Sorry I don't know latex so this may look a little messy.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I'm trying to solve the equation for null geodesics of light travelling from a rotating black hole accretion disk to an observer at r = infinity. The point of emission for each photon is given by co-ordinates r, phi where r is radial distance from centre of the black hole, phi is azimuthal angle around the accretion disk (phi = 0 is defined to be the tangent point). The problem is stated as follows:

"Light travels on null geodesics given by the solution of the equation

d^{2}u/dphi^{2}= 3u^{2}- u

where u = 1/r. The full paths can be found by integrating this from u=1/r_{em}, phi_{em}to u = 0 (r=infinity), phi = 0. This requires varying the initial gradient (du/dphi)_{em}= - u_{em}tanE until the correct solution is found for an angle E = E' + theta, where E' is the 'straight line' angle, and theta is the additional deflection from lightbending as the photon travels from r_{em}to infinity. Explore the size of theta to estimate where the straight line approximation may break down."

I've also been told that to solve the equation I need to split it into two 1st order ODEs, but I'm not sure how to do that.

2. Relevant equations

3. The attempt at a solution

I'm really struggling just to try and understand the description, let alone solve the equation. Please could someone explain to me what this means and how I can extract the light paths from the given equations?

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# Homework Help: Null geodesics of light from a black hole accretion disk

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