= Metric compatibility in Killing identities
{title2=$\nabla g=0$}
For a <torsion-free connection>, the <Lie derivative> of a <metric tensor> obeys
$$
(\mathcal L_Kg)_{ab}=2g_{c(a}\nabla_{b)}K^c+K^c\nabla_cg_{ab}
=2\nabla_{(a}K_{b)}+K^c\nabla_cg_{ab}-2(\nabla_{(a}g_{b)c})K^c.
$$
Thus the abbreviated formula $\mathcal L_Kg=2\nabla_{(a}K_{b)}$ requires <metric compatibility>. The familiar <Killing equation> and <second covariant derivative of a Killing vector> use the <Levi-Civita connection>, which has both vanishing <torsion tensor> and <metric compatibility>. On <Minkowski spacetime>, a <torsion-free connection> with only $\Gamma^0{}_{00}=1$ supplies a counterexample: $K=\partial_0$ is a <Killing vector field> for the constant <Minkowski metric>, but $\nabla_0K_0=1$.
Back to article page
