In this talk, we consider the following problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood $V$ of an observer in a Lorentzian spacetime $(M,g)$ and knowledge of $g|_V$, can we determine (up to diffeomorphism) the spacetime metric $g$ on the domain of causal influence for the set $V$?
We will show that the answer is yes. The problem we consider is a so-called inverse problem. In this talk, we will introduce the notion of an inverse problem and give an overview of three broad types. We will then present the setting of our problem. Next, we introduce the relativistic Boltzmann equation and comment on the existence of solutions to this PDE given some initial data. We then will comment on one of the key ideas in the proof of our result.
The key point presented is that the nonlinear term 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 classical microlocal techniques to determine the set of locations in $V$ where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.
The new results presented in this talk are joint work with Antti Kujanpää, Matti Lassas, and Tony Liimatainen, (University of Helsinki). https://arxiv.org/abs/2011.09312