Causal angular velocity bounds in Kerr spacetime

Outside a subextremal Kerr black hole, regular causal curves obey $(-g_{t\varphi }-\sqrt{D})/g_{\varphi \varphi }\le d\varphi /dt\le (-g_{t\varphi }+\sqrt{D})/g_{\varphi \varphi }$, where $D=g_{t\varphi }^2-g_{tt}{g}_{\varphi \varphi }>0$. Positive radial and polar metric terms establish this necessary bound. In the open Kerr ergoregion both endpoints are positive for $a>0$, forcing co-rotation.