Introduction to Cosmology

Metrics of Curved Space-time 33

wherethemetricmatrixis

g=

(

푅^20

0 푅^2 sin^2 휃

)

. (2.22)

The ‘two-volume’ or area퐴of the two-sphere in Figure 2.2 can then be written

퐴=

∫

2 휋

0

d휙

∫

휋

0

d휃

√

detg=

∫

2 휋

0

d휙

∫

휋

0

d휃푅^2 sin휃= 4 휋푅^2 , (2.23)

as expected.
In Euclidean three-space parallel lines of infinite length never cross, but this could
not be proved in Euclidean geometry, so it had to be asserted without proof, theparallel
axiom. The two-sphere belongs to the class of Riemannian curved spaces which are
locally flat: a small portion of the surface can be approximated by its tangential plane.
Lines in this plane which are parallel locally do cross when extended far enough, as
required for geodesics on the surface of a sphere.

Gaussian Curvature. The deviation of a curved surface from flatness can also be
determined from the length of the circumference of a circle. Choose a point ‘P’ on
the surface and draw the locus corresponding to a fixed distance푠from that point. If
the surface is flat, a plane, the locus is a circle and푠is its radius. On a two-sphere of
radius푅the locus is also a circle, see Figure 2.2, but the distance푠is measured along
a geodesic. The angle subtended by푠at the center of the sphere is푠∕푅, so the radius
of the circle is푟=푅sin(푠∕푅). Its circumference is then

퐶= 2 휋푅sin(푠∕푅)= 2 휋푠

(

1 − 푠

2

6 푅^2

+···

)

. (2.24)

Carl Friedrich Gauss (1777–1855) discovered an invariant characterizing the cur-
vature of two-surfaces, theGaussian curvature퐾. Although퐾can be given by a com-
pletely general formula independent of the coordinate system (see, e.g., [1]), it is most
simply described in an orthogonal system푥,푦. Let the radius of curvature along the
푥-axis be푅푥(푥)and along the푦-axis be푅푦(푦). Then the Gaussian curvature at the point
(푥 0 ,푦 0 )is

퐾= 1 ∕푅푥(푥 0 )푅푦(푦 0 ). (2.25)

On a two-sphere푅푥=푅푦=푅,so퐾=푅−^2 everywhere. Inserting this into Equa-
tion (2.24) we obtain, in the limit of small푠,

퐾=^3

휋

lim

푠→ 0

(

2 휋푠−퐶

푠^3

)

. (2.26)

This expression is true for any two-surface, and it is in fact the only invariant that can
be defined.
Whether we live in three or more dimensions, and whether our space is flat or
curved, is really a physically testable property of space. Gauss actually proceeded to
investigate this by measuring the angles in a triangle formed by three distant moun-
tain peaks. If space were Euclidean the value would be 180∘, but if the surface had