Physical Foundations of Cosmology

48 Propagation of light and horizons

η

χ

−π

π

r= H−^1

r< H−^1

t= −∞

I

II

IV

III

tˆ =const

t =const

r= H−^1

t= ∞

r> HΛ−^1

Λ Λ

Λ

ˆ

ˆ

Fig. 2.4.

Problem 2.7One can utilize for de Sitter spacetime the so called “static coordi-
nates”ˆt,r,related toη,χvia

tanh

(

Hˆt

)

=

cosη

cosχ

, Hr=

sinχ

sinη

. (2.31)

Verify that in these coordinates the metric takes the following form:

ds^2 =

[

1 −(Hr)^2

]

dˆt^2 −

dr^2

[

1 −(Hr)^2

]−r^2 d 

2. (2.32)

The hypersurfaces of constantrandtˆare shown in Figure 2.4. De Sitter horizons
correspond tor=H−^1 andˆt=±∞. The static coordinates cover only half of de
Sitter spacetime: regions I and III in Figure 2.4. They are singular on the horizons
but can be continued beyond. Forr>H−^1 ,the radial coordinaterplays the role
of time andtˆbecomes a space-like coordinate. Introduce the proper-time

dτ=dr/

[

(Hr)^2 − 1

] 1 / 2

(2.33)

and verify that in regions II and IV the “static” metric (2.32) describes contracting
and expanding space respectively. We conclude that there exists no static coordinate
system covering de Sitter spacetime on scales exceeding the curvature scale. Note
that the trajectoryr=const is a geodesic only ifr=0.