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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 ).
    Thanks
     
    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
    Last edited by a moderator: Apr 24, 2017
  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.
     
    Last edited by a moderator: Apr 24, 2017
  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

    A.T.

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  10. Oct 25, 2009 #9
    Last edited by a moderator: May 4, 2017
  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
    Thanks
     
  17. Nov 1, 2009 #16

    George Jones

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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
  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

    Nabeshin

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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?
     
  22. Nov 20, 2009 #21

    bcrowell

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    Here is some python code I wrote that does pretty much what you're talking about. It's meant to be short and easy to understand, so it uses a pretty crude method of doing the numerical integration. If you want to do accurate numerical calculations of geodesics, you'd want to substitute a better integration method. There are various general-purpose subroutines out there, e.g., in the book Numerical Recipes in C. What my code does is to calculate the deflection of a light ray that grazes the sun. Actually, it calculates half he deflection for a ray that grazes the sun, with the mass of the sun scaled up by a factor of 1000 in order to keep the result from being overwhelmed by rounding errors in my el-cheapo integration method.

    Code (Text):
    import math

    # constants, in SI units:
    G = 6.67e-11         # gravitational constant
    c = 3.00e8           # speed of light
    m_kg = 1.99e30       # mass of sun
    r_m = 6.96e8         # radius of sun

    # From now on, all calculations are in units of the
    # radius of the sun.

    # mass of sun, in units of the radius of the sun:
    m_sun = (G/c**2)*(m_kg/r_m)
    m = 1000.*m_sun

    # Start at point of closest approach.
    # initial position:
    t=0
    r=1 # closest approach, grazing the sun's surface
    phi=-math.pi/2
    # initial derivatives of coordinates w.r.t. lambda
    vr = 0
    vt = 1
    vphi = math.sqrt((1.-2.*m/r)/r**2)*vt # gives ds=0, lightlike

    l = 0    # affine parameter lambda
    l_max = 20000.
    epsilon = 1e-6 # controls how fast lambda varies
    while l<l_max:
      dl = epsilon*(1.+r**2) # giant steps when farther out
      l = l+dl
      # Christoffel symbols:
      Gttr = m/(r**2-2*m*r)
      Grtt = m/r**2-2*m**2/r**3
      Grrr = -m/(r**2-2*m*r)
      Grphiphi = -r+2*m
      Gphirphi = 1/r
      # second derivatives:
      #  The factors of 2 are because we have, e.g., G^a_{bc}=G^a_{cb}
      at   = -2.*Gttr*vt*vr
      ar   = -(Grtt*vt*vt + Grrr*vr*vr + Grphiphi*vphi*vphi)
      aphi = -2.*Gphirphi*vr*vphi
      # update velocity:
      vt = vt + dl*at
      vr = vr + dl*ar
      vphi = vphi + dl*aphi
      # update position:
      r = r + vr*dl
      t = t + vt*dl
      phi = phi + vphi*dl

    # Direction of propagation, approximated in asymptotically flat coords.
    # First, differentiate (x,y)=(r cos phi,r sin phi) to get vx and vy:
    vx = vr*math.cos(phi)-r*math.sin(phi)*vphi
    vy = vr*math.sin(phi)+r*math.cos(phi)*vphi
    prop = math.atan2(vy,vx) # inverse tan of vy/vx, in the proper quadrant
    prop_sec = prop*180.*3600/math.pi
    print "final direction of propagation = %6.2f arc-seconds" % prop_sec
     
  23. Nov 20, 2009 #22
    I actually have the numerical recipes book next to me although I may avoid solving the elliptic integral https://www.physicsforums.com/showpost.php?p=2409536&postcount=7", and just use the approximation for light deflection, i.e. 4GM/bc[tex]^{2}[/tex], to get the einstein ring effect, and focus on the trajectory of the observer descending into the black hole.
     
    Last edited by a moderator: Apr 24, 2017
  24. Nov 20, 2009 #23

    bcrowell

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    Keep in mind that 4GM/bc2 is only a weak-field approximation. It won't give you the right answer if you're close to the black hole.
     
    Last edited by a moderator: Apr 24, 2017
  25. Nov 21, 2009 #24

    George Jones

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    No, this isn't quite right. Let's start with the left side. What is

    [tex]\frac{d}{d\lambda} \left[ \left( \frac{dr}{d \lambda} \right)^2 \right] ?[/tex]
     
    Last edited: Nov 21, 2009
  26. Nov 21, 2009 #25
    Given

    [tex]\left( \frac{dr}{d \lambda} \right)^2 &= E^2 - V^2(r) \right)[/tex]
    and
    [tex]V^2(r) = \left(1 - \frac{2M}{r} \right)\frac{L^2}{r^2}[/tex]

    I have [tex] \frac{d^2r}{d\lambda^2} = -\frac{1}{2}\frac{d}{dr}V^2(r)[/tex]
     
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