Hyperbolic geometry. (English) Zbl 1304.83002

The author considers the space of relativistic velocities in the Minkowski vector space \(\mathbb{R}^{1,n}\) as the unit ball \(B^n \subset \mathbb{R}^n \) with the non-commutative non-associative invertible binary operation \(\oplus\), defined by the relativistic addition of velocities. Also a non-standard multiplication of vectors from \(B^n\) with number and new ternary operation, called gyration, are defined. Starting from these operations, the author defines axiomatically a notion of gyrovector space. The properties of this space and some geometric notions like lines and segments are defined. Imitating the standard construction of barycentric coordinates, the author defines gyrobarycentric coordinates and considers some of their applications.


83A05 Special relativity
51M10 Hyperbolic and elliptic geometries (general) and generalizations
83C10 Equations of motion in general relativity and gravitational theory