Physical Foundations of Cosmology

18 Kinematics and dynamics of an expanding universe

Three-dimensional sphere(k=+1) It follows from (1.39) that in a three-
dimensional space with positive curvature, the distance element on the surface
of a 2-sphere of radiusχis

dl^2 =a^2 sin^2 χ(dθ^2 +sin^2 θdφ^2 ). (1.41)

This expression is the same as for a sphere of radiusR=asinχ in flat three-
dimensional space, and hence we can immediately find the total surface area:

S 2 d(χ)= 4 πR^2 = 4 πa^2 sin^2 χ. (1.42)

As the radiusχincreases, the surface area first grows, reaches its maximal value
atχ=π/2, and then decreases, vanishing atχ=π(Figure 1.6).
To understand such unusual behavior of a surface area, it is useful to turn to a
low-dimensional analogy. In this analogy, the surface of the globe plays the role of
three-dimensional space with constant curvature and the two-dimensional surfaces
correspond to circles of constant latitude on the globe. Beginning from the north
pole, corresponding toθ=0, the circumferences of the circles grow as we move
southward, reach a maximum at the equator, whereθ=π/2, then decrease below
the equator and vanish at the south pole,θ=π.Asθruns from 0 toπ, it covers the
whole surface of the globe. Similarly, asχchanges from 0 toπ, it sweeps out the
whole three-dimensional space of constant positive curvature. Because the total
area of the globe is finite, we expect that the total volume of the three-dimensional
space with positive curvature is also finite.

S2d(χ)

π/ 2 πχ

4 π

k= −1

k= +1

k= 0

Fig. 1.6.