Boost Composition and Wigner Rotation in Rhodes-Semon Rapidity Space

Relativistic velocity addition can be simulated by vector addition in hyperbolic space. This Demonstration shows the Rhodes–Semon construction in the Poincaré disk that computes the composition of Lorentz transformations representing noncollinear relativistic velocity additions. Use the 2D sliders and to set the combined boosts. The Wigner rotation equals the difference between the angles of the purple arrow and its parallel-transported image, the orange arrow. Alternatively, unclick the "two points" control to set the first boost , and then the second boost can be set as a heading angle and a rapidity with the "angle" and "rapidity" controls.

A Lorentz transformation is represented by a point together with an arrow , where the defines the boost direction, the boost rapidity, and the rotation following the boost. A Lorentz transformation with boost component , followed by a second Lorentz transformation with boost component , gives a combined transformation with boost component . The Wigner rotation arising from this combined transformation is found by parallel-transporting the arrow representing the rotation component of the first transformation (the short purple arrow shown above) along the geodesic arc joining and the final state . This geodesic arc connects the points of the unique circle that both joins the points and meets the unit circle at right angles.

Reference

[1] J. A. Rhodes and M. D. Semon, "Relativistic Velocity Space, Wigner Rotation, and Thomas Precession," American Journal of Physics, 72(7), 2004 pp. 943–960. doi:10.1119/1.1652040.