What are extremal Kerr Killing vectors up to?

TL;DR

This paper investigates the properties of horizon Killing vectors in extremal Kerr spacetime, explaining their behavior and relation to nearby spacetimes with similar hypersurfaces.

Contribution

It clarifies how the null and spacelike nature of horizon Killing vectors in extremal Kerr spacetime arise and relate to the existence of timelike Killing vectors off the horizon.

Findings

Horizon Killing vector is null on a hypersurface crossing the horizon.

Killing vector is spacelike on both sides of the horizon in the equatorial plane.

Behavior is consistent with the existence of timelike Killing vectors off the horizon.

Abstract

In the extremal Kerr spacetime the horizon Killing vector field is null on a timelike hypersurface crossing the horizon at a fixed latitude, and spacelike on both sides of the horizon in the equatorial plane. We explain in some detail how this behaviour is consistent with the existence of timelike Killing vectors everywhere off the horizon, and how it arises in a limit from the Kerr spacetime where there is a similar hypersurface strictly outside the horizon.