Are Killing Horizon and Stationary Limit Surface the same?

[Elnur Hajiyev]

Yes, v is a vector field that is null on a timelike surface!

Let's consolidate two points made in the last few posts.

1) Even though a Killing horizon is null, a Killing horizon is not necessarily an event horizon. This is illustrated by the example given by martinbn, and by the example on page 245 of Carroll.

2) If a hypersurface is defined by a vector field being null, the hypersurface itself does not have to be null! This true for the vector field v cooked up by me above, and it is true for Killing vector field $\partial_t$ of Kerr spacetime. The stationary limit for Kerr is defined by $\partial_t$ being null, but this hypersurface is actually timilike! Consequently, the stationary limit is not a Killing horizon. On page 244, Carroll state this:

"In Kerr, the hypersurface on which $\partial_t$ becomes null is actually timelike, so is not a Killing horizon."

Thank you. Now I know stationary limit surface is not a killing horizon and verified the definition of a killing horizon thanks to you(have learned it doesn't have to be an event horizon). But now I am trying to understand, why is "killing horizon" so important? Beside being an event horizon in some metrics what does it mean geometrically(or physically or intuitively). Or was it killing horizon to mean "once crossed it is impossible to return" thing since beginning and when we move avay from Schwarzschild metric, begin to analyze Kerr metric, these notations have became different and this statement have not been true for event horizon? Did I get it right?