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 stop rotating. 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.
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 pseudovectorx × p, or alternatively as the exterior product to obtain a second order antisymmetric tensorx ∧ p. 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.
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 x = (x, y, z) and momentum vector p = (px, py, pz), is defined as the axial vector
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)
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 related to the boost (relative velocity) of the centre of mass (COM) of the particle or system of particles, 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
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 rotations.
The resultant Ntotal 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 motion of all the particles are constrained in relation to the centre of mass.
In special relativity, if the particle moves with velocity u relative to the lab frame, then
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 m, u, p, E, in the same lab frame is
Coordinate transformations for a boost in the x direction
Consider a coordinate frame F′ which moves with velocity v = (v, 0, 0) relative to another frame F, along the direction of the coincident xx′ axes. The origins of the two coordinate frames coincide at times t = t′ = 0. The mass–energy E = mc2 and momentum components p = (px, py, pz) of an object, as well as position coordinates x = (x, y, z) and time t in frame F are transformed to E′ = m′c2, p′ = (px′, py′, pz′), x′ = (x′, y′, z′), and t′ in F′ 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
For the x-component
In the second terms of Ly′ and Lz′, the y and z components of the cross productv×N can be inferred by recognizing cyclic permutations of vx = v and vy = vz = 0 with the components of N,
Now, Lx is parallel to the relative velocity v, and the other components Ly and Lz are perpendicular to v. 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 allv, not just for motion along the xx′ axes.
Considering L as a tensor, we get a similar result
The boost of the dynamic mass moment along the x direction is
For the x-component
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 L is a pseudovector for concreteness and compatibility with vector algebra).
Introduce a unit vector in the direction of v, given by n = v/v. The parallel components are given by the vector projection of L or N into n
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-positionX and the four-momentumP of the object
which are six independent quantities altogether. Since the components of X and P are frame-dependent, so is M. Three components
are those of the familiar classical 3-space orbital angular momentum, and the other three
are the relativistic mass moment, multiplied by −c. The tensor is antisymmetric;
The components of the tensor can be systematically displayed as a matrix
which cannot exceed a magnitude of c, since in SR the translational velocity of any massive object cannot exceed the speed of lightc. Mathematically this constraint is 0 ≤ |u| < c, the vertical bars denote the magnitude of the vector. If the angle between ω and x is θ (assumed to be nonzero, otherwise u would be zero corresponding to no motion at all), then |u| = |ω||x|sin θ and the angular velocity is restricted by
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 ω and x are perpendicular, so that θ = π/2 and sin θ = 1.
For a rotating rigid body rotating with an angular velocity ω, the u is tangential velocity at a point x 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 I
(the dot · denotes tensor contraction on one index). The relativistic angular momentum is also limited by the size of the object.
The extension to special relativity is straightforward. 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 β = u/c. As a four-vector in special relativity, the four-spin S generally takes the usual form of a four-vector with a timelike component st and spatial components s, in the lab frame
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
Equating norms leads to the invariant relation
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 st is given in the lab frame also.
Vector transformations derived from the tensor transformations
The boosted components of the four spin relative to the lab frame are
Here γ = γ(u). S′ is in the rest frame of the particle, so its timelike component is zero, S′0 = 0, not S0. Also, the first is equivalent to the inner product of the four-velocity (divided by c) and the four-spin. Combining these facts leads to
which is an invariant. Then this combined with the transformation on the timelike component leads to the perceived component in the lab frame;
The inverse relations are
The covariant constraint on the spin is orthogonality to the velocity vector,
In 3-vector notation for explicitness, the transformations are
The inverse relations
are 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.
Angular momentum from the mass–energy–momentum tensor
The following is a summary from MTW. 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 tensorTβγ (a second order tensor field depending on space and time). 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
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 X,
where dΣγ 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 X. The integral within a spacelike surface of constant time is
which collectively form the angular momentum tensor.
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
Throughout this section, see (for example) B.R. Durney (2011), and H.L. Berk et al. and references therein.
The angular momentum tensor is the generator of boosts and rotations for the Lorentz group. Lorentz boosts can be parametrized by rapidity, and a 3d unit vectorn pointing in the direction of the boost, which combine into the "rapidity vector"
where β = v/c 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 θ and a unit vector a pointing in the direction of the axis, which combine into an "axis-angle vector"
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 K and rotation generators J can be combined into one generator for Lorentz transformations; M the antisymmetric angular momentum tensor, with components
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:
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 similar to an angular momentum.
^R. Penrose (2005). The Road to Reality. vintage books. pp. 437–438, 566–569. ISBN978-0-09-944068-0. 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.
^Jackson, J. D. (1975) . "Chapter 11". Classical Electrodynamics (2nd ed.). John Wiley & Sons. pp. 556–557. ISBN0-471-43132-X. 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