Inverse problems in spacetime I: Inverse problems for Einstein equations - Extended preprint version

We consider inverse problems for the coupled Einstein equations and the matter field equations on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We give a positive answer to the question: Do the active measurements, done in a neighborhood $U\subset M$ of a freely falling observed $\mu=\mu([s_-,s_+])$, determine the conformal structure of the spacetime in the minimal causal diamond-type set $V_g=J_g^+(\mu(s_-))\cap J_g^-(\mu(s_+))\subset M$ containing $\mu$? More precisely, we consider the Einstein equations coupled with the scalar field equations and study the system $Ein(g)=T$, $T=T(g,\phi)+F_1$, and $\square_g\phi-\mathcal V^\prime(\phi)=F_2$, where the sources $F=(F_1,F_2)$ correspond to perturbations of the physical fields which we control. The sources $F$ need to be such that the fields $(g,\phi,F)$ are solutions of this system and satisfy the conservation law $\nabla_jT^{jk}=0$. Let $(\hat g,\hat \phi)$ be the background fields corresponding to the vanishing source $F$. We prove that the observation of the solutions $(g,\phi)$ in the set $U$ corresponding to sufficiently small sources $F$ supported in $U$ determine $V_{\hat g}$ as a differentiable manifold and the conformal structure of the metric $\hat g$ in the domain $V_{\hat g}$. The methods developed here have potential to be applied to a large class of inverse problems for non-linear hyperbolic equations encountered e.g. in various practical imaging problems.