Ergosphere

Remarks

At a point of the ergoregion, $\xi$ is spacelike, so no observer, whose world line is timelike, can follow an orbit of $\xi$. Every observer there must rotate relative to infinity.

Example: in the Kerr metric of mass $M$ and spin $a$ with $0 < \abs{a} < M$, in Boyer–Lindquist coordinates, $$g(\partial_t, \partial_t) = -\left(1 - \frac{2Mr}{r^2 + a^2 \cos^2\theta}\right).$$ This is positive exactly when $r^2 - 2Mr + a^2\cos^2\theta < 0$, that is when $r < M + \sqrt{M^2 - a^2 \cos^2\theta}$. The horizon is at $r_+ = M + \sqrt{M^2 - a^2}$. So the ergoregion is $r_+ < r < M + \sqrt{M^2 - a^2\cos^2\theta}$, which is nonempty for $\theta \ne 0, \pi$ and reaches $r = 2M$ at the equator.

Non-example: for the Schwarzschild metric ($a = 0$), $g(\partial_t, \partial_t) = -(1 - 2M/r) < 0$ for all $r > 2M$. The set where $\xi$ is spacelike lies inside the horizon, so the ergoregion is empty.