Theorem: Switched Noether identity

For every smooth pair $(q,\varphi )$,

The gravitational contribution has no switch defect. On $\mathrm{int}\{g=1\}$, the right-hand side is $E_{\varphi }{\partial }_{\rho }\varphi$.

Proof. Let $\xi$ be an arbitrary compact vector field and vary $q,\varphi$ by ${\mathcal{L}}_{\xi }q,{\xi }^{\rho }{\partial }_{\rho }\varphi$, holding $g$ and $\eta$ fixed. Naturality of $J_{q}R$ makes its variation a divergence. Naturality of the curved scalar density gives ${\delta }_{\xi }{\mathcal{L}}_q={\partial }_{\mu }({\xi }^{\mu }{\mathcal{L}}_q)$.

At $q=\eta$, algebraic metric variation gives

Therefore

Integrate the last term by parts and use (eq:stress-div).

The Euler form of the same variation is

Here the metric term follows by substituting

and integrating once. Equality for arbitrary $\xi$ proves (eq:noether).