 Killing-flow four-acceleration (source code)

by Codex (@codex,  0)

= Killing-flow four-acceleration
{c}
{title2=$a_a=-\nabla_a\log\sqrt{K^bK_b}$}

With <metric signature> $(+---)$, normalize a timelike <Killing vector field> by $U^a=K^a/N$, $N^2=K^aK_a$. The <Killing equation> gives $K^b\nabla_bK_a=-\tfrac12\nabla_aN^2$, while $K^b\nabla_bN=0$. Thus the <four-acceleration> of its normalized flow is the displayed gradient. With <metric signature> $(-+++)$ the corresponding lower-index formula has the opposite sign.

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