# Worldlines in Schwarzschild geometry

I have an observer and a photon on a hypersurface $\theta=\pi/2$ . My observer has $e, l$ constants of motion (energy and angular momentum divided by mass) and photon has $e',l'$. What conditions must these constants of motion satisfy for worldlines of a photon and an observer to meet at a point $r=r_0$ ($r_0$ is arbitrary point)? Schwarzschild metric in my notation is $g = -(1-\frac{2M}{r}) dt^2 + (1-\frac{2M}{r})^{-1} dr^2 + r^2 d\Omega^2$ I can write down equations for geodesics for the observer and the photon by using Lagrangian method and using that $ds^2 = 0$ for a photon and $ds^2=-1$ for the observer but I don't think I can solve them. Is there a simpler way? What am I missing?

• You should at least define everything so that your notation makes sense. – Danu Jan 31 '16 at 23:03
• I have edited my post. Is everything clear now? – Caims Jan 31 '16 at 23:09
• Could you please make your question a bit clearer like @Danu requested by defining your notation and quantities more explicitly? – user106422 Feb 10 '16 at 23:27