Relativistic angular momentum

In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity and general relativity. 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 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 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.

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
which has three components, that are systematically given by cyclic permutations of Cartesian directions
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
or writing and momentum vector, the components can be compactly abbreviated in tensor index notation
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
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 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, 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
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" 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, then
where
is the Lorentz factor and m is the mass of the particle. The corresponding relativistic mass moment in terms of,,,, in the same lab frame is
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,
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
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.
Introduce a unit vector in the direction of, given by. The parallel components are given by the vector projection of or into
while the perpendicular component by vector rejection of L or N from n
and the transformations are
or 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 definitions
gives 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
In components
which are six independent quantities altogether. Since the components of and are frame-dependent, so is. Three components
are those of the familiar classical 3-space orbital angular momentum, and the other three
are the relativistic mass moment, multiplied by. The tensor is antisymmetric;
The components of the tensor can be systematically displayed as a matrix
in 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:
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 notation
where, for a boost with normalized velocity, the Lorentz transformation matrix elements are
and 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.