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In Pedro Lauridsen Ribeiro's answer to deriving the improved stress-energy tensor using the improved Noether current, the variational equation for the improved stress-energy tensor is given by: \begin{equation} 2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\nabla_\mu X_\nu+\nabla_\mu(T^{\mu\nu}X_\nu)=\left(2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}+T^{\mu\nu}\right)\nabla_\mu X_\nu+X_\nu\nabla_\mu T^{\mu\nu}=0.\ \end{equation} Here $X$ is an arbitrary vector field $X$ on $M$ (i.e. an infinitesimal diffeomorphism).

In the above example, where the covariant derivative is defined with the levi-civita connection and the stress tensor is conserved, $X_{\nu}$ and its derivative act trivially and do not have physical significance. We see that $X_{\nu}$ is contracted with the conserved divergence of the stress tensor in the last term. In a hypothetical world, if the stress tensor was not conserved what would the physical meaning of $X_{\nu}$ be?

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Whatever I understand most of the cases, we do not really know whether the $T^{\mu\nu}$ is actually symmetric or not, but, it should be conserved covariantly because improvement terms are in general, covariantly conserved. But they can alter the symmetry property of the stress energy tensor.

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  • $\begingroup$ I am not asking about $T^{\mu\nu}$, I am asking about $X_{\nu}$. To get intuition of $X_{\nu}$, I am asking what its physical significance would be if it were not arbitrary and not independently defined from its derivative. This would happen in a made up world where $T^{\mu\nu}$ is not conserved and/or using a different connection. $\endgroup$
    – B K
    Feb 28, 2023 at 15:28

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