= Derivative-order ambiguity in covariant Maxwell potential equations
{title2=$\nabla_b\nabla^a A^b-\nabla^a\nabla_bA^b=-C^a{}_bA^b$}
Use the negative curvature convention defined in <curvature-sign convention in Killing derivative identities>, and $C_{ab}=C^c{}_{acb}$. Covariant derivatives of a vector obey $[\nabla_b,\nabla^a]A^b=-C^a{}_bA^b$. Hence the flat-space equivalent expressions $\Box A^a-\partial_b\partial^aA^b$ and $\Box A^a-\partial^a\partial_bA^b$ acquire a difference $+C^a{}_bA^b$ under the <comma-to-semicolon prescription>. Covariantizing $F_{ab}=2\nabla_{[a}A_{b]}$ and its <Maxwell equations> gives $\Box A^a-\nabla^a(\nabla_bA^b)+C^a{}_bA^b=\mu_0J^a$. This form respects <gauge invariance>. Dropping the Ricci term is generally not gauge invariant. The two expressions coincide in a Ricci-flat <vacuum spacetime>; vanishing matter alone with nonzero <cosmological constant> does not imply Ricci flatness.
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