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I Computational physics - Light trajectory near black hole

  1. Nov 12, 2017 #1
    Dear all,

    I am currently doing a project about the light trajectory near Schwarzschild black hole. I wrote down a couple of differential equation and I have created a C++ program hoping to solve the orbit of light. However, the program results turn out to be quite weird.

    The differential equation needed to be solved:
    upload_2017-11-12_15-25-58.png

    Where r is the radial position and phi is the azimuthal angle, L is the angular momentum of the photon.

    The first thing I noticed is that sometimes, depending on initial conditions, the results sometimes turns out to be negative in r, even if r is positive, the change in r is so small that it remains in order of 10^-5, and the change in phi is also small. Nevertheless the program always produce diverging results in r. I think I may be misunderstanding the physical picture of the equation.

    I would like to ask does the equation describing the motion of a photon launching at a arbitrary position? Furthermore, what is the typical value of L in the Geometrized unit system? Furthermore, I know that the impact parameter b is related to L and energy of photon e by b=L/e, is there any ways to simplify the above differential equation so that I could get a differential equation only depends on b? Thank you!
     
  2. jcsd
  3. Nov 12, 2017 #2

    Ibix

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    See https://www.physicsforums.com/threads/null-geodesics-in-schwarzschild-spacetime.895174/

    Note that my initial post is incorrect - do read all the way down. As you say, you can write L in terms of E, which is the photon energy at infinity (which has no effect on the trajectory - it just scales ##\lambda##), and the angle at which it is launched.
     
  4. Nov 12, 2017 #3

    mfb

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    I'm not sure if the equations are fully correct but they should reproduce all the interesting features of the motion of light, so it looks like they are right.
    I don't think "typical value" is a useful concept for the actual angular momentum (a photon with a different energy will follow the same trajectory while having a different angular momentum), and in the way it appears in the equations it looks like an overall scaling factor only. To get different trajectories, start with different r and/or r'.
     
  5. Nov 12, 2017 #4
    Thanks for the reply Ibix, I have read through the passage, I think I understand how the initial conditions on L could be altered. In the post, you stated that E is just a scaling factor which do not affect the orbit, and that the ratio of L/E is the one that is important. Also, George also state that it is easier to divide the equation into two coupled differential equation, combining results, I get the following:
    upload_2017-11-12_18-16-10.png
    Where theta is the initial value of launching angle, R0 is the initial radial position from the center

    Where I could substitute L/E back into the ode, and since E is just a scaling factor, Could I just set E=1? Thanks!


    Thanks for the reply mfb. Indeed, as stated in the book of Hartle, the trajectory of photon could only depends on the ratio L/E but not L or E individually, so I was very confused when I saw the factor L in the differential equation in the op. But after getting the advise from Ibix, I think I could get rid of it, do you think this modification is reasonable? Thanks!
     
  6. Nov 12, 2017 #5

    mfb

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    Now you have L/E and E in the equations. I think you have to rescale ##\lambda## to get rid of it.
     
  7. Nov 12, 2017 #6
    Thanks for the relpy mfb. I think I can get rid of L/E by definition(the 3rd row),but I am not able to get rid of E in this case.
     
  8. Nov 12, 2017 #7

    mfb

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    Well, the third row gives you L/E as function of the initial conditions.
    Changing E shouldn't change the photon trajectory, so you can probably set it to whatever you like.
     
  9. Nov 13, 2017 #8

    Ibix

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    I regarded E as a scale factor, basically. Changing it can be countered by changing the affine parameter by the same factor, which tells you it doesn't matter. The point about L/E is that it is simply a measure of what fraction of the light's momentum is pointing in the tangential direction. In flat space it would simply be ##\sin\theta## times appropriate constants for the units (note that in my thread I used ##\psi## for this angle because ##\theta## is already in use as a coordinate). In curved spacetime, however, you also need an extra factor to relate local angle measures to the coordinates, which is where the square root prefactor comes in.
     
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