Thomas precession

Not to be confused with Thomas rotation.

Spacetime
Part of a series on

Special relativity General relativity
Introduction Equivalence principle Differential geometry Einstein field equations Introduction to the mathematics
Types Spacetime Minkowski spacetime Spacetime diagrams
Mathematics Mathematics of general relativity Lorentz transformations Four-vector Derivations Spacetime topology Einstein field equations
Relation to gravity Gravity Newton's Law of Gravity Curved spacetime

Llewellyn Thomas (1903 – 1992)

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.

Two successive Lorentz boosts lead to the Wigner rotation of the lab frame and doubly boosted frame. 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 is 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.

History

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".

Introduction

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.^[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]

Statement

Value of

γ 2 /(γ + 1)

β = v/c

increases, with v the instantaneous magnitude of the particle's velocity. The Thomas rotation is negligible for

β < 0.5

, increases steadily for

0.5 < β < 0.8

, then rapidly shoots to infinity as

β

tends to 1. The "Thomas half" is evident in the low-speed limit, and the rotation is only very clear for speeds approaching that of light.

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 $0 \leq v < c$ . 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 angular velocity^[6]^[7]^[8]^[9]

Thomas precession

$\boldsymbol{\omega}_T = \frac{1}{c^2}\left(\frac{\gamma^2}{\gamma+1}\right)\mathbf{a}\times\mathbf{v}$

where × is the cross product and

\gamma = \dfrac{1}{\sqrt{1-\dfrac{|\mathbf{v}(t)|^2}{c^2}}}

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, $v \to 0$ so $γ \to 1$ , and the angular velocity is approximately

\boldsymbol{\omega}_T \approx \frac{1}{2c^2}\mathbf{a}\times\mathbf{v}

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".

Mathematical explanation

Lorentz transformations

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 $0 \leq β < 1$ .

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

X = \begin{bmatrix} c t \\ x \\ y \\ z \end{bmatrix}\,,\quad B(\boldsymbol{\beta}) = \begin{bmatrix} \gamma&-\gamma\beta_x &-\gamma\beta_y&-\gamma\beta_z\\ -\gamma\beta_x &1+(\gamma-1)\dfrac{\beta_x^2}{\beta^2} &(\gamma-1)\dfrac{\beta_x \beta_y}{\beta^2} &(\gamma-1)\dfrac{\beta_x \beta_z}{\beta^2} \\ -\gamma\beta_y &(\gamma-1)\dfrac{\beta_y \beta_x}{\beta^2} &1+(\gamma-1)\dfrac{\beta_y^2}{\beta^2} &(\gamma-1)\dfrac{\beta_y \beta_z}{\beta^2} \\ -\gamma\beta_z&(\gamma-1)\dfrac{\beta_z \beta_x}{\beta^2} &(\gamma-1)\dfrac{\beta_z \beta_y}{\beta^2} & 1+(\gamma-1)\dfrac{\beta_z^2}{\beta^2} \\ \end{bmatrix}

and turn

\gamma = \frac{1}{\sqrt{1-|\boldsymbol{\beta}|^2}}

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

$X' = B(\boldsymbol{\beta})X$

(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

$X'' = B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta})X$

(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]}

\Delta\boldsymbol{\beta} = \Delta\boldsymbol{\beta}_\parallel + \Delta\boldsymbol{\beta}_\perp

Combining (1) and (2) obtains the Lorentz transformation between $Σ'$ and $Σ''$ ,

$X'' = B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta})B(-\boldsymbol{\beta})X' \,,$

(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;

$\Lambda = B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta})B(-\boldsymbol{\beta}) = R(\Delta\boldsymbol{\theta})B(\Delta\mathbf{b})$

(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 $(- β)\times(β + Δ β) = - β \timesΔ β$ . 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

$B(\Delta\mathbf{b})X' = R(-\Delta\boldsymbol{\theta}) X'' = X''' \,,$

(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);

$X''' = R(-\Delta\boldsymbol{\theta}) X'' = R(-\Delta\boldsymbol{\theta}) B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta}) X \,,$

(6)

the frame $Σ'''$ is rotated in the negative sense.

The rotation is between two instants of lab time. As $Δ t \to 0$ , 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 $Δ t \to 0$ , the angular velocity is by definition

$\boldsymbol{\omega}_T = -\lim_{\Delta t \rightarrow 0 }\frac{\Delta\boldsymbol{\theta}}{\Delta t}$

(7)

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

Extracting the formula

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" $| Δ β | 2$ , $(Δ β x) 2$ , $(Δ β y) 2$ , $Δ β x Δ β y$ and higher are negligible. Taking advantage of this fact the magnitude squared of the vector is

|\boldsymbol{\beta}+\Delta\boldsymbol{\beta}|^2=|\boldsymbol{\beta}|^2+2\boldsymbol{\beta}\cdot\Delta\boldsymbol{\beta}

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

\begin{align}\frac{1}{\sqrt{1-|\boldsymbol{\beta}+\Delta\boldsymbol{\beta}|^2}} & =1+\frac{1}{2}|\boldsymbol{\beta}+\Delta\boldsymbol{\beta}|^2+\frac{3}{8}|\boldsymbol{\beta}+\Delta\boldsymbol{\beta}|^4+\cdots \\ & = \left(1+\frac{|\boldsymbol{\beta}|^2}2+\frac{3}{8}|\boldsymbol{\beta}|^4+\cdots\right)+\left(1+\frac{3}2|\boldsymbol{\beta}|^2+\cdots\right)\boldsymbol{\beta}\cdot\Delta\boldsymbol{\beta} \\ & \approx \gamma + \gamma^3\boldsymbol{\beta}\cdot\Delta\boldsymbol{\beta} \end{align}

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

\boldsymbol{\beta} = \beta\mathbf{e}_x \,, \quad \Delta\boldsymbol{\beta}_\parallel = \Delta\beta_x\mathbf{e}_x \,, \quad \Delta\boldsymbol{\beta}_\perp = \Delta\beta_y\mathbf{e}_y \,,\quad \boldsymbol{\beta}\times\Delta\boldsymbol{\beta}= \beta\Delta\beta_y\mathbf{e}_z

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

B(-\boldsymbol{\beta}) = \begin{bmatrix} \gamma& \gamma\beta & 0 & 0 \\ \gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}

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

B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta})= \begin{bmatrix} \gamma + \gamma^3 \beta \Delta \beta_x & -(\gamma\beta + \gamma^3 \Delta \beta_x) & -\gamma\Delta\beta_y & 0 \\ -(\gamma\beta + \gamma^3 \Delta \beta_x) & \gamma + \gamma^3 \beta \Delta \beta_x & \left(\dfrac{\gamma-1}{\beta}\right)\Delta\beta_y & 0 \\ -\gamma\Delta\beta_y & \left(\dfrac{\gamma-1}{\beta}\right)\Delta\beta_y & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

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

\Lambda = B(\boldsymbol{\beta} + \Delta\boldsymbol{\beta})B(-\boldsymbol{\beta})= \begin{bmatrix} 1 & -\gamma^2 \Delta \beta_x & -\gamma\Delta\beta_y & 0 \\ -\gamma^2 \Delta \beta_x & 1 & \left(\dfrac{\gamma-1}{\beta}\right)\Delta\beta_y & 0 \\ -\gamma\Delta\beta_y & - \left(\dfrac{\gamma-1}{\beta}\right)\Delta\beta_y & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

Introducing the boost generators

K_x = \begin{bmatrix} 0 &1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}\,,\quad K_y = \begin{bmatrix}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\,,\quad K_z = \begin{bmatrix}0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 \end{bmatrix}

and rotation generators

J_x = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}\,,\quad J_y = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{bmatrix}\,,\quad J_z = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}

along with the dot product · facilitates the coordinate independent expression

\Lambda = I - \left(\frac{\gamma-1}{\beta^2}\right)(\boldsymbol{\beta}\times\Delta\boldsymbol{\beta})\cdot\mathbf{J} - \gamma(\gamma\Delta\boldsymbol{\beta}_\parallel+\Delta\boldsymbol{\beta}_\perp)\cdot\mathbf{K}

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]}

\Lambda = I - \Delta\boldsymbol{\theta}\cdot\mathbf{J} - \Delta\mathbf{b}\cdot\mathbf{K}

where

\Delta\boldsymbol{\theta} = \left(\frac{\gamma-1}{\beta^2}\right)\boldsymbol{\beta}\times\Delta\boldsymbol{\beta} = \frac{1}{c^2}\left(\frac{\gamma^2}{\gamma+1}\right)\mathbf{v}\times\Delta\mathbf{v}

\Delta\mathbf{b} = \gamma(\gamma\Delta\boldsymbol{\beta}_\parallel+\Delta\boldsymbol{\beta}_\perp)

After dividing $Δ θ$ by $Δ t$ and taking the limit as in (7), one obtains the instantaneous angular velocity

\boldsymbol{\omega}_T = \frac{1}{c^2}\left(\frac{\gamma^2}{\gamma+1}\right)\mathbf{a}\times\mathbf{v}

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.

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.^[10] 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}.

Remarks

↑ Explicitly, using vector projection and rejection relative to the direction of $β$ gives
$\Delta\boldsymbol{\beta}_\parallel = \frac{\Delta\boldsymbol{\beta}\cdot\boldsymbol{\beta}}{\beta^2}\boldsymbol{\beta}\,,\quad \Delta\boldsymbol{\beta}_\perp = \Delta\boldsymbol{\beta} - \frac{\Delta\boldsymbol{\beta}\cdot\boldsymbol{\beta}}{\beta^2}\boldsymbol{\beta}$
but it is easier to simply use the parallel-perpendicular components.
↑ The rotation and boost matrices (each infinitesimal) are given by
$R(\Delta\boldsymbol{\theta}) = I - \Delta\boldsymbol{\theta}\cdot\mathbf{J} \,,\quad B(\Delta\mathbf{b}) = I - \Delta\mathbf{b}\cdot\mathbf{K} \,,$
At the infinitesimal level, they commute with each other
$\Lambda = B(\Delta\mathbf{b})R(\Delta\boldsymbol{\theta}) = R(\Delta\boldsymbol{\theta})B(\Delta\mathbf{b})$
because the products $(Δ θ \cdot J)(Δ b \cdot K)$ and $(Δ b \cdot K)(Δ θ \cdot J)$ are negligible. The full boost and rotations do not commute in general.

Notes

References

Ben-Menahem, A. (1985). "Wigner's rotation revisited". Am. J. Phys. 53: 62–66. doi:10.1119/1.13953.
Ben-Menahem, S. (1986). "The Thomas precession and velocityspace curvature". J. Math. Phys. 27: 1284 pp. doi:10.1063/1.527132.
Cushing, J. T. (1967). "Vector Lorentz Transformations". Am. J. Phys. (AAPT) 35: 858–862. doi:10.1119/1.1974267. (subscription required (help)).
Einstein, A. (1922), (Ebook) Sidelights on relativity Check |url= value (help), London: Methuen
Eddington, A.S. (1924). (Introduction) The Mathematical Theory of Relativity Check |url= value (help). Cambridge University Press.
Krivoruchenko, M. I. (2009). "Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin". Phys. Usp. (IOPScience) 52 (8): 821–829. arXiv:0805.1136v2. doi:10.3367/UFNe.0179.200908e.0873.
Malykin, G. B. (2006). "Thomas precession: correct and incorrect solutions". Phys. Usp. 49 (8): 83 pp. doi:10.1070/PU2006v049n08ABEH005870. (subscription required (help)).
Mocanu, C.I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation". Found. Phys. Lett. (Kluwer Academic Publishers-Plenum Publishers) 5 (5): 443–456. doi:10.1007/BF00690425. ISSN 0894-9875. (subscription required (help)).
Rebilas, K. (2013). "Comment on Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession". Eur. J. Phys. (IOPScience) 34 (3): L55. doi:10.1088/0143-0807/34/3/L55. (free access)
Rhodes, J. A.; Semon, M. D. (2005). "Relativistic velocity space, Wigner rotation and Thomas precession". Am. J. Phys. 72: 943 pp. arXiv:gr-qc/0501070v1. Bibcode:2005APS..NES..R001S. doi:10.1119/1.1652040.
Silberstein, L. (1914). The Theory of Relativity. London: Macmillan Publishers.
Thomas, L. H. (1926). "Motion of the spinning electron". Nature 117: 514. doi:10.1038/117514a0. (subscription required (help)).
Ungar, A. A. (1988). "Thomas rotation and parameterization of the Lorentz group". Foundations of Physics Letters (Springer) 1 (1): 57–81. doi:10.1007/BF00661317. ISSN 0894-9875. (subscription required (help)).
Weinberg, S. (2002), The Quantum Theory of Fields 1, Cambridge University Press, ISBN 0-521-55001-7
Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics 40 (1): 149–204, Bibcode:1939AnMat..40..922E, doi:10.2307/1968551, MR 1503456 .

Textbooks

Goldstein, H. (1980) [1950]. "Chapter 7". Classical Mechanics (2nd ed.). Reading MA: Addison-Wesley. ISBN 0-201-02918-9.
Jackson, J. D. (1999) [1962]. "Chapter 11". Classical Electrodynamics (3d ed.). John Wiley & Sons. ISBN 0-471-30932-X.
Jackson, J. D. (1975) [1962]. "Chapter 11". Classical Electrodynamics (2nd ed.). John Wiley & Sons. pp. 542–545. ISBN 0-471-43132-X.
Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics 2 (4th ed.). Butterworth–Heinemann. p. 38. ISBN 0 7506 2768 9.
Rindler, W. (2006) [2001]. "Chapter 9". Relativity Special, General and Cosmological (2nd ed.). Dallas: Oxford University Press. ISBN 978-0-19-856732-5.
Ryder, L. H. (1996) [1985]. Quantum Field Theory (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0521478144.
Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.
R. U. Sexl, H. K. Urbantke (2001) [1992]. Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer. pp. 38–43. ISBN 978-3211834435.

External links

Mathpages article on Thomas Precession
Alternate, detailed derivation of Thomas Precession (by Robert Littlejohn)
Short derivation of the Thomas precession

Relativity

Special
relativity

Background	Principle of relativity Introduction to special relativity Special relativity

Foundations	Relative motion Frame of reference Speed of light Maxwell's equations

Formulation	Galilean relativity Galilean transformation Lorentz transformation

Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks

Spacetime	Minkowski spacetime World line Spacetime diagrams Light cone

General
relativity

Background	Introduction Mathematical formulation Resources

Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Minkowski diagram

Phenomena	Two-body problem Lenses Waves Frame-dragging Geodetic effect Event horizon Singularity Black hole

Equations	Linearized gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism Hamilton–Jacobi–Einstein equation

Advanced theories	Kaluza–Klein Quantum gravity Brans–Dicke theory

Solutions	Schwarzschild Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub–NUT Milne Robertson–Walker pp-wave van Stockum dust

Scientists

This article is issued from Wikipedia - version of the Friday, February 05, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.