Optimal version of the fundamental theorem of chronogeometry

Mori, Michiya (author), Šemrl, Peter (author)

We study lightlikeness preserving mappings from the $4$-dimensional Minkowski spacetime $\mathcal{M}_4$ to itself under no additional regularity assumptions like continuity, surjectivity, or injectivity. We prove that such a mapping $\phi$ satisfies one of the following three conditions. (1) The mapping $\phi$ can be written as a composition of a Lorentz transformation, a multiplication by a positive scalar, and a translation. (2) There is an event $r\in \mathcal{M}_4$ such that $\phi(\mathcal{M}_4\setminus\{r\})$ is contained in one light cone. (3) There is a lightlike line $\ell$ such that $\phi(\mathcal{M}_4\setminus \ell)$ is contained in another lightlike line. Here, a line that is contained in some light cone in $\mathcal{M}_4$ is called a lightlike line. We also give several similar results on mappings defined on a certain subset of $\mathcal{M}_4$ or the compactification of $\mathcal{M}_4$.

• Language: English
• Keywords: fundamental theorem of chronogeometry, special relativity, coherency preserving mapping
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.11.2025
• Year: 2025
• Number of pages: 85 str.
• Numbering: Vol. 480, part C, art. no. 110528
• PID: 20.500.12556/RUL-175109
• UDC: 530.12:514.7
• ISSN on article: 0001-8708
• DOI: 10.1016/j.aim.2025.110528
• COBISS.SI-ID: 253408515
• Publication date in RUL: 16.10.2025

Record is a part of a journal

• Title: Advances in mathematics
• Shortened title: Adv. math.
• Publisher: Elsevier
• ISSN: 0001-8708
• COBISS.SI-ID: 24891904

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.