Proposition: Plateau action and Euler covariance

For a diffeomorphism $F$ supported in a compact subset of $U$, the density difference

has compact support and integrates to zero. The exact Euler densities obey

The first pullback is that of a contravariant symmetric tensor density, and the second that of a scalar density.

Proof. The curved scalar and Einstein–Hilbert densities are natural. Their pullbacks have unchanged integrals under a compactly supported change of variables. Moreover, $g\circ F^{-1}=g$, since $g=1$ near the support of $F-\mathrm{id}$. The remaining flat scalar term is unchanged pointwise where $1-g\ne 0$.

Differentiate this action equality after arbitrary compact variations of $q,\varphi$. Change variables in the resulting invariant tensor-density pairings. Arbitrariness of the variations proves Euler covariance.