Relativistic angular momentum

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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

.

.

Relativistic angular momentum is less obvious. The classical definition of angular momentum is the

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

axial vector

, of a particle of mass m with instantaneous 3-position x and 3-momentum p.

Orbital 3d angular momentum

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

In

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. This can be achieved by replacing the cross product by the

$${\displaystyle \mathbf {L} =\mathbf {x} \wedge \mathbf {p} }$$

or writing x = (x₁, x₂, x₃) = (x, y, z) and momentum vector p = (p₁, p₂, p₃) = (p_x, p_y, p_z), the components can be compactly abbreviated in

tensor index notation

$${\displaystyle L^{ij}=x^{i}p^{j}-x^{j}p^{i}}$$

where the indices i and j take the values 1, 2, 3. On the other hand, the components can be systematically displayed fully in a 3 × 3

antisymmetric matrix

$${\displaystyle {\begin{aligned}\mathbf {L} &={\begin{pmatrix}L^{11}&L^{12}&L^{13}\\L^{21}&L^{22}&L^{23}\\L^{31}&L^{32}&L^{33}\\\end{pmatrix}}={\begin{pmatrix}0&L_{xy}&L_{xz}\\L_{yx}&0&L_{yz}\\L_{zx}&L_{zy}&0\end{pmatrix}}={\begin{pmatrix}0&L_{xy}&-L_{zx}\\-L_{xy}&0&L_{yz}\\L_{zx}&-L_{yz}&0\end{pmatrix}}\\&={\begin{pmatrix}0&xp_{y}-yp_{x}&-(zp_{x}-xp_{z})\\-(xp_{y}-yp_{x})&0&yp_{z}-zp_{y}\\zp_{x}-xp_{z}&-(yp_{z}-zp_{y})&0\end{pmatrix}}\end{aligned}}}$$

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]

has the 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

$${\displaystyle \sum _{n}\mathbf {N} _{n}=\sum _{n}m_{n}\left(\mathbf {x} _{n}-t\mathbf {u} _{n}\right)=\left(\mathbf {x} _{\mathrm {COM} }\sum _{n}m_{n}-t\sum _{n}m_{n}\mathbf {u} _{n}\right)=M_{\text{tot}}(\mathbf {x} _{\mathrm {COM} }-\mathbf {u} _{\mathrm {COM} }t)}$$

where the system's

centre of mass

position and velocity and total mass are respectively

$${\displaystyle {\begin{aligned}\mathbf {x} _{\mathrm {COM} }&={\frac {\sum _{n}m_{n}\mathbf {x} _{n}}{\sum _{n}m_{n}}},\\[3pt]\mathbf {u} _{\mathrm {COM} }&={\frac {\sum _{n}m_{n}\mathbf {u} _{n}}{\sum _{n}m_{n}}},\\[3pt]M_{\text{tot}}&=\sum _{n}m_{n}.\end{aligned}}}$$

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, then

$${\displaystyle {\begin{aligned}E&=\gamma (\mathbf {u} )m_{0}c^{2},&\mathbf {p} &=\gamma (\mathbf {u} )m_{0}\mathbf {u} \end{aligned}}}$$

where

$${\displaystyle \gamma (\mathbf {u} )={\frac {1}{\sqrt {1-{\frac {\mathbf {u} \cdot \mathbf {u} }{c^{2}}}}}}}$$

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

The Cartesian components are

Special relativity

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 = (p_x, p_y, p_z) 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′ = (p_x′, p_y′, p_z′), 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

In the second terms of L_y′ and L_z′, the y and z components of the cross product v × N can be inferred by recognizing cyclic permutations of v_x = v and v_y = v_z = 0 with the components of N,

Now, L_x is parallel to the relative velocity v, and the other components L_y and L_z 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 all v, not just for motion along the xx′ axes.

Considering L as a tensor, we get a similar result

where

The boost of the dynamic mass moment along the x 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 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

while the perpendicular component by of L or N from n and the transformations are or reinstating v = vn,

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

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

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^[4][5]

In components

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

in which the last array is a . 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

where, for a boost (without rotations) with normalized velocity β = v/c, 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.

Rigid body rotation

For a particle moving in a curve, the cross product of its angular velocity ω (a pseudovector) and position x give its tangential velocity

which cannot exceed a magnitude of c, since in SR the translational velocity of any massive object cannot exceed the speed of light c. 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.

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

), the direction of the particle's spin vector will change, but its magnitude will be constant.

The extension to special relativity is straightforward.

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

$${\displaystyle \mathbf {S} \equiv \left(S^{0},S^{1},S^{2},S^{3}\right)=(s_{t},s_{x},s_{y},s_{z})}$$

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

$${\displaystyle \mathbf {S} '\equiv \left({S'}^{0},{S'}^{1},{S'}^{2},{S'}^{3}\right)=\left(0,s_{x}',s_{y}',s_{z}'\right)}$$

Equating norms leads to the invariant relation

$${\displaystyle s_{t}^{2}-\mathbf {s} \cdot \mathbf {s} =-\mathbf {s} '\cdot \mathbf {s} '}$$

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.

Vector transformations derived from the tensor transformations

The boosted components of the four spin relative to the lab frame are

$${\displaystyle {\begin{aligned}{S'}^{0}&={\Lambda ^{0}}_{\alpha }S^{\alpha }={\Lambda ^{0}}_{0}S^{0}+{\Lambda ^{0}}_{i}S^{i}=\gamma \left(S^{0}-\beta _{i}S^{i}\right)\\&=\gamma \left({\frac {c}{c}}S^{0}-{\frac {u_{i}}{c}}S^{i}\right)={\frac {1}{c}}U_{0}S^{0}-{\frac {1}{c}}U_{i}S^{i}\\[3pt]{S'}^{i}&={\Lambda ^{i}}_{\alpha }S^{\alpha }={\Lambda ^{i}}_{0}S^{0}+{\Lambda ^{i}}_{j}S^{j}\\&=-\gamma \beta ^{i}S^{0}+\left[\delta _{ij}+{\frac {\gamma -1}{\beta ^{2}}}\beta _{i}\beta _{j}\right]S^{j}\\&=S^{i}+{\frac {\gamma ^{2}}{\gamma +1}}\beta _{i}\beta _{j}S^{j}-\gamma \beta ^{i}S^{0}\end{aligned}}}$$

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

$${\displaystyle {S'}^{0}={\frac {1}{c}}U_{\alpha }S^{\alpha }=0}$$

which is an invariant. Then this combined with the transformation on the timelike component leads to the perceived component in the lab frame;

$${\displaystyle S^{0}=\beta _{i}S^{i}}$$

The inverse relations are

$${\displaystyle {\begin{aligned}S^{0}&=\gamma \left({S'}^{0}+\beta _{i}{S'}^{i}\right)\\S^{i}&={S'}^{i}+{\frac {\gamma ^{2}}{\gamma +1}}\beta _{i}\beta _{j}{S'}^{j}+\gamma \beta ^{i}{S'}^{0}\end{aligned}}}$$

The covariant constraint on the spin is orthogonality to the velocity vector,

$${\displaystyle U_{\alpha }S^{\alpha }=0}$$

In 3-vector notation for explicitness, the transformations are

$${\displaystyle {\begin{aligned}s_{t}&={\boldsymbol {\beta }}\cdot \mathbf {s} \\\mathbf {s} '&=\mathbf {s} +{\frac {\gamma ^{2}}{\gamma +1}}{\boldsymbol {\beta }}\left({\boldsymbol {\beta }}\cdot \mathbf {s} \right)-\gamma {\boldsymbol {\beta }}s_{t}\end{aligned}}}$$

The inverse relations

$${\displaystyle {\begin{aligned}s_{t}&=\gamma {\boldsymbol {\beta }}\cdot \mathbf {s} '\\\mathbf {s} &=\mathbf {s} '+{\frac {\gamma ^{2}}{\gamma +1}}{\boldsymbol {\beta }}\left({\boldsymbol {\beta }}\cdot \mathbf {s} '\right)\end{aligned}}}$$

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.

The Pauli–Lubanski pseudovector

The Pauli–Lubanski pseudovector

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,

This applies to a particle, a mass–energy–momentum distribution, or field.

Angular momentum of a mass–energy–momentum distribution

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 X^β 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 dΣ_γ is the volume 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

$${\displaystyle M^{ij}=\oint _{\partial \Omega }{\mathcal {M}}^{ij0}d\Sigma _{0}=\oint _{\partial \Omega }\left[\left(X^{i}-Y^{i}\right)T^{j0}-\left(X^{j}-Y^{j}\right)T^{i0}\right]dx\,dy\,dz}$$

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

$${\displaystyle \mathbf {X} _{\text{COM}}=\left(X_{\text{COM}}^{0},X_{\text{COM}}^{1},X_{\text{COM}}^{2},X_{\text{COM}}^{3}\right)}$$

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

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

Angular momentum conservation

The

$${\displaystyle \partial _{\gamma }T^{\beta \gamma }=0}$$

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

The integral equations use

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]

or in tensor components: 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

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

n pointing in the direction of the boost, which combine into the "rapidity vector"

$${\displaystyle {\boldsymbol {\zeta }}=\zeta \mathbf {n} =\mathbf {n} \tanh ^{-1}\beta }$$

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: where the

$${\displaystyle \Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})=\exp \left({\frac {1}{2}}\omega _{\alpha \beta }M^{\alpha \beta }\right)=\exp \left({\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} \right)}$$

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

$${\displaystyle \mathbf {A} '=\Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})\mathbf {A} }$$

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.

See also

• Thomas precession – Relativistic correction
• Angular momentum of light – Physical quantity carried in photons
• Two-body problem in general relativity – Interaction of two bodies in general relativity
• Relativistic mechanics – Theory of motion and forces for objects close to the speed of light
• Mathisson–Papapetrou–Dixon equations – General relativity equation

References

• C. Chryssomalakos; H. Hernandez-Coronado; E. Okon (2009). "Center of mass in special and general relativity and its role in an effective description of spacetime". J. Phys. Conf. Ser. 174 (1). Mexico: 012026.
  S2CID 17734387
  .
• U.E. Schroder (1990). Special relativity. Lecture Notes in Physics Series. Vol. 33.
  ISBN 981-02-0132-X
  .

Further reading

Special relativity

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

General relativity

• L. Blanchet; A. Spallicci; B. Whiting (2011). Mass and motion in general relativity. Fundamental theories of physics. Vol. 162. Springer. p. 87.
  ISBN 978-90-481-3015-3
  .
• M. Ludvigsen (1999). General Relativity: A Geometric Approach.
  ISBN 0-521-63976-X
  .
• N. Ashby, D.F. Bartlett, W.Wyss (1990). General Relativity and Gravitation 1989: Proceedings of the 12th International Conference on General Relativity and Gravitation. Cambridge University Press.
  ISBN 0-521-38428-1.{{cite book}}: CS1 maint: multiple names: authors list (link
  )
• B.L. Hu; M.P. Ryan; M.P. Ryan; C.V. Vishveshwara (2005). Directions in General Relativity: Volume 1: Proceedings of the 1993. Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner. Vol. 1. Cambridge University Press. p. 347.
  ISBN 0-521-02139-1
  .
• A. Papapetrou (1974). Lectures on General Relativity. Springer.
  ISBN 90-277-0514-3
  .

External links

• N. Menicucci (2001). "Relativistic Angular Momentum" (PDF).
• "Special Relativity" (PDF). Archived from the original (PDF) on 2013-11-04. Retrieved 2013-10-30.