Thomas precession

In physics the Thomas precession, named after Llewellyn Thomas, is a special 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. It 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-associativity of the relativistic velocity-addition formula.

It 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.

The composition of two Lorentz boosts which are non-collinear, results in a Lorentz transformation that is not a pure boost but is the product of a boost and a rotation. This rotation is called Thomas rotation, Thomas-Wigner rotation or Wigner rotation. The rotation was discovered by Thomas in 1926[1], and derived by Wigner in 1939[2]. If a sequence of non-collinear boosts returns the spatial origins of a sequence of inertial frame to the starting point, then the sequence of Wigner rotations combine to produce a net rotation called the Thomas precession[3].

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 starting point) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some centripetal force causing the curvilinear motion such as that caused by an electromagnetic field, gravitational field or mechanical force so Thomas precession is always accompanied by dynamical effects. [4] A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to Thomas precession) is present in non-dynamical situations whenever you have 3 or more inertial frames in non-collinear motion - see the velocity composition section below.

To calculate the spin of a particle in a magnetic field, one must also take into account Larmor precession.

Contents

History

Thomas precession in relativity was already known to Ludwik Silberstein,[5] 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.[6]

In 1925 Thomas relativistically recomputed the 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 therefore named Thomas precession.

Applications

In Quantum Mechanics

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.

In a Foucault pendulum

The rotation of the swing plane of Foucault pendulum can be treated as a result of parallel transport of the pendulum in a 2-dimensional sphere of Euclidean space. The hyperbolic space of velocities in Minkowski spacetime represents a 3-dimensional (pseudo-) sphere with imaginary radius and imaginary timelike coordinate. Parallel transport of spinning particle in the relativistic velocity space leads to Thomas precession, which is similar to the rotation of the swing plane of Foucault pendulum. [7] The angle of rotation in both cases is determined by the area integral of curvature in agreement with the Gauss-Bonnet theorem.

Thomas precession gives a correction to the precession of a Foucault pendulum. For a Foucault pendulum located in the city of Nijmegen in the Netherlands the correction is:

\omega \approx 9.5 \cdot 10^{-7}\, \mathrm{arcseconds} / \mathrm{day}.

Velocity composition

Einstein velocity addition is associative and commutative only when \mathbf{v} and \mathbf{u} are parallel. In fact,

\mathbf{v} \oplus \mathbf{u}=\text{gyr}[\mathbf{v},\mathbf{u}](\mathbf{u} \oplus \mathbf{v}),

where gyr is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by

\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))

for all w.

The gyr operator forms the foundation of gyrovector spaces.[8]

References

  1. ^ L. H. Thomas, "Motion of the spinning electron", Nature 117, 514, 1926
  2. ^ E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40, 149–204 (1939).
  3. ^ Relativistic velocity space, Wigner rotation and Thomas precession, John A. Rhodes, Mark D. Semon (2005)
  4. ^ G. B. Malykin, Thomas precession: correct and incorrect solutions, Phys. Usp. 49, 83 (2006).
  5. ^ L. Silberstein, The Theory of Relativity (MacMillan London 1914), page 169
  6. ^ A.S. Eddington, The Mathematical Theory of Relativity (Cambridge 1924)
  7. ^ M. I. Krivoruchenko, Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin, Phys. Usp. 52, 821-829 (2009).
  8. ^ A.A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, Kluwer, 2002.

Textbooks

External links