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I am trying to determine the method to calculate proper distance with constant time and radius in Schwarzschild Geometry. With only $\theta$ and $\phi$ being variable. I think it involves integrating the line element (metric) using two variables but an example of this is hard to find. If I can find the integral; I can probably look up a solution in a table or use Maple?

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For a fixed $r$ and $t$, $$ds^2 = r^2(d\theta^2 +\sin^2\theta\ d\phi^2)\ ,$$ which is the same arc length as in Euclidean space. Then if $\phi = f(\theta)$ then $$s =r \int \left(1 + \sin^2\theta\ \left(\frac{df}{d\theta}\right)^2\right)^{1/2}\ d\theta\ .$$

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  • $\begingroup$ could (df/dtheta) be related to a geometric projection? I can see that it would be useful to reduce the # of variables (phi) for integration purposes. Can the line element be integrated including theta and phi? $\endgroup$ May 23 at 0:07

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