While reading the paper "Disorder horizons: Holography of randomly disordered fixed points" by Hartnoll and Santos, I came across this:

We are interested in solutions with a zero temperature horizon. This not only means that there is one time-like Killing vector field $\partial_t$, but also that the norm of $\partial_t$ must vanish at least quadratically at the extremal horizon.

What is a zero temperature horizon and how is it related to the time-like Killing vector field?

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    $\begingroup$ I haven't read the paper but I suspect this refers to the link between surface gravity and temperature as discussed in Derivation for the temperature of Reissner-Nordström (charged) black hole $\endgroup$ – John Rennie Mar 30 '16 at 17:01
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    $\begingroup$ As John Rennie mentions temperature of a null horizon is defined as the surface gravity of the horizon divided by $2\pi$. The surface gravity itself is related to the first normal derivative of the norm of the time like killing vector that generates the horizon. If this vanishes then the first non-trivial term that appears in the near Horizon expansion of the norm must be quadratic $\endgroup$ – Prahar Mar 31 '16 at 5:39

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