Thomas precession
Encyclopedia
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, and derived by Wigner in 1939. 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.
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. 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
.
, 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 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.
, 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.
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. 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:
and 
are parallel. In fact,
,
where gyr is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by
for all w.
The gyr operator forms the foundation of gyrovector space
s.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
the Thomas precession, named after Llewellyn Thomas
Llewellyn Thomas
Llewellyn Hilleth Thomas was a British physicist and applied mathematician. He is best known for his contributions to atomic physics, in particular:...
, is a special relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
correction that applies to the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
of an elementary particle or the rotation of a macroscopic gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
and relates the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
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
Mathematical space
In mathematics, a space is a set with some added structure.Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space...
of velocities in relativity is hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, and so parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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
Velocity-addition formula
In physics, a velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.- Galilean addition of velocities :...
.
It gives a correction to the spin-orbit interaction
Spin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, which takes into account the relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
between the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
and the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
of an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
.
The composition of two Lorentz boosts which are non-collinear, results in a Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
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, and derived by Wigner in 1939. 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.
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
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...
causing the curvilinear motion such as that caused by an electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
, gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
or mechanical force so Thomas precession is always accompanied by dynamical effects. 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
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, one must also take into account Larmor precession
Larmor precession
In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...
.
History
Thomas precession in relativity was already known to Ludwik SilbersteinLudwik Silberstein
Ludwik Silberstein was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework...
, 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
Arthur Stanley Eddington
Sir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
.
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.
In Quantum Mechanics
In quantum mechanics Thomas precession is a correction to the spin-orbit interactionSpin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
, which takes into account the relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
between the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
and the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
in hydrogenic atoms
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
.
Basically, it states that spinning objects precess when they accelerate in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
because Lorentz boosts do not commute with each other.
In a Foucault pendulum
The rotation of the swing plane of Foucault pendulumFoucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...
can be treated as a result of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
of the pendulum in a 2-dimensional sphere of Euclidean space. The hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
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. 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:
Velocity composition
Einstein velocity addition is associative and commutative only when and are parallel. In fact,,where gyr is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by
for all w.
The gyr operator forms the foundation of gyrovector space
Gyrovector space
A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. This vector-based approach has been developed by Abraham Albert Ungar from the late 1980s onwards...
s.
External links
- Mathpages article on Thomas Precession
- Alternate, detailed derivation of Thomas Precession (by Robert Littlejohn)