Lemma: Detectable Lie derivative of a causal solution

There exist real compact tests $a,g$, a compact spacetime vector field $\xi$, and a real spacelike-compact solution $U=Eg$ such that

Proof. Fix $t_0\in I$. Use the normally hyperbolic Cauchy problem for $\mathcal{L}$ to choose compact Cauchy data with scalar component $\phi (t_0)=0$ and $\dot{\phi }(t_0)=b$, where $b$ is a nonzero real compact spatial bump. The other initial components can be chosen zero. No physical subsidiary condition is required for this gauge-fixed comparison.

Finite propagation gives a real solution $U$ with uniformly compact spatial support on compact time intervals. Choose compact $\xi$ whose time component is nonzero where $b\ne 0$. Then the scalar component of $A_{\xi }U$ is nonzero at $t_0$, and hence on a spacetime neighborhood. A real compact test $a$ supported there detects it under the nondegenerate pairing.

Choose a smooth time function $\eta$ which is zero before an interior time and one after a later interior time, both chosen before $t_0$, with compact derivative support in $I$. Put $g=\mathcal{B}(\eta U)$. Since $\mathcal{B}U=0$, $g$ is supported in the transition region; finite propagation makes it spacetime compact.

Cauchy uniqueness gives

Their difference is $Eg=U$. Formal adjunction proves (eq:witness).