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 ⊂ M of a timelike path µ 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) 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. Contents 1. Introduction 1.1. Theorem 1.3 proof summary 2. Preliminaries 2.1. Notation 2.2. Lagrangian distributions and Fourier integral operators 3. Vlasov and Boltzmann Kinetic Models 3.1. The Vlasov model 3.2. The Boltzmann model 4. Microlocal analysis of particle interactions 4.1. Existence of transversal collisions 5. Proof of Theorem 1.3 5.1. Delta distribution of a submanifold 5.2. Nonlinear interaction in the inverse problem 5.3. Separation time functions 5.4. Source-to-Solution map determines earliest light observation sets