= Euclidean temperature and signed horizon surface gravity
{c}
{title2=$T=|\kappa|/(2\pi)$}
For a normalized static metric $ds^2=-Vdt^2+dr^2/V+r^2d\Omega^2$ with a simple outer zero at $r_0$, set $b=V'(r_0)>0$. The <Wick rotation> and $r-r_0=b\rho^2/4$ give $d\rho^2+(b\rho/2)^2dt_E^2$ to leading order, so smoothness fixes period $4\pi/b$. In an ingoing null chart, $\nabla_k k=(b/2)k$ on the future horizon; the outgoing chart gives the opposite sign on the past branch. The <Hawking temperature> is $b/(4\pi)=|\kappa|/(2\pi)$ and uses the time generator normalized at infinity.
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