Lemma: Detection by spatially compact solutions

For $d\in \mathcal{D}$, the following are equivalent:

1. $d\in P_m\mathcal{D}$;

2. ${\Delta }_{\mathrm{S}}d=0$;

3. $⟨d,u⟩=0$ for every $u\in {\mathrm{Sol}}_{\mathrm{sc}}(P_m)$;

4. $⟨d,u⟩=0$ for every smooth free solution.

Also

Proof. Green identities immediately give (i)$\Rightarrow$(ii) and integration by parts gives (i)$\Rightarrow$(iv). The implication (iv)$\Rightarrow$(iii) is immediate.

For completeness, the image statement follows from Cauchy uniqueness. If $u$ has compact Cauchy data, choose a temporal cutoff $\chi$ which is zero early and one late. The source

is compact, because its time support is bounded and $u$ is spatially compact. Retarded and advanced uniqueness give

Thus ${\Delta }_{\mathrm{S}}(-a)=u$. The converse image inclusion is finite propagation.

If (iii) holds, then for every $a\in \mathcal{D}$,

so (ii) follows. If (ii) holds, let $v=\text{EAd}=\text{ERd}$. Its support is contained in

a compact causal diamond on Minkowski space. Hence $v\in \mathcal{D}$ and $P_{m}v=d$, proving (i). This is the usual Green exactness mechanism BGP (Section 3.4).