Thomas-Wigner rotation

In special relativity the combination of two non-collinear pure Lorentz transformations (boosts) is not a boost. Rather, it is a boost combined with a spatial rotation, the Thomas-Wigner rotation. We visualize this relativistic effect by moving a Born-rigid object in two spatial dimensions (e.g. within the xy-plane) on a closed trajectory.

For simplicity, the object's path is split into a finite number of segments. Within each segment the object's proper acceleration is taken to be constant. In addition to the acceleration's magnitude and direction also the boost's (proper time) duration is assumed to be the same in each section. Finally, the object is initially at rest and returns to its starting position after completion of its trajectory.

Visualization of Thomas-Wigner rotation

The animation above visualizes the Thomas-Wigner rotation by boosting a square object consisting 21 x 21 points five times with a proper acceleration of 1 ls/s2. Here, the unit of length is light-seconds (ls). The color code indicates the individual point's current segment number (ranging from boost no. 1 to boost no. 5, top left). Born-rigidity implies that the switchover between one boost segment to the next is synchronous in the comoving inertial frame. Hence, the switchover is asynchronous in the laboratory frame and at certain instances in (laboratory) time one part of the object appears to be in one, the other part in the next boost section. This asynchronicity produces not only a shortening, but also a shear effect, which eventually adds up to the object's rotation.

The red point marks the reference vertex. The clock in the top right corner displays coordinate time in the laboratory frame. The proper time duration for each boost is chosen such that the objects lab frame velocity is β = 0.7 at the end of boost no. 1. With these parameter settings the resulting Thomas-Wigner rotation angle is 33.7°.

This movie shows the same simulation as the animinated GIF above, however in higher resolution.

Technical details are given in this paper. The SymPy source files
are used in the derivation and evaluation of several relations used in the simulation. The MATLAB script
creates the simulation graphics.


M. Born (1909): Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips.
Annalen der Physik, 335(11):1-56.

A. Einstein (1905): Zur Elektrodynamik bewegter Körper.
Annalen der Physik, 322(10):891-921.

G. Herglotz (1909): Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper.
Annalen der Physik, 336(2):393-415.

F. Noether (1910): Zur Kinematik des starren Körpers in der Relativtheorie.
Annalen der Physik, 336(5):919-944.

L. H. Thomas (1926): The motion of the spinning electron.
Nature, 117(2945):514-514,

L. H. Thomas (1927): The kinematics of an electron with an axis.
Philos. Mag., 3(13):1-22,

E. P. Wigner (1939): On unitary representations of the inhomogeneous Lorentz group.
Ann. Math. 40(1):149-204,