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Time dilation at the ISCO

  1. Dec 6, 2015 #1
    Hello,

    I like to calculate the time dilatation at the ISCO of a Kerr black hole:

    According to general relativity the time dilation is given by following formular:

    [tex]d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}[/tex]

    If I'm interestet in the time dilation at the ISCO I set [tex]\Theta = \frac{\pi}{2} , dr=0[/tex] so I get:

    [tex]d\tau = \sqrt{g_{tt} + g_{\phi t} \dot{\phi} + g_{\phi \phi} \dot{\phi}^2} dt[/tex]

    But now I need [tex]\dot{\phi}[/tex] at the ISCO of a Kerr black hole but I only know the angular momentum:

    [tex]L= \frac{2mMar}{\rho²}sin²(\theta) \frac{dt}{d\tau}- \frac{m(r²+a²)²-m \Delta a²sin²(\theta)}{\rho²}sin²(\theta) \frac{d \phi}{d\tau}[/tex]

    which includes only the derivation with respect to τ not t.

    So how is it possible to calculate the angular velocity?

    Thanks
    Neutrino
     
  2. jcsd
  3. Dec 6, 2015 #2

    bcrowell

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  4. Dec 8, 2015 #3
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