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Source: 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

Mathematics of gyrovector spaces

= Gyrogroups

= Beltrami–Klein disc/ball model and Einstein addition

space

gyrovector

space

gyrovector space (Vs,

⊕

{\displaystyle \oplus }

,

⊗

{\displaystyle \otimes }

) is an Einstein gyrogroup (Vs,

⊕

{\displaystyle \oplus }

) with scalar multiplication given by r

⊗

{\displaystyle \otimes }

v = s tanh(r tanh−1(|v|/s))v/|v| where r is any real number, v ∈ Vs, v ≠ 0 and r

⊗

{\displaystyle \otimes }

0 = 0 with the notation v

⊗

{\displaystyle \otimes }

r = r

⊗

{\displaystyle \otimes }

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:

= Poincaré disc/ball model and Möbius addition

=
The Möbius transformation of the open unit disc in the complex plane is given by the polar decomposition

z
→

e

i
θ

a
+
z

1
+
a

z
¯

{\displaystyle z\to {e^{i\theta }}{\frac {a+z}{1+a{\bar {z}}}}}

which can be written as

e

i
θ

(
a

⊕

M

z

)

{\displaystyle e^{i\theta }{(a\oplus _{M}{z})}}

which defines the Möbius addition

a

⊕

M

z

=

a
+
z

1
+
a

z
¯

{\displaystyle {a\oplus _{M}{z}}={\frac {a+z}{1+a{\bar {z}}}}}

.
To generalize this to higher dimensions the complex numbers are considered as vectors in the plane

R

2

{\displaystyle \mathbf {\mathrm {R} } ^{2}}

, and Möbius addition is rewritten in vector form as:

u

⊕

M

v

=

(
1
+

2

s

2

u

⋅

v

+

1

s

2

|

v

|

2

)

u

+
(
1
−

1

s

2

|

u

|

2

)

v

1
+

2

s

2

u

⋅

v

+

1

s

4

|

u

|

2

|

v

|

2

{\displaystyle \mathbf {u} \oplus _{M}\mathbf {v} ={\frac {(1+{\frac {2}{s^{2}}}\mathbf {u} \cdot \mathbf {v} +{\frac {1}{s^{2}}}|\mathbf {v} |^{2})\mathbf {u} +(1-{\frac {1}{s^{2}}}|\mathbf {u} |^{2})\mathbf {v} }{1+{\frac {2}{s^{2}}}\mathbf {u} \cdot \mathbf {v} +{\frac {1}{s^{4}}}|\mathbf {u} |^{2}|\mathbf {v} |^{2}}}}

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.

= Möbius gyrovector spaces

=
Let s be any positive constant, let (V,+,.) be any real inner product space and let Vs={v ∈ V :|v|gyrovector space (Vs,

⊕

{\displaystyle \oplus }

,

⊗

{\displaystyle \otimes }

) is a Möbius gyrogroup (Vs,

⊕

{\displaystyle \oplus }

) with scalar multiplication given by r

⊗

{\displaystyle \otimes }

v = s tanh(r tanh−1(|v|/s))v/|v| where r is any real number, v ∈ Vs, v ≠ 0 and r

⊗

{\displaystyle \otimes }

0 = 0 with the notation v

⊗

{\displaystyle \otimes }

r = r

⊗

{\displaystyle \otimes }

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.

= Proper velocity space model and proper velocity addition

=
A proper velocity space model of hyperbolic geometry is given by proper velocities with vector addition given by the proper velocity addition formula:

u

⊕

U

v

=

u

+

v

+

{

β

u

1
+

β

u

u

⋅

v

c

2

+

1
−

β

v

β

v

}

u

{\displaystyle \mathbf {u} \oplus _{U}\mathbf {v} =\mathbf {u} +\mathbf {v} +\left\{{\frac {\beta _{\mathbf {u} }}{1+\beta _{\mathbf {u} }}}{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}+{\frac {1-\beta _{\mathbf {v} }}{\beta _{\mathbf {v} }}}\right\}\mathbf {u} }

where

β

w

{\displaystyle \beta _{\mathbf {w} }}

is the beta factor given by

β

w

=

1

1
+

|

w

|

2

c

2

{\displaystyle \beta _{\mathbf {w} }={\frac {1}{\sqrt {1+{\frac {|\mathbf {w} |^{2}}{c^{2}}}}}}}

.
This formula provides a model that uses a whole space compared to other models of hyperbolic geometry which use discs or half-planes.
A proper velocity gyrovector space is a real inner product space V, with the proper velocity gyrogroup addition

⊕

U

{\displaystyle \oplus _{U}}

and with scalar multiplication defined by r

⊗

{\displaystyle \otimes }

v = s sinh(r sinh−1(|v|/s))v/|v| where r is any real number, v ∈ V, v ≠ 0 and r

⊗

{\displaystyle \otimes }

0 = 0 with the notation v

⊗

{\displaystyle \otimes }

r = r

⊗

{\displaystyle \otimes }

v.

= Isomorphisms

=
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

⊕

E

{\displaystyle \oplus _{E}}

and

⊕

M

{\displaystyle \oplus _{M}}

is given by the equations:

u

⊕

E

v

=
2
⊗

(

1
2

⊗

u

⊕

M

1
2

⊗

v

)

{\displaystyle \mathbf {u} \oplus _{E}\mathbf {v} =2\otimes \left({{\frac {1}{2}}\otimes \mathbf {u} \oplus _{M}{\frac {1}{2}}\otimes \mathbf {v} }\right)}

u

⊕

M

v

=

1
2

⊗

(

2
⊗

u

⊕

E

2
⊗

v

)

{\displaystyle \mathbf {u} \oplus _{M}\mathbf {v} ={\frac {1}{2}}\otimes \left({2\otimes \mathbf {u} \oplus _{E}2\otimes \mathbf {v} }\right)}

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."

Notes and references

Domenico Giulini, Algebraic and geometric structures of Special Relativity, A Chapter in "Special Relativity: Will it Survive the Next 100 Years?", edited by Claus Lämmerzahl, Jürgen Ehlers, Springer, 2006.

External links

Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint
Ungar, Abraham A. (2001). "Hyperbolic Trigonometry and its Application in the Poincaré Ball Model of Hyperbolic Geometry". pp. 6–19. CiteSeerX 10.1.1.17.6107.