A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

Vermeer, Johannes

doi:10.1016/j.topol.2004.10.012

Cited by 27 publications

(21 citation statements)

References 5 publications

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“…7,17 It is not manifest whether the present approach is involving intrinsically a rotation operator in a comparable fashion as the explicit introduction of a Thomas rotation operator in gyro-groups. 7 Alternatively, it is possible that this scheme is not involving a rotation of the inertial frames at all.…”

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confidence: 97%

Alternative realization for the composition of relativistic velocities

Fernández‐Guasti

2011

SPIE Proceedings

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The reciprocity principle requests that if an observer, say in the laboratory, sees an event with a given velocity, another observer at rest with the event must see the laboratory observer with minus the same velocity. The composition of velocities in the Lorentz-Einstein scheme does not fulfill the reciprocity principle because the composition rule is neither commutative nor associative. In other words, the composition of two non-collinear Lorentz boosts cannot be expressed as a single Lorentz boost but requires in addition a rotation. The Thomas precession is a consequence of this composition procedure. Different proposals such as gyro-groups have been made to fulfill the reciprocity principle.An alternative velocity addition scheme is proposed consistent with the invariance of the speed of light and the relativity of inertial frames. An important feature of the present proposal is that the addition of velocities is commutative and associative. The velocity reciprocity principle is then immediately fulfilled. This representation is based on a transformation of a hyperbolic scator algebra. The proposed rules become identical with the special relativity addition of velocities in one dimension. They also reduce to the Galilean transformations in the low velocity limit. The Thomas gyration needs to be revised in this nonlinear realization of the special relativity postulates. The deformed Minkowski metric presented here is compared with other deformed relativity representations.

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confidence: 97%

Alternative realization for the composition of relativistic velocities

Fernández‐Guasti

2011

SPIE Proceedings

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“…In the papers [18][19][20][21][22]the operations C −a (z), B a (z) were studied, also for higher dimensions, which we describe now as follows:…”

Section: Theorem 2 For Any Two Functionsmentioning

confidence: 99%

Quaternionic Blaschke Group

Pap

Schipp

2018

Mathematics

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In the complex case, the Blaschke group was introduced and studied. It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. A first step in this direction is to give the quaternionic analogue of the Blaschke group. In this paper we introduce the quaternionic Blaschke group and we study the properties of this group and its subgroups.

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“…For any u, v∈R n c , let gyr[u, v] : R n c → R n c be the self-map of R n c given in terms of Einstein addition ⊕, (19), by the equation [55] ( 28) gyr[u, v]w = ⊖(u⊕v)⊕{u⊕(v⊕w)} for all w∈R n c . The self-map gyr[u, v] of R n c , which takes w∈R n c into ⊖(u⊕v)⊕{u⊕(v⊕w)}∈ R n c , is the gyration generated by u and v. Being the mathematical abstraction of the relativistic Thomas precession, The gyration has an interpretation in hyperbolic geometry [76] as the negative hyperbolic triangle defect [68,Theorem 8.55].…”

Section: Einstein Gyrogroups and Gyrationsmentioning

confidence: 99%

Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

Ungar

2012

Springer Optimization and Its Applications

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In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Möbius transformation and Einstein velocity addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that Möbius addition that is extracted from Möbius transformation of the complex disc and Einstein addition from his special theory of relativity are isomorphic in the sense of gyrovector spaces.(1) Möbius addition in the ball R n c forms the algebraic setting for the Cartesian-Poincaré ball model of hyperbolic geometry, and (2) Einstein addition in the ball R n c forms the algebraic setting for the Cartesian-Beltrami-Klein ball model of hyperbolic geometry, just as the common (3) vector addition in the space R n forms the algebraic setting for the standard Cartesian model of Euclidean geometry.Remarkably, Items (1)-( 3) enable Möbius addition in R n c , Einstein addition in R n c , and the standard vector addition in R n to be studied comparatively, as in [72]. for all a, b, u, v ∈ R n c . Hence, gyr[u, v] is an isometry of R n c , keeping the norm of elements of the ball R n c invariant, (34) gyr[u, v]w = w Accordingly, gyr[u, v] represents a rotation of the ball R n c about its origin for any u, v∈R n c . The invertible self-map gyr[u, v] of R n c respects Einstein addition in R n c , (35) gyr[u, v](a⊕b) = gyr[u, v]a⊕gyr[u, v]bfor all a, b, u, v∈R n c , so that gyr [u, v] is an automorphism of the Einstein groupoid (R n c , ⊕). We recall that an automorphism of a groupoid (R n c , ⊕) is a bijective self-map of the groupoid R n c that respects its binary operation, that is, it satisfies (35). Under bijection composition the automorphisms of a groupoid (R n c , ⊕) form a group known as the automorphism group, and denoted Aut(R n c , ⊕). Being special automorphisms, gyrations gyr [u, v]∈Aut(R n c , ⊕), u, v ∈R n c , are also called gyroautomorphisms, gyr being the gyroautomorphism generator called the gyrator.

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A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

Cited by 27 publications

References 5 publications

Alternative realization for the composition of relativistic velocities

Alternative realization for the composition of relativistic velocities

Quaternionic Blaschke Group

Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

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