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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

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# Time dilation at the ISCO

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