= Static Killing field is a Ricci eigenvector
{title2=$R_{ab}k^b\ \parallel\ k_a$}
A timelike <Killing vector> with <hypersurface orthogonality> admits local coordinates with $k=\partial_t$ and $ds^2=N(\mathbf x)^2dt^2-h_{ij}(\mathbf x)dx^idx^j$. The only connection coefficients with time indices that can be nonzero are $\Gamma^0_{0i}=\partial_i\log N$ and $\Gamma^i_{00}=Nh^{ij}\partial_jN$. Substitution into the <Ricci tensor> gives $R_{0i}=0$. Thus $R_{ab}k^b$ has only its time component and is proportional to $k_a$. Equivalently $k_dR^d{}_{[a}k_{b]}=0$. This local statement does not require the <Vacuum Einstein equations> and is independent of the overall curvature-sign convention.
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