Killing-flow four-acceleration ($a_a=-{\nabla }_a\log \sqrt{K^b{K}_b}$)

With metric signature $(+---)$, normalize a timelike Killing vector field by $U^a=K^a/N$, $N^2=K^a{K}_a$. The Killing equation gives $K^b{\nabla }_b{K}_a=-\frac{1}{2}{\nabla }_a{N}^2$, while $K^b{\nabla }_{b}N=0$. Thus the four-acceleration of its normalized flow is the displayed gradient. With metric signature $(-+++)$ the corresponding lower-index formula has the opposite sign.