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?


1 Answer 1

  • $\Omega$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.