I'm trying to understand the difference between proper distance
$d\sigma$ and coordinate distance $dr$ in Schwarzschild
geometry. The bottom bit of the diagram represents flat space,
the upper bit curved space. The inner circles represent
Euclidean spheres of radius $r$, the outer circles radius
$r+dr$. Is the proper radius of these circles the same as $r$? I
think I mean if I measured the radius of these circles with a
real ruler would I get the coordinate distance $r$ of the
Schwarzschild metric. The more I think about this the more
confusing I find it.