Gauge-compensated time translation in electromagnetism ($\delta A_{\nu }=\alpha F_{0\nu }$)

With signature $(+,-,-,-)$ and $\mathcal{L}=-F_{\mu \nu }{F}^{\mu \nu }/4$, combine a time translation $\alpha {\partial }_0{A}_{\nu }$ with the gauge transformation of parameter $-\alpha A_0$. Its Noether current is $J^{\mu }=-F^{\mu \nu }{F}_{0\nu }-{\delta }_0^{\mu }\mathcal{L}$, the gauge-invariant energy current. The density is $(|\mathbf{E}{|}^2+|\mathbf{B}{|}^2)/2$. On the source-free field equations it differs from the canonical translation current by ${\partial }_{\nu }(A_0{F}^{\mu \nu })$, an antisymmetric superpotential divergence, so total energies agree when the boundary term vanishes.