Proposition: The Einstein row is a compact plateau test of Aγ

For every smooth $\gamma ,\varphi$ and $\xi \in {\mathfrak{g}}_U$,

It vanishes if $A\gamma =0$ on $U$. On the gauge-fixed free shell ${\mathcal{C}}_{\mathrm{sol}}$,

Proof. By (eq:exi) and Lemma lem:betapair, ${\mathcal{A}}_1(e_{\xi };\varphi )=-\int \beta \cdot {{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(A\gamma )$. For a covariant tensor $\beta$ and a contravariant density $v$, ${\mathcal{L}}_{\xi }(\beta \cdot v)=({\mathcal{L}}_{\xi }\beta )\cdot v+\beta \cdot {{\mathcal{L}}^{\mathrm{dens}}}_{\xi }v$, and the left side is ${\partial }_{\rho }({\xi }^{\rho }\beta \cdot v)$, which integrates to zero for compact $\xi$. This gives (eq:einsteinrow); $\beta$ is smooth, so ${\mathcal{L}}_{\xi }\beta$ is smooth and supported in $\mathrm{supp}\xi \subset U$. On ${\mathcal{C}}_{\mathrm{sol}}$, $\square \gamma =0$ and $b=D\gamma$, so $A\gamma =-\square \mathcal{P}\gamma -2D^{\ast }D\gamma =-2D^{\ast }b$; the second equality in (eq:shellrow) is the flat adjoint of $D^{\ast }$ on a compact test.