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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.

- History
- Introduction
- Definition
- Statement
- Mathematical explanation
- Lorentz transformations
- Extracting the formula
- Applications
- In electron orbitals
- In a Foucault pendulum
- See also
- Remarks
- Notes
- References
- Textbooks
- External links

For a given inertial frame, if a second frame is Lorentz-boosted relative to it, and a third boosted relative to the second, but non-colinear with the first boost, then the Lorentz transformation between the first and third frames involves a combined boost and rotation, known as the "Wigner rotation" or "Thomas rotation". For accelerated motion, the accelerated frame has an inertial frame at every instant. Two boosts a small time interval (as measured in the lab frame) apart leads to a Wigner rotation after the second boost. In the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity.

The precession 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.

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 usually accompanied by dynamical effects.^{ [1] }

If the system experiences no external torque, e.g., in external scalar fields, its spin dynamics are determined only by the Thomas precession. A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.

Thomas precession in relativity was already known to Ludwik Silberstein,^{ [2] } 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.^{ [3] }

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".

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.^{ [4] } 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 produce 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 bring the system back to its initial velocity after some finite time. Fix a Lorentz frame O such that the initial and final velocities are zero.

The **Pauli–Lubanski spin vector**S_{μ} is defined to be (0, *S*_{i}) in the system's *rest* frame, with S_{i} the angular-momentum three-vector about the center of mass. In the motion from initial to final position, S_{μ} undergoes a rotation, as recorded in O, from its initial to its final value. This continuous change is the Thomas precession.^{ [5] }

Consider the motion of a particle. Introduce a lab frame Σ in which an observer can measure the relative motion of the particle. At each instant of time the particle has an inertial frame in which it is at rest. Relative to this lab frame, the instantaneous velocity of the particle is **v**(*t*) with magnitude |**v**| = *v* bounded by the speed of light c, so that 0 ≤ *v* < *c*. Here the time t is the coordinate time as measured in the lab frame, *not* the proper time 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** = *d***v**(*t*)/*dt*. As a result of the Wigner rotation at every instant, the particle's frame precesses with an angular velocity given by the^{ [6] }^{ [7] }^{ [8] }^{ [9] }

where × is the cross product and

is the instantaneous Lorentz factor, a function of the particle's instantaneous velocity. Like any angular velocity, **ω**_{T} is a pseudovector; its magnitude is the angular speed the particle's frame precesses (in radians per second), and the direction points along the rotation axis. As is usual, the right-hand convention of the cross product is used (see right-hand rule).

The precession depends on *accelerated* motion, and the non-collinearity of the particle's instantaneous velocity and acceleration. 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 particle has to move in a curve, say an arc, spiral, helix, or a circular orbit or elliptical orbit, for its frame to precess. The angular velocity of the precession is a maximum if the velocity and acceleration vectors are perpendicular throughout the motion (a circular orbit), and is large if their magnitudes are large (the magnitude of **v** is almost c).

In the non-relativistic limit, **v** → **0** so *γ* → 1, and the angular velocity is approximately

The factor of 1/2 turns out to be the critical factor to agree with experimental results. It is informally known as the "Thomas half".

The description of relative motion involves Lorentz transformations, and it is convenient to use them in matrix form; symbolic matrix expressions summarize the transformations and are easy to manipulate, and when required the full matrices can be written explicitly. Also, to prevent extra factors of c cluttering the equations, it is convenient to use the definition **β**(*t*) = **v**(*t*)/*c* with magnitude |**β**| = *β* such that 0 ≤ *β* < 1.

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**. In other frames, the corresponding coordinates are also arranged into column vectors. The inverse matrix of the boost corresponds to a boost in the opposite direction, and is given by *B*(**β**)^{−1} = *B*(−**β**).

At an instant of lab-recorded time t measured in the lab frame, the transformation of spacetime coordinates from the lab frame Σ to the particle's frame Σ′ is

**(1)**

and at later lab-recorded time *t* + Δ*t* we can define a new frame Σ′′ for the particle, which moves with velocity **β** + Δ**β** relative to Σ, and the corresponding boost is

**(2)**

The vectors **β** and Δ**β** are two separate vectors. 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′′,

**(3)**

and this composition contains all the required information about the motion between these two lab times. Notice *B*(**β** + Δ**β**)*B*(−**β**) and *B*(**β** + Δ**β**) are infinitesimal transformations because they involve a small increment in the relative velocity, while *B*(−**β**) is not.

The composition of *two* boosts equates to a single boost combined with a Wigner rotation about an axis perpendicular to the relative velocities;

**(4)**

The rotation is given by is a 4×4 rotation matrix *R* in the axis–angle representation, and coordinate systems are taken to be right-handed. This matrix rotates 3d vectors anticlockwise about an axis (active transformation), or equivalently rotates coordinate frames clockwise about the same axis (passive transformation). The axis-angle vector Δ**θ** parametrizes the rotation, its magnitude Δ*θ* is the angle Σ′′ has rotated, and direction is parallel to the rotation axis, in this case the axis is parallel to the cross product (−**β**)×(**β** + Δ**β**) = −**β**×Δ**β**. If the angles are negative, then the sense of rotation is reversed. The inverse matrix is given by *R*(Δ**θ**)^{−1} = *R*(−Δ**θ**).

Corresponding to the boost is the (small change in the) boost vector Δ**b**, with magnitude and direction of the relative velocity of the boost (divided by *c*). The boost *B*(Δ**b**) and rotation *R*(Δ**θ**) here are infinitesimal transformations because Δ**b** and rotation Δ**θ** are small.

The rotation gives rise to the Thomas precession, but there is a subtlety. To interpret the particle's frame as a co-moving inertial frame relative to the lab frame, and agree with the non-relativistic limit, we expect the transformation between the particle's instantaneous frames at times *t* and *t* + Δ*t* to be related by a boost *without* rotation. Combining (** 3 **) and (** 4 **) and rearranging gives

**(5)**

where another instantaneous frame Σ′′′ is introduced with coordinates *X*′′′, to prevent conflation with Σ′′. To summarize the frames of reference: in the lab frame Σ an observer measures the motion of the particle, and three instantaneous inertial frames in which the particle is at rest are Σ′ (at time t), Σ′′ (at time *t* + Δ*t*), and Σ′′′ (at time *t* + Δ*t*). The frames Σ′′ and Σ′′′ are at the same location and time, they differ only by a rotation. By contrast Σ′ and Σ′′′ differ by a boost and lab time interval Δ*t*.

Relating the coordinates *X*′′′ to the lab coordinates *X* via (** 5 **) and (** 2 **);

**(6)**

the frame Σ′′′ is rotated in the negative sense.

The rotation is between two instants of lab time. As Δ*t* → 0, the particle's frame rotates at every instant, and the continuous motion of the particle amounts to a continuous rotation with an angular velocity at every instant. Dividing −Δ**θ** by Δ*t*, and taking the limit Δ*t* → 0, the angular velocity is by definition

**(7)**

It remains to find what Δ**θ** precisely is.

The composition can be obtained by explicitly calculating the matrix product. The boost matrix of **β** + Δ**β** will require the magnitude and Lorentz factor of this vector. Since Δ**β** is small, terms of "second order" |Δ**β**|^{2}, (Δ*β _{x}*)

and expanding the Lorentz factor of **β** + Δ**β** as a power series gives to first order in Δ**β**,

using the Lorentz factor *γ* of **β** as above.

Composition of boosts in the xy plane

To simplify the calculation without loss of generality, take the direction of **β** to be entirely in the *x* direction, and Δ**β** in the *xy* plane, so the parallel component is along the *x* direction while the perpendicular component is along the *y* direction. The axis of the Wigner rotation is along the *z* direction. In the Cartesian basis **e**_{x}, **e**_{y}, **e**_{z}, 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. In general, of course, **β** and Δ**β** can be in any plane, the final result given later will not be different.

Explicitly, at time *t* the boost is in the negative *x* direction

and the boost at the time *t* + Δ*t* is

where *γ* is the Lorentz factor of **β**, *not***β** + Δ**β**. The composite transformation is then the matrix product

Introducing the boost generators

and rotation generators

along with the dot product · facilitates the coordinate independent expression

which holds if **β** and Δ**β** lie in any plane. This is an *infinitesimal* Lorentz transformation in the form of a combined boost and rotation^{ [nb 2] }

where

After dividing Δ**θ** by Δ*t* 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. However, forces cause accelerations, so the Thomas precession is observed if the particle is subject to forces.

Thomas precession can also be derived using the Fermi-Walker transport equation.^{ [10] } One assumes uniform circular motion in flat Minkowski spacetime. The spin 4-vector is orthogonal to the velocity 4-vector. Fermi-Walker transport preserves this relation. One finds that the dot product of the acceleration 4-vector with the spin 4-vector varies sinusoidally with time with an angular frequency Ύ ω, where ω is the angular frequency of the circular motion and Ύ=1/√⟨1-v^2/c^2). This is easily shown by taking the second time derivative of that dot product. Because this angular frequency exceeds ω, the spin precesses in the retrograde direction. 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.

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.^{ [11] } 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:

Note that it is more than two orders of magnitude smaller than the precession due to the general-relativistic correction arising from frame-dragging, the Lense–Thirring precession.

- ↑ Explicitly, using vector projection and rejection relative to the direction of
**β**gives - ↑ The rotation and boost matrices (each infinitesimal) are given by
**θ**·**J**)(Δ**b**·**K**) and (Δ**b**·**K**)(Δ**θ**·**J**) are negligible. The full boost and rotations*do not*commute in general.

- ↑ Malykin 2006
- ↑ Silberstein 1914 , p. 169
- ↑ Eddington 1924
- ↑ Goldstein 1980
- ↑ Ben-Menahem 1986
- ↑ Jackson 1975 , p. 543–546
- ↑ Goldstein 1980 , p. 288
- ↑ Sard 1970 , p. 280
- ↑ Sexl & Urbantke 1992 , p. 42
- ↑ Misner, Thorne, and Wheeler, Gravitation, p 165, pp 175-176
- ↑ Krivoruchenko 2009

In physics, the **Lorentz transformations** are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

In mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In physics, the **special theory of relativity**, or **special relativity** for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:

- The laws of physics are invariant in all inertial frames of reference.
- The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or observer.

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

In special relativity, a **four-vector** is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

The **Lorentz factor** or **Lorentz term** is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.

In relativistic physics, a **velocity-addition formula** is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In physics, **relativistic mechanics** refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light *c*. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at *any* speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics.

In relativity, **rapidity** is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

The **covariant formulation of classical electromagnetism** refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

In relativity, **proper velocity****w** of an object relative to an observer is the ratio between observer-measured displacement vector and proper time τ elapsed on the clocks of the traveling object:

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. It gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. It sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. It motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

In physics, **relativistic quantum mechanics** (**RQM**) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light *c*, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. *Non-relativistic quantum mechanics* refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. *Relativistic quantum mechanics* (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

In theoretical physics, 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 and proved by Ludwik Silberstein in his 1914 book 'Relativity', rediscovered by Llewellyn Thomas in 1926, and rederived by Wigner in 1939. Wigner acknowledged Silberstein. 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.

In theoretical physics, **relativistic Lagrangian mechanics** is Lagrangian mechanics applied in the context of special relativity and general relativity.

In physics, **relativistic angular momentum** refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.

There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.

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