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

TL;DR

This paper investigates the linearization stability of solutions to the Einstein-scalar field equations, establishing conditions under which linearized solutions can be extended to actual solutions, with a focus on microlocal analysis for certain geometric configurations.

Contribution

It proves a microlocal version of linearization stability for the Einstein-scalar field equations when the number of scalar fields is at least five, linking linearized solutions to actual solutions via conormal distributions.

Findings

Linearized solutions can be extended to actual solutions under certain conditions.

A microlocal version of linearization stability is established for specific geometric settings.

The results depend on the number of scalar fields and the geometric configuration of the problem.

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.