Conventions and ambient propagators

Indices are moved with $\eta$; parentheses include $\frac{1}{2}$; $k\cdot y={\eta }_{\mu \nu }{k}^{\mu }{y}^{\nu }$. A covector $p$ is null if ${\eta }^{\mu \nu }{p}_{\mu }{p}_{\nu }=0$; ${\tilde{V}}_{+}$ (${\tilde{V}}_{-}$) is the closed cone of future (past) directed causal covectors with zero included (XI-E); for a null covector $p\ne 0$, $p\in {\tilde{V}}_{+}$ iff $p_0>0$. The ramp ${\sigma }_{\epsilon }$, switch $g$, plateau $U$, ${\mathfrak{g}}_U$, and the observation data $f\in {C_c}^{\infty }({M_T}^{\circ })$, ${\mathcal{D}}_f=J^{-}(\mathrm{supp}f)\cap M_T$, are XI-F's. Trace reversal $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$, ${\mathcal{P}}^2=\mathrm{id}$; $D^{\mu }r={\partial }_{\nu }(\mathcal{P}r{)}^{\nu \mu }$; $(Kv{)}_{\mu \nu }={\partial }_{\mu }{v}_{\nu }+{\partial }_{\nu }{v}_{\mu }$; $A=2G_{\mathrm{lin}}$;

For $\mu \in \{0,m\}$ let ${{\mathsf{E}}_{\mu }}^{\mathrm{R}},{{\mathsf{E}}_{\mu }}^{\mathrm{A}}$ be the Minkowski retarded and advanced inverses of $\square +{\mu }^2$ on $M$; on compact interior sources they restrict to XI-B's slab inverses ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ XIB. Put

which is XI-B's causal kernel ${\delta }_{\mu }$. The operators ${{\mathsf{E}}_{\mu }}^{\mathrm{R}/A}$ act on distributions with past/future compact support, satisfy $(\square +{\mu }^2){{\mathsf{E}}_{\mu }}^{\mathrm{R}}w=w$ and ${{\mathsf{E}}_{\mu }}^{\mathrm{R}}(\square +{\mu }^2)r=r$ for $r$ compactly supported, $\mathrm{supp}{{\mathsf{E}}_{\mu }}^{\mathrm{R}}w\subset J^{+}(\mathrm{supp}w)$, and the retarded solution with past-compact support of $(\square +{\mu }^2)\Phi =w$ is unique BGP (Ch. 3). For a distribution $w$ on $M^3$ whose support is compact in the slot-$i$ direction, $i\in \{1,2,3\}$, the slot operators ${{\mathsf{E}}^{\mathrm{R}}}_{\mu ,i}$, ${{\mathsf{E}}^{\mathrm{A}}}_{\mu ,i}$, ${\mathsf{E}}_{\mu ,i}$ are defined by transposition in slot $i$, for instance $⟨{\mathsf{E}}_{\mu ,i}w,\phi ⟩=-⟨w,{\mathsf{E}}_{\mu ,i}\phi ⟩$ for $\phi \in {C_c}^{\infty }(M^3)$; they commute with multiplication by smooth functions of, and differential operators in, the other slots. We write ${\mathsf{E}}_0$ for the metric slots and ${\mathsf{E}}_m$ for the scalar slot, and ${\mathcal{E}}^{\prime }$ for compactly supported distributions on $M^3$ with the slot type of $u$ below.