Tuesday, 20 August 2019

general relativity - Difference between coordinate and proper distance in Schwarzschild geometry


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.


Schwarzschild radial distances



The more I think about this the more confusing I find it.


Thank you




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