A1 Journal article (refereed)
An Inverse Problem for the Relativistic Boltzmann Equation (2022)

Balehowsky, T., Kujanpää, A., Lassas, M., & Liimatainen, T. (2022). An Inverse Problem for the Relativistic Boltzmann Equation. Communications in Mathematical Physics, 396(3), 983-1049. https://doi.org/10.1007/s00220-022-04486-8

JYU authors or editors

Publication details

All authors or editors: Balehowsky, Tracey; Kujanpää, Antti; Lassas, Matti; Liimatainen, Tony

Journal or series: Communications in Mathematical Physics

ISSN: 0010-3616

eISSN: 1432-0916

Publication year: 2022

Publication date: 10/09/2022

Volume: 396

Issue number: 3

Pages range: 983-1049

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s00220-022-04486-8

Publication open access: Not open

Publication channel open access:

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/83258

Publication is parallel published: https://arxiv.org/abs/2011.09312


We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood V⊂M of a timelike path μ that connects a point x− to a point x+. The measurements are modelled by a source-to-solution map, which maps a source supported in V to the restriction of the solution to the Boltzmann equation to the set V. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set
I+(x−)∩I−(x+)⊂M. The set I+(x−)∩I−(x+) is the intersection of the future of the point x− and the past of the point x+, and hence is the maximal set to where causal signals sent from x− can propagate and return to the point x+. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

Keywords: inverse problems; equations; mathematics; particle physics; condensed matter physics; modelling (representation)

Free keywords: Boltzmann equation

Contributing organizations

Related projects

Ministry reporting: Yes

Reporting Year: 2022

JUFO rating: 3

Last updated on 2023-03-10 at 12:58