Kerr black hole

The Kerr black hole is the asymptotically flat stationary axisymmetric vacuum black hole with mass $M$ and angular momentum $J=Ma$.

Boyer-Lindquist coordinates adapt the Kerr metric to its stationary and axial symmetries. The horizon radii are the roots $r_{\pm }=M\pm \sqrt{M^2-a^2}$ of $\Delta =r^2-2Mr+a^2$.

The Kerr ergoregion is the region where the asymptotically timelike stationary Killing field becomes spacelike. Its outer boundary is $r=M+\sqrt{M^2-a^2{\cos }^2\theta }$.

The Penrose process extracts rotational energy by allowing a fragment with negative stationary Killing energy to enter the black hole while another fragment escapes with more energy than the original body carried.

On the Kerr symmetry axis, the radial metric has $F=\Delta /(r^2+a^2)$ and the tortoise coordinate satisfies $d{r}_{\ast }/dr=F^{-1}=(r^2+a^2)/\Delta$.