Conventions

Indices are moved with $\eta$; parentheses include the factor $\frac{1}{2}$. Classical fields are smooth on a neighbourhood of the closed slab; no spatial decay is imposed. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$, $0<\epsilon <T$, with the explicit $\chi$ of X-A, satisfies ${\sigma }_{\epsilon }=0$ on $\{t\le 0\}$, ${\sigma }_{\epsilon }=1$ on $\{t\ge \epsilon \}$, ${\sigma }_{\epsilon }>0$ on $\{t>0\}$, and $\chi (1-s)=1-\chi (s)$. The switch $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ has $P_g=\mathrm{int}\{g=1\}\ne \varnothing$. The plateau $U$ is open, relatively compact, with $\overline{U}\subset P_g\cap \{\epsilon <t<T\}$, and ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$. The observation test is a real $f\in {C_c}^{\infty }({M_T}^{\circ })$ with

a compact set that is closed under $J^{-}$ inside $M_T$.

For a symmetric tensor $r$,

With $A=2G_{\mathrm{lin}}$,

Here ${{\mathsf{E}}_0}^{\mathrm{R}}$ is the zero-initial-data inverse of $\square$ on the slab, ${{\mathsf{E}}_0}^{\mathrm{A}}$ the zero-final-data inverse, and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ the Klein–Gordon analogues. For $a\in {C_c}^{\infty }({M_T}^{\circ })$ and $s$ smooth on $M_T$, $\int a{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}s=\int ({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}a)s$ (X-A, transposition lemma); the proof only needs ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}a$ to vanish near $t=T$, so it applies verbatim to $a$ smooth on $M_T$ with $\mathrm{supp}a\subset {\mathcal{D}}_f$. For $s$ smooth with zero Cauchy data and $\mathrm{supp}s\subset \{t>0\}$, ${{\mathsf{E}}_0}^{\mathrm{R}}\square s=s$ and ${{\mathsf{E}}_0}^{\mathrm{R}}{\partial }_{\mu }s={\partial }_{\mu }{{\mathsf{E}}_0}^{\mathrm{R}}s$.