The concepts of Distance on a 2D curved space are here extended
to a 3D space that might be curved in a 4D space. We can start
with the generic infinitesimal distance in a static 3D universe, but
this time not in spherical coordinates instead of polar coordinates.
\(\theta\) is shown in Figure 9.2, but here we are 3D
beings, positioned on \(O\) (the center of the sphere) and the
point \(O\) is tangent to a 4D-sphere. In our 3D space, a generic
infinitesimal displacement will have the distance:
$$ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2).$$Like
our 2D friend before, we now have to assume an abstract dimension
which we cannot visualize. Let’s call the fourth dimension \(w\),
then the general change in coordinates in the full four
dimensional space will be:
$$ds_s^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)+dw^2.$$But we
can only work on a 3D curved space, so following exactly the same
steps and conventions as our 2D friend, we arrive at:
$$ds_s^2={dr^2\over
1-Kr^2}+r^2(d\theta^2+\sin^2{\theta}d\phi^2).$$In a non-static universe
(with a scale factor a(t), the distance can be written as:
$$ds^2=c^2dt^2-a^2(t)[d\chi^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)].$$