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I need to calculate the apparent speed of a beam of light as it passes along a defined path near a massive object, as observed by a distant observer.

For this, can I use the following equation derived from the Schwarzschild metric, $$ d\tau^2 = g_{00}dt^2 +\frac{1}{c^2}(g_{11}dr^2 + g_{22}d\theta^2 + g_{33}d\phi^2) $$ by setting $d\tau^2=0$ , as the proper time for a beam of light is always zero?

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Yes you are on the right way!

The Schwarzschild metric describes spacetime outside some spherical object, which I assume corresponds to your case. And since you consider the motion of light you have a null line element so that $d\tau^2 = 0$.

Then, as a hint to continue your exercise, you can consider a radial motion (which means $\theta=constant$ and $\phi=constant$) and remember that the coordinate velocity is given by $$\frac{dr}{dt}=\frac{\frac{dr}{d\tau}}{\frac{dt}{d\tau}}$$

Good luck

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