Thomas precession
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 and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.
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.
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.
[3] In 1925 Thomas recomputed the relativistic precessional frequency of the doublet separation in the fine structure of the atom.
Let the system be subject to external forces that produce no torque with respect to its center of mass in its (instantaneous) rest frame.
As a simplifying assumption one assumes that the external forces bring the system back to its initial velocity after some finite time.
Introduce a lab frame Σ in which an observer can measure the relative motion 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[6][7][8][9]
No precession occurs if the particle moves with uniform velocity (constant v so a = 0), or accelerates in a straight line (in which case v and a are parallel or antiparallel so their cross product is zero).
The spacetime coordinates of the lab frame are collected into a 4×1 column vector, and the boost is represented as a 4×4 symmetric matrix, respectively and turn is the Lorentz factor of β.
The latter is a small increment, and can be conveniently split into components parallel (‖) and perpendicular (⊥) to β[nb 1] Combining (1) and (2) obtains the Lorentz transformation between Σ′ and Σ′′, and this composition contains all the required information about the motion between these two lab times.
The boost matrix of β + Δβ will require the magnitude and Lorentz factor of this vector.
Since Δβ is small, terms of "second order" |Δβ|2, (Δβx)2, (Δβy)2, ΔβxΔβy and higher are negligible.
In the Cartesian basis ex, ey, ez, a set of mutually perpendicular unit vectors in their indicated directions, we have This simplified setup allows the boost matrices to be given explicitly with the minimum number of matrix entries.
This is an infinitesimal Lorentz transformation in the form of a combined boost and rotation[nb 2] where After dividing Δθ by Δt and taking the limit as in (7), one obtains the instantaneous angular velocity where a is the acceleration of the particle as observed in the lab frame.
No forces were specified or used in the derivation so the precession is a kinematical effect - it arises from the geometric aspects of motion.
The difference (γ-1)ω is the Thomas precession angular frequency already given, as is simply shown by realizing that the magnitude of the 3-acceleration is ω v. In quantum mechanics Thomas precession is a correction to the spin-orbit interaction, which takes into account the relativistic time dilation between the electron and the nucleus in hydrogenic atoms.
Basically, it states that spinning objects precess when they accelerate in special relativity because Lorentz boosts do not commute with each other.
To calculate the spin of a particle in a magnetic field, one must also take into account Larmor precession.
Parallel transport of a spinning particle in relativistic velocity space leads to Thomas precession, which is similar to the rotation of the swing plane of a Foucault pendulum.
[11] The angle of rotation in both cases is determined by the area integral of curvature in agreement with the Gauss–Bonnet theorem.
