# Symmetric energy-momentum tensor using derivative wrt. metric [duplicate]

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I can find the Noether current for space time translation symmetry by demanding that the first order correction to the Lagrangian vanishes upon infinitesimal translations of coordinates. But in cases like Maxwell Lagrangian I get a non symmetric energy momentum tensor when I do this. I have seen places that mention that a symmetric energy-momentum tensor can be obtained if one takes the derivative wrt the metric. But if I am given a Lagrangian without the metric explicitly written in, how do I rewrite the Lagrangian with the correct metric dependence? For example say I am thinking of the the Free boson Lagrangian or the Maxwell Lagrangian? Is there a unique way to write the Lagrangian with the space time metric? I presume there has to be metric written in such a way that the action is an integral of a d+1 form. Is that correct?