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We consider inverse problems for the coupled Einstein equations and thematter field equations on a 4-dimensional globally hyperbolic Lorentzianmanifold $(M,g)$. We give a positive answer to the question: Do the activemeasurements, done in a neighborhood $U\subset M$ of a freely falling observed$\mu=\mu([s_-,s_+])$, determine the conformal structure of the spacetime in theminimal causal diamond-type set $V_g=J_g^+(\mu(s_-))\cap J_g^-(\mu(s_+))\subsetM$ containing $\mu$? More precisely, we consider the Einstein equations coupledwith 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 thesources $F=(F_1,F_2)$ correspond to perturbations of the physical fields whichwe control. The sources $F$ need to be such that the fields $(g,\phi,F)$ aresolutions of this system and satisfy the conservation law $\nabla_jT^{jk}=0$.Let $(\hat g,\hat \phi)$ be the background fields corresponding to thevanishing 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 structureof the metric $\hat g$ in the domain $V_{\hat g}$. The methods developed herehave potential to be applied to a large class of inverse problems fornon-linear hyperbolic equations encountered e.g. in various practical imagingproblems.

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