In this talk, we consider the following question: Given the source to solution map for a relativistic Boltzmann equation on a known open set $V$ of a Lorentzian spacetime $(\mathbb{R}\times N,g)$, can we use this data to uniquely determine the spacetime metric on an unknown region of $\mathbb{R}\times N$? We will show that the answer is yes. Precisely, we determine the metric up to conformal factor on the domain of causal influence for the set $V$. Key to our proof is that the nonlinearity in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by microlocal techniques to determine the set of locations in $V$ where we first receive light signals from collisions in the unknown domain. From this data we obtain the desired diffeomorphism. The technique of using the nonlinearity in a partial differential equation as a feature with which to gain knowledge of a related linearized problem first appeared in [1] in the context of a wave equation with a quadratic nonlinearity. We will briefly survey this and related work as they provide context for our result. The new results presented in this talk is joint work with Antti Kujanapää, Matti Lassas, and Tony Liimatainen (University of Helsinki).

[1] Kuylev Y., Lassas M., Uhlmann G., Inventiones mathematicae 212.3 (2018): 781-857.