# Physical significance of angular velocity of orbits around Kerr black holes

For the Kerr metric $$ds^2=\left(g_{tt}-\frac{g_{t\phi}^2}{g_{\phi\phi}}\right)dt^2+g_{\phi\phi}\left(d\phi-\omega dt\right)^2+g_{rr}dr^2+g_{\theta\theta}d\theta^2$$ the angular momentum is defined as $$\Omega=\frac{u^\phi}{u^t}=-\frac{g_{t\phi}+lg_{tt}}{g_{\phi\phi}+lg_{t\phi}}$$ where $$l$$ is the specific angular momentum.

However, in the metric we have a term $$\omega$$ defined as $$\omega=\frac{\dot{\phi}}{\dot{t}}=-\frac{g_{t\phi}}{g_{\phi\phi}}$$

Observing the two expressions for $$\Omega$$ and $$\omega$$, we can see that when $$l=0$$, then $$\Omega=\omega$$. These expressions are mainly used in the study of orbits in Kerr geometry.

$$\textbf{Question 1: }$$ So I could not understand the difference between these two terms, $$\Omega$$ and $$\omega$$. What are their physical significance?

$$\textbf{Question 2: }$$ What does the specific angular momentum $$l$$ mean?

• $$\Omega$$ is the angular velocity of a (general) geodesic orbit, where $$l$$ is the ratio of the angular momentum and energy of the orbit. (Both of which are constants of motion).
• $$\omega$$ is the angular velocity of a Zero Angular Momentum Observer (ZAMO) orbiting the Kerr black hole. That is, it is the angular momentum of a particular type of geodesic orbit, namely those with $$l=0$$. Particularly, it is the limit of $$\Omega$$ as $$l$$ goes to zero. The limit of $$\omega$$ as $$r$$ goes to the outer horizon $$r_{+}$$ is usually interpreted as the angular velocity of the horizon.