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

Abstract

We study the Einstein equations coupled with the scalar field equations, $\hbox{Ein}(g)=T$, $T=T(g,\phi)+F^1$, and $\square_g\phi^\ell-m^2\phi^\ell= F^2$, where the sources $F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. Here $\phi=(\phi^\ell)_{\ell=1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,\phi,F)$ satisfy the conservation law $\hbox{div}_g(T)=0$. If $(g_\epsilon,\phi_\epsilon)$ solves the above equations, $\dot g=\partial_\epsilon g_\epsilon|_{\epsilon=0}$, $\dot\phi=\phi_\epsilon|_{\epsilon=0}$, and $f=(f^1,f^2)= \partial_\epsilon F_\epsilon|_{\epsilon=0}$ solve the linearized Einstein equations and the linearized conservation law $$ \frac 12 \hat g^{pk}\hat \nabla_p f^1_{kj}+ \sum_{\ell=1}^L f^2_\ell \, \partial_j\hat\phi_\ell=0, $$ where $\hat g= g_\epsilon|_{\epsilon=0}$ and $\hat \phi= \phi_\epsilon|_{\epsilon=0}$. Then $(\hat g,\hat \phi)$ and $f$ have the linearization stability property. Here ask the converse: If $\dot g$, $\dot \phi$, and $f$ solve the linearized Einstein equations and the linearized conservation law, are there $F_\epsilon=(F^1_\epsilon,F^2_\epsilon)$ and $(g_\epsilon,\phi_\epsilon)$ depending on $\epsilon\in [0,\epsilon_0)$, $\epsilon_0>0$, such that $(g_\epsilon,\phi_\epsilon)$ solves the Einstein-scalar field equations and the conservation law. When $\hat g$ and $\hat \phi$ vary enough and $L\geq 5$, we prove a microlocal version of this: When $Y\subset M$ is a 2-surface and $(y,\eta)\in N^*Y$, 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.