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Light ray paths near schwarzschild blackhole

  1. Oct 22, 2009 #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 ).
    Last edited: Oct 23, 2009
  2. jcsd
  3. Oct 24, 2009 #2
    Anyone have any advice?
    I'm currently going over lagrangian mechanics
  4. Oct 24, 2009 #3
    You might start with this paper.
  5. Oct 24, 2009 #4
  6. Oct 24, 2009 #5
    Thanks for the paper, all the others I have read from arxiv have not been very non-physicist friendly.
  7. Oct 25, 2009 #6

    George Jones

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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.
  8. Oct 25, 2009 #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
  9. Oct 25, 2009 #8


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  10. Oct 25, 2009 #9
    Last edited: Oct 25, 2009
  11. Oct 25, 2009 #10

    George Jones

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  12. Oct 25, 2009 #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.
  13. Oct 25, 2009 #12

    George Jones

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    Actually, my Exercise 6 is about lightlike (null) geodesics, which is what you want.
  14. Oct 25, 2009 #13
    George, it is wonderful.
    Do you know any animations about what observer would see inside the second horizon, near loop singularity?
  15. Oct 26, 2009 #14
    Would I still use geodesics for photon paths in polar coordinates?
  16. Nov 1, 2009 #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
  17. Nov 1, 2009 #16

    George Jones

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

    \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), \\

    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
  18. Nov 1, 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]
  19. Nov 4, 2009 #18
    what does W(r) define?
  20. Nov 4, 2009 #19


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    W(r) should be the effective potential of the system.
  21. Nov 20, 2009 #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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