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

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### Abstract

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.

### Author and article information

###### Journal
2014-05-18
###### Article
1405.4503