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Homework Help: Impact parameter of a photon in Schwarzchild metric

  1. Oct 19, 2015 #1
    Hi, I'm having trouble answering Question 9.20 in Hobson's book (Link: http://tinyurl.com/pjsymtd). This asks to prove that a photon will just graze the surface of a massive sphere if the impact parameter is [tex]b = r(\frac{r}{r-2\mu})^\frac{1}{2}[/tex]

    So far I have used the geodeisic equations [tex](1-\frac{2\mu}{r})\dot{t} = k[/tex] and [tex]r^2\dot{\phi} = h[/tex] to give [tex]\frac{d\phi}{dt} = \frac{b(1-\frac{2\mu}{r})}{r^2}[/tex] and b = h/k due to the argument given here http://www.physicspages.com/2013/06/13/photon-equations-of-motion/

    This is extremely close to the actual result but I can't figure out why [tex]\frac{d\phi}{dt}=\frac{1}{b}[/tex].

    Any help? Thank you!
  2. jcsd
  3. Oct 24, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
  4. Oct 28, 2015 #3
    I solved it myself. The metric for lightlike separation implies [tex] g_{00}\dot{t}^2 +g_{11}\dot{r}^2+g_{22}\dot{\phi}^2 =0[/tex] and we have expressions for phi dot and t dot from the OP. Just plug them in and since the expression is true everywhere we evaluate it on the surface of the star i.e where motion is purely tangential -> r dot is zero. So we just arrange the above equation for b = h/k to get the required answer.
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