= Signed surface gravity of a de Sitter horizon
{title2=$\kappa_{\rm future}=1/R,\quad\kappa_{\rm past}=-1/R$}
For the future-directed central-time <Killing vector field> in a <static patch of de Sitter spacetime>, the outgoing null chart gives the future branch coefficient $+1/R$ in $\nabla_k k=\kappa k$. The ingoing chart gives $-1/R$ on the past branch. Both follow from $d(k^2)=-2\kappa k^\flat$; the cross-term sign changes the horizon covector. The physical <Hawking temperature> uses $|\kappa|$, not a signed negative temperature.
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