Proposition: Off-shell reconstruction

For every smooth $h$, on $M_{+}$,

Consequently,

is an exact off-shell identity.

For $K=\mathrm{supp}f$, set

Then ${\mathcal{D}}_K$ is compact, and $B_f$ depends on $h$ only in ${\mathcal{D}}_K$ and on $\varphi$ only in $K$.

Proof. Let $A=\sqrt{-{\Delta }_{\mathbf{x}}}$. In spatial Fourier notation,

and

The quotients at zero spatial frequency are interpreted by their smooth limits.

The difference between the two sides of (eq:reconstruction) solves the homogeneous wave equation with zero Cauchy data, and therefore vanishes. Applying ${\mathsf{E}}_{0}C$ gives (eq:wsplit), and substituting into (eq:B) gives the claimed split.

Finite propagation speed implies that $w[h](x)$, for $x\in K$, depends only on the restriction of $h$ to the causal segment between ${\Sigma }_0$ and $x$. The same holds for its derivatives. The union of these segments is contained in ${\mathcal{D}}_K$. Compactness follows directly from compactness of $K$ and the Minkowski causal bound.