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Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem to consider this particular case... and some actually said "wlog". But why? I don't think the motion is necessarily confined to a plane?

Correct me if I'm wrong, but isn't the Euler-Lagrange equations for the coordinate $\theta$ $$\ddot\theta +\frac{2\dot r}{r}\dot\theta-\dot\phi\sin\theta\cos\theta=0?$$ I don't see why the motion can be "wlog" in $\theta=\frac{\pi}{2}$.

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The metric is spherically symmetric. This means that angular momentum of the system is conserved (you can show this directly using the metric by computing the three killing vectors associated with spatial rotation and their corresponding conserved quantities) and therefore that the motion is contained to lie in a plane. If the motion is in a given plane, then there always exist rotated coordinates in which that plane corresponds to the $\theta = \pi/2$ equatorial plane, so we lose no generality by originally assuming that the geodesic satisfies this constraint.

If you have the time/energy, I would highly recommend actually trying to work out the killing vectors associated with spatial rotations and to explicitly demonstrate what I claim amount angular momentum conservation. You could also do it using Noether's theorem. In my opinion, there's really nothing like getting your hands dirty when learning this stuff.

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