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