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Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 47b) regarding the gravity contribution to the stress-energy-momentum-tensor through the Hamiltonian:

$$ \begin{align} \frac{\partial}{\partial x_{\alpha}} \left( \frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right ) - \frac{\partial H}{\partial g^{\mu\nu}} = 0 \tag{47b} \end{align} $$

According to the discussion in the following paragraphs, this should be equivalent to:

$$ \frac{\partial}{\partial x_{\alpha}} \left( g^{\mu\nu}_{,\sigma}\frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right ) - \frac{\partial H}{\partial x_{\sigma}} = 0 $$

but you can see that in the original equation there is a term:

$$ - \frac{\partial H}{\partial g^{\mu\nu}} $$

not appearing in the modified equation, which doesn't seem to be vanishing.
Can you explain why?

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