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{{short description|Mathematical space used to study hyperbolic geometry}}
{{Primary sources|date=August 2021}}
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A '''gyrovector space''' is a [[mathematics|mathematical]] concept proposed by Abraham A. Ungar for studying [[hyperbolic geometry]] in analogy to the way [[vector space]]s are used in [[Euclidean geometry]].<ref name=anhyp>Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Foundations and Applications", Published by World Scientific, {{ISBN|981-256-457-8}}, {{ISBN|978-981-256-457-3}}</ref>  Ungar introduced the concept of  gyrovectors that have addition based on gyrogroups instead of vectors which have  addition based on [[group (mathematics)|group]]s.  Ungar developed his concept as a tool for the formulation of [[special relativity]] as an alternative to  the use of [[Lorentz transformation]]s to represent compositions of velocities (also called [[Lorentz boost|boost]]s – "boosts" are aspects of [[relative velocity|relative velocities]], and should not be conflated with "[[Translation (geometry)|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 group]]s. Gyrogroups are a type of [[Bol loop]]. Gyrocommutative gyrogroups are equivalent to ''K-loops''<ref>Hubert Kiechle (2002), "Theory of K-loops", Published by Springer,{{ISBN|3-540-43262-0}}, {{ISBN|978-3-540-43262-3}}</ref> although defined differently. The terms ''Bruck loop''<ref>Larissa Sbitneva (2001), Nonassociative Geometry of Special Relativity, International Journal of Theoretical Physics, Springer, Vol.40, No.1 / Jan 2001 {{doi|10.1023/A:1003764217705}}</ref> and ''dyadic symset''<ref>J lawson Y Lim (2004), Means on dyadic symmetrie sets and polar decompositions, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Springer, Vol.74, No.1 / Dec 2004 {{doi|10.1007/BF02941530}}</ref> are also in use.

==Mathematics of gyrovector spaces==

===Gyrogroups===

====Axioms====
A ''gyrogroup'' (''G'', <math>\oplus</math>) consists of an underlying set ''G'' and a [[binary operation]] <math>\oplus</math> satisfying the following axioms:
#In ''G'' there is at least one element 0 called a left identity with 0 <math>\oplus</math> ''a'' = ''a'' for all ''a'' in ''G''.
#For each ''a'' in ''G'' there is an element <math>\ominus</math>''a'' in ''G'' called a left inverse of a with (<math>\ominus</math>''a'') <math>\oplus</math> ''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'' <math>\oplus</math> (''b'' <math>\oplus</math> ''c'') = (''a'' <math>\oplus</math> ''b'') <math>\oplus</math> gyr[''a'',''b'']''c''
#The map gyr[''a'',''b'']: ''G'' → ''G'' given by ''c'' ↦ gyr[''a'',''b'']''c'' is an [[automorphism]] of the [[magma (algebra)|magma]] (''G'', <math>\oplus</math>) – that is, gyr[''a'',''b''] is a member of Aut(''G'', <math>\oplus</math>) and the automorphism gyr[''a'',''b''] of ''G'' is called the gyroautomorphism of ''G'' generated by ''a'',&nbsp;''b'' in ''G''.  The operation gyr: ''G''&nbsp;×&nbsp;''G''&nbsp;→&nbsp;Aut(''G'',&nbsp;<math>\oplus</math>) is called the gyrator of ''G''.
#The gyroautomorphism gyr[''a'',''b''] has the left [[loop (algebra)|loop]] property gyr[''a'',''b''] = gyr[''a'' <math>\oplus</math> ''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 (algebra)|loop]].

Gyrogroups are a generalization of [[group (mathematics)|group]]s. 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 <ref>{{cite journal | doi = 10.1016/S0898-1221(00)00163-2 |url=http://www.ndsu.edu/pubweb/~ungar/dir_webpapers/hyptrig01.ps | volume=40 | issue=2–3 | title=Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry | year=2000 | journal=Computers & Mathematics with Applications | pages=313–332 [317] | last1 = Ungar | first1 = A.A.| doi-access= }}</ref><!-- -->.<!-- the empty comment keeps bots from moving the punctuation. -->

====Identities====

Some identities which hold in any gyrogroup (''G'', <math>\oplus</math>) are:

#<math>\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))</math> (gyration)
#<math>\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}) = (\mathbf{u} \oplus \mathbf{v})\oplus \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}</math> (left associativity)
#<math>(\mathbf{u} \oplus \mathbf{v}) \oplus \mathbf{w} = \mathbf{u} \oplus (\mathbf{v}\oplus \mathrm{gyr}[\mathbf{v},\mathbf{u}]\mathbf{w})</math> (right associativity)

Furthermore, one may prove the Gyration inversion law, which is the motivation for the definition of gyrocommutativity below:
#<math> \ominus (\mathbf{u} \oplus \mathbf{v}) = \mathrm{gyr}[\mathbf{u},\mathbf{v}] (\ominus \mathbf{v} \ominus \mathbf{u}) </math> (gyration inversion law)

Some additional theorems satisfied by the Gyration group of any gyrogroup include:
#<math>\mathrm{gyr}[\mathbf{0},\mathbf{u}] = \mathrm{gyr}[\mathbf{u},\mathbf{u}] = \mathrm{gyr}[\ominus \mathbf{u},\mathbf{u}] = I </math> (identity gyrations)
#<math>\mathrm{gyr}^{-1}[\mathbf{u},\mathbf{v}] =\mathrm{gyr}[\mathbf{v},\mathbf{u}] </math> (gyroautomorphism inversion law)
#<math>\mathrm{gyr}[\ominus \mathbf{u},\ominus  \mathbf{v}] =\mathrm{gyr}[\mathbf{u},\mathbf{v}] </math> (gyration even property)
#<math>\mathrm{gyr}[\mathbf{u},\mathbf{v}] =\mathrm{gyr}[\mathbf{u},\mathbf{v} \oplus \mathbf{u}] </math> (right loop property)
#<math>\mathrm{gyr}[\mathbf{u},\mathbf{v}] =\mathrm{gyr}[\mathbf{u} \oplus \mathbf{v},\mathbf{v}] </math> (left loop property)

More identities given on page 50 of <ref name=ung2008>[https://books.google.com/books?id=p-d6MozY3GUC Analytic hyperbolic geometry and Albert Einstein's special theory of relativity], Abraham A. Ungar, World Scientific, 2008, {{ISBN|978-981-277-229-9}}</ref><!-- -->.<!--The empty comment keeps bots from moving the punctuation.--> One particularly useful consequence of the above identities is that Gyrogroups satisfy the [[Bol loop|left Bol property]]

#<math>(\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{u})) \oplus \mathbf{w} = \mathbf{u} \oplus (\mathbf{v} \oplus (\mathbf{u} \oplus \mathbf{w})) </math>

====Gyrocommutativity====

A gyrogroup (G,<math>\oplus</math>) is ''gyrocommutative'' if its binary operation obeys the gyrocommutative law: ''a'' <math>\oplus</math> ''b'' = gyr[''a'',''b''](''b'' <math>\oplus</math> ''a''). For relativistic velocity addition, this formula showing the role of rotation relating ''a''&nbsp;+&nbsp;''b'' and ''b''&nbsp;+&nbsp;''a'' was published in 1914 by [[Ludwik Silberstein]].<ref>Ludwik Silberstein, The theory of relativity, Macmillan, 1914</ref><ref>Page 214, Chapter 5, Symplectic matrices: first order systems and special relativity, Mark Kauderer, World Scientific, 1994, {{ISBN|978-981-02-1984-0}}</ref>

====Coaddition====

In every gyrogroup, a second operation can be defined called ''coaddition'': ''a'' <math>\boxplus</math> ''b'' = ''a'' <math>\oplus</math> gyr[''a'',<math>\ominus</math>''b'']''b'' for all ''a'', ''b''&nbsp;∈&nbsp;''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|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 formula|velocity addition]] of two velocities <math>\mathbf{u}</math> and <math>\mathbf{v}</math> is given in coordinate-independent form as:

:<math>\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\}</math>

where <math>\gamma_\mathbf{u}</math> is the gamma factor given by the equation <math>\gamma_\mathbf{u}=\frac{1}{\sqrt{1-\frac{|\mathbf{u}|^2}{c^2}}}</math>.

Using coordinates this becomes:

:<math>\begin{pmatrix}w_1\\ w_2\\ w_3\\ \end{pmatrix}=\frac{1}{1+\frac{u_1v_1+u_2v_2+u_3v_3}{c^2}}\left\{\left[1+\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{1+\gamma_\mathbf{u}}(u_1v_1+u_2v_2+u_3v_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\}</math>

where <math>\gamma_\mathbf{u}=\frac{1}{\sqrt{1-\frac{u_1^2+u_2^2+u_3^2}{c^2}}}</math>.

Einstein velocity addition is [[commutative]] and [[associative]] ''only'' when <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are ''parallel''.  In fact

:<math>\mathbf{u} \oplus \mathbf{v}=\mathrm{gyr}[\mathbf{u},\mathbf{v}](\mathbf{v} \oplus \mathbf{u})
</math>
and 
:<math>\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}) = (\mathbf{u} \oplus \mathbf{v})\oplus \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}</math>
where "gyr" is the mathematical abstraction of [[Thomas precession]] into an operator called Thomas gyration and given by
:<math>\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))</math>
for all '''w'''. Thomas precession has an interpretation in hyperbolic geometry as the negative [[hyperbolic triangle]] defect.

====Lorentz transformation composition====

If the 3&nbsp;×&nbsp;3 matrix form of the rotation applied to 3-coordinates is given by gyr['''u''','''v'''], then the 4&nbsp;×&nbsp;4 matrix rotation applied to 4-coordinates is given by:
:<math>
\mathrm{Gyr}[\mathbf{u},\mathbf{v}]=
\begin{pmatrix}
1 & 0 \\
0 & \mathrm{gyr}[\mathbf{u},\mathbf{v}]
\end{pmatrix}
</math>.<ref name="relcompara"/>

The composition of two [[Lorentz boost]]s B('''u''') and B('''v''') of velocities '''u''' and '''v''' is given by:<ref name="relcompara">Ungar, A. A: The relativistic velocity composition paradox and the Thomas rotation. Found. Phys. 19, 1385–1396 (1989) {{doi|10.1007/BF00732759}}</ref><ref>{{cite journal | citeseerx = 10.1.1.35.1131 | title = The relativistic composite-velocity reciprocity principle | first = A. A. | last = Ungar | journal = Foundations of Physics | year = 2000 | publisher = Springer |volume=30|issue=2| pages = 331–342 |doi=10.1023/A:1003653302643| bibcode = 2000FoPh...30..331U | s2cid = 118634052 }}</ref>

:<math>B(\mathbf{u})B(\mathbf{v})=B(\mathbf{u}\oplus\mathbf{v})\mathrm{Gyr}[\mathbf{u},\mathbf{v}]=\mathrm{Gyr}[\mathbf{u},\mathbf{v}]B(\mathbf{v}\oplus\mathbf{u})</math>

This fact that either B('''u'''<math>\oplus</math>'''v''') or B('''v'''<math>\oplus</math>'''u''') can be used depending whether you write the rotation before or after explains the [[Velocity-addition formula#Velocity composition paradox|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:<ref>eq. (55), Thomas rotation and the parametrization of the Lorentz transformation group, AA Ungar – Foundations of Physics Letters, 1988</ref>

:<math>L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v}, \mathrm{gyr}[\mathbf{u},U\mathbf{v}]UV)</math>

In the above, a boost can be represented as a 4&nbsp;×&nbsp;4 matrix. The boost matrix B('''v''') means the boost B that uses the components of '''v''', i.e. ''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub> in the entries of the matrix, or rather the components of '''v'''/''c'' in the representation that is used in the section [[Lorentz transformation#Matrix forms]].  The matrix entries depend on the components of the 3-velocity '''v''', and that's what the notation B('''v''') means. It could be argued that the entries depend on the components of the 4-velocity because 3 of the entries of the 4-velocity are the same as the entries of the 3-velocity, but the usefulness of parameterizing the boost by 3-velocity is that the resultant boost you get from the composition of two boosts uses the components of the 3-velocity composition '''u'''<math>\oplus</math>'''v''' in the 4&nbsp;×&nbsp;4 matrix B('''u'''<math>\oplus</math>'''v'''). But the resultant boost also needs to be multiplied by a rotation matrix because boost composition (i.e. the multiplication of two 4&nbsp;×&nbsp;4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4&nbsp;×&nbsp;4 matrix that corresponds to the rotation Gyr['''u''','''v'''] to get B('''u''')B('''v''') = B('''u'''<math>\oplus</math>'''v''')Gyr['''u''','''v'''] = Gyr['''u''','''v''']B('''v'''<math>\oplus</math>'''u''').

====Einstein gyrovector spaces====

Let s be any positive constant, let (V,+,.) be any real [[inner product space]] and let V<sub>s</sub>={'''v''' &nbsp;∈&nbsp; V :|'''v'''|<s}.  An Einstein gyrovector space (''V''<sub>''s''</sub>,&nbsp;<math>\oplus</math>,&nbsp;<math>\otimes</math>) is an Einstein gyrogroup (''V''<sub>''s''</sub>,&nbsp;<math>\oplus</math>) with scalar multiplication given by ''r''<math>\otimes</math>'''v'''&nbsp;=&nbsp;''s''&nbsp;tanh(''r''&nbsp;tanh<sup>&minus;1</sup>(|'''v'''|/''s''))'''v'''/|'''v'''| where ''r'' is any real number, '''v''' &nbsp;∈&nbsp;''V''<sub>''s''</sub>, '''v'''&nbsp;≠&nbsp;'''0''' and ''r''&nbsp;<math>\otimes</math>&nbsp;'''0'''&nbsp;=&nbsp;'''0''' with the notation '''v'''&nbsp;<math>\otimes</math>&nbsp;''r''&nbsp;=&nbsp;''r''&nbsp;<math>\otimes</math>&nbsp;'''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'',''r''<sub>1</sub>,''r''<sub>2</sub> and '''v''' &nbsp;∈&nbsp;''V''<sub>''s''</sub>:
{| class="wikitable"
|-
| ''n''&nbsp;<math>\otimes</math>&nbsp;'''v'''&nbsp;=&nbsp;'''v'''&nbsp;<math>\oplus</math>&nbsp;...&nbsp;<math>\oplus</math>&nbsp;'''v'''
| ''n'' terms
|-
| (''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>)&nbsp;<math>\otimes</math>&nbsp;'''v'''&nbsp;=&nbsp;''r''<sub>1</sub>&nbsp;<math>\otimes</math>&nbsp;'''v'''&nbsp;<math>\oplus</math>&nbsp;''r''<sub>2</sub>&nbsp;<math>\otimes</math>&nbsp;'''v'''
| Scalar distributive law
|-
| (''r''<sub>1</sub>''r''<sub>2</sub>)&nbsp;<math>\otimes</math>&nbsp;'''v'''&nbsp;=&nbsp;''r''<sub>1</sub>&nbsp;<math>\otimes</math>&nbsp;(''r''<sub>2</sub>&nbsp;<math>\otimes</math>&nbsp;'''v''')
| Scalar associative law
|-
| ''r''&nbsp;<math>\otimes</math>(''r''<sub>1</sub>&nbsp;<math>\otimes</math>&nbsp;'''a'''&nbsp;<math>\oplus</math>&nbsp;''r''<sub>2</sub>&nbsp;<math>\otimes</math>&nbsp;'''a''')&nbsp;=&nbsp;''r''&nbsp;<math>\otimes</math>(''r''<sub>1</sub>&nbsp;<math>\otimes</math>&nbsp;'''a''')&nbsp;<math>\oplus</math>&nbsp;''r''&nbsp;<math>\otimes</math>(''r''<sub>2</sub>&nbsp;<math>\otimes</math>&nbsp;'''a''') 
| Monodistributive law
|}

==={{anchor|Möbius addition}} 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
:<math>z\to {e^{i\theta}}{\frac{a+z}{1+a\bar{z}}}</math>{{citation needed|reason=Why is the addition defined like this? Isn't this supposed to be a conformal mapping of z? This is not.|date=February 2023}}{{clarify|reason=Where does the angle theta come from? This is unspecified! Please explain what the variables in formulas that you use mean before you use them in articles.|date=February 2023}} which can be written as <math>e^{i\theta} {(a\oplus_M {z})} </math> which defines the Möbius addition <math>{a\oplus_M {z}}= \frac{a+z}{1+a\bar{z}}</math>.

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

:<math>\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}</math>

This gives the vector addition of points in the [[Poincaré disk model|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 V<sub>s</sub>={'''v''' &nbsp;∈&nbsp; V :|'''v'''|<s}.  A Möbius gyrovector space (''V''<sub>''s''</sub>,&nbsp;<math>\oplus</math>,&nbsp;<math>\otimes</math>) is a Möbius gyrogroup (''V''<sub>''s''</sub>,&nbsp;<math>\oplus</math>) with scalar multiplication given by ''r''&nbsp;<math>\otimes</math>'''v'''&nbsp;=&nbsp;''s''&nbsp;tanh(''r''&nbsp;tanh<sup>&minus;1</sup>(|'''v'''|/''s''))'''v'''/|'''v'''| where ''r'' is any real number, '''v''' &nbsp;∈&nbsp;''V''<sub>''s''</sub>, '''v'''&nbsp;≠&nbsp;'''0''' and ''r''&nbsp;<math>\otimes</math>&nbsp;'''0'''&nbsp;=&nbsp;'''0''' with the notation '''v'''&nbsp;<math>\otimes</math>&nbsp;''r''&nbsp;=&nbsp;''r''&nbsp;<math>\otimes</math>&nbsp;'''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 velocity|proper velocities]] with vector addition given by the proper velocity addition formula:<ref name=ung2008/><ref name=ung1997>Thomas Precession: Its Underlying Gyrogroup Axioms and Their Use in Hyperbolic Geometry and Relativistic Physics, Abraham A. Ungar, Foundations of Physics, Vol. 27, No. 6, 1997 {{doi|10.1007/BF02550347}}</ref><ref>Ungar, A. A. (2006), [http://www.jpier.org/PIER/pier60/04.0512151.Ungar.pdf "The relativistic proper-velocity transformation group"] {{Webarchive|url=https://web.archive.org/web/20171025235947/http://www.jpier.org/PIER/pier60/04.0512151.Ungar.pdf |date=2017-10-25 }}, ''Progress in Electromagnetics Research'', PIER '''60''', pp. 85–94, equation (12)</ref>

:<math>\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} </math>

where <math>\beta_\mathbf{w}</math> is the beta factor given by <math>\beta_\mathbf{w}=\frac{1}{\sqrt{1+\frac{|\mathbf{w}|^2}{c^2}}}</math>.

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 <math>\oplus_U</math> and with scalar multiplication defined by ''r''&nbsp;<math>\otimes</math>'''v'''&nbsp;=&nbsp;''s''&nbsp;sinh(''r''&nbsp;sinh<sup>&minus;1</sup>(|'''v'''|/''s''))'''v'''/|'''v'''| where ''r'' is any real number, '''v''' &nbsp;∈&nbsp;''V'', '''v'''&nbsp;≠&nbsp;'''0''' and ''r''&nbsp;<math>\otimes</math>&nbsp;'''0'''&nbsp;=&nbsp;'''0''' with the notation '''v'''&nbsp;<math>\otimes</math>&nbsp;''r''&nbsp;=&nbsp;''r''&nbsp;<math>\otimes</math>&nbsp;'''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'''<sub>m</sub>, '''v'''<sub>e</sub> and '''v'''<sub>u</sub> then the isomorphisms are given by:
{| class="wikitable"
| E<math>\rightarrow</math>U by <math>\gamma_{\mathbf{v}_e} \mathbf{v}_e</math>
|-
| U<math>\rightarrow</math>E by <math>\beta_{\mathbf{v}_u} \mathbf{v}_u</math>
|-
| E<math>\rightarrow</math>M by <math>\frac{1}{2} \otimes_E \mathbf{v}_e</math>
|-
| M<math>\rightarrow</math>E by <math>2 \otimes_M \mathbf{v}_m</math>
|-
| M<math>\rightarrow</math>U by <math>2 {{{\gamma}^{2}}_{\mathbf{v}_m}} \mathbf{v}_m</math>
|-
| U<math>\rightarrow</math>M by <math>\frac{\beta_{\mathbf{v}_u}}{1+\beta_{\mathbf{v}_u}}\mathbf{v}_u</math>
|}

From this table the relation between <math>\oplus_E</math> and <math>\oplus_M</math> is given by the equations:
 
<math>\mathbf{u}\oplus_E\mathbf{v}=2\otimes\left({\frac{1}{2}\otimes\mathbf{u}\oplus_M\frac{1}{2}\otimes\mathbf{v}}\right)</math>

<math>\mathbf{u}\oplus_M\mathbf{v}=\frac{1}{2}\otimes\left({2\otimes\mathbf{u}\oplus_E 2\otimes\mathbf{v}}\right)</math>

This is related to the [[Möbius transformation#Lorentz transformations|connection between Möbius transformations and Lorentz transformations]].

===Gyrotrigonometry===

Gyrotrigonometry is the use of gyroconcepts to study [[hyperbolic triangle]]s.

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 trigonometry#Identities|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 center]]s 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.<ref>[http://ajmaa.org/searchroot/files/pdf/v6n1/v6i1p18.pdf Hyperbolic Barycentric Coordinates], 
Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp. 1–35, 2009</ref><ref>[https://www.springer.com/astronomy/cosmology/book/978-90-481-8636-5 Hyperbolic Triangle Centers: The Special Relativistic Approach], Abraham Ungar, Springer, 2010</ref><ref name="barycalc">[http://www.worldscibooks.com/mathematics/7740.html Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction] {{Webarchive|url=https://web.archive.org/web/20120519111614/http://www.worldscibooks.com/mathematics/7740.html |date=2012-05-19 }}, Abraham Ungar, World Scientific, 2010</ref>

====Gyroparallelogram addition====

Using gyrotrigonometry, a gyrovector addition can be found which operates according to the gyroparallelogram law. This is the [[#Coaddition|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.<ref>Abraham A. Ungar (2009), "A Gyrovector Space Approach to Hyperbolic Geometry", Morgan & Claypool, {{ISBN|1-59829-822-4}}, {{ISBN|978-1-59829-822-2}}</ref>

===Bloch vectors===

[[Bloch vector]]s which belong to the open unit ball of the Euclidean 3-space, can be studied with Einstein addition<ref>[https://arxiv.org/abs/quant-ph/0112169 Geometric observation for the Bures fidelity between two states of a qubit], Jing-Ling Chen, Libin Fu, Abraham A. Ungar, Xian-Geng Zhao, Physical Review A, vol. 65, Issue 2</ref> or Möbius addition.<ref name=ung2008/>

==Book reviews==
A review of one of the earlier gyrovector books<ref>Abraham A. Ungar (2002), "Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces", Kluwer, {{ISBN|1-4020-0353-6}}, {{ISBN|978-1-4020-0353-0}}</ref> says the following:

<blockquote>
"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."<ref>Scott Walter, Foundations of Physics 32:327–330 (2002).  [http://www.univ-nancy2.fr/DepPhilo/walter/papers/fop32.htm  A book review] {{Webarchive|url=https://web.archive.org/web/20110516125133/http://www.univ-nancy2.fr/DepPhilo/walter/papers/fop32.htm |date=2011-05-16 }},</ref>
</blockquote>

==Notes and references==
{{reflist}}
*Domenico Giulini, [https://arxiv.org/abs/math-ph/0602018v2 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.

==Further reading==
{{refbegin|2}}
*{{cite book|title=Gyrovectors: an Approach to Hyperbolic Geometry|url=https://books.google.com/books?id=dlFTgrhqm3IC&pg=PR12|author=A. A. Ungar|publisher=Morgan & Claypool Publishers|year=2009|isbn=978-159-829-822-2|issue=4|series=Synthesis lectures on mathematics and statistics}}
*{{cite book|title=Mathematical Analysis and Applications|url=https://books.google.com/books?id=rhjvAAAAMAAJ&q=gyrovector|author=T. M. Rassias|pages=307, 326, 336|publisher=Hadronic Press|year=2000|isbn=157-485-045-8|series=Collection of Articles in Mathematics}}
*Maks A. Akivis And Vladislav V. Goldberg (2006), [https://www.ams.org/journals/bull/2006-43-02/home.html Local Algebras Of A Differential Quasigroup], Bulletin of the AMS, Volume 43, Number 2
*Oğuzhan Demirel, Emine Soytürk (2008), [http://www.emis.de/journals/NSJOM/Papers/38_2/NSJOM_38_2_033_039.pdf The Hyperbolic Carnot Theorem In The Poincare Disc Model Of Hyperbolic Geometry], Novi Sad J. Math. Vol. 38, No. 2, 2008, 33–39
*M Ferreira (2008), Spherical continuous wavelet transforms arising from sections of the Lorentz group, Applied and Computational Harmonic Analysis, Elsevier {{arxiv|0706.1956}}
*T Foguel (2000), Comment. Math. Univ. Carolinae, [http://www.emis.de/journals/CMUC/pdf/cmuc0002/foguel.pdf Groups, transversals, and loops]
*Yaakov Friedman (1994), "Bounded symmetric domains and the JB*-triple structure in physics", Jordan Algebras: Proceedings of the Conference Held in Oberwolfach, Germany, August 9–15, 1992, By Wilhelm Kaup, Kevin McCrimmon, Holger P. Petersson, Published by Walter de Gruyter, {{ISBN|3-11-014251-1}}, {{ISBN|978-3-11-014251-8}}
*Florian Girelli, Etera R. Livine (2004), [https://arxiv.org/abs/gr-qc/0407098 Special Relativity as a non commutative geometry: Lessons for Deformed Special Relativity], Phys. Rev. D 81, 085041 (2010)
*Sejong Kim, Jimmie Lawson (2011), [https://web.archive.org/web/20120424035311/https://www.math.lsu.edu/~ksejong/Smooth_Loops.pdf Smooth Bruck Loops, Symmetric Spaces, And Nonassociative Vector Spaces], Demonstratio Mathematica, Vol. XLIV, No 4
* Peter Levay (2003), [https://arxiv.org/abs/quant-ph/0312023v1 Mixed State Geometric Phase From Thomas Rotations]
*Azniv Kasparian, Abraham A. Ungar, (2004) Lie Gyrovector Spaces, J. Geom. Symm. Phys
*R Olah-Gal, J Sandor (2009), [http://forumgeom.fau.edu/FG2009volume9/FG200914.pdf On Trigonometric Proofs of the Steiner–Lehmus Theorem], Forum Geometricorum, 2009 – forumgeom.fau.edu
*Gonzalo E. Reyes (2003), On the law of motion in Special Relativity {{arxiv|physics/0302065}}
*Krzysztof Rozga (2000), Pacific Journal of Mathematics, Vol. 193, No. 1,[https://web.archive.org/web/20100701073205/http://pjm.math.berkeley.edu/pjm/2000/193-1/pjm-v193-n1-p.pdf#page=203 On Central Extensions Of Gyrocommutative Gyrogroups]
*L.V. Sabinin (1995), [http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=2628 "On the gyrogroups of Hungar"] {{Webarchive|url=https://web.archive.org/web/20090830002544/http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=2628 |date=2009-08-30 }}, RUSS MATH SURV, 1995, 50 (5), 1095–1096. <!-- “Hungar” is an alternate spelling? or, probably, incorrect translation? have such note a Russian title? --Incnis Mrsi -->
*L.V. Sabinin, L.L. Sabinina, Larissa Sbitneva (1998), ''[[Aequationes Mathematicae]]'', [https://web.archive.org/web/20210421151854/https://www.mathnet.or.kr/mathnet/thesis_content.php?no=331803 On the notion of gyrogroup]
*L.V. Sabinin, Larissa Sbitneva, I.P. Shestakov (2006), "Non-associative Algebra and Its Applications", CRC Press,{{ISBN|0-8247-2669-3}}, {{ISBN|978-0-8247-2669-0}}
*F. Smarandache, C. Barbu (2010), [http://fs.gallup.unm.edu/MenelausInPoincareDiscModel.pdf The Hyperbolic Menelaus Theorem in The Poincaré Disc Model of Hyperbolic Geometry]
*Roman Ulrich Sexl, Helmuth Kurt Urbantke, (2001), "Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics", pages 141–142, Springer, {{ISBN|3-211-83443-5}}, {{ISBN|978-3-211-83443-5}}
{{refend}}

==External links==
*[https://www.researchgate.net/profile/Abraham_Ungar/publication/235726885_Einstein%27s_Special_Relativity_The_Hyperbolic_Geometric_Viewpoint/links/00b49532babe63b291000000.pdf Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint]
*{{cite web|author=Ungar, Abraham A. |url=https://citeseerx.ist.psu.edu/doc/10.1.1.17.6107 | title = Hyperbolic Trigonometry and its Application in the Poincaré Ball Model of Hyperbolic Geometry | year = 2001 | pages = 6–19 |citeseerx=10.1.1.17.6107 }}

[[Category:Euclidean geometry]]
[[Category:Hyperbolic geometry]]
[[Category:Non-associative algebra]]
[[Category:Special relativity]]
[[Category:Quantum mechanics]]