Non UBC
DSpace
Tracey Balehowsky
2019-10-14T09:26:59Z
2019-04-16T15:34
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 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 strategy of using the nonlinearity of the inverse problem as a feature with which to gain knowledge of a related linearized problem is classical (see for example [2]). In a Lorentzian setting, this technique combined with microlocal analysis first appeared in [1] in the context of a wave equation with a quadratic nonlinearity and source-to-solution data. We will briefly survey this and later related work as they provide context for our result. We will also provide some physical motivation and context for the problem we consider.
The new results presented in this talk is joint work with Antti KujanapÃ¤Ã¤, Matti Lassas, and Tony Liimatainen (University of Helsinki).
[1] Kuylev Y., Lassas M., Uhlmann G., Inventiones mathematicae 212.3 (2018): 781-857.
[2] Sun Z., Mathematische Zeltschrift 221 (1996): 293-305.
https://circle.library.ubc.ca/rest/handle/2429/71905?expand=metadata
39.0 minutes
video/mp4
Author affiliation: University of Helsinki
Banff (Alta.)
10.14288/1.0383387
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Postdoctoral
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Global analysis, analysis on manifolds
Dynamical systems and ergodic theory
Global analysis
Determining a Lorentzian metric from the source-to-solution map for the relativistic Boltzmann equation
Moving Image
http://hdl.handle.net/2429/71905