Thomas Precession in Accelerated Planar Motion

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)


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
[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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+