# On the relativistic velocity composition paradox and the Thomas rotation

@article{Mocanu1992OnTR, title={On the relativistic velocity composition paradox and the Thomas rotation}, author={Costel Iulian Mocanu}, journal={Foundations of Physics Letters}, year={1992}, volume={5}, pages={443-456} }

The non-commutativity and the non-associativity of the composition law of the non-colinear velocities lead to an apparent paradox, which in turn is solved by the Thomas rotation. A 3×3 parametric, unimodular and orthogonal matrix elaborated by Ungar is able to determine the Thomas rotation. However, the algebra involved in the derivation of the Thomas rotation matrix is overwhelming. The aim of this paper is to present a direct derivation of the Thomas angle as the angle between the composite… Expand

#### 26 Citations

Thomas rotation: a Lorentz matrix approach

- Physics
- 2002

The composition of two pure Lorentz transformations (boosts) parametrized by non-parallel velocities is equivalent to a boost combined with a pure spatial rotation - the Thomas rotation. Thirty years… Expand

Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

- Mathematics, Physics
- 2021

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic… Expand

Vectorial Form of the Successive Lorentz Transformations. Application: Thomas Rotation

- Mathematics
- 2012

A complete treatment of the Thomas rotation involves algebraic manipulations of overwhelming complexity. In this paper, we show that a choice of convenient vectorial forms for the relativistic… Expand

The Quartet of Eigenvectors for Quaternionic Lorentz Transformation

- Mathematics
- 2020

In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term… Expand

Visualization of Thomas–Wigner Rotations

- Physics
- 2017

It is well known that a sequence of two non-collinear Lorentz boosts (pure Lorentz transformations) does not correspond to a Lorentz boost, but involves a spatial rotation, the Wigner or… Expand

Composition of velocities and momentum transformations in a scator-deformed Lorentz metric

- Physics
- 2020

The transformations of coordinates, velocities and momenta between inertial frames are described via real scator algebra. This algebra imposes a hyperbolic metric that is identical to the Lorentz… Expand

THE LORENTZ BOOST-LINK IS NOT UNIQUE. Relative velocity as a morphism in a connected groupoid category of null objects

- Mathematics, Physics
- 2006

The isometry-link problem is to determine all isometry transformations among given pair of vectors with the condition that if these initial and final vectors coincide, the transformation-link must be… Expand

Successive Lorentz transformations for energy and momentum. Application: relativistic elastic scattering of two particles having non-collinear velocities and its dependence on Thomas rotation

- Physics
- 2021

Successive Lorentz transformations (SLT) for energy and momentum are derived, and the involved Thomas rotation angle (TRA) is highlighted. In particular, if the axes of the coordinate systems related… Expand

Uniqueness of the Isotropic Frame and Usefulness of the Lorentz Transformation

- Physics
- 2018

According to the postulates of the special theory of relativity (STR), physical quantities such as proper times and Doppler shifts can be obtained from any inertial frame by regarding it as… Expand

Einstein's special relativity: Unleashing the power of its hyperbolic geometry

- Physics
- 2005

We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special… Expand

#### References

SHOWING 1-10 OF 28 REFERENCES

The Relativistic Noncommutative Nonassociative Group of Velocities and the Thomas Rotation

- Mathematics
- 1989

The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of special theory… Expand

The relativistic velocity composition paradox and the Thomas rotation

- Physics
- 1989

The relativistic velocity composition paradox of Mocanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non-communtativity of the relativistic velocity… Expand

Successive Lorentz transformations of the electromagnetic field

- Physics
- 1991

A velocity-orientation formalism to deal with compositions of successive Lorentz transformations, emphasizing analogies shared by Lorentz and Galilean transformations, has recently been developed.… Expand

Thomas rotation and the parametrization of the Lorentz transformation group

- Physics
- 1988

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation… Expand

The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities

- Mathematics
- 1988

Physics is a major external source of mathematical theories. It is shown here that the Thomas rotation effect of special theory of relativity (STR) gives rise to a “group” structure for the set of… Expand

The Thomas precession and velocity-space curvature

- Physics
- 1986

The motion of a physical system acted upon by external torqueless forces causes the relativistic Thomas precession of the system’s spin vector, relative to an inertial frame. A time‐dependent force… Expand

Axiomatic approach to the nonassociative group of relativistic velocities

- Physics
- 1989

The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of noncollinear velocities is attributed to the presence of the Thomas rotation. The Thomas… Expand

Weakly Associative Groups

- Mathematics
- 1990

The space ℝc3 of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the… Expand

The Thomas Precession

- Physics
- 1972

This paper is intended to give a simple physical understanding of the kinematic effect referred to as the Wigner rotation or, when applied to an orbiting object, the Thomas precession. Since this is… Expand

Thomas precession and its associated grouplike structure

- Physics
- 1991

Mathematics phenomena and discovers the secret analogies which unite them. Joseph Fourier. Where there is physical significance, there is pattern and mathematical regularity. The aim of this article… Expand