Introduction to Cosmology

Metrics of Curved Space-time 35

An observer located at any one point would see all the other points recede radially.
This is exactly how we see distant galaxies except that we are not on a two-sphere but,
as we shall see, on a spatially curved three-surface with the cosmic scale factor푎(푡).
Suppose this observer wants to make a map of all the points on the expanding
surface. It is then no longer convenient to use coordinates dependent on푎(푡)as in
Equations (2.12) and (2.20), because the map would quickly be outdated. Instead it
is convenient to factor out the cosmic expansion and replace푎by푎(푡)휎,where휎is a
dimensionlesscomovingcoordinate, thus

d푙^2 =푎^2 (푡)(d휎^2 +휎^2 d휃^2 +휎^2 sin^2 휃d휙^2 ). (2.27)

Returning to the space we inhabit, we manifestly observe that there are three
spatial coordinates, so our space must have at least one dimension more than a
two-sphere. It is easy to generalize from the curved two-dimensional manifold (sur-
face) [Equation (2.18)] embedded in three-space to the curved three-dimensional
manifold (hypersurface)

푥^2 +푦^2 +푧^2 +푤^2 =푅^2 (2.28)

of a three-sphere (hypersphere) embedded in Euclidean four-space with coordinates
푥,푦,푧and a fourth fictitious space coordinate푤.
Just as the metric in Equation (2.19) could be written without explicit use of푧,the
metric on the three-sphere in Equation (2.28) can be written without use of푤,

d푙^2 =d푥^2 +d푦^2 +d푧^2 +

(푥d푥+푦d푦+푧d푧)^2

푅^2 −푥^2 −푦^2 −푧^2

, (2.29)

or, in the more convenient spherical coordinates used in Equation (2.27),

d푙^2 =푅^2 (푡)

(

푅^2 d휎^2

푅^2 −(푅휎)^2

+휎^2 d휃^2 +휎^2 sin^2 휃d휙^2

)

. (2.30)

Note that the introduction of the comoving coordinate휎in Equation (2.27) does not
affect the parameter푅defining the hypersurface in Equation (2.28). No point is pre-
ferred on the manifold [Equation (2.28)], and hence it can describe a spatially homo-
geneous and isotropic three-dimensional universe in accord with the cosmological
principle.
Another example of a curved Riemannian two-space is the surface of a hyperboloid
obtained by changing the sign of푅^2 in Equation (2.18). The geodesics are hyperbolas,
the surface is also unbounded, but in contrast to the spherical surface it is infinite
in extent. It can also be generalized to a three-dimensional curved hypersurface, a
three-hyperboloid, defined by Equation (2.28) with푅^2 replaced by−푅^2.
The Gaussian curvature of all geodesic three-surfaces in Euclidean four-space is

퐾=푘∕푅^2 , (2.31)

where thecurvature parameter푘can take the values+1, 0,−1, corresponding to the
three-sphere, flat three-space, and the three-hyperboloid, respectively. Actually,푘can
take any positive or negative value, because we can always rescale휎to take account
of different values of푘.