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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?

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