# Thomas Precession in Accelerated Planar Motion

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Gyroscopes aligned with the coordinate axes of a relativistically accelerated observer are seen by an inertial observer to precess even in torque-free acceleration. Increase the values to simulate precession and use the rapidity control () to vary the speed of the body; slow the simulation down for high rapidity values to see the precession more clearly. You can unclick the "show contraction" checkbox to suppress the Lorentz–Fitzgerald contraction of the gyroscope lengths to focus their directions.

Contributed by: Rod Vance (a.k.a. The Wet Savanna Animal) (September 2015)

Open content licensed under CC BY-NC-SA

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## Details

Thomas precession is a consequence of Wigner rotation, which arises because there is no group of three-dimensional Lorentz boosts: two noncollinear boosts generally compose to a boost *plus* a rotation, and the latter cannot be eliminated. This is in contrast with the group of Galilean transformations, in which there is a three-dimensional group of Galilean boosts: two successive Galilean boosts compose to another pure Galilean boost.

In the Lorentz case, the coordinate axes of a rigid body undergoing relativistic circular motion under the action of a torque-free force precess in the opposite sense to the circular motion through an angle of for each circuit of the motion, where is the Lorentz factor. This angle is negative, showing the sense of precession *counter to* the sense of circular motion.

In contrast, in Newtonian torque-free circular motion (i.e., with the net force directed toward the center of the circle through the center of mass), the body's axes maintain a constant direction. Reference [2] presents a simple, intuitive derivation of this phenomenon.

References

[1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, *Gravitation*, San Francisco: W. H. Freeman, 1973 Section 6.6.

[2] A. Dragan and T. Odrzygóźdź, "A Half-Page Derivation of the Thomas Precession," *American Journal of Physics*, 81, 2013 p. 631–632. doi:10.1119/1.4807564.

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