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Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial Riemannian metric $$\mathrm{d}s^2=\left(1-\frac{r_s}{r}\right)^{-1}\mathrm{d}r^2+r^2\left(\mathrm{d}\theta^2+\sin^2\theta \,\mathrm{d}\varphi^2 \right)$$

Now I would like to write down the set points of my perfect rectangle. But I need some help. Although I asked about the notion of angles in curved space I still feel unsure to find the set points of the rectangle. Of course the position of the rectangle shall be such that it is easy to solve the problem.

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  • $\begingroup$ Can you clarify what you're asking? Do you want your rectangle to have sides that are geodesics, or are you asking what the coordinates of a rectangle constructed in flat space would be in curved space? $\endgroup$ – John Rennie Apr 15 '14 at 14:53

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