I am reading a cosmology textbook, and the distance metrics for three dimensional spaces exhibiting various curvatures are being presented. My question is about their treatment of a three dimensional space under unifom positive curvature:
In polar coordinates, on the two dimensional surface of a sphere, we can express the distance $d\ell$ between two points as a function in their separation in the radial coordinate $r$, and
$d\ell^2 = dr^2 + R^2\sin^2(r/R)d\theta^2$
In three dimensions, this extends to
$d\ell^2 = dr^2 + R^2\sin^2(r/R)[d\theta^2 + \sin^2\theta d\phi^2]$.
Now, my texbook asserts that when the two points whose separation we are measuring are at antipolar locations, we have $r = \pi R \rightarrow r/R=\pi$, which gives
$sin^2(r/R) = 0\rightarrow d\ell^2 = dr^2$.
But this makes no sense to me. This isn't how spherical coordinates work at all, right...? If I have two point at antipolar points on a sphere, and I measure each of their $r$ coordinates (the length to the point along a line forming angles $\theta$ and $\phi$ from the $z$ and $x$ axes, respectively) as $r_1$ and $r_2$, then shoudln't their separation $dr = |r_1-r_2|$ simply be $2R$? This would mean that $r/R = 2 \neq \pi$
Claiming that $r/R = \pi$ implies that $r$ refers to the actual path length between the points along the surface of the sphere, which is not how spherical coordinates work.
Is my error in applying these spherical coordinates to a two dimensional sphere surface, when I am supposed to be thinking about the three dimensional surface under uniform positive curvature? I.e. the image I have in my head of how spherical coordinates work in this context is all wrong, or at least my placement of the points?
I am picturing a "three dimensional space under uniform positive curvature" as a sphere, like the Earth. But that isn't right, is it? That is just me imagining a 2D surface being curved into a third dimension, when the more accurate analog is to somehow imagine a 3D space being "curved" into a fourth dimension?