# The relativistic velocity composition paradox and the Thomas rotation

@article{Ungar1989TheRV, title={The relativistic velocity composition paradox and the Thomas rotation}, author={A. Ungar}, journal={Foundations of Physics}, year={1989}, volume={19}, pages={1385-1396} }

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 composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are “some difficulties within the framework of relativistic electrodynamics.” The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by… Expand

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#### References

SHOWING 1-10 OF 39 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

Lorentz Transformation and the Thomas Precession

- Physics
- 1955

The various kinematic effects of special relativity are here treated by considering iterations of the infinitesimal Lorentz transformation. There is only one relativistic effect that appears as a… 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 Motion of the Spinning Electron

- Physics
- Nature
- 1926

IN a letter published in NATURE of February 20, p. 264, Messrs. Uhlenbeck and Goudsmit have shown how great difficulties which atomic theory had met in the attempt to explain spectral structure and… Expand

Relativistic motion of a charged particle, the Lorentz group, and the Thomas precession

- Physics
- 1984

The equation of motion of a classical charged particle in a homogeneous, static electromagnetic field is solved exactly in terms of a Lorentz transformation. All higher‐order corrections to the… 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

Derivation of the Spin-Orbit Interaction

- Physics
- 1967

Many students find it paradoxical that spin-orbit coupling is discussed in nonrelativistic quantum mechanics although it is usually said to be a consequence of relativity. The resolution of this… Expand

Special relativity completed: The source of some 2s in the magnitude of physical phenomena

- Physics
- 1986

Some factors of 2 relate predictions made of relativistic systems by the special and the general theory. Conventional wisdom often assigns the source of the factors of 2 to complicated kinematical… Expand

The Lorentz group and the Thomas precession. II. Exact results for the product of two boosts

- Physics
- 1986

The product of two Lorentz boosts in different directions is equal to the product of a pure boost and a spatial rotation. To second order, the resulting boost is simply the sum of the individual… 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