An Inverse Problem for the Relativistic Boltzmann Equation

Abstract

We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $\mu$ that connects a point $x^{-}$ to a point $x^{+}$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^{+}(x^{-})\cap I^{-}(x^{+})\subset M$. The set $I^{+}(x^{-})\cap I^{-}(x^{+})$ is the intersection of the future of the point $x^{-}$ and the past of the point $x^{+}$, and hence is the maximal set to where causal signals sent from $x^{-}$ can propagate and return to the point $x^{+}$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

60 pages. 4 figures. In this version, we have have improved the presentation of Section 4, which includes added motivation for Definition 4.1. We have also added more exposition in the introduction, in particular a more thorough discussion of Definition 1.2. Minor typos were fixed throughout the paper. This version coincides with the published version of our work, to appear in CMP

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Cited by in corpus (1)

• Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds