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

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 $Ein(g)=T$, $T=T(g,\varphi )+F_1$, and ${\square }_g\varphi -{\mathcal{V}}^{\prime }(\varphi )=F_2$, 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 $(g,\varphi ,F)$ are solutions of this system and satisfy the conservation law ${\nabla }_j{T}^{jk}=0$. Let $(\hat{g},\hat{\varphi })$ be the background fields corresponding to the vanishing source $F$. We prove that the observation of the solutions $(g,\varphi )$ 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.

This is an extended preprint version of the paper Inverse problems in spacetime I: Inverse problems for Einstein equations. The supplementary video can be downloaded at the page http://www.rni.helsinki.fi/~mjl/publications_time.html. arXiv admin note: text overlap with arXiv:1405.3384, arXiv:1405.3386

References in corpus (1)

• Full-wave invisibility of active devices at all frequencies

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