝑠2 = 0
Note that a radial null ray can be either incoming or outgoing:
𝑐𝑑𝑡 = ±𝑎(𝑡)
𝑑𝑟
1 − 𝑘𝑟2
𝑑𝑠2
> 0
𝑑𝑠2
< 0
`null’ – this describes the path of a light ray
`timelike’
`spacelike’

3. Proper Distance
There are various ways of defining proper distances in cosmology.
The simplest is the proper distance, defined as a radial interval
with 𝑑𝑡 = 0 in the Robertson-Walker metric:
𝐷 𝑡 = 𝑎 𝑡
𝑜
𝑟
𝑑𝑟
1 − 𝑘𝑟2
= 𝑎 𝑡 𝑓 𝑟
where
𝑓 𝑟 =
sin−1
𝑟 (𝑘 = +1)
𝑟 (𝑘 = 0)
sinh−1
𝑟 (𝑘 = −1)
Proper distances are useful conceptually but are not
measurable directly. We usually deal with information carried
light rays with 𝑑𝑠 = 0 rather than intervals with 𝑑𝑡 = 0.

4. The observed redshift 𝑧 of a cosmological source relates the
scale factor at the cosmic time a light signal was emitted to the
scale factor when it received by an observer
𝜆 𝑡 ∝ 𝑎 𝑡 ⇒ 1 + 𝑧 =
𝑎(𝑡0)
𝑎(𝑡 𝑒)
≡
𝑎0
𝑎 𝑒

5. Luminosity Distance - 1
One way of defining a distance is through the behaviour of
luminous flux. In a (static) Euclidean space, the flux 𝑆 density
arriving at distance 𝐷 from a source of luminosity 𝐿 is
𝑆 =
𝐿
𝐴
=
𝐿
4𝜋𝐷2
⟹ 𝐷 =
𝐿
4𝜋𝑆
1/2
In an expanding space-time described by the R-W metric
this relationship changes.
First the (proper) surface area of a sphere must be written
𝐴 = 4𝜋𝑎0
2
𝑟2.
In addition photons are redshifted by a factor 𝑎0
𝑎 𝑒 ,
reducing their energy at arrival, and also the time between
each photon is increases by the same factor. Here

6. Luminosity Distance - 2
Putting these effects together we find that
𝑆 =
𝐿
4𝜋𝑎0
2
𝑟2
𝑎 𝑒
𝑎0
2
This suggests that we define a luminosity distance
𝐷𝐿 =
𝐿
4𝜋𝑆
1/2
= 𝑎0
2
𝑟
𝑎 𝑒
Note that if 𝑟 ⟶ 0, 𝑎 𝑒 ⟶ 𝑎 𝑜, 𝐷𝐿 ⟶ 𝐷. At larger
distances, though, these measures diverge in a way that
depends on cosmology.

7. Angular-Diameter Distance
If we know the physical size of an object rather than its luminosity
we can work out an angular-diameter distance derived as in the
case of the luminosity distance from what one would expect in a
static Euclidean space. In the R-W metric (assuming small angles)
we have
𝑥
𝐷
≈ 𝛿𝜃,
for an object of proper length 𝑥 at distance D. According to
the R-W metric, 𝑥 ≈ 𝑎 𝑡 𝑒 𝑟𝛿𝜃 so that defining an angular
diameter distance
𝐷𝐴 =
𝑥
𝛿𝜃
⇒ 𝐷𝐴 = 𝑎 𝑡 𝑒 𝑟

8. The Etherington Reciprocity Theorem
Putting the two results for angular diameter distance and
luminosity distance together we have that
𝐷𝐴 = 𝑎 𝑒 𝑟
𝐷𝐿 = 𝑎 𝑜
2
𝑟
𝑎 𝑒
which means that
𝐷 𝐿
𝐷 𝐴
=
𝑎0
2
𝑎 𝑒
2 = (1 + 𝑧)2
since 1 + 𝑧 = 𝑎0
𝑎 𝑒 defines the redshift 𝑧.