In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope. It relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion. For a given inertial frame, if a second frame is Lorentz-boosted relative to it, and a third boosted relative to the second, but non-colinear with the first boost, then the Lorentz transformation between the first and third frames involves a combined boost and rotation, known as the "Wigner rotation" or "Thomas rotation". For accelerated motion, the accelerated frame has an inertial frame at every instant. Two boosts a small time interval (as measured in the lab frame) apart leads to a Wigner rotation after the second boost. In the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity. The precession can be understood geometrically as a consequence of the fact that the space of velocities in relativity is hyperbolic, and so parallel transport of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the non-commutativity of Lorentz transformations. Thomas precession gives a correction to the spin–orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom. Thomas precession is a kinematic effect in the flat spacetime of special relativity. In the curved spacetime of general relativity, Thomas precession combines with a geometric effect to produce de Sitter precession. Although Thomas precession (net rotation after a trajectory that returns to its initial velocity) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some external force causing the curvilinear motion such as that caused by an electromagnetic field, a gravitational field or a mechanical force, so Thomas precession is usually accompanied by dynamical effects. If the system experiences no external torque, e.g., in external scalar fields, its spin dynamics are determined only by the Thomas precession. A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.
History Thomas precession in relativity was already known to Ludwik Silberstein in 1914. But the only knowledge Thomas had of relativistic precession came from de Sitter's paper on the relativistic precession of the moon, first published in a book by Eddington. In 1925 Thomas recomputed the relativistic precessional frequency of the doublet separation in the fine structure of the atom. He thus found the missing factor 1/2, which came to be known as the Thomas half. This discovery of the relativistic precession of the electron spin led to the understanding of the significance of the relativistic effect. The effect was consequently named "Thomas precession".
Introduction
Definition Consider a physical system moving through Minkowski spacetime. Assume that there is at any moment an inertial system such that in it, the system is at rest. This assumption is sometimes called the third postulate of relativity. This means that at any instant, the coordinates and state of the system can be Lorentz transformed to the lab system through some Lorentz transformation. Let the system be subject to external forces that produce no torque with respect to its center of mass in its (instantaneous) rest frame. The condition of "no torque" is necessary to isolate the phenomenon of Thomas precession. As a simplifying assumption one assumes that the external forces bring the system back to its initial velocity after some finite time. Fix a Lorentz frame O such that the initial and final velocities are zero. The Pauli–Lubanski spin vector Sμ is defined to be (0, Si) in the system's rest frame, with Si the angular-momentum three-vector about the center of mass. In the motion from initial to final position, Sμ undergoes a rotation, as recorded in O, from its initial to its final value. This continuous change is the Thomas precession.
Statement
Consider the motion of a particle. Introduce a lab frame Σ in which an observer can measure the relative motion of the particle. At each instant of time the particle has an inertial frame in which it is at rest. Relative to this lab frame, the instantaneous velocity of the particle is v(t) with magnitude |v| = v bounded by the speed of light c, so that 0 ≤ v < c. Here the time t is the coordinate time as measured in the lab frame, not the proper time of the particle. Apart from the upper limit on magnitude, the velocity of the particle is arbitrary and not necessarily constant; its corresponding vector of acceleration is a = dv(t)/dt. As a result of the Wigner rotation at every instant, the particle's frame precesses with an angular velocity given by the equation
where × is the cross product and
γ = 1 1 − | v ( t ) | 2 c 2 {\displaystyle \gamma ={\dfrac {1}{\sqrt {1-{\dfrac {|\mathbf {v} (t)|^{2}}{c^{2}}}}}}}
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