= Solution
The symmetrized covariant <derivative> equation is $\nabla_a\xi_b+\nabla_b\xi_a=\psi g_{ab}$. Lower $\xi_b=g_{bc}\xi^c$ and expand both <derivatives>. Using the <Christoffel symbols> of the metric and their symmetry, the coefficient of $\xi^c$ simplifies by
$$
\partial_ag_{bc}+\partial_bg_{ac}-2\Gamma^d_{ab}g_{dc}=\partial_cg_{ab}.
$$
Hence
$$
\boxed{\xi^c\partial_cg_{ab}+g_{bc}\partial_a\xi^c+g_{ac}\partial_b\xi^c=\psi g_{ab}}.
$$
This is the coordinate formula for the <Lie derivative> $\mathcal L_\xi g=\psi g$, proving equivalence with the <Conformal Killing vector field> condition.
Back to article page
