Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 17

Problem 1.9By embedding a three-dimensional sphere (pseudo-sphere) in a
four-dimensional Euclidean (Lorentzian) space, verify that the metric of a three-
dimensional space of constant curvature can be written as

dl 32 d=a^2

(

dr^2

1 −kr^2

+r^2 (dθ^2 +sin^2 θdφ^2 )

)

, (1.34)

wherea^2 is positive andk= 0 ,± 1 .Introduce the rescaled radial coordinater ̄,
defined by

r=

r ̄

1 +kr ̄^2 / 4

, (1.35)

and show that this metric can then be rewritten in explicitly isotropic form:

dl 32 d=a^2

(dx ̄^2 +d ̄y^2 +d ̄z^2 )

(1+k ̄r^2 /4)^2

, (1.36)

where

x ̄=r ̄sinθcosφ, ̄y=r ̄sinθsinφ, z ̄=r ̄cosθ.

In many applications, instead of the radial coordinater, it is convenient to use
coordinateχdefined via the relation

dχ^2 =

dr^2

1 −kr^2

. (1.37)

It follows that

χ=

⎧

⎨

⎩

arcsinhr, k=−1;

r, k=0;

arcsinrk=+ 1.

(1.38)

The coordinateχvaries between 0 and+∞in flat and hyperbolic spaces, while
π≥χ≥0 in spaces with positive curvature(k=+ 1 ).In this last case, to every
particularrcorrespond two differentχ.Thus, introducingχremoves the coordinate
degeneracy mentioned above. In terms ofχ, metric (1.34) takes the form

dl 32 d=a^2 (dχ^2 + 2 (χ)d 

2 )≡a^2

⎡

⎣dχ^2 +

⎛

⎝

sinh^2 χ

χ^2

sin^2 χ

⎞

⎠d 

2

⎤

⎦

k=−1;

k=0;

k=+ 1 ,

(1.39)

where

d 

2 =(dθ^2 +sin^2 θdφ^2 ). (1.40)

Let us now take a closer look at the properties of the constant curvature spaces.