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Sorry I don't know latex so this may look a little messy.

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

where u = 1/r. The full paths can be found by integrating this from u=1/r

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.

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?

## Homework Statement

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}- uwhere 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.

## Homework Equations

## 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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