Null geodesics of light from a black hole accretion disk

[Favicon]

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

Homework Statement

I'm trying to solve the equation for null geodesics of light traveling 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

d²u/dphi² = 3u² - 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_emtanE 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?

 

[George Jones]

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

I learned latex when I joined Physics Forums. See

https://www.physicsforums.com/showthread.php?t=8997.

I'm trying to solve the equation for null geodesics of light traveling from a rotating black hole

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

d²u/dphi² = 3u² - u

where u = 1/r.

But this equation is for null geodesics of non-rotating black holes. Is this what you want to do?

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.

This second-order equation can be reduced to a pair of first-order equations by setting [itex]p = du/d\phi[/itex], so that [itex]dp/d\phi = d^2 u/d\phi^2[/itex]. Consequently,
the second-order equation is eqiuvalent to

[tex]
\begin{equation*}
\begin{split}
\frac{du}{d\phi} &= p \\
\frac{dp}{d \phi} &= 3u^2 - u.\\
\end{split}
\end{equation*}
[/tex]

 

[Favicon]

Thanks for the latex tip.

But this equation is for null geodesics of non-rotating black holes. Is this what you want to do?

Yes it probably is a non-rotating black hole. I'm actually writing a program to produce the expected line spectrum from a black hole, but the description I've been given (to explain the physics of relativistic line smearing) isn't very clear so when it talked about the motion of the accretion disk I assumed it meant the black hole itself was rotating.

[tex]\frac{du}{d\phi} = p[/tex]
[tex]\frac{dp}{d\phi} = 3u^{2} - u[/tex]

So now I have
[tex]\frac{du}{d\phi} = p_{em} = -u_{em}tan(E)[/tex]
and have to vary E? But I still can't see how I'll know when I've found the correct value for E.