I'm trying to understand proper distance equation in Schwarzschild spacetime.
I'm sure I'm missing something really obvious here, but how do I use this to find the coordinate distance $r$ for a particular proper distance $\sigma$ . For example, if I found the proper circumference of circle going round the Sun that roughly coincides with the Earth's orbit. Then I move radially inwards one proper mile, how would I then find the circumference of the circle I now find myself on. This example is also in the context of trying to understand the spacing of concentric circles in embedding diagrams.