Proposition: Preflight identity

Let $\xi \in {C_c}^{\infty }({M_T}^{\circ };TM)$ with $\mathrm{supp}\xi \subset \{t\ge \epsilon \}$. Then

In particular (eq:preflight) holds for every $\xi \in {\mathfrak{g}}_U$. In the notation of the plan, ${\mathcal{E}}_{g,\epsilon }\square \xi =\xi$ with the prescribed data.

Proof. By (eq:a), $a_1[K\xi ]=({\partial }_{\rho }{\xi }^{\rho })\eta -K{\xi }^{\cdot \cdot }$ is supported in $\mathrm{supp}\xi \subset \{t\ge \epsilon \}$, where ${\sigma }_{\epsilon }=1$ and ${\sigma }_{\epsilon }^{\prime }=0$. Hence, by (eq:V) and (eq:C),

Both ${\mathsf{X}}_1[K\xi ]$ and $\xi$ have zero Cauchy data on $\Sigma$, since $\xi$ vanishes on $\{t<\epsilon \}$. Uniqueness in Lemma lem:cauchy gives (eq:preflight).