I Confusion about killing horizon in Carroll

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In the opening paragraph of section 6.3 Carroll defines a killing horizon to be a null hypersurface Σ where some killing vector field χμ becomes null. Later (on page 247 if you have the book) when distinguishing between static and stationary space times, he says that in a stationary, but not static spacetime, we still have the killing field Kμ = (∂t)μ , but it won't become null at the killing horizon. How is that possible? By definition that seems wrong. How do you have a killing horizon if that's not where your killing field becomes null?

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