Relativistic angular momentum
- "Angular momentum tensor" redirects to here.
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 a dynamical quantity derived from position and momentum, and is important; angular momentum is a measure of an object's "amount of 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 Poincaré group and Lorentz group.
Physical quantities which remain separate in classical physics are naturally combined in SR and GR by enforcing the postulates of relativity, an appealing characteristic. Most notably; space and time coordinates combine into the four-position, and energy and momentum combine into the four-momentum. 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 x×p, or alternatively as the exterior product to obtain a second order antisymmetric tensor x∧p. What does this combine with, if anything? There is another vector quantity not often discussed – it is the time-varying moment of mass (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 to form an antisymmetric tensor of second order. 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 Special relativity
- 1.1 Orbital 3d angular momentum
- 1.2 Dynamic mass moment
- 1.3 Intertwine of L and N: Lorentz transformations
- 1.4 4d Angular momentum as a bivector
- 1.5 Lorentz transformation
- 1.6 Rigid body rotation
- 2 Spin, orbital, and total angular momentum in special relativity
- 3 Torque in special relativity
- 4 Angular momentum in general relativity
- 5 Angular momentum as the generator of spacetime boosts and rotations
- 6 See also
- 7 References
- 8 External links
Orbital 3d angular momentum
The classical definition of angular momentum can be used in SR and GR, but this needs some consideration, as outlined below.
Cross product definition: pseudovector
In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector x = (x1, x2, x3) = (x, y, z) and momentum vector p = (p1, p2, p3) = (px, py, pz), is defined as the axial vector
which has three components:
This quantity is additive, and for an isolated system, the total angular momentum of a system is conserved. However, this definition can be used in three dimensions only – considering that the cross product in the definition defines an axial vector perpendicular to the plane spanned by x and p. In four dimensions, there is not one axis uniquely perpendicular to a two-dimensional plane, but two such axes, allowed by the additional dimension.
Exterior product definition: antisymmetric tensor
An alternative 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:
where the indices i and j take the values 1, 2, 3. The components can be systematically collected into a 3 × 3 antisymmetric matrix:
Outer product definition: bivector
although in this context the product ∧ is the outer product of geometric algebra, which happens to have the same symbol and properties as the exterior product in standard exterior algebra.
Dynamic mass moment
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 denote it by various other symbols (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 is:
For an isolated system, N is conserved in time, apparent by differentiating with respect to time. Unlike L, N is a (polar) vector, not a pseudovector, 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 doesn't necessarily mean all particles "follow" the COM, nor that all particles all move in almost the same direction simultaneously, just that the motion of each particle is coupled with respect to the COM.
In special relativity, if the particle moves with velocity v relative to the lab frame, then
The corresponding relativistic mass moment in terms of m0, m, v, p, E, in the same lab frame is:
defined here so that the relativistic equation in terms of the relativistic mass, and classical definition, have the same form. The relativistic mass simplifies the expressions in this context as it removes extra Lorentz factors. However relativistic mass is discouraged by some authors since it can be a misleading quantity to apply in certain equations. In the following, N is expressed in terms of the rest and relativistic masses.
Intertwine of L and N: Lorentz transformations
Consider the Lorentz boost in standard setup with velocity V = (V, 0, 0) in the direction of the coincident xx′ axes. 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 velocity V here is the relative velocity between the frames, not necessarily of the object relative to F: i.e. neither F nor F′ is the rest frame of the object.
For the orbital 3-angular momentum L as a pseudovector, we have:
In the second terms of Ly′ and Lz′, there are cyclic permutations in the components of V and N', that are entirely in the y and z directions perpendicular to v in the x direction, and hence no components actually in the x direction. The cross product of the vectors V and N can be inferred:
Since Lx is parallel to the relative velocity V, and the other components Ly and Lz are perpendicular to V, we can collect the components into the pseudovector equations:
using the decomposition of 3-angular momentum in each frame into components parallel and perpendicular to V, respectively subscripted by ∥ and ⊥:
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 exterior product term has the factor of two for the antisymmetrization of position and momentum components:
Due to length contraction in the plane perpendicular to L, the component of L parallel to the Lorentz boost remains unaffected, while the components of L perpendicular to the boost include an angular momentum contribution V × N from the relative motion and are "warped" by the Lorentz factor γ(V).
For the dynamic mass moment:
and collecting parallel and perpendicular components as before:
as before, the mass moment in the direction perpendicular to the boost gains a contribution due to the motion of the COM under a Lorentz boost and warps by the Lorentz factor γ(V), while in the direction parallel to the boost it remains the same.
4d Angular momentum as a bivector
In relativistic mechanics, the COM boost and orbital 3-angular momentum of a rotating object are combined into a four-dimensional bivector in terms of the 4-position X and the 4-momentum P of the object:
which are six independent quantities altogether. Since X and P are frame-dependent, so is M. Three components:
are the components of the familiar classical 3-orbital angular momentum, and the other three:
correspond to the relativistic mass moment given above, multiplied by −c. The components of the tensor can be systematically displayed as a matrix:
The components of the angular momentum pseudovector enter the angular momentum tensor in the same way as if it were a 3d bivector.
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.
In fact, 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
and 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 translates to:
The maximum angular velocity of any massive object therefore depends on the size of the object. 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, orbital, and total angular momentum in special relativity
The total angular momentum density about the position 4-vector Yβ (an event), in terms of the stress–energy tensor Tβγ (a second order tensor field depending on space and time) and the position 4-vector Xα, is given by a 3rd order tensor:
Note it is antisymmetric in α and β. The integral over a 3d spacetime hypersurface denoted by , a boundary (indicated by the ∂ symbol) of a region of 4d spacetime , yields the total angular momentum tensor:
where Σγ is the volume 1-form, analogous to a normal unit vector on a 2d surface in ordinary 3d Euclidean space. The integral within a spacelike surface of constant time is:
which has spacelike components Jij that collect together into the spatial part of the angular momentum tensor, since Tj0 for j = 1, 2, 3 is the jth component of the object's 3d momentum per unit volume.
Angular momentum conservation
where ∂γ is the four gradient. (In non-Cartesian coordinates and general relativity this would be replaced by the covariant derivative). Then, angular momentum conservation is given by another continuity equation:
The integral equations use Gauss' theorem in spacetime:
Intrinsic angular momentum
In the rest frame of the object, the 4-momentum reduces to:
where the integrals are over a purely spacelike 3d hypersurface. There is an intrinsic angular momentum in this frame, in other words, the angular momentum about any event
in which Sjk form the spacelike components of the spin tensor and Sl form the spacelike components of the intrinsic angular momentum pseudovector, recalling that T0n is the nth component of momentum of the object per unit volume. The entries of the spacelike spin tensor can be arranged into a matrix:
Collecting these components into a 4-vector with a zero timelike component S0 = 0 and spacelike components are (S1, S2, S3), the above equations form part of:
in which the 4-momentum Pν of the center of mass related to the 4-velocity of the center of mass Uν according to the usual definition:
and the spin pseudovector is orthogonal to Sα:
For some displacement 4-vector (Xα − Yα) orthogonal to the 4-momentum,
the total angular momentum tensor Jμν about the event Yα:
is the sum of the orbital angular momentum tensor as introduced earlier, with (Xα − Yα) replacing Xα:
and the spin tensor determined previously. Contracting Jμν with Pν on ν leads to:
and contracting Jμν with Uλελμνρ on μν yields:
which is the Pauli-Lubanski pseudovector.
Torque in special relativity
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 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 similar to an angular momentum.
Angular momentum as the generator of spacetime boosts and rotations
The angular momentum tensor is the generator of boosts and rotations in spacetime, in the Lorentz group. Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector , and a rotation angle θ about a three-dimensional unit vector defining an axis, so and are together six parameters of the Lorentz group (three for rotations and three for boosts). The 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 Λ matrices act on any four vector A = (A0, A1, A2, A3) and mix the time-like and the space-like components, according to:
The angular momentum tensor forms 6 of the 10 generators of the Poincaré group, the other four being the components of the four-momentum for spacetime translations.
- Thomas precession
- Angular momentum of light
- Two-body problem in general relativity
- Kepler problem in general relativity
- Relativistic mechanics
- Center of mass (relativistic)
- Mathisson–Papapetrou–Dixon equations
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