Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$? 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 $\theta=\frac{\pi}{2}$.
Thanks.
You can check (explicitly) that $\theta=\pi/2$ satisfies that equation for $\theta$. Since the solution for the given initial conditions of a geodesic should be unique, it must be that $\theta=\pi/2$ over the entire geodesic. (So the geodesic will necessarily lie in a plane.)
If you have a geodesic with $\theta(0)\neq\pi/2$ or $\dot\theta(0)\neq0$, you can rotate your system of coordinates to get $\theta(0)=\pi/2$ and $\dot\theta(0)=0$, for which $\theta=\pi/2$ will then solve your equation of motion.