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Frequency of Photon in Schwarzschild Metric?

  1. May 16, 2015 #1
    1. The problem statement, all variables and given/known data

    The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##.

    (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it is at ##r##, ##\omega_r##.
    (b) Find the range of frequencies.

    2007_B5_Q3.png
    2. Relevant equations


    3. The attempt at a solution
    I have found the geodesic equations:
    [tex]\frac{d}{ds}\left[A \dot t \right] = 0 [/tex]
    [tex]0 = -Ac^2 \dot {t}^2 + \frac{1}{A} \dot {r}^2 + r^2 \dot {\phi}^2 [/tex]
    [tex]\frac{d}{ds} \left[ r^2 \dot \phi \right] = 0 [/tex]

    I have also found the christoffel symbols: ##\Gamma^r_{rr}=-\frac{A'}{2A}## and ##\Gamma^r_{tt}=-\frac{1}{2}A A'c^2## and ##\Gamma^r_{\theta \theta} = -Ar## and ##\Gamma^r_{\phi\phi}=-Ar## and ##\Gamma^t_{rt}=\Gamma^r_{tr}=\frac{A'}{2A}##.

    Part(a)
    The differential equation is simply the geodesic equation for a photon:
    [tex]\dot p^\mu + \frac{1}{\hbar} \Gamma^\mu_{\alpha \beta} p^\alpha p^\beta = 0 [/tex]
    For the temporal component we have exactly what the question wants:
    [tex]\dot p^0 = - \frac{A'}{A}p^0 \dot r [/tex]

    The energy of the photon is given by ##E = p_\mu U^\mu =p_0 U^0+p_1 U^1+p_2 U^2+p_3 U^3 = \hbar \omega##.

    How do I find ##p^\mu## and ##U^\mu##?
     
  2. jcsd
  3. May 18, 2015 #2
  4. May 22, 2015 #3
    bumpp
     
  5. May 23, 2015 #4
  6. May 25, 2015 #5
    bumppp
     
  7. May 31, 2015 #6
    bumpp on redshift
     
  8. Jun 1, 2015 #7
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