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 group-like 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 gyrogroup (G, ) consists of an underlying set G and a binary operation satisfying the following axioms: In G there is at least one element 0 called a left identity with 0 a = a for all a in G. For each a in G there is an element a in G called a left inverse of a with (a) a = 0. For any a, b, c in G there exists a unique element gyr[a,b]c in G such that the binary operation obeys the left gyroassociative law: a (b c) = (a b) gyr[a,b]c The map gyr[a,b]: G → G given by c ↦ gyr[a,b]c is an automorphism of the magma (G, ) – that is, gyr[a,b] is a member of Aut(G, ) and the automorphism gyr[a,b] of G is called the gyroautomorphism of G generated by a, b in G. The operation gyr: G × G → Aut(G, ) is called the gyrator of G. The gyroautomorphism gyr[a,b] has the left loop property gyr[a,b] = gyr[a b,b]
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[a,b] defined as the identity map for all a and b in G.
An example of a finite gyrogroup is given in .
Identities
Some identities which hold in any gyrogroup (G, ) are:
(gyration) (left associativity) (right associativity)
Furthermore, one may prove the Gyration inversion law, which is the motivation for the definition of gyrocommutativity below: (gyration inversion law)
Some additional theorems satisfied by the Gyration group of any gyrogroup include: (identity gyrations) (gyroautomorphism inversion law) (gyration even property) (right loop property) (left loop property)
More identities given on page 50 of . One particularly useful consequence of the above identities is that Gyrogroups satisfy the left Bol property
Gyrocommutativity
A gyrogroup (G,) is gyrocommutative if its binary operation obeys the gyrocommutative law: a b = gyr[a,b](b 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: a b = a gyr[a,b]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 and is given in coordinate-independent form as:
where is the gamma factor given by the equation .
Using coordinates this becomes:
where .
Einstein velocity addition is commutative and associative only when and are parallel. In fact
and
where "gyr" is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by
for all w. Thomas precession has an interpretation in hyperbolic geometry as the negative hyperbolic triangle defect.