Gyrogroups

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 ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ is given in coordinate-independent form as:

${\displaystyle \mathbf {u} \oplus _{E}\mathbf {v} ={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left\{\mathbf {u} +{\frac {1}{\gamma _{\mathbf {u} }}}\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{1+\gamma _{\mathbf {u} }}}(\mathbf {u} \cdot \mathbf {v} )\mathbf {u} \right\}}$

where ${\displaystyle \gamma _{\mathbf {u} }}$ is the gamma factor given by the equation ${\displaystyle \gamma _{\mathbf {u} }={\frac {1}{\sqrt {1-{\frac {|\mathbf {u} |^{2}}{c^{2}}}}}}}$.

Using coordinates this becomes:

${\displaystyle {\begin{pmatrix}w_{1}\\w_{2}\\w_{3}\\\end{pmatrix}}={\frac {1}{1+{\frac {u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}}{c^{2}}}}}\left\{\left[1+{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{1+\gamma _{\mathbf {u} }}}(u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3})\right]{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\\\end{pmatrix}}+{\frac {1}{\gamma _{\mathbf {u} }}}{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\\\end{pmatrix}}\right\}}$

where ${\displaystyle \gamma _{\mathbf {u} }={\frac {1}{\sqrt {1-{\frac {u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}{c^{2}}}}}}}$.

Einstein velocity addition is commutative and associative only when ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ are parallel. In fact

${\displaystyle \mathbf {u} \oplus \mathbf {v} =\mathrm {gyr} [\mathbf {u} ,\mathbf {v} ](\mathbf {v} \oplus \mathbf {u} )}$

and

${\displaystyle \mathbf {u} \oplus (\mathbf {v} \oplus \mathbf {w} )=(\mathbf {u} \oplus \mathbf {v} )\oplus \mathrm {gyr} [\mathbf {u} ,\mathbf {v} ]\mathbf {w} }$

where "gyr" is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by

${\displaystyle \mathrm {gyr} [\mathbf {u} ,\mathbf {v} ]\mathbf {w} =\ominus (\mathbf {u} \oplus \mathbf {v} )\oplus (\mathbf {u} \oplus (\mathbf {v} \oplus \mathbf {w} ))}$

for all w. Thomas precession has an interpretation in hyperbolic geometry as the negative hyperbolic triangle defect.

Einstein gyrovector spaces

Let s be any positive constant, let (V,+,.) be any real inner product space and let V_s={v  ∈  V :|v|<s}. An Einstein gyrovector space (V_s, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$) is an Einstein gyrogroup (V_s, ${\displaystyle \oplus }$) 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  ∈ V_s, 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  ∈ V_s:

n ${\displaystyle \otimes }$ v = v ${\displaystyle \oplus }$ ... ${\displaystyle \oplus }$ v	n terms
(r1 + r2) ${\displaystyle \otimes }$ v = r1 ${\displaystyle \otimes }$ v ${\displaystyle \oplus }$ r2 ${\displaystyle \otimes }$ v	Scalar distributive law
(r1r2) ${\displaystyle \otimes }$ v = r1 ${\displaystyle \otimes }$ (r2 ${\displaystyle \otimes }$ v)	Scalar associative law
r ${\displaystyle \otimes }$(r1 ${\displaystyle \otimes }$ a ${\displaystyle \oplus }$ r2 ${\displaystyle \otimes }$ a) = r ${\displaystyle \otimes }$(r1 ${\displaystyle \otimes }$ a) ${\displaystyle \oplus }$ r ${\displaystyle \otimes }$(r2 ${\displaystyle \otimes }$ a)	Monodistributive law

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

${\displaystyle z\to {e^{i\theta }}{\frac {a+z}{1+a{\bar {z}}}}}$ which can be written as ${\displaystyle e^{i\theta }{(a\oplus _{M}{z})}}$ which defines the Möbius addition ${\displaystyle {a\oplus _{M}{z}}={\frac {a+z}{1+a{\bar {z}}}}}$.

To generalize this to higher dimensions the complex numbers are considered as vectors in the plane ${\displaystyle \mathbf {\mathrm {R} } ^{2}}$, and Möbius addition is rewritten in vector form as:

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

This gives the vector addition of points in the Poincaré ball model of hyperbolic geometry where 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 V_s={v  ∈  V :|v|<s}. A Möbius gyrovector space (V_s, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$) is a Möbius gyrogroup (V_s, ${\displaystyle \oplus }$) 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  ∈ V_s, 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:^[6][12][13]

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

where ${\displaystyle \beta _{\mathbf {w} }}$ is the beta factor given by ${\displaystyle \beta _{\mathbf {w} }={\frac {1}{\sqrt {1+{\frac {|\mathbf {w} |^{2}}{c^{2}}}}}}}$.

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 ${\displaystyle \oplus _{U}}$ 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 v_m, v_e and v_u then the isomorphisms are given by:

E${\displaystyle \rightarrow }$U by ${\displaystyle \gamma _{\mathbf {v} _{e}}\mathbf {v} _{e}}$

U${\displaystyle \rightarrow }$E by ${\displaystyle \beta _{\mathbf {v} _{u}}\mathbf {v} _{u}}$

E${\displaystyle \rightarrow }$M by ${\displaystyle {\frac {1}{2}}\otimes _{E}\mathbf {v} _{e}}$

M${\displaystyle \rightarrow }$E by ${\displaystyle 2\otimes _{M}\mathbf {v} _{m}}$

M${\displaystyle \rightarrow }$U by ${\displaystyle 2{{{\gamma }^{2}}_{\mathbf {v} _{m}}}\mathbf {v} _{m}}$

U${\displaystyle \rightarrow }$M by ${\displaystyle {\frac {\beta _{\mathbf {v} _{u}}}{1+\beta _{\mathbf {v} _{u}}}}\mathbf {v} _{u}}$

From this table the relation between ${\displaystyle \oplus _{E}}$ and ${\displaystyle \oplus _{M}}$ is given by the equations:

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

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

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.

Bloch vectors

Bloch vectors which belong to the open unit ball of the Euclidean 3-space, can be studied with Einstein addition^[18] or Möbius addition.^[6]