Electromagnetic stress-energy divergence identity (${\nabla }_b{T}^{ab}=F^a{}_c{\nabla }_b{F}^{cb}/(4\pi )$)

For signature $(+---)$ in Gaussian electromagnetic units, $T^{ab}=(4\pi {)}^{-1}[-F^a{}_c{F}^{bc}+g^{ab}{F}_{cd}{F}^{cd}/4]$. The Electromagnetic Bianchi identity gives $F^{bc}{\nabla }_b{F}^a{}_c={\nabla }^a(F_{cd}{F}^{cd})/4$, 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.