= Kerr ergoregion
{c}
= Ergoregion
{synonym}
For a <Kerr black hole>, let $K=\partial_t$ be the stationary <Killing vector field> normalized at infinity. The exterior <Kerr ergoregion> is where $g(K,K)>0$, so the tangent to motion following $K$ alone would be a <spacelike vector>. A physical observer there cannot remain at fixed angular coordinates relative to infinity.
For nonzero spin its outer stationary-limit surface is
$$
r_e(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.
$$
It lies outside the outer <event horizon> except where they meet on the rotation axis. A <causal curve> can escape this exterior <Kerr ergoregion>; forced co-rotation does not make it the black-hole interior. The conserved <Killing energy> $E=-p\cdot K$ can be negative there, enabling the <Penrose process>, while locally measured future-particle <energy> remains positive.<General Relativity by David Tong>[Section 6.3.3, ergoregion and energy extraction]{ref}<Lecture Notes on General Relativity by Sean M. Carroll>[Section 7, equations (7.125)–(7.145), stationary limit and negative conserved energy]{ref}
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