Perpendicular-Boost Wigner Rotation

worked-example

Consider two rapidities a = 1 and b = 1. The velocities are \beta_a ≈ 0.76159 and \beta_b ≈ 0.76159. The rotation magnitude Ω can be computed as follows: This requires solving the nonlinear combination of the two rapidities using the exact formula provided above, ensuring the correct result when dealing with pure special relativistic effects. The following arithmetic explicit trace shows the exact evaluation to sufficient precision, avoiding rounding errors:

Omega = 2 * atan(tanh(1/2) * tanh(1/2)) = 0.42078 radians.

experiment

Use the interactive module to set the first and second rapidity and observe the Wigner rotation. We will trace how changing inputs affect the rotation axis. You can find the module at the Interactive Module link.

1. Set the first rapidity control to exactly 1 to configure the primary boost along the x-axis.
2. Set the second rapidity control to exactly 1 to configure the secondary boost, and observe the rotation arc.

limitations

This module assumes flat spacetime (special relativity) and standard non-commuting pure boosts along orthogonal axes. It reports only the magnitude of the rotation, not the full matrix. Further analysis is required to model relativistic velocity additions with generic non-orthogonal boosts or involving accelerations. The provided formula is specific to two purely orthogonal boosts with the specified rapidities, lacking generalization to arbitrary directions.

references

For more details on the kinematics of Wigner rotation, see the provided reference. https://arxiv.org/abs/physics/0211022This paper provides an exhaustive derivation of the Wigner rotation formula, examining how successive Lorentz boosts in perpendicular directions result in a net spatial rotation rather than just a simple translational boost.