# Derivation of Maxwell's Equations using the Energy-Momentum tensor [duplicate]

If the energy momentum tensor is related to the EM field tensor by $$T^{\mu v}=F^{\mu \sigma}F^v_\sigma-\frac{1}{4}\eta^{\mu v}F^{\sigma \tau}F_{\sigma \tau}$$ Is it possible to derive Maxwell's Equation $$\partial_v F^{\mu v}=J^\mu$$ with the energy momentum tensor without using the EM field tensor lagrangian $$-\frac{1}{4}F^{\mu v}F_{\mu v} + A_\mu J^\mu$$ alone?