Thomas precession: Difference between revisions

From Wikipedia, the free encyclopedia
Browse history interactively
← Previous editNext edit →
Content deleted Content added
VisualWikitext
Line 98: Line 98:


*{{cite journal|ref=harv|first=L. H.|last=Thomas|authorlink=Llewellyn Thomas|title=Motion of the spinning electron|journal=Nature|volume=117|pages=514|year=1926|url=http://www.nature.com/nature/journal/v117/n2945/abs/117514a0.html|doi=10.1038/117514a0|subscription=yes}}
*{{cite journal|ref=harv|first=L. H.|last=Thomas|authorlink=Llewellyn Thomas|title=Motion of the spinning electron|journal=Nature|volume=117|pages=514|year=1926|url=http://www.nature.com/nature/journal/v117/n2945/abs/117514a0.html|doi=10.1038/117514a0|subscription=yes}}

*{{cite journal|ref=harv|title=Thomas rotation and parameterization of the Lorentz group|journal=Foundations of physics letters|volume=1|issue=1|author=A. A. Ungar|pages=57–81|year=1988|url=http://link.springer.com/article/10.1007/BF00661317|subscription=yes|doi=10.1007/BF00661317|publisher=[[Springer Science+Business Media|Springer]]|issn=0894-9875}}


*{{citation|first=E. P.|last=Wigner|authorlink=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]|issue=1|volume=40|pages=149–204|year=1939|doi=10.2307/1968551|mr=1503456|bibcode=1939AnMat..40..922E}}.
*{{citation|first=E. P.|last=Wigner|authorlink=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]|issue=1|volume=40|pages=149–204|year=1939|doi=10.2307/1968551|mr=1503456|bibcode=1939AnMat..40..922E}}.

Revision as of 11:58, 12 July 2015

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. 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-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.

The composition of two non-collinear Lorentz boosts, results in a Lorentz transformation that is not a pure boost but is the composition 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 an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.[3]

There are still ongoing discussions about the correct form of equations for the Thomas precession in different reference systems with contradicting results.[4] Goldstein:[5]

The spatial rotation resulting from the successive application of two non-collinear Lorentz transformations have been declared every bit as paradoxical as the more frequently discussed apparent violations of common sense, such as the twin paradox.

Einstein's principle of velocity reciprocity (EPVR) reads[6]

We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v

With less careful interpretation, the EPVR is seemingly violated in some models.[7] There is, of course, no true paradox present.

History

Thomas precession in relativity was already known to Ludwik Silberstein,[8] 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.[9]

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 consequently named "Thomas precession".

Introduction

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 always accompanied by dynamical effects.[10]

If the system experiences no external torque, its spin dynamics is determined only by the Thomas precession. 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 three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.

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.[11] 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 produces 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 brings the system back to its initial velocity after some finite time. Fix a Lorentz frame O such that this initial and final velocity is 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.[12]

Basic phenomena, their cause and explanation

Velocity composition and Thomas rotation in xy plane, velocities u and v separated by angle θ. Left: As measured in Σ′, the orientations of Σ and Σ′′ appear parallel to Σ′. Right: In frame Σ, Σ′′ moves with velocity w relative to Σ and is rotated through angle ε about an axis parallel to u×v.[13]

When studying Thomas precession at the fundamental level, one typically uses a setup with three coordinate frames, Σ, Σ′ Σ′′. Frame Σ′ has velocity u relative to frame Σ, and frame Σ′′ has velocity v relative to frame Σ′. The axes are oriented as follows. Viewed from Σ′, the axes of Σ′ and Σ are parallel (the same holds true for the pair of frames when viewed from Σ.) Also viewed from Σ′, the axes of Σ′ and Σ′′ are parallel (and the same holds true for the pair of frames when viewed from Σ′′.)

This is a "discrete" version of the continuous Thomas precession tailored to study the Thomas rotation resulting from two consecutive finite boosts, whereas in the continuous process, the system can be thought of as being "continually boosted".

The velocity of Σ′′ as seen in Σ is denoted wd = u ⊕ v, where ⊕ refers to the relativistic addition of velocity (and not ordinary vector addition), to be formally introduced at a later stage.

Consider the reversed configuration, namely, frame Σ moves with velocity −u relative to frame Σ′, and frame Σ′, in turn, moves with velocity −v relative to frame Σ′′. In short, u → − u and v → −v by velocity reciprocity. Then the velocity of Σ relative to Σ′′ is (−v) ⊕ (−u) ≡ −v ⊕ u. By velocity reciprocity again, the velocity of Σ′′ relative to Σ is then wi = v ⊕ u. (A)

But wd ≠ wi. While they are equal in magnitude, there is an angle between them. Which is the correct velocity of Σ′′ relative to Σ? Since this inequality may be somewhat unexpected, this question is warranted.[nb 1]

The answer to the question lies in the Thomas precession, and that one must be careful in specifying which coordinate system is involved at each step. When viewed from Σ, the coordinate axes of Σ and Σ′′ are not parallel. While this can be hard to imagine since both pairs (Σ, Σ′) and (Σ′, Σ′′) have parallel coordinate axes, it is easy to explain qualitatively mathematically. The generators of boosts, K1, K2, K3, in different directions do not commute. This has the effect that two consecutive boosts is not a pure boost in general, but a boost followed by a rotation equivalently, a rotation preceding a boost. Formally,

where, as the notation indicates, the B′s are boosts depending on their argument, and R(u, v) is a rotation. Now, realizing that wd is calculated in Σ and wi is calculated in Σ′′, one finds that wd and wi are supposed to be numerically different. Both are correct! In system Σ the velocity wd applies, and in system Σ′′ it is wi. The fallacious point in the original discussion is (A). While everything is in order with reciprocity, and the result for wi is true, the coordinates in which it is true are those of Σ′′.

Returning to the continuous case, the effect of the torqueless forces on Sμ between two instants of proper system time t, t + Δt, is to multiply it by a pure boost matrix (in an appropriate representation). The product of these matrices, a limiting case as Δt → 0, for the entire trajectory corresponds to a pure spatial rotation, and this is the common explanation of the precession.[14]

Applications

In electron orbitals

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.

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 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.[15] 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:

See also

Remarks

  1. ^ This is sometimes called the Mocanu paradox. Mocanu himself didn't name it a paradox, but rather a "difficulty" within the framework of relativistic electrodynamics in a 1986 paper. He was also quick to acknowledge that the problem is explained by Thomas precession Mocanu (1992), but the name lingers on.

Notes

  1. ^ Thomas 1926
  2. ^ Wigner 1939
  3. ^ Rhodes & Semon 2005
  4. ^ Rebilas 2013
  5. ^ Goldstein 1980, p. 287
  6. ^ Einstein 1922
  7. ^ Mocanu 1992
  8. ^ Silberstein 1914, p. 169
  9. ^ Eddington 1924
  10. ^ Malykin 2006
  11. ^ Goldstein 1980
  12. ^ Ben-Menahem 1986
  13. ^ Ungar 1988, p. 61
  14. ^ Ben-Menahem 1986
  15. ^ Krivoruchenko 2009

References

  • Ben-Menahem, S. (1986). "The Thomas precession and velocityspace curvature". J. Math. Phys. 27: 1284 pp. doi:10.1063/1.527132. {{cite journal}}: Invalid |ref=harv (help)

Textbooks

External links

Retrieved from "https://en.wikipedia.org/w/index.php?title=Thomas_precession&oldid=671098388"