Gyrovector space

A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups. Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts - "boosts" are aspects of relative velocities, and should not be conflated with "translations"). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity.
Name.
Gyrogroups are weakly associative grouplike structures. Ungar proposed the term gyrogroup for what he called a gyrocommutative-gyrogroup, with the term gyrogroup being reserved for the non-gyrocommutative case, in analogy with groups vs. abelian groups. Gyrogroups are a type of Bol loop. Gyrocommutative gyrogroups are equivalent to "K-loops" although defined differently. The terms "Bruck loop" and "dyadic symset" are also in use.
Mathematics of gyrovector spaces.
Gyrogroups.
Axioms.
A magma ("G", formula_1) is a gyrogroup if its binary operation satisfies the following axioms:
The first pair of axioms are like the group axioms. The last pair present the gyrator axioms and the middle axiom links the two pairs.
Since a gyrogroup has inverses and an identity it qualifies as a quasigroup and a loop.
Gyrogroups are a generalization of groups. Every group is an example of a gyrogroup with gyr defined as the identity map.
An example of a finite gyrogroup is given in .
Identities.
Some identities which hold in any gyrogroup (G,formula_1):
Furthermore, one may prove the Gyration inversion law, which is the motivation for the definition of gyrocommutativity below:
Some additional theorems satisfied by the Gyration group of any gyrogroup include:
More identities given on page 50 of .
Gyrocommutativity.
A gyrogroup (G,formula_1) is gyrocommutative if its binary operation obeys the gyrocommutative law: a formula_1 b = gyr[a, b](b formula_1 a). For relativistic velocity addition, this formula showing the role of rotation relating a+b and b+a was published in 1914 by Ludwik Silberstein
Coaddition.
In every gyrogroup, a second operation can be defined called "coaddition": aformula_27 b = aformula_1 gyr[a,formula_3b]b for all a, b  ∈  G. Coaddition is commutative if the gyrogroup addition is gyrocommutative.
Beltrami–Klein disc/ball model and Einstein addition.
Relativistic velocities can be considered as points in the Beltrami–Klein model of hyperbolic geometry and so vector addition in the Beltrami–Klein model can be given by the velocity addition formula. In order for the formula to generalize to vector addition in hyperbolic space of dimensions greater than 3, the formula must be written in a form that avoids use of the cross product in favour of the dot product.
In the general case, the Einstein velocity addition of two velocities formula_30 and formula_31 is given in coordinate-independent form as:
where formula_33 is the gamma factor given by the equation formula_34 and formula_35 is given by the equations:
formula_36
formula_37
This is related to the connection between Möbius transformations and Lorentz transformations.
Gyrotrigonometry.
Gyrotrigonometry is the use of gyroconcepts to study hyperbolic triangles.
Hyperbolic trigonometry as usually studied uses the hyperbolic functions cosh, sinh etc., and this contrasts with spherical trigonometry which uses the Euclidean trigonometric functions cos, sin, but with spherical triangle identities instead of ordinary plane triangle identities. Gyrotrigonometry takes the approach of using the ordinary trigonometric functions but in conjunction with gyrotriangle identities.
Triangle centers.
The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must "not" encapsulate the specification of the anglesum being 180 degrees.
Gyroparallelogram addition.
Using gyrotrigonometry, a gyrovector addition can be found which operates according to the gyroparallelogram law. This is the coaddition to the gyrogroup operation. Gyroparallelogram addition is commutative.
The gyroparallelogram law is similar to the parallelogram law in that a gyroparallelogram is a hyperbolic quadrilateral the two gyrodiagonals of which intersect at their gyromidpoints, just as a parallelogram is a Euclidean quadrilateral the two diagonals of which intersect at their midpoints.
Bloch vectors.
Bloch vectors which belong to the open unit ball of the Euclidean 3-space, can be studied with Einstein addition or Möbius addition.
Book reviews.
A review of one of the earlier gyrovector books says the following:
"Over the years, there have been a handful of attempts to promote the non-Euclidean style for use in problem solving in relativity and electrodynamics, the failure of which to attract any substantial following, compounded by the absence of any positive results must give pause to anyone considering a similar undertaking. Until recently, no one was in a position to offer an improvement on the tools available since 1912. In his new book, Ungar furnishes the crucial missing element from the panoply of the non-Euclidean style: an elegant nonassociative algebraic formalism that fully exploits the structure of Einstein’s law of velocity composition."