Covariance of the invariant metric Euler operator

If covariant indices are preferred, define

It transforms as a covariant rank-two density; its two indices also acquire Jacobian factors. Treating it as four-by-four scalar densities would be incorrect.

For a general compact $F$ which need not preserve $g$, the exact metric covariance defect is

Its infinitesimal form involves $X_{\xi }g$. The divergence defect involves $({\partial }_{\mu }g)T^{\mu \nu }$, as in (eq:N1); these two defects are related but not identical formulas.

In particular, $\partial g$ is a source-conservation issue, not an extra term in the minimally coupled metric Euler derivative. The restriction to ${\mathcal{G}}_U$ removes the covariance defect. It does not imply global solvability of the metric equation in the switching region.