 Solution

Differentiate the electromagnetic stress-energy tensor. The current term isThe other two terms cancel by the homogeneous Maxwell equations. To see the contraction explicitly, multiply ${\partial }_{\mu }{F}_{\beta \alpha }+{\partial }_{\beta }{F}_{\alpha \mu }+{\partial }_{\alpha }{F}_{\mu \beta }=0$ by $F^{\mu \alpha }$. The first and third contracted terms are equal after swapping indices and using antisymmetry; the middle is $-\frac{1}{2}{\partial }_{\beta }(F^{\mu \alpha }{F}_{\mu \alpha })$. ThereforeIt cancels differentiation of the trace term, leavingThe sign is tied to the printed convention ${\partial }_{\mu }{F}^{\mu \nu }=-{\mu }_0{J}^{\nu }$ and the displayed field-strength tensor.