1.3 From Newtonian to relativistic cosmology 19
In fact, since the physical width of an infinitesimal shell isdl=adχ, the volume
element between two spheres with radiiχandχ+dχis
dV=S 2 dadχ= 4 πa^3 sin^2 χdχ.Therefore, the volume within the sphere of radiusχ 0 is
V(χ 0 )= 4 πa^3∫χ^00sin^2 χdχ= 2 πa^3(
χ 0 −^12 sin 2χ 0)
. (1.43)
Forχ 0 1, the volume,
V(χ 0 )= 4 π(aχ 0 )^3 / 3 +···,grows in the same way as in Euclidean space. The total volume, obtained by sub-
stitutingχ 0 =πin (1.43), is finite and equal to
V= 2 π^2 a^3. (1.44)The other distinguishing property of a space of constant positive curvature is that
the sum of the angles of a triangle constructed from geodesics (curves of minimal
length) is larger than 180 degrees.
Three-dimensional pseudo-sphere(k=−1) The metric on the surface of a 2-
sphere of radiusχ in a three-dimensional space of constant negative curvature
is
dl^2 =a^2 sinh^2 χ(dθ^2 +sin^2 θdφ^2 ), (1.45)and the area of the sphere,
S 2 d(χ)= 4 πa^2 sinh^2 χ, (1.46)increases exponentially forχ 1 .Since the coordinateχvaries from 0 to+∞,
the total volume of the hyperbolic space is infinite. The sum of angles of a triangle
is less than 180 degrees.
Problem 1.10Calculate the volume of a sphere with radiusχ 0 in a space with
constant negative curvature.
1.3.2 The Einstein equations and cosmic evolution
The only way to preserve the homogeneity and isotropy of space and yet incorporate
time evolution is to allow the curvature scale, characterized bya, to be time-
dependent. The scale factora(t) thus completely describes the time evolution of