Relativistic angular momentum explained

See main article: Angular momentum.

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.

Angular momentum is an important dynamical quantity derived from position and momentum. It is a measure of an object's rotational motion and resistance to changes in its rotation. Also, in the same way momentum conservation corresponds to translational symmetry, angular momentum conservation corresponds to rotational symmetry – the connection between symmetries and conservation laws is made by Noether's theorem. While these concepts were originally discovered in classical mechanics, they are also true and significant in special and general relativity. In terms of abstract algebra, the invariance of angular momentum, four-momentum, and other symmetries in spacetime, are described by the Lorentz group, or more generally the Poincaré group.

Physical quantities that remain separate in classical physics are naturally combined in SR and GR by enforcing the postulates of relativity. Most notably, the space and time coordinates combine into the four-position, and energy and momentum combine into the four-momentum. The components of these four-vectors depend on the frame of reference used, and change under Lorentz transformations to other inertial frames or accelerated frames.

Relativistic angular momentum is less obvious. The classical definition of angular momentum is the cross product of position x with momentum p to obtain a pseudovector, or alternatively as the exterior product to obtain a second order antisymmetric tensor . What does this combine with, if anything? There is another vector quantity not often discussed – it is the time-varying moment of mass polar-vector (not the moment of inertia) related to the boost of the centre of mass of the system, and this combines with the classical angular momentum pseudovector to form an antisymmetric tensor of second order, in exactly the same way as the electric field polar-vector combines with the magnetic field pseudovector to form the electromagnetic field antisymmetric tensor. For rotating mass–energy distributions (such as gyroscopes, planets, stars, and black holes) instead of point-like particles, the angular momentum tensor is expressed in terms of the stress–energy tensor of the rotating object.

In special relativity alone, in the rest frame of a spinning object, there is an intrinsic angular momentum analogous to the "spin" in quantum mechanics and relativistic quantum mechanics, although for an extended body rather than a point particle. In relativistic quantum mechanics, elementary particles have spin and this is an additional contribution to the orbital angular momentum operator, yielding the total angular momentum tensor operator. In any case, the intrinsic "spin" addition to the orbital angular momentum of an object can be expressed in terms of the Pauli–Lubanski pseudovector.^[1]

Definitions

Orbital 3d angular momentum

For reference and background, two closely related forms of angular momentum are given.

In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector and momentum vector, is defined as the axial vector$$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$$which has three components, that are systematically given by cyclic permutations of Cartesian directions (e.g. change to, to, to, repeat)

A related definition is to conceive orbital angular momentum as a plane element. This can be achieved by replacing the cross product by the exterior product in the language of exterior algebra, and angular momentum becomes a contravariant second order antisymmetric tensor^[2] $$\backslash mathbf\; =\; \backslash mathbf\backslash wedge\backslash mathbf$$

or writing and momentum vector, the components can be compactly abbreviated in tensor index notation$$L^\; =\; x^i\; p^j\; -\; x^j\; p^i$$where the indices and take the values 1, 2, 3. On the other hand, the components can be systematically displayed fully in a 3 × 3 antisymmetric matrix

This quantity is additive, and for an isolated system, the total angular momentum of a system is conserved.

Dynamic mass moment

In classical mechanics, the three-dimensional quantity for a particle of mass m moving with velocity u^{[2] [3]} $$\backslash mathbf\; =\; m\; \backslash left(\backslash mathbf\; -\; t\; \backslash mathbf\; \backslash right)\; =\; m\; \backslash mathbf\; -\; t\; \backslash mathbf$$has the dimensions of mass moment – length multiplied by mass. It is equal to the mass of the particle or system of particles multiplied by the distance from the space origin to the centre of mass (COM) at the time origin, as measured in the lab frame. There is no universal symbol, nor even a universal name, for this quantity. Different authors may denote it by other symbols if any (for example μ), may designate other names, and may define N to be the negative of what is used here. The above form has the advantage that it resembles the familiar Galilean transformation for position, which in turn is the non-relativistic boost transformation between inertial frames.

This vector is also additive: for a system of particles, the vector sum is the resultant$$\backslash sum\_n\; \backslash mathbf\_n\; =\; \backslash sum\_n\; m\_n\; \backslash left(\backslash mathbf\_n\; -\; t\; \backslash mathbf\_n\; \backslash right)\; =\; \backslash left(\backslash mathbf\_\backslash mathrm\backslash sum\_n\; m\_n\; -\; t\; \backslash sum\_n\; m\_n\; \backslash mathbf\_n\; \backslash right)=\; M\_\backslash text(\backslash mathbf\_-\backslash mathbf\_t)$$where the system's centre of mass position and velocity and total mass are respectively

For an isolated system, N is conserved in time, which can be seen by differentiating with respect to time. The angular momentum L is a pseudovector, but N is an "ordinary" (polar) vector, and is therefore invariant under inversion.

The resultant N_tot for a multiparticle system has the physical visualization that, whatever the complicated motion of all the particles are, they move in such a way that the system's COM moves in a straight line. This does not necessarily mean all particles "follow" the COM, nor that all particles all move in almost the same direction simultaneously, only that the collective motion of the particles is constrained in relation to the centre of mass.

In special relativity, if the particle moves with velocity u relative to the lab frame, thenwhere$$\backslash gamma\; (\backslash mathbf)\; =\; \backslash frac$$is the Lorentz factor and m is the mass (i.e. the rest mass) of the particle. The corresponding relativistic mass moment in terms of,,,, in the same lab frame is$$\backslash mathbf\; =\; \backslash frac\backslash mathbf\; -\; \backslash mathbft\; =\; m\backslash gamma(\backslash mathbf)(\backslash mathbf\; -\; \backslash mathbft).$$

The Cartesian components are

Special relativity

Coordinate transformations for a boost in the x direction

Consider a coordinate frame which moves with velocity relative to another frame F, along the direction of the coincident axes. The origins of the two coordinate frames coincide at times . The mass–energy and momentum components of an object, as well as position coordinates and time in frame are transformed to,,, and in according to the Lorentz transformations

The Lorentz factor here applies to the velocity v, the relative velocity between the frames. This is not necessarily the same as the velocity u of an object.

For the orbital 3-angular momentum L as a pseudovector, we have

In the second terms of and, the and components of the cross product can be inferred by recognizing cyclic permutations of and with the components of,

Now, is parallel to the relative velocity, and the other components and are perpendicular to . The parallel–perpendicular correspondence can be facilitated by splitting the entire 3-angular momentum pseudovector into components parallel (∥) and perpendicular (⊥) to v, in each frame,$$\backslash mathbf\; =\; \backslash mathbf\_\backslash parallel\; +\; \backslash mathbf\_\backslash perp\; \backslash ,,\backslash quad\; \backslash mathbf\text{'}\; =\; \backslash mathbf\_\backslash parallel\text{'}\; +\; \backslash mathbf\_\backslash perp\text{'}\backslash ,.$$

Then the component equations can be collected into the pseudovector equations

Therefore, the components of angular momentum along the direction of motion do not change, while the components perpendicular do change. By contrast to the transformations of space and time, time and the spatial coordinates change along the direction of motion, while those perpendicular do not.

These transformations are true for all, not just for motion along the axes.

Considering as a tensor, we get a similar result$$\backslash mathbf\_\backslash perp\text{'}\; =\; \backslash gamma(\backslash mathbf)\backslash left(\backslash mathbf\_\backslash perp\; +\; \backslash mathbf\; \backslash wedge\; \backslash mathbf\; \backslash right)$$where

The boost of the dynamic mass moment along the direction is

Collecting parallel and perpendicular components as before

Again, the components parallel to the direction of relative motion do not change, those perpendicular do change.

Vector transformations for a boost in any direction

So far these are only the parallel and perpendicular decompositions of the vectors. The transformations on the full vectors can be constructed from them as follows (throughout here is a pseudovector for concreteness and compatibility with vector algebra).

Introduce a unit vector in the direction of, given by . The parallel components are given by the vector projection of or into $$\backslash mathbf\_\backslash parallel\; =\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf\; \backslash ,,\; \backslash quad\; \backslash mathbf\_\backslash parallel\; =\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf$$while the perpendicular component by vector rejection of L or N from n$$\backslash mathbf\_\backslash perp\; =\; \backslash mathbf\; -\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf\; \backslash ,,\; \backslash quad\; \backslash mathbf\_\backslash perp\; =\; \backslash mathbf\; -\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf$$and the transformations areor reinstating,

These are very similar to the Lorentz transformations of the electric field and magnetic field, see Classical electromagnetism and special relativity.

Alternatively, starting from the vector Lorentz transformations of time, space, energy, and momentum, for a boost with velocity,inserting these into the definitionsgives the transformations.

4d angular momentum as a bivector

In relativistic mechanics, the COM boost and orbital 3-space angular momentum of a rotating object are combined into a four-dimensional bivector in terms of the four-position X and the four-momentum P of the object^{[4] [5]} $$\backslash mathbf\; =\; \backslash mathbf\backslash wedge\backslash mathbf$$

In components$$M^\; =\; X^\backslash alpha\; P^\backslash beta\; -\; X^\backslash beta\; P^\backslash alpha$$which are six independent quantities altogether. Since the components of and are frame-dependent, so is . Three components$$M^\; =\; x^i\; p^j\; -\; x^j\; p^i\; =\; L^$$are those of the familiar classical 3-space orbital angular momentum, and the other three$$M^\; =\; x^0\; p^i\; -\; x^i\; p^0\; =\; c\backslash ,\backslash left(t\; p^i\; -\; x^i\; \backslash frac\; \backslash right)\; =\; -\; c\; N^i$$are the relativistic mass moment, multiplied by . The tensor is antisymmetric;$$M^\; =\; -M^$$

The components of the tensor can be systematically displayed as a matrixin which the last array is a block matrix formed by treating N as a row vector which matrix transposes to the column vector N^T, and as a 3 × 3 antisymmetric matrix. The lines are merely inserted to show where the blocks are.

Again, this tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system:$$\backslash mathbf\_\backslash text\; =\; \backslash sum\_n\; \backslash mathbf\_n\; =\; \backslash sum\_n\; \backslash mathbf\_n\; \backslash wedge\; \backslash mathbf\_n\; \backslash ,.$$

Each of the six components forms a conserved quantity when aggregated with the corresponding components for other objects and fields.

The angular momentum tensor M is indeed a tensor, the components change according to a Lorentz transformation matrix Λ, as illustrated in the usual way by tensor index notationwhere, for a boost (without rotations) with normalized velocity, the Lorentz transformation matrix elements areand the covariant β_i and contravariant βⁱ components of β are the same since these are just parameters.

In other words, one can Lorentz-transform the four position and four momentum separately, and then antisymmetrize those newly found components to obtain the angular momentum tensor in the new frame.

Rigid body rotation

See also: Ehrenfest paradox.

For a particle moving in a curve, the cross product of its angular velocity (a pseudovector) and position give its tangential velocity$$\backslash mathbf\; =\; \backslash boldsymbol\; \backslash times\; \backslash mathbf$$

which cannot exceed a magnitude of, since in SR the translational velocity of any massive object cannot exceed the speed of light c. Mathematically this constraint is, the vertical bars denote the magnitude of the vector. If the angle between and is (assumed to be nonzero, otherwise u would be zero corresponding to no motion at all), then and the angular velocity is restricted by

\sin\theta

The maximum angular velocity of any massive object therefore depends on the size of the object. For a given |x|, the minimum upper limit occurs when and are perpendicular, so that and .

For a rotating rigid body rotating with an angular velocity, the is tangential velocity at a point inside the object. For every point in the object, there is a maximum angular velocity.

The angular velocity (pseudovector) is related to the angular momentum (pseudovector) through the moment of inertia tensor $$\backslash mathbf\; =\; \backslash mathbf\backslash cdot\backslash boldsymbol\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; L\_i\; =\; I\_\; \backslash omega\_j$$(the dot denotes tensor contraction on one index). The relativistic angular momentum is also limited by the size of the object.

Spin in special relativity

Four-spin

A particle may have a "built-in" angular momentum independent of its motion, called spin and denoted s. It is a 3d pseudovector like orbital angular momentum L.

The spin has a corresponding spin magnetic moment, so if the particle is subject to interactions (like electromagnetic fields or spin-orbit coupling), the direction of the particle's spin vector will change, but its magnitude will be constant.

The extension to special relativity is straightforward.^[6] For some lab frame F, let F′ be the rest frame of the particle and suppose the particle moves with constant 3-velocity u. Then F′ is boosted with the same velocity and the Lorentz transformations apply as usual; it is more convenient to use . As a four-vector in special relativity, the four-spin S generally takes the usual form of a four-vector with a timelike component s_t and spatial components s, in the lab frame$$\backslash mathbf\; \backslash equiv\; \backslash left(S^0,\; S^1,\; S^2,\; S^3\backslash right)\; =\; (s\_t,\; s\_x,\; s\_y,\; s\_z)$$although in the rest frame of the particle, it is defined so the timelike component is zero and the spatial components are those of particle's actual spin vector, in the notation here s′, so in the particle's frame$$\backslash mathbf\text{'}\; \backslash equiv\; \backslash left(^0,\; ^1,\; ^2,\; ^3\backslash right)\; =\; \backslash left(0,\; s\_x\text{'},\; s\_y\text{'},\; s\_z\text{'}\backslash right)$$

Equating norms leads to the invariant relation$$s\_t^2\; -\; \backslash mathbf\backslash cdot\backslash mathbf\; =\; -\backslash mathbf\text{'}\backslash cdot\backslash mathbf\text{'}$$so if the magnitude of spin is given in the rest frame of the particle and lab frame of an observer, the magnitude of the timelike component s_t is given in the lab frame also.

The covariant constraint on the spin is orthogonality to the velocity vector,$$U\_\backslash alpha\; S^\backslash alpha\; =\; 0$$

In 3-vector notation for explicitness, the transformations are

The inverse relationsare the components of spin the lab frame, calculated from those in the particle's rest frame. Although the spin of the particle is constant for a given particle, it appears to be different in the lab frame.

The Pauli–Lubanski pseudovector

The Pauli–Lubanski pseudovector$$S\_\backslash mu\; =\; \backslash frac\backslash varepsilon\_\; J^\; P^\backslash sigma,$$applies to both massive and massless particles.

Spin–orbital decomposition

In general, the total angular momentum tensor splits into an orbital component and a spin component,$$J^\; =\; M^\; +\; S^\; ~.$$This applies to a particle, a mass–energy–momentum distribution, or field.

Angular momentum of a mass–energy–momentum distribution

See main article: Spin tensor and Pauli–Lubanski pseudovector.

Angular momentum from the mass–energy–momentum tensor

The following is a summary from MTW.^[7] Throughout for simplicity, Cartesian coordinates are assumed.In special and general relativity, a distribution of mass–energy–momentum, e.g. a fluid, or a star, is described by the stress–energy tensor T^βγ (a second order tensor field depending on space and time). Since T⁰⁰ is the energy density, T^j0 for j = 1, 2, 3 is the jth component of the object's 3d momentum per unit volume, and T^ij form components of the stress tensor including shear and normal stresses, the orbital angular momentum density about the position 4-vector ^β is given by a 3rd order tensor$$\backslash mathcal^\; =\; \backslash left(X^\backslash alpha\; -\; \backslash bar^\backslash alpha\backslash right)\; T^\; -\; \backslash left(X^\backslash beta\; -\; \backslash bar^\backslash beta\backslash right)\; T^$$

This is antisymmetric in α and β. In special and general relativity, T is a symmetric tensor, but in other contexts (e.g., quantum field theory), it may not be.

Let Ω be a region of 4d spacetime. The boundary is a 3d spacetime hypersurface ("spacetime surface volume" as opposed to "spatial surface area"), denoted ∂Ω where "∂" means "boundary". Integrating the angular momentum density over a 3d spacetime hypersurface yields the angular momentum tensor about,$$M^\backslash left(\backslash bar\backslash right)\; =\; \backslash oint\_\; \backslash mathcal^\; d\; \backslash Sigma\_\backslash gamma$$where dΣ_γ is the volume 1-form playing the role of a unit vector normal to a 2d surface in ordinary 3d Euclidean space. The integral is taken over the coordinates X, not . The integral within a spacelike surface of constant time is$$M^\; =\; \backslash oint\_\; \backslash mathcal^\; d\; \backslash Sigma\_0\; =\; \backslash oint\_\; \backslash left[\backslash left(X^i\; -\; Y^i\backslash right)T^\{j0\}\; -\; \backslash left(X^j\; -\; Y^j\backslash right)\; T^\{i0\}\backslash right]\; dx\; \backslash ,\; dy\; \backslash ,\; dz$$which collectively form the angular momentum tensor.

Angular momentum about the centre of mass

There is an intrinsic angular momentum in the centre-of-mass frame, in other words, the angular momentum about any event$$\backslash mathbf\_\backslash text\; =\; \backslash left(X^0\_\backslash text,\; X^1\_\backslash text,\; X^2\_\backslash text,\; X^3\_\backslash text\backslash right)$$on the wordline of the object's center of mass. Since T⁰⁰ is the energy density of the object, the spatial coordinates of the center of mass are given by$$X^i\_\backslash text\; =\; \backslash frac\; \backslash int\_\; X^i\; T^\; dxdydz$$

Setting Y = X_COM obtains the orbital angular momentum density about the centre-of-mass of the object.

Angular momentum conservation

The conservation of energy–momentum is given in differential form by the continuity equation$$\backslash partial\_\backslash gamma\; T^\; =\; 0$$where ∂_γ is the four-gradient. (In non-Cartesian coordinates and general relativity this would be replaced by the covariant derivative). The total angular momentum conservation is given by another continuity equation$$\backslash partial\_\backslash gamma\; \backslash mathcal^\; =\; 0$$

The integral equations use Gauss' theorem in spacetime

Torque in special relativity

The torque acting on a point-like particle is defined as the derivative of the angular momentum tensor given above with respect to proper time:^{[8] [9]} $$\backslash boldsymbol\; =\; \backslash frac\; =\; \backslash mathbf\backslash wedge\; \backslash mathbf$$or in tensor components:$$\backslash Gamma\_\; =\; X\_\backslash alpha\; F\_\backslash beta\; -\; X\_\backslash beta\; F\_\backslash alpha$$where F is the 4d force acting on the particle at the event X. As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.

Angular momentum as the generator of spacetime boosts and rotations

See also: Symmetry in quantum mechanics.

The angular momentum tensor is the generator of boosts and rotations for the Lorentz group.^{[10] [11]} Lorentz boosts can be parametrized by rapidity, and a 3d unit vector pointing in the direction of the boost, which combine into the "rapidity vector"$$\backslash boldsymbol\; =\; \backslash zeta\backslash mathbf\; =\; \backslash mathbf\backslash tanh^\backslash beta$$where is the speed of the relative motion divided by the speed of light. Spatial rotations can be parametrized by the axis–angle representation, the angle and a unit vector pointing in the direction of the axis, which combine into an "axis-angle vector"$$\backslash boldsymbol\; =\; \backslash theta\backslash mathbf$$

Each unit vector only has two independent components, the third is determined from the unit magnitude. Altogether there are six parameters of the Lorentz group; three for rotations and three for boosts. The (homogeneous) Lorentz group is 6-dimensional.

The boost generators and rotation generators can be combined into one generator for Lorentz transformations; the antisymmetric angular momentum tensor, with components$$M^\; =\; -M^\; =\; K\_i\; \backslash ,,\backslash quad\; M^\; =\; \backslash varepsilon\_\; J\_k\; \backslash ,.$$and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix, with entries:$$\backslash omega\_\; =\; -\; \backslash omega\_\; =\; \backslash zeta\_i\; \backslash ,,\backslash quad\; \backslash omega\_\; =\; \backslash varepsilon\_\; \backslash theta\_k\; \backslash ,,$$where the summation convention over the repeated indices i, j, k has been used to prevent clumsy summation signs. The general Lorentz transformation is then given by the matrix exponential$$\backslash Lambda(\backslash boldsymbol,\backslash boldsymbol)\; =\; \backslash exp\backslash left(\backslash frac\backslash omega\_M^\backslash right)\; =\; \backslash exp\; \backslash left(\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; +\; \backslash boldsymbol\backslash cdot\backslash mathbf\backslash right)$$and the summation convention has been applied to the repeated matrix indices α and β.

The general Lorentz transformation Λ is the transformation law for any four vector A = (A₀, A₁, A₂, A₃), giving the components of this same 4-vector in another inertial frame of reference$$\backslash mathbf\text{'}\; =\; \backslash Lambda(\backslash boldsymbol,\backslash boldsymbol)\; \backslash mathbf$$

The angular momentum tensor forms 6 of the 10 generators of the Poincaré group, the other four are the components of the four-momentum for spacetime translations.

Angular momentum in general relativity

The angular momentum of test particles in a gently curved background is more complicated in GR but can be generalized in a straightforward manner. If the Lagrangian is expressed with respect to angular variables as the generalized coordinates, then the angular momenta are the functional derivatives of the Lagrangian with respect to the angular velocities. Referred to Cartesian coordinates, these are typically given by the off-diagonal shear terms of the spacelike part of the stress–energy tensor. If the spacetime supports a Killing vector field tangent to a circle, then the angular momentum about the axis is conserved.

One also wishes to study the effect of a compact, rotating mass on its surrounding spacetime. The prototype solution is of the Kerr metric, which describes the spacetime around an axially symmetric black hole. It is obviously impossible to draw a point on the event horizon of a Kerr black hole and watch it circle around. However, the solution does support a constant of the system that acts mathematically similarly to an angular momentum.

References

• Center of mass in special and general relativity and its role in an effective description of spacetime. C. Chryssomalakos . H. Hernandez-Coronado . E. Okon . 2009. Mexico. J. Phys. Conf. Ser.. 0901.3349. 10.1088/1742-6596/174/1/012026 . 174 . 1 . 012026. 2009JPhCS.174a2026C . 17734387 .
• Book: Special relativity. U.E. Schroder. 139. World Scientific. 1990. 33. Lecture Notes in Physics Series. 981-02-0132-X.

Further reading

Special relativity

• Book: Relativity and Geometry. Courier Dover Publications. 1996. R. Torretti. Dover Books on Physics Series. 0-486-69046-6.

General relativity

• Book: Mass and motion in general relativity. L. Blanchet . A. Spallicci . B. Whiting . 87. Springer . 2011. 978-90-481-3015-3. 162. Fundamental theories of physics.
• Book: General Relativity: A Geometric Approach. M. Ludvigsen. Cambridge University Press. 1999 . 77. 0-521-63976-X.
• Book: General Relativity and Gravitation 1989: Proceedings of the 12th International Conference on General Relativity and Gravitation . 1990. Cambridge University Press. 0-521-38428-1. N. Ashby, D.F. Bartlett, W.Wyss.
• Book: Directions in General Relativity: Volume 1: Proceedings of the 1993. Cambridge University Press. 2005 . 1 . Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner . 0-521-02139-1. B.L. Hu . M.P. Ryan . M.P. Ryan . C.V. Vishveshwara . 347 .
• Book: Lectures on General Relativity. A. Papapetrou. 1974. 90-277-0514-3. Springer .

External links

• News: Relativistic Angular Momentum. N. Menicucci. 2001.
• Web site: Special Relativity. 2013-10-30. dead. https://web.archive.org/web/20131104054917/http://www.physics.usu.edu/Wheeler/EM/EMRelativity.pdf. 2013-11-04.

Notes and References

1. Book: Geometry and quantum field theory. 2nd. D.S.A. Freed . K.K.A. Uhlenbeck . 1995. American Mathematical Society. Institute For Advanced Study (Princeton, N.J.). 0-8218-8683-5.
2. Book: R. Penrose. The Road to Reality. vintage books. 433. 2005 . 978-0-09-944068-0. Penrose includes a factor of 2 in the wedge product, other authors may also.
3. Book: Special Relativity and How it Works. M. Fayngold. 2008. John Wiley & Sons. 138. 978-3-527-40607-4.
4. Book: R. Penrose. The Road to Reality. vintage books. 437–438, 566–569. 2005 . 978-0-09-944068-0. Note: Some authors, including Penrose, use Latin letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.
5. Book: Special Relativity and How it Works. M. Fayngold. 2008. John Wiley & Sons. 137–139. 978-3-527-40607-4.
6. Book: Jackson, J. D.. John David Jackson (physicist). Classical Electrodynamics. 556–557. 2nd. 1975. 1962. 0-471-43132-X. John Wiley & Sons. Chapter 11. registration. https://archive.org/details/classicalelectro00jack_0/page/556. Jackson's notation: S (spin in F, lab frame), s (spin in F′, rest frame of particle), S⁰ (timelike component in lab frame), S′⁰ = 0 (timelike component in rest frame of particle), no symbol for 4-spin as a 4-vector
7. Book: 156–159, §5.11. J.A. Wheeler . C. Misner . K.S. Thorne . Gravitation. W.H. Freeman & Co. 1973 . 0-7167-0344-0.
8. Torque and angular momentum on a system at equilibrium in special relativity. S. Aranoff. American Journal of Physics. 37. 1969. 4. 453–454. 10.1119/1.1975612. 1969AmJPh..37..453A. This author uses T for torque, here we use capital Gamma Γ since T is most often reserved for the stress–energy tensor.
9. Equilibrium in special relativity. S. Aranoff. Nuovo Cimento. 10. 1972. 1. 159. 10.1007/BF02911417. 1972NCimB..10..155A. 117291369. 2013-10-27. 2012-03-28. https://web.archive.org/web/20120328060625/http://www.analysis-knowledge.com/Physics/Equilibrium%20in%20Special%20Relativity.pdf. dead.
10. Book: Quantum Mechanics. E. Abers. Addison Wesley. 2004. 11, 104, 105, 410–411. 978-0-13-146100-0.
11. The Proper Homogeneous Lorentz Transformation Operator e^L = e^{− ω·S − ξ·K}, Where's It Going, What's the Twist. H.L. Berk . K. Chaicherdsakul . T. Udagawa . American Journal of Physics. 10.1119/1.1371919. 2001. 69. 996.