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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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Perhaps a question would sharpen your understanding: how do you propose to measure the radius? –  genneth Dec 19 '11 at 17:17
Well, if r was Euclidean I'd use a ruler. –  Peter4075 Dec 19 '11 at 17:31
That's kind of the point --- you couldn't use a ruler to measure $r$. Rulers would measure $\sigma$. –  genneth Dec 19 '11 at 18:33
@genneth - thanks for that. I'm assuming there's no problem in measuring r with my ruler in the "flat space" circles at the bottom of the diagram? –  Peter4075 Dec 19 '11 at 18:39
In the flat space, you can do that because $r = \sigma_\text{flat}$. But those measurements don't then automatically apply to the curved space. You would have to specify how points in the flat space correspond to points in the curved space, i.e. define a map from the flat space to the curved space, before you can transfer your measurements of $r$. In a sense, this corresponds to specifying the meaning of the dotted lines in your picture. –  David Z Dec 20 '11 at 21:00
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1 Answer

In the Schwartzchild coordinates the r co-ordinate is the value you get by dividing the circumference of the circle by 2$\pi$. That is, it's the radius of the circles you've projected onto the base of your diagram.

As others have mentioned in the comments, deciding what you mean by the "real" distance from the circle to the singularity is a vexed issue. Far away from the singularity r agrees with what we think of as the radius, but of course that's only because space is (nearly) flat far away from the singularity. Once the curvature becomes significant r will not be the same value as you get by integrating the metric from the singularity out to your circle.


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