Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 15

x

y

z

a

r′

φ

Fig. 1.4.

Substituting this expression into the three-dimensional Euclidean metric,

dl^2 =dx^2 +dy^2 +dz^2 , (1.29)

gives

dl^2 =dx^2 +dy^2 +

(xdx+ydy)^2

a^2 −x^2 −y^2

. (1.30)

In this way, the distance between a pair of points located on the 2-sphere is expressed
entirely in terms of two independent coordinatesxandy, which are bounded,
x^2 +y^2 ≤a^2. These coordinates, however, are degenerate in the sense that to every
given (x,y) there correspond two different points on the sphere located in the
northern and southern hemispheres. It is convenient to introduce instead ofxand
ythe angular coordinatesr′,φdefined in the standard way:

x=r′cosφ,y=r′sinφ. (1.31)

Differentiating the relationx^2 +y^2 =r′^2 ,wehave

xdx+ydy=r′dr′.

Combining this with

dx^2 +dy^2 =dr′^2 +r′^2 dφ^2 ,