Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 3 / b / iii / Solution

In the open exterior Kerr ergoregion, $g_{tt}>0$ andSince $D=g_{t\varphi }^2-g_{tt}{g}_{\varphi \varphi }<g_{t\varphi }^2$,so even the lower permitted angular velocity is positive. Every regular causal trajectory in the ergoregion has $d\varphi /dt>0$: it must co-rotate with the Kerr black hole. At the outer stationary-limit surface $g_{tt}=0$, the lower bound is zero; strict positivity is asserted in the open ergoregion. The sign reverses if the orientation of $a$ is reversed.