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

For the Maxwell field, the variation $\delta A_{\nu }=\alpha F_{0\nu }=\alpha ({\partial }_0{A}_{\nu }-{\partial }_{\nu }{A}_0)$ is time translation combined with a gauge transformation. With constant $\alpha$, $\delta F_{\mu \nu }=\alpha {\partial }_0{F}_{\mu \nu }$, and the Lagrangian changes by $\alpha {\partial }_0\mathcal{L}$. Its Noether current is directly the time component of the gauge-invariant Maxwell stress-energy tensor, providing the physical energy density without a potential-dependent improvement step.