Lemma: Pure gauge with zero data: coordinate and switching terms

Let $\zeta$ be smooth on $M_T$ with zero Cauchy data and $\mathrm{supp}\zeta \subset \{t\ge \epsilon \}$, and let ${\varphi }_0$ be free. Then

where $X_{\zeta }={\zeta }^{\mu }{\partial }_{\mu }$ and $[P_m,g]u=(\square g)u+2({\partial }^{\mu }g){\partial }_{\mu }u$. Every term carries a factor $g-1$ or $\partial g$ evaluated on $\mathrm{supp}\zeta \cap {\mathcal{D}}_f$. In particular

Proof. (eq:preflight): $a_1[K\zeta ]$ is supported in $\mathrm{supp}\zeta$, where ${\sigma }_{\epsilon }=1$ and ${\sigma }_{\epsilon }^{\prime }=0$, so $\square {\mathsf{X}}_1[K\zeta ]=-{\partial }_{\alpha }{a_1}^{\alpha \cdot }[K\zeta ]=C[K\zeta ]=\square \zeta$, and both sides have zero data; this is X-A's preflight argument, which uses only the support and the data, not compactness. (eq:switch): by X-A's lemma on the pure-gauge variation, ${\mathsf{V}}_{1,g}[K\zeta ]{\varphi }_0=g(D_{\zeta }{P}_m{\varphi }_0-P_m{X}_{\zeta }{\varphi }_0)+(\partial g)a_1[K\zeta ]\partial {\varphi }_0=-g{P}_m{X}_{\zeta }{\varphi }_0+(\partial g)a_1[K\zeta ]\partial {\varphi }_0$. Since $g{X}_{\zeta }{\varphi }_0$ has zero data, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}(g{P}_m{X}_{\zeta }{\varphi }_0)=g{X}_{\zeta }{\varphi }_0-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}[P_m,g]X_{\zeta }{\varphi }_0$. Subtract the coordinate term $\int f\zeta \cdot \partial {\varphi }_0$. The support statement follows by transposing ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$ onto $f$: $\int f{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}(\cdot )=\int ({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)(\cdot )$ with ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ supported in ${\mathcal{D}}_f$. If $g=1$ on a neighbourhood of ${\mathcal{D}}_f\cap \mathrm{supp}\zeta$, then $\partial g=0$ there as well, and all terms vanish.