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.

Topics
Thomas precession, General relativity, Special relativity
1.
The concepts of COM and rest frame for a spatially extended relativistic system are somewhat subtle, and not uniquely defined. See Refs. 2 and 3.
2.
A. Schild, in Relativity Theory and Astrophysics, 1. Relativity and Cosmology, Lectures in Applied Mathematics, Vol. 8, edited by J. Ehlers (Am. Math. Soc., Providence, RI, 1967), pp. 17–20.
3.
H. P. Robertson and T. W. Noonan, Relativity and Cosmology (Saunders, Philadelphia, 1968), §5.4 and 5.5, pp. 139–144.
4.
It will be seen below that this assumption involves no loss of generality.
5.
We use Latin letters for spatial indices, and Greek for Minkowski indices. Our Minkowski metric shall be (l,−1,−1,−1), and we use the normal convention ε123 = −1, εijk = −εjik, and εijk = εjik.
6.
Reference 3, §3.11, pp. 66–69. These authors derive the precession as an infinitesimal Wigner rotation, and do not treat the spin vector specifically. A reference that introduces the spin precession in particular, in the context of atomic physics, where it was discovered, is J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), §11.5, pp. 364–369. Our formula is a more formal version of that appearing in Robertson and Noonan [see our Eq. (6)].
7.
From this point on, all quantities are implicitly understood to be those of the physical system: velocity, acceleration, spin, etc.
8.
Alternatively, one may boost to a frame R(τ+ds), to find (Sμ(τ) in R(τ)) = (Sμ(τ+ds) in R(τ+ds)). Since the physical space axes and clocks, used to operationally define the frames R(s), indeed undergo this boost [of velocity (+dυi)] due to the forces, it follows that the Thomas precession is unobservable via internal measurements in the physical system.
9.
This is possible because V is topologically equivalent to R3
10.
Since the problem is now effectively that of two spatial dimensions, the Wigner rotations are characterized by a single angle.
11.
Reference 3, §7.1 and §8.12. Gauss’ theorem applies to geodesic triangles, but the area enclosed by the trajectory can be divided into an infinite number of such triangles.
12.
That is, (ξ1,ξ2,υ3) is a right‐handed systsm in V.
13.
On maximally symmetric spaces, see, e.g., Ref. 3, Chap. 13.
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© 1986 American Institute of Physics.
1986
American Institute of Physics
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