Gyrovector space

A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry^[1]. This vector-based approach has been developed by Abraham Albert Ungar from the late 1980s onwards. These gyrovectors can be used to unify the study of Euclidean and hyperbolic geometry.

Soon after special relativity was developed in 1905 it was realized that Einstein's velocity addition law could be interpreted in terms of hyperbolic geometry (see History of special relativity). Only colinear velocities are commutative and associative, but in general, addition of non-colinear velocities is non-associative and non-commutative. The set of admissible velocities forms a hyperbolic space. The gyrovector approach introduces the concepts of gyroassociativity and gyrocommutativity. The use of the prefix gyro comes from Thomas gyration which is the mathematical abstraction of the gyroscopic Thomas precession into an operator called a gyrator and denoted gyr.

Gyrovectors can be used in the study of the Bloch vectors of quantum computation.

Different models of hyperbolic geometry are regulated by different gyrovector spaces. The Beltrami-Klein model is regulated by gyrovector spaces based on relativistic velocity addition^[2]. The Poincaré ball model is regulated by gyrovector spaces based on part of the formula for Möbius transformations.^[3]

Gyrovectors

A gyrovector is just a fancy name for a vector, except that instead of addition being componentwise e.g. instead of (a, b, c) + (d, e, ƒ) = (a + d, b + e, c + ƒ) you have addition defined by a formula that satisfies the axioms for a gyrogroup (see section below) such as the relativistic velocity-addition formula

The addition of vectors using ordinary addition forms a group, but relativistic addition does not form a group as it is not associative: a + (b + c) ≠ (a + b) + c. Relativistic addition is not commutative either: a + b ≠ b + a. But Ungar has shown that relativistic addition has a weaker form of associativity, which he calls gyroassociativity, involving an operator called gyr based on Thomas precession: u + (v + w) = (u + v) + gyr(w). The gyr operator also appears in a weaker form of commutativity, called gyrocommutativity: u + v = gyr(v + u). (gyr isn't a single operator but depends on v and u, i.e. gyr = gyr[v, u] )

So relativistic addition does not form a group, but what to call the structure? Well it is similar to a group except that the gyr operator appears in the associativity and commutativity expressions, so Ungar called this structure a gyrogroup. Vectors are based on groups. Gyrovectors are just vectors based on gyrogroups and so gyrovector spaces are a generalization of vector spaces.

In euclidean geometry the 'gyr' operator is just the identity map, and the gyroassociativity and gyrocommutativity laws reduce to the usual laws of associativity and commutativity.

Gyrogroups

Axioms

A groupoid (G, $\oplus$ ) is a gyrogroup if its binary operation satisfies the following axioms:

In G there is at least one element 0 called a left identity with 0 $\oplus$ a = a for all a ∈ G.
For each a ∈ G there is an element $\ominus$ a in G called a left inverse of a with $\ominus$ a $\oplus$ 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 $\oplus$ (b $\oplus$ c) = (a $\oplus$ b) $\oplus$ gyr[a, b]c
The map gyr[a, b]:G → G given by c → gyr[a, b]c is an automorphism of the groupoid (G, $\oplus$ ). That is gyr[a, b] is a member of Aut(G, $\oplus$ ) 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, $\oplus$ ) is called the gyrator of G.
The gyroautomorphism gyr[a, b] has the left loop property gyr[a, b] = gyr[a $\oplus$ 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 defined as the identity map.

An example of a finite gyrogroup is given in^[4].

Identities

Some identities which hold in any gyrogroup (G, $\oplus$ ):

$\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))$ (gyration)
$\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}) = (\mathbf{u} \oplus \mathbf{v})\oplus \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}$ (left associativity)
$(\mathbf{u} \oplus \mathbf{v}) \oplus \mathbf{w} = \mathbf{u} \oplus (\mathbf{v}\oplus \mathrm{gyr}[\mathbf{v},\mathbf{u}]\mathbf{w})$ (right associativity)

More identities given on page 50 of ^[5].

Gyrocommutativity

A gyrogroup (G, $\oplus$ ) is gyrocommutative if its binary operation obeys the gyrocommutative law: a $\oplus$ b = gyr[a, b](b $\oplus$ 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^[6]^[7]

Coaddition

In every gyrogroup, a second operation can be defined called coaddition: a $\boxplus$ b = a $\oplus$ gyr[a, $\ominus$ b]b for all a, b ∈ G. Coaddition is commutative if the gyrogroup addition is gyrocommutative.

Alternative terminology

Gyrogroups are a type of Bol loop^[8]. Gyrocommutative gyrogroups are equivalent to K-loops^[9] although defined differently. The terms Bruck loop^[10] and dyadic symset^[11] are also in use.

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.

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^[12].

Poincaré disc/ball model and Möbius addition

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

$z\to {e^{i\theta}}{\frac{a%2Bz}{1%2Ba\bar{z}}}$ which can be written as $e^{i\theta} {(a\oplus_M {z})}$ which defines the Möbius addition ${a\oplus_M {z}}= \frac{a%2Bz}{1%2Ba\bar{z}}$ .

To generalize this to higher dimensions the complex numbers are considered as vectors in the plane R^2, and Möbius addition is rewritten in vector form as:

$\mathbf{u} \oplus_M \mathbf{v}=\frac{(1%2B\frac{2}{s^2}\mathbf{u}\cdot\mathbf{v}%2B\frac{1}{s^4}|\mathbf{v}|^2)\mathbf{u}%2B(1-\frac{1}{s^2}|\mathbf{u}|^2)\mathbf{v}}{1%2B\frac{2}{s^2}\mathbf{u}\cdot\mathbf{v}%2B\frac{1}{s^4}|\mathbf{u}|^2|\mathbf{v}|^2}$

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, $\oplus$ , $\otimes$ ) is a Möbius gyrogroup (V_s, $\oplus$ ) with scalar multiplication given by r $\otimes$ v = s tanh(r tanh⁻¹(|v|/s))v/|v| where r is any real number, v ∈ V_s, v ≠ 0 and r $\otimes$ 0 = 0 with the notation v $\otimes$ r = r $\otimes$ v.

Möbius scalar multiplication coincides with Einstein scalar multiplication (see section below) and this stems from Möbius addition and Einstein addition coinciding for vectors that are parallel.

Special relativity

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

$\mathbf{u} \oplus_E \mathbf{v}=\frac{1}{1%2B\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left\{\mathbf{u}%2B\frac{1}{\gamma_\mathbf{u}}\mathbf{v}%2B\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{1%2B\gamma_\mathbf{u}}(\mathbf{u}\cdot\mathbf{v})\mathbf{u}\right\}$

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

Using coordinates this becomes:

$\begin{pmatrix}w_1\\ w_2\\ w_3\\ \end{pmatrix}=\frac{1}{1%2B\frac{u_1v_1%2Bu_2v_2%2Bu_3v_3}{c^2}}\left\{\left[1%2B\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{1%2B\gamma_\mathbf{u}}(u_1v_1%2Bu_2v_2%2Bu_3v_3)\right]\begin{pmatrix}u_1\\ u_2\\ u_3\\ \end{pmatrix}%2B\frac{1}{\gamma_\mathbf{u}}\begin{pmatrix}v_1\\ v_2\\ v_3\\ \end{pmatrix}\right\}$

where $\gamma_\mathbf{u}=\frac{1}{\sqrt{1-\frac{u_1^2%2Bu_2^2%2Bu_3^2}{c^2}}}$ .

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

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

and

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

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

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

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

Lorentz transformation composition

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:

$\mathrm{Gyr}[\mathbf{u},\mathbf{v}]= \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{gyr}[\mathbf{u},\mathbf{v}] \end{pmatrix}$ .^[13]

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:^[13]^[14]

$B(\mathbf{u})B(\mathbf{v})=B(\mathbf{u}\oplus\mathbf{v})\mathrm{Gyr}[\mathbf{u},\mathbf{v}]=\mathrm{Gyr}[\mathbf{u},\mathbf{v}]B(\mathbf{v}\oplus\mathbf{u})$

This fact that either B(u $\oplus$ v) or B(v $\oplus$ 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:^[15]

$L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v}, \mathrm{gyr}[\mathbf{u},U\mathbf{v}]UV)$

In the above, a boost can be represented as a 4 × 4 matrix. The boost matrix B(v) means the boost B that uses the components of v, i.e. v₁, v₂, v₃ 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 form. 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 $\oplus$ v in the 4 × 4 matrix B(u $\oplus$ v). But the resultant boost also needs to be multiplied by a rotation matrix because boost composition (i.e. the multiplication of two 4 × 4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4 × 4 matrix that corresponds to the rotation Gyr[u,v] to get B(u $\oplus$ v) = B(u $\oplus$ v)Gyr[u,v] = Gyr[u,v]B(v $\oplus$ u).

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, $\oplus$ , $\otimes$ ) is an Einstein gyrogroup (V_s, $\oplus$ ) with scalar multiplication given by r $\otimes$ v = s tanh(r tanh⁻¹(|v|/s))v/|v| where r is any real number, v ∈ V_s, v ≠ 0 and r $\otimes$ 0 = 0 with the notation v $\otimes$ r = r $\otimes$ 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 $\otimes$ v = v $\oplus$ ... $\oplus$ v	n terms
(r₁ + r₂) $\otimes$ v = r₁ $\otimes$ v $\oplus$ r₂ $\otimes$ v	Scalar distributive law
(r₁r₂) $\otimes$ v = r₁ $\otimes$ (r₂ $\otimes$ v)	Scalar associative law
r $\otimes$ (r₁ $\otimes$ a $\oplus$ r₂ $\otimes$ a) = r $\otimes$ (r₁ $\otimes$ a) $\oplus$ r $\otimes$ (r₂ $\otimes$ a)	Monodistributive law

Coaddition and stellar aberration

The formulae and calculations of relativistic dynamics can be made to appear similar to those of Newtonian dynamics when a quantity called coaddition is used in place of ordinary vector addition. The coaddition formula is given by u $\boxplus$ v = u $\oplus$ gyr[u, $\ominus$ v]v. Coaddition simplifies to^[16]:

$\mathbf{u}\boxplus \mathbf{v}=2\otimes{\frac{\gamma_\mathbf{u}\mathbf{u}%2B\gamma_\mathbf{v}\mathbf{v}}{\gamma_\mathbf{u}%2B\gamma_\mathbf{v}}}$

where $\otimes$ is the scalar multiplication given by 2 $\otimes$ 'v =' v $\oplus$ v. The coaddition is commutative and is called the gyroparallelogram addition of velocity composition in analogy with the ordinary parallelogram law of vector addition for Galilean velocities. So u $\boxplus$ v = v $\boxplus$ u, but if a third vector w is added, there will be another rotation to consider, so a formula is derived for $\mathbf{u} \boxplus_3 \mathbf{v} \boxplus_3 \mathbf{w}$ (called the gyroparallelepiped law), which is symmetric in all 3 velocities, and co-addition $\boxplus$ is denoted $\boxplus_2$ to distinguish it from $\boxplus_3$ . This technique is extended again to derive a formula for $\boxplus_n$ .^[5]^[1] For colinear velocities there is no rotation, the gyr operator becomes the identity map, and $\boxplus$ coincides with $\oplus$ .

The classical particle aberration formulas can be derived by employing trigonometry, the triangle equality, the triangle addition law, and the parallelogram addition law of Newtonian velocities. In full analogy, relativistic particle aberration formulas can be derived by employing gyrotrigonometry, the gyrotriangle equality, the gyrotriangle addition law, and the gyroparallelogram addition of Einsteinian velocities. In the literature, relativistic particle aberration formulas are usually obtained by employing the Lorentz transformation group.

Dark matter

Ungar takes a system of N particles each with their own invariant masses and then defines a quantity called the 'invariant mass of the system' which depends on all the relative velocities. This invariant mass m0 of the system is invariant only so long as the particles are freely moving about, but if any particles stick to each other then this m0 changes. The formula for m0 includes two terms. The first which could be called the 'newtonian mass of the system' is just the sum of the invariant masses of each particle. The second term in the formula which depends on all the velocities is given the name 'dark mass' and when this changes, m0 changes. The 'relativistic mass of the system' is the final quantity which is defined in terms of m0 and this changes in line with m0 and the dark mass. This 'relativistic mass of the system' meshes well with the Minkowski 4-vector formalism of special relativity. Moreover, this 'relativistic mass of the system' is additive in the sense that it is equal to the sum of the relativistic masses of its constituent particles.

Ungar speculates that the quantity called dark mass in these formulae contributes to the dark matter that astronomers are looking for, but cautions that this is a special-relativistic effect and need not account for all of the dark matter, there may also be a general-relativistic effect, plus other sources.^[17]^[18]^[5]. The dark mass of a galaxy increases when there is a supernova spewing out fast particles and the dark mass of a galaxy decreases whenever stars form because all the relativistic mass of the collapsing cloud particles disappears.

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^[19]^[5]^[20]:

$\mathbf{u} \oplus_U \mathbf{v}=\mathbf{u}%2B\mathbf{v}%2B\left\{ {\frac{\beta_\mathbf{u}}{1%2B\beta_\mathbf{u}}}{\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}} %2B {\frac{1 - \beta_\mathbf{v}}{\beta_\mathbf{v}}} \right\} \mathbf{u}$

where $\beta_\mathbf{w}$ is the beta factor given by $\beta_\mathbf{w}=\frac{1}{\sqrt{1%2B\frac{|\mathbf{w}|^2}{c^2}}}$ .

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 $\oplus_U$ and with scalar multiplication defined by r $\otimes$ v = s sinh(r sinh⁻¹(|v|/s))v/|v| where r is any real number, v ∈ V, v ≠ 0 and r $\otimes$ 0 = 0 with the notation v $\otimes$ r = r $\otimes$ 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 $\rightarrow$ U by $\gamma_{\mathbf{v}_e} \mathbf{v}_e$
U $\rightarrow$ E by $\beta_{\mathbf{v}_u} \mathbf{v}_u$
E $\rightarrow$ M by $\frac{1}{2} \otimes_E \mathbf{v}_e$
M $\rightarrow$ E by $2 \otimes_M \mathbf{v}_m$
M $\rightarrow$ U by $2 {{{\gamma}^{2}}_{\mathbf{v}_m}} \mathbf{v}_m$
U $\rightarrow$ M by $\frac{\beta_{\mathbf{v}_u}}{1%2B\beta_{\mathbf{v}_u}}\mathbf{v}_u$

From this table the relation between $\oplus_E$ and $\oplus_M$ is given by the equations:

$\mathbf{u}\oplus_E\mathbf{v}=2\otimes\left({\frac{1}{2}\otimes\mathbf{u}\oplus_M\frac{1}{2}\otimes\mathbf{v}}\right)$

$\mathbf{u}\oplus_M\mathbf{v}=\frac{1}{2}\otimes\left({2\otimes\mathbf{u}\oplus_E 2\otimes\mathbf{v}}\right)$

Bloch vectors

Bloch vectors which belong to the open unit ball of the Euclidean 3-space, can be studied with Einstein addition^[21] or Möbius addition.^[5]

Book reviews

A review of one of the earlier gyrovector books^[22] 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."^[23]

Notes and references

^ ^a ^b Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Foundations and Applications", Published by World Scientific, ISBN 9812564578, 9789812564573
^ Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint
^ Hyperbolic Trigonometry and its Application in the Poincaré Ball Model of Hyperbolic Geometry
^ Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry, AA Ungar - Computers & Mathematics with Applications, 2000 - Elsevier, Page 5, Postscript version
^ ^a ^b ^c ^d ^e Analytic hyperbolic geometry and Albert Einstein's special theory of relativity, Abraham A. Ungar, World Scientific, 2008, ISBN 9789812772299
^ Ludwik Silberstein, The theory of relativity, Macmillan, 1914
^ Page 214, Chapter 5, Symplectic matrices: first order systems and special relativity, Mark Kauderer, World Scientific, 1994, ISBN 9789810219840
^ A. Nourou Issa (1998), Gyrogroups and homogeneous loops
^ Hubert Kiechle (2002), "Theory of K-loops",Published by Springer,ISBN 3540432620, 9783540432623
^ Larissa Sbitneva (2001), Nonassociative Geometry of Special Relativity, International Journal of Theoretical Physics, Springer, Vol.40, No.1 / Jan 2001
^ 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
^ Abraham A. Ungar (2009), "A Gyrovector Space Approach to Hyperbolic Geometry", Morgan & Claypool, ISBN 1598298224, 9781598298222
^ ^a ^b Ungar, A. A: The relativistic velocity composition paradox and the Thomas rotation. Found. Phys. 19, 1385–1396 (1989)
^ The relativistic composite-velocity reciprocity principle, AA Ungar – Foundations of Physics, 2000 – Springer
^ eq. (55), Thomas rotation and the parametrization of the Lorentz transformation group, AA Ungar – Foundations of Physics Letters, 1988
^ Page 10 eq(29) of Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint
^ Page 81, Hyperbolic Triangle Centers: The Special Relativistic Approach, Abraham Ungar, Springer, 2010
^ , Abraham A. Ungar (2008), "On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly", Progress in Physics vol 3
^ 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
^ Ungar, A. A. (2006), "The relativistic proper-velocity transformation group", Progress in Electromagnetics Research, PIER 60, pp. 85–94, equation (12)
^ 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
^ Abraham A. Ungar (2002), "Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces", Kluwer, ISBN 1402003536, 9781402003530
^ Scott Walter, Foundations of Physics 32:327-330 (2002). A book review,

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.
Hyperbolic Triangle Centers: The Special Relativistic Approach, Abraham Ungar, Springer, 2010
Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction, Abraham Ungar, World Scientific, 2010