Maxwell's equations in GR with the potential as a Killing Vector I am trying to prove
$g^{\mu\nu}∇_{\mu}F_{\nu\rho} = 0$, with $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$ and $A^\mu=K^\mu$ is a Killing vector.
I tried using the following relation $\nabla_\mu\nabla_\nu K_\rho=[\nabla_\rho,\nabla_\nu]K_\mu$, but can't figure out how to complete the proof.
What I tried to do:
$g^{\mu\nu}∇_{\mu}F_{\nu\rho}=2g^{\mu\nu}∇_{\mu}\nabla_\nu K_\rho=g^{\mu\nu}(∇_{\mu}\nabla_\nu K_\rho+[\nabla_\rho,\nabla_\mu]K_\nu)$
But I never get to something that becomes zero.
 A: Well, it seems to me that the assertion to be proved is false without other hypotheses like Ricci flatness.
Here there is an elementary counter example.
Consider a $2$-sphere of radius $1$ immersed in $\mathbb{R}^3$ and thus equipped with the natural metric
$$g = d\theta \otimes d\theta + \sin^2 \theta d\phi \otimes d\phi\tag{1}$$
in polar coordinates. The vector field
$$K = \partial_\phi$$
is Killing as the components  of the metric in (1) do not depend  on $\phi$.
We know that, in a generic (semi-)Riemannian manifold with Killing vector field $K$,
$$\nabla_a \nabla_b K^c = {R^c}_{bad} K^d$$
so that
$$\nabla_a \nabla^a K_c = g^{ba}R_{cbad} K^d =  -g^{ba}R_{bcad} K^d =-R_{cd}K^d\:.$$
That is
$$\nabla_a \nabla^a K^c = -R^c_dK^d\:.$$
In our case, since the space is homogeneous $R_{cd}= \kappa g_{ab}$ for a constant $\kappa \neq 0$ (I do not remember the value which depends on the radius of the sphere, but it does not matter). Therefore,
$$(\nabla_a \nabla^a \partial_\phi)^c = -\kappa \delta^c_d  (\partial_\phi)^d = -\kappa \delta_\phi^c \neq 0\:.\tag{2}$$
On the other hand, with your notations, using the Killing equation
$$\nabla_cK_d + \nabla_d K_c =0$$ and the fact that
$$\nabla_cK_d - \nabla_d K_c =\partial_cK_d - \partial_d K_c\:,$$
we have that
$$g^{ab}\nabla_aF_{bc} = g^{ab}\nabla_a(\nabla_b K_c - \nabla_cK_b) = 
2 g^{ab}\nabla_a\nabla_b K_c = 2 \nabla_a\nabla^a K_c\:.$$
The initial assertion $g^{ab}\nabla_aF_{bc} = 0$ is equivalent to
$$\nabla_a\nabla^a K^c =0\:.$$
But this is false for the $2$-sphere and choosing the Killing field $K = \partial_\phi$ as found in (2).
