Proposition: Sourced conformal diagnostic

Let $m=0$, ${\varphi }_0=t$, $h_{\mu \nu }=\psi {\eta }_{\mu \nu }$, $k=0$, for a real compact interior bump $\psi$. Then

There are supported choices for which $A_1\ne 0$ and its smearing is nonzero.

If instead $\mathrm{supp}\psi \subset P_g\cap \{t>\epsilon \}$, then $A_1=0$, while ${\rho }_{1,g}$ and $w^0$ are nonzero when ${\partial }_t\psi \not\equiv 0$.

Proof. In four dimensions $\mathrm{tr}h=4\psi$, $h:h=4{\psi }^2$. Substitution in (eq:densitycoeff) gives the first line. In fact the conformal identity $A_{(1+\kappa \psi )\eta }=(1+\kappa \psi ){\eta }^{-1}$ checks $a_2[h]=0$ directly.

The remaining formulae follow from the coefficient equations and ${\partial }_{\nu }t={{\delta }^0}_{\nu }$. Choose $\psi$ at positive post-ramp time outside $\mathrm{supp}g$, with ${\partial }_t\psi \ne 0$. Then $A_1={\mathsf{E}}_0{\partial }_t\psi \ne 0$, since applying $\square$ gives the nonzero source. A bump in an open set where $A_1$ has fixed sign gives a nonzero observation. All supports and their pasts can be chosen within the retained laboratory.

In the plateau alternative $g\psi =\sigma \psi =\psi$, so $A_1=0$. Applying $\square$ to either displayed nontrivial response proves that it is not identically zero.