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I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself.

I just need equations which I can integrate for:

$$ r(\lambda) = \text{function(initial direction, initial position)} $$ $$ \theta(\lambda) = \text{function(initial direction, initial position)} $$ $$ \phi(\lambda) = \text{function(initial direction, initial position)} $$ $$ t(\lambda) = \text{function(initial direction, initial position)} $$

given the initial conditions of the light ray.

Surely this has been solved before?

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You can find a treatment of this topic in any book on GR, in particular I liked Schutz's book, where he does this in chapter 11. IIRC, the idea is to recognize and exploit the constants of motion. –  dayareishq Jun 2 '13 at 2:55
    
Thanks, I will look into it, but immediately, is there a way you could write the equations in an answer? –  user912 Jun 2 '13 at 3:14
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1 Answer

As requested in the comments by the user, I will write down the differential equation satisfied by the trajectory of a photon that travels along a radial path (no $\theta$ or $\phi$) dependence. By properties of the Schwarszchild metric, $p_0 = p_t$ and $p_{\phi}$ are conserved quantities along the path followed. Then define $-p_0 = E$ and $p_{\phi} = L$. Then by using $\overrightarrow{p}.\overrightarrow{p} = 0,$ we get the equation

$$(\frac{dr}{d\lambda})^2 = E^2 - (1-\frac{2M}{r})\frac{L^2}{r^2}.$$

I (along with Schutz) am using units where $c,G =1$

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Where would I input the initial conditions? And does this apply to an arbitrary direction (not necessarily a radial path)? –  user912 Jun 2 '13 at 3:56
    
@user912: you're going to have to input something yourself. Do you know anything of the Lagrange formulation? –  Jerry Schirmer Nov 29 '13 at 5:53
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