Proposition: Linear time-slice construction

Let $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$, and suppose $u=E_{+}c$ is smooth. Choose a smooth temporal cutoff $\rho$ equal to zero sufficiently far in the past and one sufficiently far in the future. Then

is smooth and compact, and

for some compact tensor distribution $r$.

Proof. The field $u$ solves $\square u=0$ and is spacelike compact. The derivatives of $\rho$ are supported in a bounded time slab, whose intersection with the causal hull of a compact set is compact. Thus (eq:our-wave-representative) is smooth and compact.

Since $c_{\mathrm{sm}}=\square (\rho u)$, uniqueness gives

Therefore $E_{+}{c}_{\mathrm{sm}}=u=E_{+}c$. Lemma lem:compact-primitive supplies the compact $r$.