Canonical stress-energy tensor of the Maxwell field ($T^{\mu \nu }=-F^{\mu \rho }{\partial }^{\nu }{A}_{\rho }+\frac{1}{4}{\eta }^{\mu \nu }{F}_{\rho \sigma }{F}^{\rho \sigma }$)

For signature $(+,-,-,-)$, translation invariance of the Maxwell Lagrangian gives this tensor. It is generally neither symmetric nor gauge invariant. The antisymmetric superpotential improvement $-{\partial }_{\rho }(F^{\rho \mu }{A}^{\nu })$ produces, on the source-free equations of motion, the symmetric electromagnetic stress-energy tensor ${\Theta }^{\mu \nu }=-F^{\mu \rho }{F}^{\nu }{}_{\rho }+{\eta }^{\mu \nu }{F}^2/4$, which is traceless in four spacetime dimensions.

In signature $+---$, the Maxwell Lagrangian yieldsIt is symmetric after raising its second index, gauge-invariant, and conserved on the source-free Maxwell equations. Its energy density is $\frac{1}{2}({\mathbf{E}}^2+{\mathbf{B}}^2)$ and its energy flux is the Poynting vector. Its difference from the canonical stress-energy tensor is an antisymmetric superpotential divergence plus an equation-of-motion term, so integrated charges coincide when the constraints and vanishing boundary terms are imposed.