Theorem: Plateau metric Euler covariance

For $F\in {\mathcal{G}}_U$, put $q^{\prime }=F^{\ast }q$, ${\varphi }^{\prime }=F^{\ast }\varphi$, and ${\gamma }^{\prime }=(q^{\prime }-\eta )/\kappa$. Then

The identity is formal and holds after expansion through ${\kappa }^2$, provided all coefficients needed before projection are retained. It does not hold with $e$ replaced by $e+e_{\mathrm{gf}}$ in general.

Proof. The Einstein tensor-density in (eq:eexact) is natural. The scalar stress density $\sqrt{-q}{T_q}^{\mu \nu }$ is natural as well. Since $F^{\ast }g=g$, its switched multiple transforms by (eq:tensorpull). This proves (eq:covariance). The counterexample in Proposition prop:gfbad, or the explicit formula (eq:egf), excludes the gauge-fixed assertion.