- #1
Neutrinos02
- 43
- 0
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 formula:
d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}
If I'm interestet in the time dilation at the ISCO I set \Theta = \frac{\pi}{2} , dr=0 so I get:
d\tau = \sqrt{g_{tt} + g_{\phi t} \dot{\phi} + g_{\phi \phi} \dot{\phi}^2} dt
But now I need \dot{\phi} at the ISCO of a Kerr black hole but I only know the angular momentum:
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}
which includes only the derivation with respect to τ not t.
So how is it possible to calculate the angular velocity?
Thanks
Neutrino
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 formula:
d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}
If I'm interestet in the time dilation at the ISCO I set \Theta = \frac{\pi}{2} , dr=0 so I get:
d\tau = \sqrt{g_{tt} + g_{\phi t} \dot{\phi} + g_{\phi \phi} \dot{\phi}^2} dt
But now I need \dot{\phi} at the ISCO of a Kerr black hole but I only know the angular momentum:
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}
which includes only the derivation with respect to τ not t.
So how is it possible to calculate the angular velocity?
Thanks
Neutrino