A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry, using so-called gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups.
Soon after special relativity was developed in 1905 it was realized that Einstein's velocity addition law could be interpreted in terms of hyperbolic geometry (see History of special relativity). Only colinear velocities are commutative and associative, but in general, addition of non-colinear velocities is non-associative and non-commutative. Relativistic velocity addition does however have a weaker form of associativity, called gyroassociativity: u + (v + w) = (u + v) + gyr(w), and a weaker form of commutativity, called gyrocommutativity: u + v = gyr(v + u). The use of the prefix gyro comes from Thomas gyration which is the mathematical abstraction of the gyroscopic Thomas precession into an operator called a gyrator and denoted gyr.
Different models of hyperbolic geometry are regulated by different gyrovector spaces. The Beltrami–Klein model is regulated by gyrovector spaces based on relativistic velocity addition. The Poincaré ball model is regulated by gyrovector spaces based on part of the formula for Möbius transformations.
Read more about Gyrovector Space: Beltrami–Klein Disc/ball Model and Einstein Addition, Poincaré Disc/ball Model and Möbius Addition, Proper Velocity Space Model and Proper Velocity Addition, Isomorphisms, Gyrotrigonometry, Bloch Vectors, Book Reviews
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