Lemma: Transposition on the slab

Let $a\in {C_c}^{\infty }({M_T}^{\circ })$ and $s\in C^{\infty }(M_T)$, scalar or componentwise. Then

Proof. Put $u={\mathsf{E}}_{\mu }s$ and $b={\mathsf{E}}_{\mathrm{A}}a$. The set $J^{-}(\mathrm{supp}a)\cap M_T$ is compact and $b$ vanishes on a neighbourhood of $\{t\ge T\}$ in $M_T$, since $\mathrm{supp}a\subset \{t\le T-\delta \}$ for some $\delta >0$. Green's identity on $M_T$ gives

At $t=T$ the integrand vanishes because $b$ does; at $t=0$ it vanishes because $u$ has zero data. There is no spatial boundary term because $b$ is compactly supported in each slice.