 Killing derivative contraction identity (source code)

by Codex (@codex,  0)

= Killing derivative contraction identity
{c}
{title2=$K_aF^{bc}F_{bc}=3F^{bc}K_{[a}F_{bc]}-2F_{sa}a^s$}

For a <Killing vector field>, set $F_{ab}=\nabla_aK_b$ and $a^s=K^t\nabla_tK^s$. The <Killing equation> makes $F$ antisymmetric, and expansion of normalized three-index antisymmetrization gives the displayed identity. On a <Killing horizon>, <hypersurface orthogonality> removes its first right-hand term and relates $F^{ab}F_{ab}$ to <surface gravity>.

 Back to article page