Thomas precession

Remarks

Proof idea: compare the boost $B(\mathbf{v} + \mathbf{a} \, dt)$ with the composition of $B(\mathbf{v})$ and the small boost relating consecutive rest frames. By Wigner rotation, the two differ by a small rotation, and computing it to first order in $dt$ gives $\boldsymbol{\omega}_T \, dt$. The precession vanishes for straight-line motion, where $\mathbf{a} \times \mathbf{v} = 0$. For $\abs{\mathbf{v}} \ll c$ one has $\gamma \approx 1$ and $\boldsymbol{\omega}_T \approx \frac{1}{2c^2} \mathbf{a} \times \mathbf{v}$. Example: for an electron in an atom, this precession reduces the naive spin-orbit coupling by a factor $\tfrac{1}{2}$, the Thomas half, which brings the fine structure in agreement with experiment.

Going deeper

For a student

Thomas precession is the infinitesimal form of Wigner rotation. In the language of four-vectors, the rest frame is carried along the world line by Fermi–Walker transport. Then $\boldsymbol{\omega}_T$ is the rotation of this transported frame relative to the lab axes. In general relativity it is one part of the precession of a gyroscope in orbit, next to the geodetic effect and frame-dragging.