= Electromagnetic stress-energy divergence identity
{title2=$\nabla_bT^{ab}=F^a{}_c\nabla_bF^{cb}/(4\pi)$}
For signature $(+---)$ in Gaussian electromagnetic units, $T^{ab}=(4\pi)^{-1}[-F^a{}_cF^{bc}+g^{ab}F_{cd}F^{cd}/4]$. The <Electromagnetic Bianchi identity> gives $F^{bc}\nabla_bF^a{}_c=\nabla^a(F_{cd}F^{cd})/4$, so differentiating the stress tensor proves the displayed identity. Stress conservation implies the source-free <Maxwell equations> if the matrix $F^a{}_c$ is invertible; otherwise it only forces the field divergence into that matrix's kernel. Conversely source-free Maxwell equations always imply stress conservation.
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