# Thomas rotation and the parametrization of the Lorentz transformation group

@article{Ungar1988ThomasRA, title={Thomas rotation and the parametrization of the Lorentz transformation group}, author={Abraham Albert Ungar}, journal={Foundations of Physics Letters}, year={1988}, volume={1}, pages={57-89} }

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 axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical…

## 133 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…

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.…

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…

Graphical Representations for the Successive Lorentz Transformations. Application: Lorentz Contraction and Its Dependence on Thomas Rotation

- Mathematics
- 2016

A new vectorial representation for the successive Lorentz transformations (SLT) has recently been proved very convenient to achieve a straightforward treatment of the Thomas rotation effect. Such a…

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…

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…

On the relativistic velocity composition paradox and the Thomas rotation

- Physics
- 1992

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,…

From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups

- Physics
- 2016

The Lorentz transformation of order $(m=1,n)$, $ninNb$, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the…

Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces

- Physics, Mathematics
- 2015

The Lorentz transformation group $SO(m,n)$ is a group of Lorentz transformations of order $(m,n)$, that is, a group of special linear transformations in a pseudo-Euclidean space of signature $(m,n)$…

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…

## References

SHOWING 1-10 OF 65 REFERENCES

Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two‐component spinors

- Mathematics
- 1986

A set of rotations and Lorentz boosts is presented for studying the three‐parameter little groups of the Poincare group. This set constitutes a Lorentz generalization of the Euler angles for the…

Representation of the active Lorentz transformation for particle dynamics

- Physics
- 1982

The rank‐2 tensor representation of a passive Lorentz transformation constructed by Krause solely from the four‐velocities of two inertial observers is shown to lack sufficient generality to describe…

Applications of the Lorentz Transformation Properties of Canonical Spin Tensors

- Physics
- 1964

Some applications of the Lorentz transformations of relativistic spin tensors in the canonical representation are discussed. The problem of precession of polarization is discussed in Sec. 2. It is…

Lorentz transformations as space‐time reflections

- Physics
- 1977

A rank‐two tensor is built out of the 4‐velocities of two inertial observers, which corresponds precisely to the most general Lorentz matrix connecting the two Cartesian frames of the observers. The…

Vector Lorentz Transformations

- Physics
- 1967

A derivation of the vector Lorentz transformation is given by explicitly compounding the pure Lorentz transformation along one spatial axis with pure spatial rotations. The orthogonal matrix is found…

Lorentz transformations in terms of initial and final vectors

- Mathematics
- 1986

Given arbitrary initial vector(s) and their final vector(s) in a Lorentz transformation, the problem is to determine the operator of the transformation. The solution presented here consists of…

On Unitary Representations of the Inhomogeneous Lorentz Group

- Mathematics
- 1939

It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally…

Galilei Group and Galilean Invariance

- Physics
- 1971

Publisher Summary Galileo Galilei explicitly introduced the principle of relativity in physics. He was the first one to recognize the existence of inertial transformations, connecting various frames…

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…

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…