Light ray paths near schwarzschild blackhole

  1. hey there, i'm interested in (eventually) simulating light ray paths near black holes, starting with schwarzschild blackholes and working my way to kerr-newman blackholes.
    I have a good understanding of the nature of blackholes but have trouble when it comes to the equations.
    My background is in computer science and I was wondering if anyone here could put a non-physicist on the right track to plotting the trajectory of a light ray near a black hole. ( in 2d for now ).
    Thanks
     
    Last edited: Oct 23, 2009
  2. jcsd
  3. Anyone have any advice?
    I'm currently going over lagrangian mechanics
     
  4. You might start with this paper.
     
  5. Thanks for the paper, all the others I have read from arxiv have not been very non-physicist friendly.
     
  6. George Jones

    George Jones 6,358
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    Are you interested in what an observer would see with his eyes, or in plotting the trajectory of a photon on a coordinate map, or maybe both? The first thing is somewhat hard, and the second thing is easier.

    Let me use an example to illustrate what I mean. Suppose a ship makes a long ocean voyage. The voyage could be watched from a telescope on a satellite in geosynchronous orbit, or the voyage could be plotted as a moving, glowing dot on a map.
     
  7. My end goal will be to render what an observer would see , but for now I would like to simply plot the trajectory of several photons on a coordinate map.

    I've been studying geodesics (not with much luck) and reading up on equations for the impact parameter and deflection angle in the schwarzschild metric which I have as :
    [tex]\Delta\phi = \int^{r_{observed}}_{r_{omitted}} \stackrel{dr}{r\sqrt{r^{2}/b^{2} - 1 + R_{s}/r}} [/tex]
     
    Last edited: Oct 25, 2009
  8. A.T.

    A.T. 5,302
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  9. Last edited: Oct 25, 2009
  10. George Jones

    George Jones 6,358
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  11. Yes something like this is exactly what i'm aiming for at the moment. I noticed you focused on timelike geodesics for the app. Am I right in thinking that I would be using null geodesics for photon trajectories and timelike for the trajectory of an observer.
     
  12. George Jones

    George Jones 6,358
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    Actually, my Exercise 6 is about lightlike (null) geodesics, which is what you want.
     
  13. George, it is wonderful.
    Do you know any animations about what observer would see inside the second horizon, near loop singularity?
     
  14. Would I still use geodesics for photon paths in polar coordinates?
     
  15. Could you give me some more information on plotting the trajectory on a coordinate map? I've been toying with equations for a while now and not making much progress
    Thanks
     
  16. George Jones

    George Jones 6,358
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    Yes, exactly. The geodesic equation when [itex]\theta = \pi /2[/itex] then is

    [tex]\begin{equation*}
    \begin{split}
    \frac{d \phi}{d \lambda} &= \frac{L}{r^2} \\
    \left( \frac{dr}{d \lambda} \right)^2 &= E^2 - L^2 W \left( r \left( \lambda \right) \right), \\
    \end{split}
    \end{equation*}
    [/tex]

    where [itex]W \left( r \right)[/itex] is function that I'll specify later, and [itex]E[/itex] and [itex]L[/itex] are constants of motion.

    In its present form, the second equation is a little difficult to implement on a computer since sometimes a positive square root is needed (increasing [itex]r[/itex]) and sometimes a negative square root is needed (decreasing [itex]r[/itex]) in the same photon orbit. To get aorund this, differentiate the second equation with respect to the affine parameter [itex]\lambda[/itex] taking into account that the [itex]r[/itex] in [itex]W \left( r \right)[/itex] is itself a function of [itex]\lambda[/itex].

    What do you get for the second equation after this differentiation?
     
    Last edited: Nov 21, 2009
  17. Given it's been a while since i've done any differentiation, is it simply:

    [tex]\frac{dr}{d \lambda} \right) &= \frac{ E - L}{W \left( r \right)}[/tex]
     
  18. what does W(r) define?
     
  19. Nabeshin

    Nabeshin 2,200
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    W(r) should be the effective potential of the system.
     
  20. Could anyone expand on this please? I would appreciate the help.
    I have the the effective potential in the schwarzschild metric as (L being angular momentum)
    [tex]V_{eff} = ( 1 - \frac{r_{s}}{r})(mc^{2} + \frac{L^{2}}{mr^{2}})[/tex]

    Would this be enough information to solve for r and [tex]\Phi[/tex] so that I could plot the trajectories.

    Also could I use the same equation for plotting the course of an observer descending into the black hole?
     
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