In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
,
gyrovectors are a tool for studying
hyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
in analogy to the way vector spaces are used in
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
. 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 relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
was developed in 1905 it was realized that
Einstein's velocity addition lawIn physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...
could be interpreted in terms of hyperbolic geometry (see Non-euclidean reformulations of special relativity). Only colinear velocites 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 spaceIn mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry...
. There were various attempts to formulate special relativity based on hyperbolic geometry but these were unpopular due to lack of major new insight compared to the effort involved, and complicated by the non-associativity. The gyrovector approach tackles the issue by introducing the concepts of gyroassociativity and gyrocommutativity. The use of the prefix
gyro comes from
Thomas gyration which is the mathematical abstraction of
Thomas precessionIn physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction to the precession of a gyroscope in a rotating non-inertial frame...
into an operator called a
gyrator and denoted
gyr.
The Bloch vector of quantum computation is not really a vector but can be seen as an example of a gyrovector and the geometry of quantum computation is really hyperbolic geometry and its algebra is the algebra of gyrovector spaces.
Different models of hyperbolic geometry are regulated by different gyrovector spaces. The Beltrami-Klein model is regulated by gyrovector spaces based on
relativistic velocity additionIn physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...
. The Poincaré ball model is regulated by gyrovector spaces based on
Möbius transformationIn geometry, a Möbius transformation of the plane is a rational function of the formof one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0...
s.
Gyrovectors
A
gyrovector is just a fancy name for a
vector-Vectors:*Euclidean vector, a geometric entity endowed with both length and direction; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities which have both magnitude and direction, such as force, in contrast to scalar quantities, which have...
, 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 the relativistic
velocity-addition formulaIn physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...
.
The addition of vectors using ordinary addition forms a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, but relativistic addition does not form a group as it is not associative i.e.
a + (
b +
c) ≠ (
a +
b) +
c. Relativistic addition is not commutative either i.e.
a +
b ≠
b +
a. So you don't have a group, but Ungar has shown that relativistic addition involves a weaker form of associativity involving an operator called gyr based on
Thomas precessionIn physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction to the precession of a gyroscope in a rotating non-inertial frame...
. The weaker form which he calls gyroassociativity is
u + (
v +
w) = (
u +
v) + gyr(
w). The same operator that appears in the gyroassociativity law also appears in a weaker form of commutativity called gyrocommutativity which is
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.
The vectors in
hyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
are gyrovectors. Gyrovectors are equivalence classes that add according to the
gyroparallelogram law, just like vectors, which are equivalence classes that add according to the
parallelogram lawIn mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...
. A gyroparallelogram, in turn, 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.
Axioms
A
groupoidIn abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M...
(
G, ) is a
gyrogroup if its
binary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
satisfies the following axioms:
- In G there is at least one element 0 called a left identity with 0a = a for all a ∈ G.
- For each a ∈ G there is an element a in G called a left inverse of a with a'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(b'c) = (a'b)gyr[a, b]c
- The map gyr[a, b]:G → G given by c → gyr[a, b]c is an automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of the groupoid (G, ). That is gyr[a, b] is a member of Aut(G, ) 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, ) is called the gyrator of G.
- The gyroautomorphism gyr[a, b] has the left loop property gyr[a, b] = gyr[a'b, b]
The first pair of axioms are like the
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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
quasigroupIn mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
and a loop.
Gyrogroups are a generalization of
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s. Every group is an example of a gyrogroup with gyr defined as the identity map.
Gyrocommutativity
A gyrogroup (G,) is gyrocommutative if its binary operation obeys the gyrocommutative law: a b = gyr[a, b](b a).
Coaddition
In every gyrogroup, a second operation can be defined called
coaddition: a b = a gyr[a,b]b for all a, b ∈ G. Coaddition is commutative if the gyrogroup addition is gyrocommutative.
Alternative terminology
Certain types of gyrogroups are equivalent to some older terms:
Bol loopIn mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of semigroup.A loop, L, is said to be a left Bol loop if it satisfies the identity, for every a,b,c in L,...
and
K-loop. The terms
Bruck loop and
dyadic symset are also in use.
Gyrovector spaces
Gyrovector spaces are a generalization of
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s. Whereas vector spaces are based on commutative groups, gyrovector spaces are based on gyrocommutative gyrogroups. Since groups are a special case of gyrogroups, vector spaces are special cases of gyrovector spaces.
Gyrotrigonometry
Gyrotrigonometry is the use of gyroconcepts to study
hyperbolic triangle-Euclidean geometry:In the foundations of the hyperbolic functions sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant of the Cartesian plane,...
s.
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.
Einstein velocity addition
Let c be any positive constant, let (V,+,.) be any real
inner product spaceIn mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
and let V
c={
v ∈ V :|
v|
In the general case, the Einstein
velocity additionIn physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...
of two velocities and is given by
where and are the components of parallel and perpendicular, respectively, to , and .
For any
u,
v ∈ V
c, let for all
w ∈ V
c. The self-map gyr[
u,
v] of V
c is called the
Thomas gyration generated by
u and
v. Thomas gyration is the abstraction of
Thomas precessionIn physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction to the precession of a gyroscope in a rotating non-inertial frame...
and it has an interpretation in hyperbolic geometry as the negative
hyperbolic triangle-Euclidean geometry:In the foundations of the hyperbolic functions sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant of the Cartesian plane,...
defect.
Einstein velocity addition is commutative
only when and are
parallel. In fact,
Einstein gyrovector spaces
An Einstein gyrovector space (
Vs, , ) is an Einstein gyrogroup (
Vs, ) with scalar multiplication given by
rv = s
tanh(r
tanh−1(|v|/s
))v/|v| where r
is any real number, v ∈ V
s
, v ≠ 0 and r
0 = 0 with the notation v r
= r
v.
Einstein scalar multiplication does not distribute over Einstein addition, but it has other properties of vector spaces: For any positive integer n
and for all real numbers r
,r
1,r
2 and v ∈ V
s':
| n v = v ... v |
n terms |
| (r1 + r2) v = r1 v r2 v |
Scalar distributive law |
| (r1r2) v = r1 (r2 v) |
Scalar associative law |
| r (r1 a r2 a) = rx(r1 a) r (r2 a) |
Monodistributive law |
Stellar aberration
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. Ungar says this means that Einsteinian relativistic velocities add not according to Einstein's 1905 velocity addition law, but are gyrovectors that add according to the gyroparallelogram addition law, which is commutative, just as Newtonian, classical velocities are vectors that add according to the common parallelogram addition law. Einstein's velocity addition law and the gyroparallelogram addition law only disagree for non-colinear velocities, but for colinear velocities the two laws coincide. Einstein's addition of colinear velocites is consistent with the
Fizeau experimentThe Fizeau experiment was carried out by Hippolyte Fizeau in the 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity.-The experiment:...
which determined the speed of light in a fluid moving parallel to the light.
Dark matter
Ungar takes a system of N particles each with their own
invariant massThe invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference...
es and then defines a quantity called the 'invariant mass of the system' which depends on all the relative velocites. 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 the 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 unlike the usual relativistic mass of a particle which does not mesh with the Minkowski formalism.
Ungar speculates that the quantity called dark mass in these formulae is the
dark matterIn astronomy and cosmology, dark matter is hypothetical matter that is undetectable by its emitted radiation, but whose presence can be inferred from gravitational effects on visible matter...
that astronomers are looking for.. The dark mass of a
galaxyA galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The name is from the Greek root galaxias [γαλαξίας], meaning "milky," a reference...
increases when there is a
supernovaA supernova is a stellar explosion. Supernovae are extremely luminous and cause a burst of radiation that often briefly outshines an entire galaxy, before fading from view over several weeks or months. During this short interval, a supernova can radiate as much energy as the Sun could emit over...
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.
Quantum computation
Bloch vectors, which belong to the open unit ball of the Euclidean 3-space, are an example of gyrovectors and the geometry of quantum computation is hyperbolic geometry and its algebra is the algebra of gyrovector spaces.
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."
Further reading
- Oğuzhan Demirel, Emine Soytürk (2008), 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
- T Foguel (2000), Comment. Math. Univ. Carolinae, 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 3110142511, 9783110142518
- Peter Levay (2003), Mixed State Geometric Phase From Thomas Rotations
- R Olah-Gal, J Sandor (2009), On Trigonometric Proofs of the Steiner–Lehmus Theorem, Forum Geometricorum, 2009 – forumgeom.fau.edu
- Zbigmiew Oziewicz (2006), "Relativity groupoid instead of relativity group"
- Gonzalo E. Reyes (2003), On the law of motion in Special Relativity
- Krzysztof Rozga (2000), Pacific Journal Of Mathematics, Vol. 193, No. 1,On Central Extensions Of Gyrocommutative Gyrogroups
- L.V. Sabinin (1995), "On the gyrogroups of Hungar", RUSS MATH SURV, 1995, 50 (5), 1095–1096.
- L.V. Sabinin, L.L. Sabinina, Larissa Sbitneva (1998), Aequationes Mathematicae, On the notion of gyrogroup
- L.V. Sabinin, Larissa Sbitneva, I.P. Shestakov (2006), "Non-associative Algebra and Its Applications",CRC Press,ISBN 0824726693, 9780824726690
- 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 3211834435, 9783211834435
External links