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

Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann
2014 arXiv   pre-print
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⊂ M of a freely falling observed μ=μ([s_-,s_+]), determine the conformal structure of the spacetime in the minimal causal diamond-type set V_g=J_g^+(μ(s_-))∩ J_g^-(μ(s_+))⊂ M containing μ? More precisely, we consider the Einstein equations
more » ... led with the scalar field equations and study the system Ein(g)=T, T=T(g,ϕ)+F_1, and _gϕ- V^'(ϕ)=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,ϕ,F) are solutions of this system and satisfy the conservation law ∇_jT^jk=0. Let (ĝ,ϕ̂) be the background fields corresponding to the vanishing source F. We prove that the observation of the solutions (g,ϕ) in the set U corresponding to sufficiently small sources F supported in U determine V_ĝ as a differentiable manifold and the conformal structure of the metric ĝ in the domain V_ĝ. 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.
arXiv:1405.4503v1 fatcat:mproukbnizalzgagymykxaoeoe