Lemma: Support saturation and independence of extension

Let $h\in C^{\infty }(U;S^2{T}^{\ast }U)$. Choose a compact cutoff in $U$ equal to one near $K_{\ast }$, extend the cutoff field by zero to auxiliary Minkowski space, and solve the zero-data wave problem there. Its value on $V$, when used in the response below, is independent of the cutoff.

The same statement holds for its smooth derivatives and its pairings with observation tests supported in $V$.

Proof. Two such extensions agree on a neighbourhood of $K_{\ast }$, hence their first differential expressions agree on every $D_f$. The difference of their zero-data solutions has zero source in the truncated past of $\mathrm{supp}f$ and zero Cauchy data there. Finite propagation and local uniqueness give zero difference near $\mathrm{supp}f$. The strict support margin gives the same conclusion for derivatives. These are local consequences of ordinary Minkowski Cauchy theory BGP (Chapter 3).