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Let $g_{\mu\nu}$ be a metric on a manifold with a time direction $x^0$ singled out. I'm wondering if there exists a function $F(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots)$ that transforms under spatial diffeomorphisms as \begin{align*} F(g_{\mu\nu}'(x'),\partial_\rho g_{\mu\nu}'(x'),\ldots)=F(g_{\mu\nu}(x),\partial_\rho g_{\mu\nu}(x),\ldots)+ \nabla_\mu \Lambda^{\mu}(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots,x'), \end{align*} where $\Lambda$ is some functional of the metric and $x'$. This would imply that the integral \begin{align*} \int d^dx\, \sqrt{-g}F \end{align*} is invariant under spatial diffs.

Any ideas?

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