Relativistic angular momentum

While these concepts were originally discovered in classical mechanics, they are also true and significant in special and general relativity.

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.

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 any case, the intrinsic "spin" addition to the orbital angular momentum of an object can be expressed in terms of the Pauli–Lubanski pseudovector.

which has three components, that are systematically given by cyclic permutations of Cartesian directions (e.g. change x to y, y to z, z to x, repeat)

A related definition is to conceive orbital angular momentum as a plane element.

or writing x = (x1, x2, x3) = (x, y, z) and momentum vector p = (p1, p2, p3) = (px, py, pz), the components can be compactly abbreviated in tensor index notation

In classical mechanics, the three-dimensional quantity for a particle of mass m moving with velocity u[2][3]

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 angular momentum L is a pseudovector, but N is an "ordinary" (polar) vector, and is therefore invariant under inversion.

Alternatively, starting from the vector Lorentz transformations of time, space, energy, and momentum, for a boost with velocity v,

where, for a boost (without rotations) with normalized velocity β = v/c, the Lorentz transformation matrix elements are

Then F′ is boosted with the same velocity and the Lorentz transformations apply as usual; it is more convenient to use β = u/c.

Since T00 is the energy density, Tj0 for j = 1, 2, 3 is the jth component of the object's 3d momentum per unit volume, and Tij form components of the stress tensor including shear and normal stresses, the orbital angular momentum density about the position 4-vector Xβ is given by a 3rd order tensor

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

where dΣγ is the volume 1-form playing the role of a unit vector normal to a 2d surface in ordinary 3d Euclidean space.

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

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]

As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.

The angular momentum tensor is the generator of boosts and rotations for the Lorentz group.

and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:

and the summation convention has been applied to the repeated matrix indices α and β.

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.

The angular momentum of test particles in a gently curved background is more complicated in GR but can be generalized in a straightforward manner.

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.