I'm trying to understand the difference between proper distance dσ 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.
Thank you
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