Given the Lagrangian density for a real scalar field $\mathcal{L}(\phi, \partial_\mu \phi)$, we obtain from Noether's theorem the canonical stress-energy tensor $$ T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\nu \phi - g^{\mu \nu} \mathcal{L} $$ Can anyone tell me, or provide a (hopefully detailed) reference, why there is always (I think?) a rank $3$ tensor $X^{\lambda \mu \nu}$ such that $$\widetilde{T}^{\mu \nu} = T^{\mu \nu} + \partial_{\lambda} X^{\lambda \mu \nu}$$ is symmetric? I'm fine with assuming $g$ is the Minkowski metric, but I can only find this done in particular cases of $\mathcal{L}$ and no general argument.
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