The Thomas precession and velocity-space curvature

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Abstract
The motion of a physical system acted upon by external torqueless forces causes the relativistic Thomas precession of the system’s spin vector, relative to an inertial frame. A time-dependent force that returns the system to its initial velocity is considered. The precession accumulates to become a finite rotation of the final spin vector, relative to its initial value. This rotation is commonly explained as the Wigner rotation due to the sequence of pure boosts caused by the force. An alternative interpretation is presented here: The rotation is due to the change of the spin vector as it is parallel-transported around the closed trajectory described by the system in hyperbolic three-velocity space. As an application, the angle of precession for a planar motion is shown to be equal to the area enclosed by the trajectory in velocity space.

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