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 $V\subset M$ 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 $I^+(x^-)\cap I^-(x^+)$istheintersectionofthefutureofthepoint$x^-$ 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.