Linearization stability results and active measurements for the Einstein-scalar field equations

We study the Einstein equations coupled with the scalar field equations, $\text{Ein}(g)=T$, $T=T(g,\varphi )+F^1$, and $\square_g\phi^\ell-m^2\phi^\ell= F^2$,wherethesources$F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. Here $\varphi =({\varphi }^{\mathrm{\ell }}{)}_{\mathrm{\ell }=1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,\varphi ,F)$ satisfy the conservation law ${\text{div}}_g(T)=0$. If $(g_{ϵ},{\varphi }_{ϵ})$ solves the above equations, $\dot{g}={\mathrm{\partial }}_{ϵ}{g}_{ϵ}{\mathrm{\mid }}_{ϵ=0}$, $\dot{\varphi }={\varphi }_{ϵ}{\mathrm{\mid }}_{ϵ=0}$, and $f=(f^1,f^2)= \partial_\epsilon F_\epsilon|{\epsilon=0}$ solve the linearized Einstein equations and the linearized conservation law $$\frac{1}{2}{\widehat{g}}^{pk}{\widehat{\mathrm{\nabla }}}_p{f}_{kj}^1+\sum _{\mathrm{\ell }=1}^L{f}_{\mathrm{\ell }}^2{\mathrm{\partial }}_j{\widehat{\varphi }}_{\mathrm{\ell }}=0,$$ where $\hat g= g\epsilon|{\epsilon=0}$and$\hat \phi= \phi\epsilon|_{\epsilon=0}$\mathrm{.}Then$(\hat g,\hat \phi)$and$f$ have the linearization stability property. Here ask the converse: If $\dot{g}$, $\dot{\varphi }$, and $f$ solve the linearized Einstein equations and the linearized conservation law, are there $F_{ϵ}=(F_{ϵ}^1,F_{ϵ}^2)$ and $(g_{ϵ},{\varphi }_{ϵ})$ depending on $ϵ\in [0,{ϵ}_0)$, ${ϵ}_0>0$, such that $(g_{ϵ},{\varphi }_{ϵ})$ solves the Einstein-scalar field equations and the conservation law. When $\widehat{g}$ and $\widehat{\varphi }$ vary enough and $L\ge 5$, we prove a microlocal version of this: When $Y\subset M$ is a 2-surface and $(y,\eta )\in N^{\ast }Y$, there is $f$ that is a conormal distibutions wrt. the surface $Y$ with a given principal symbol at $(y,\eta )$ such that $(\hat g,\hat \phi)$and$f$ have the linearization stability property.