# Time dilation at the ISCO

1. Dec 6, 2015

### Neutrinos02

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:

$$d \tau = \sqrt{g_{\mu \nu} \dot{x^{\mu}} \dot{x^{\nu}}}$$

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

2. Dec 6, 2015

### bcrowell

Staff Emeritus
3. Dec 8, 2015