Relativistic angular momentum

In <a href="/facts/Physics/a7aH2y08">physics</a>, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> in <a href="/facts/Special_relativity/vsFFc63E">special relativity</a> (SR) and <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> (GR). The relativistic quantity is subtly different from the <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> quantity in <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>.
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 <a href="/facts/Symmetry_(physics)/gJmgsooI">symmetries</a> and <a href="/facts/Conservation_law_(physics)/6g1Ay2FR">conservation laws</a> is made by <a href="/facts/Noether%27s_theorem/VFjqrEvz">Noether's theorem</a>. 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 <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a>, or more generally the <a href="/facts/Poincar%C3%A9_group/3IerRYG1">Poincaré group</a>.
<a href="/facts/Physical_quantity/S8pWBEpK">Physical quantities</a> 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 <a href="/facts/Four-vector/89GpELMA">four-position</a>, and energy and momentum combine into the <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a>. The components of these <a href="/facts/Four-vector/89GpELMA">four-vectors</a> depend on the <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> used, and change under <a href="/facts/Lorentz_transformation/jotJL06G">Lorentz transformations</a> to other <a href="/facts/Inertial_frame/1U6TrdG3">inertial frames</a> or <a href="/facts/Non-inertial_reference_frame/UGfKCASo">accelerated frames</a>.
Relativistic angular momentum is less obvious. The classical definition of angular momentum is the <a href="/facts/Cross_product/uRBJzkcW">cross product</a> of position x with momentum p to obtain a <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a> x × p, or alternatively as the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a> to obtain a second order <a href="/facts/Antisymmetric_tensor/atcSaPST">antisymmetric tensor</a> 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 polar-vector (not the <a href="/facts/Moment_of_inertia/TpLLAVl7">moment of inertia</a>) related to the boost of the <a href="/facts/Centre_of_mass/n4Lr9GAz">centre of mass</a> 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 <a href="/facts/Gyroscopes/UjXGsfMq">gyroscopes</a>, <a href="/facts/Planet/rbPjE6Bx">planets</a>, <a href="/facts/Star/PFbd6ICx">stars</a>, and <a href="/facts/Black_hole/1r2lmkcF">black holes</a>) instead of point-like particles, the angular momentum tensor is expressed in terms of the <a href="/facts/Stress%E2%80%93energy_tensor/wWKrzjfI">stress–energy tensor</a> of the rotating object.
In special relativity alone, in the <a href="/facts/Rest_frame/gFkIWgXk">rest frame</a> of a spinning object, there is an intrinsic angular momentum analogous to the "spin" in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> and <a href="/facts/Relativistic_quantum_mechanics/DhkDNnHf">relativistic quantum mechanics</a>, although for an extended body rather than a point particle. In relativistic quantum mechanics, <a href="/facts/Elementary_particle/CTFDM97U">elementary particles</a> have <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a> 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 <a href="/facts/Pauli%E2%80%93Lubanski_pseudovector/r8Tnsnbc">Pauli–Lubanski pseudovector</a>.

Relativistic angular momentum open-in-new

Relativistic angular momentum