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 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").
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.
Gyrocommutative gyrogroups are equivalent to K-loops[2] although defined differently.
The terms Bruck loop[3] and dyadic symset[4] are also in use.
Since a gyrogroup has inverses and an identity it qualifies as a quasigroup and a loop.
) are: 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 [6].
For relativistic velocity addition, this formula showing the role of rotation relating a + b and b + a was published in 1914 by Ludwik Silberstein.
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 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.
If the 3 × 3 matrix form of the rotation applied to 3-coordinates is given by gyr[u,v], then the 4 × 4 matrix rotation applied to 4-coordinates is given by: The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:[9][10] This fact that either B(u
u) can be used depending whether you write the rotation before or after explains the velocity composition paradox.
The composition of two Lorentz transformations L(u,U) and L(v,V) which include rotations U and V is given by:[11] In the above, a boost can be represented as a 4 × 4 matrix.
The matrix entries depend on the components of the 3-velocity v, and that's what the notation B(v) means.
v. Einstein scalar multiplication does not distribute over Einstein addition except when the gyrovectors are colinear (monodistributivity), but it has other properties of vector spaces: For any positive integer n and for all real numbers r,r1,r2 and v ∈ Vs: The Möbius transformation of the open unit disc in the complex plane is given by the polar decomposition To generalize this to higher dimensions the complex numbers are considered as vectors in the plane
, and Möbius addition is rewritten in vector form as: This gives the vector addition of points in the Poincaré ball model of hyperbolic geometry where radius s=1 for the complex unit disc now becomes any s>0.
v. Möbius scalar multiplication coincides with Einstein scalar multiplication (see section above) and this stems from Möbius addition and Einstein addition coinciding for vectors that are parallel.
v. A gyrovector space isomorphism preserves gyrogroup addition and scalar multiplication and the inner product.
The three gyrovector spaces Möbius, Einstein and Proper Velocity are isomorphic.
If M, E and U are Möbius, Einstein and Proper Velocity gyrovector spaces respectively with elements vm, ve and vu then the isomorphisms are given by: From this table the relation between
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.
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.
[14][15][16] Using gyrotrigonometry, a gyrovector addition can be found which operates according to the gyroparallelogram law.
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.
[6] A review of one of the earlier gyrovector books[19] 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.
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.