# Mathematical Sciences Research Institute

Home » MLA - Regular Seminar (Part 2): Determining a Lorentzian metric from the source-to-solution map for the relativistic Boltzmann equation

# Seminar

MLA - Regular Seminar (Part 2): Determining a Lorentzian metric from the source-to-solution map for the relativistic Boltzmann equation October 24, 2019 (02:00 PM PDT - 03:00 PM PDT)
Parent Program: Microlocal Analysis MSRI: Simons Auditorium
Speaker(s) Tracey Balehowsky (University of Helsinki)
Description No Description
Video
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 on the domain of causal influence for the set $V$. Key to our proof 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 signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.