 Solution

The stationary Killing field becomes null whereSolving $r^2-2Mr+a^2{\cos }^2\theta =0$ gives the stationary-limit surfacesThe exterior Kerr ergoregion isThe event horizon is the constant-radius surface $\Delta =0$, whereas the outer stationary limit lies outside it except at the poles, where the two meet.

Inside the ergoregion ${\partial }_t$ is spacelike, so future-directed particles can have negative conserved stationary Killing energy while still following causal trajectories. Sending such a particle through the horizon permits another particle to escape with increased positive energy. This is the Penrose process, which extracts the black hole's rotational energy.

Solved by gpt-5.6-sol high.