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

Abstract

We study the Einstein equations coupled with the scalar field equations, $Ein(g)=T$, $T=T(g,\varphi )+F^1$, and ${\square }_g{\varphi }^{\ell }-m^2{\varphi }^{\ell }=F^2$, where the sources $F=(F^1,F^2)$ correspond to perturbations of the physical fields which we control. Here $\varphi =({\varphi }^{\ell }{)}_{\ell =1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,\varphi ,F)$ satisfy the conservation law ${div}_g(T)=0$. If $(g_{\epsilon },{\varphi }_{\epsilon })$ solves the above equations, $\dot{g}={\partial }_{\epsilon }{g}_{\epsilon }{|}_{\epsilon =0}$, $\dot{\varphi }={\varphi }_{\epsilon }{|}_{\epsilon =0}$, and $f=(f^1,f^2)={\partial }_{\epsilon }{F}_{\epsilon }{|}_{\epsilon =0}$ solve the linearized Einstein equations and the linearized conservation law $$\frac{1}{2}{\hat{g}}^{pk}{\hat{\nabla }}_p{f}_{kj}^1+\sum _{\ell =1}^L{f}_{\ell }^2{\partial }_j{\hat{\varphi }}_{\ell }=0,$$ where $\hat{g}=g_{\epsilon }{|}_{\epsilon =0}$ and $\hat{\varphi }={\varphi }_{\epsilon }{|}_{\epsilon =0}$. Then $(\hat{g},\hat{\varphi })$ 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 $F_{\epsilon }=(F_{\epsilon }^1,F_{\epsilon }^2)$ and $(g_{\epsilon },{\varphi }_{\epsilon })$ depending on $\epsilon \in [0,{\epsilon }_0)$, ${\epsilon }_0>0$, such that $(g_{\epsilon },{\varphi }_{\epsilon })$ solves the Einstein-scalar field equations and the conservation law. When $\hat{g}$ and $\hat{\varphi }$ 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 $(\hat{g},\hat{\varphi })$ and $f$ have the linearization stability property.

arXiv commentarXiv admin note: text overlap with arXiv:1305.1739

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Cited by in corpus (1)

• Inverse problems for semilinear wave equations on Lorentzian manifolds