Lemma: Exact patch agreement

Under Definition def:completion, compact pairings satisfy

The corresponding tensor identity also holds. Local differential equations and linearized curvature identities pull back through $D$. A flat zero-data wave calculation agrees with its local calculation whenever the relevant causal domain is contained in $D(U)$.

Proof. The first assertions are change of variables and naturality of differential operators under an isometry. The volume form is $\mathrm{d}{\mathrm{vol}}_g=c\mathrm{d}T{\mathrm{d}}^{3}y=\mathrm{d}t{\mathrm{d}}^{3}y$. For the final assertion, subtract the two solutions on their common domain. The difference solves the homogeneous equation with zero Cauchy data; the local energy identity gives uniqueness.