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In [[physics]] the '''Thomas precession''', named after [[Llewellyn Thomas]], is a [[Theory of relativity|relativistic]] correction that applies to the [[Spin (physics)|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 [[mathematical space|space]] of velocities in relativity is [[hyperbolic geometry|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 [[Atomic nucleus|nucleus]] of an [[atom]]. | |||
The composition of two [[Lorentz boost]]s 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,<ref>L. H. Thomas, "Motion of the spinning electron", Nature 117, 514, 1926</ref> and derived by Wigner in 1939.<ref>E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. | |||
40, 149–204 (1939).</ref> 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'''.<ref>[http://arxiv.org/abs/gr-qc/0501070v1 Relativistic velocity space, Wigner rotation and Thomas precession], John A. Rhodes, Mark D. Semon (2005)</ref> | |||
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.<ref>G. B. Malykin, ''Thomas precession: correct and incorrect solutions'', Phys. Usp. 49, 83 (2006).</ref> In the Lorentz scalar field, spin of the particle does not feel the torque, resulting in the 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 3 or more inertial frames in non-collinear motion - see the [[#Velocity composition|velocity composition]] section below. | |||
To calculate the spin of a particle in a [[magnetic field]], one must also take into account [[Larmor precession]]. | |||
== History == | |||
Thomas precession in relativity was already known to [[Ludwik Silberstein]],<ref>L. Silberstein, ''The Theory of Relativity'' (MacMillan London 1914), page 169</ref> 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 [[Arthur Stanley Eddington|Eddington]].<ref>A.S. Eddington, The Mathematical Theory of Relativity (Cambridge 1924)</ref> | |||
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 [[special relativity|relativistic]] [[time dilation]] between the [[electron]] and the [[Atomic nucleus|nucleus]] in [[Hydrogen atom|hydrogenic atoms]]. | |||
Basically, it states that spinning objects [[precess]] when they accelerate in [[special relativity]] because [[Lorentz boost]]s 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 geometry|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.<ref>M. I. Krivoruchenko, [http://arxiv.org/abs/0805.1136 ''Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin''], Phys. Usp. 52, 821-829 (2009).</ref> 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: | |||
:<math>\omega \approx 9.5 \cdot 10^{-7}\, \mathrm{arcseconds} / \mathrm{day}.</math> | |||
==See also== | |||
* [[Velocity-addition formula]] | |||
* [[Relativistic angular momentum]] | |||
== References == | |||
<references /> | |||
== Textbooks == | |||
* {{cite book | first = Wolfgang | last = Rindler | authorlink = Wolfgang Rindler | coauthors = | year = 2006 | month = | title = Relativity Special, General and Cosmological | chapter = 9 | editor = | others = | edition = second edition | pages = | publisher = Oxford University Press | location = Dallas | isbn = 978-0-19-856732-5 | url = }} | |||
== External links == | |||
*[http://www.mathpages.com/rr/s2-11/2-11.htm Mathpages article on Thomas Precession] | |||
*[http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf Alternate, detailed derivation of Thomas Precession] (by Robert Littlejohn) | |||
{{Relativity}} | |||
[[Category:Special relativity]] | |||
[[Category:Atomic physics]] | |||
[[Category:Precession]] | |||
Revision as of 06:28, 14 March 2013
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-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] In the Lorentz scalar field, spin of the particle does not feel the torque, resulting in the 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 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.
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:
See also
References
- ↑ L. H. Thomas, "Motion of the spinning electron", Nature 117, 514, 1926
- ↑ E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40, 149–204 (1939).
- ↑ Relativistic velocity space, Wigner rotation and Thomas precession, John A. Rhodes, Mark D. Semon (2005)
- ↑ G. B. Malykin, Thomas precession: correct and incorrect solutions, Phys. Usp. 49, 83 (2006).
- ↑ L. Silberstein, The Theory of Relativity (MacMillan London 1914), page 169
- ↑ A.S. Eddington, The Mathematical Theory of Relativity (Cambridge 1924)
- ↑ 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).
Textbooks
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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External links
- Mathpages article on Thomas Precession
- Alternate, detailed derivation of Thomas Precession (by Robert Littlejohn)