Theorem: No support- or past-preserving embedding of relative ghosts

Let $\iota :{\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}\to {\mathcal{G}}_0$ be a linear map satisfying at least one of:

1. $\iota (c)=c$ on a neighbourhood of $\overline{W}$ for every $c$;

2. $\iota (c)$ has zero Cauchy data on some surface $\{t=t_0\}$ with $0\le t_0<{\min }_{\overline{W}}t$, for every $c$;

3. $\mathrm{supp}\iota (c)\subset J^{+}(\mathrm{supp}c)\cap M_T$ for every $c$.

Then $\iota =0$. In particular the inclusion ${\mathfrak{F}}_W\hookrightarrow \mathfrak{F}$ restricts to the zero map on the ghost sector of ${\mathcal{C}}_{\mathrm{sol}}$.

Proof. (a) On the neighbourhood $N$ of $\overline{W}$, $\square c=\square \iota (c)=0$; on the complement of $\overline{W}$, $c=0$. Thus $\square c=0$ on all of $M_T$ and $c=0$ by Theorem thm:ghost(i); then $\iota (c)=\iota (0)=0$. (b) A free solution with zero Cauchy data on $\{t=t_0\}$ vanishes on $[t_0,T]\times {\mathbb{R}}^3$ by uniqueness and on $[0,t_0]$ by the backward problem with the same data; hence $\iota (c)=0$. (c) $J^{+}(\mathrm{supp}c)\cap M_T\subset \{t\ge {\min }_{\overline{W}}t\}$, so $\iota (c)$ has zero Cauchy data on any earlier surface, and (b) applies.