M\"obius Transformation and Einsten Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky
Abraham A. Ungar
TL;DR
This paper explores the connection between M"obius transformations and Einstein velocity addition within the hyperbolic geometry framework of Bolyai and Lobachevsky, revealing their isomorphic relationship in gyrovector spaces.
Contribution
It demonstrates that M"obius addition and Einstein addition are isomorphic in gyrovector spaces, linking complex analysis and relativistic velocity addition in hyperbolic geometry.
Findings
M"obius addition derived from complex disc transformations
Einstein addition from special relativity
Isomorphism between M"obius and Einstein addition in gyrovector spaces
Abstract
In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, M\"obius transformation and Einstein velocity addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that M\"obius addition that is extracted from M\"obius transformation of the complex disc and Einstein addition from his special theory of relativity are isomorphic in the sense of gyrovector spaces.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematics and Applications · Advanced Theoretical and Applied Studies in Material Sciences and Geometry · Medical and Biological Sciences