Lemma: A compact causal source with a local nonzero Hessian

For every nonempty $W$ satisfying (eq:W), there are real compact $q,h$ as above such that $C(q,h)>0$.

Proof. Choose $(t_0,{\mathbf{x}}_0)\in W$ and a small open dependence diamond $V$ whose closure is contained in $W$. Consider the polynomial

It solves $\square {\chi }_{\mathrm{loc}}=0$ and has ${{\partial }_t}^2{\chi }_{\mathrm{loc}}=2$.

At $t=t_0$, multiply its spatial Cauchy value by a smooth compact cutoff equal to one on a ball large enough to contain the Cauchy domain of dependence of $\overline{V}$; use zero time derivative. The smooth wave Cauchy problem produces a real solution $\chi$ with compact Cauchy data. By finite propagation and uniqueness, it agrees with ${\chi }_{\mathrm{loc}}$ on $V$. On every bounded time interval its spatial support is bounded.

Choose a smooth temporal function $\theta$ which is zero before one time and one after a later time, with its transition compactly inside $(0,T)$. Set

This is a smooth compact test in ${M_T}^{\circ }$. The functions $\theta \chi$ and $(\theta -1)\chi$ solve the sourced equation with respectively zero past and zero future data. Uniqueness gives

This is the compact time-slice construction in the supplied wave conventions XIIA,BGP.

Choose a nonzero nonnegative $\omega \in {C_c}^{\infty }(V)$ and put $q^{00}=\omega$, with all other components zero. Then

This construction is independent of the observation $f$ and of the scalar mass.