An Inverse Problem for the Relativistic Boltzmann Equation

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 I + ( x - ) ∩ I - ( x + ) ⊂ M . The set I + ( x - ) ∩ 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.