Angular Momentum  

In Classical Theory > s.a. conservation laws; light [orbital angular momentum]; physics teaching [lab]; rotation; spin and spinors.
* Particles in 3D flat space: The angular momentum is defined by L:= r × p, or Li(x, p):= xj pkxk pj, with {Li, Lj} = εijk Lk.
* Particles in axially symmetric spacetime: L = gab ψa ub, with ψa:= (∂/∂φ)a, ua = particle 4-velocity.
* Question: How could you make the Earth spin faster?
@ Textbooks, II/III: Thompson 94.
@ General references: Dain PRL(14)-a1305 [inequality between size and angular momentum]; Pleitez a1508 [massless fields]; Sparavigna IJS(15)-a1511 [pedagogical, historical discussion and Euler equation]; Barrow & Gibbons PRD(17)-a1701 [conjectured upper bound on (magnetic moment)/(angular momentum)].
@ Electromagnetic field: Stewart EJP(05) [waves, and plane wave paradox]; Białynicki-Birula & Białynicka-Birula JO(11)-a1105 [splitting into orbital and spin parts]; Philbin & Allanson PRA(12) [optical, in dispersive media]; Bliokh et al NJP(14)-a1404 [spin and orbital angular momenta of an electromagnetic field in free space]; Beskin & Zheltoukhov PU(14)-a1411 [torque on a rotating magnetized sphere].

Gravitational, In the ADM Framework > s.a. ADM canonical gravity.
$ Def: Given a spacelike surface Σ in spacetime which is asymptotically flat, with induced metric hab, extrinsic curvature Kab and some flat metric f on it, the total angular momentum component with respect to a rotational Killing vector field ξ of the flat metric is

QN := \(1\over8\pi G\)limr → ∞  (KabK hab) ξa dsb = \(1\over8\pi G\) Kab ξa dS .

* Problem: This definition has supertranslation ambiguities, and the angular momentum can be made to take on any value we like, except if the trace of the extrinsic curvature K vanishes faster than r−2; This difficulty can be removed only using stronger boundary conditions than in the usual definition of asymptotic flatness, under which the magnetic part of the asymptotic Weyl curvature vanishes (see below).
@ References: in Misner et al 73; Ashtekar & Streubel JMP(79) [relationship with null infinity]; Chruściel CQG(87), and GR11-05:05; Chen et al AHP-a1401.

Gravitational, In the Spi Framework
$ Def: If we impose that the magnetic part of the Weyl tensor Bab = 0 on the hyperboloid \(\cal D\) of unit spacelike vectors at i0, we can define

Mab Fab := \(1\over8\pi G\) βab ζa dsb ,

for any skew tensor Fab at i0, where ζa:= Fab Xb is the Lorentz Killing vector field in the tangent space at i0, and βab is the piece of the Weyl curvature of order ρ−4 in the physical spacetime.
@ Gravitational radiation: Nesterov PLA(98)gq/04; Randono & Sloan PRD(09)-a0905 [internal spin angular momentum].

Gravitational, Related Topics > s.a. black-hole geometry [area–angular-momentum inequality]; kerr metric; kerr-newman solution; teleparallel gravity.
$ Komar integral: For an axisymmetric asymptotically flat spacetime with axial Killing vector field ψa (tangent to a hypersurface Σ),

J = \(1\over16\pi G\)S εabcdcψd = − Σ Tab na ψb dv ,

where S is a 2-sphere on which the matter stress-energy vanishes.
@ General references: Nahmad-Achar & Schutz CQG(87); Detweiler PRD(94)gq/93 [approximate solution]; Zhang CMP(99) [and mass]; Garecki G&C(01)gq [Bergmann-Thomson]; Rizzi gq/02 [and linear momentum]; Hayward PRD(06)gq [conservation, general black holes]; Maluf & Ulhoa GRG(09); Jaramillo & Gourgoulhon in(10)-a1001-proc [and energy, rev]; Flanagan & Nichols PRD(15)-a1411 [observer dependence, and the gravitational memory effect]; Flanagan et al PRD(16)-a1602 [measuring and transporting local angular momenta].
@ At null infinity: Rizzi PRL(98), PRD(01); Moreschi gq/03 [no supertranslation ambiguity]; Chruściel & Tod a0706 [inequality]; Helfer GRG(07)-a0709 [twistorial approach], a0903-conf [for non-specialists]; Wang & Wu a1310 [gravitational radiation and angular momentum flux]; Gallo & Moreschi PRD(14)-a1404 [and the Komar integral]; Bonga et al PRD(20)-a1911 [in Einstein-Maxwell theory]; Helfer PRD-a2105 [fluxes]; > s.a. asymptotic flatness.
@ Quasi-local: Penrose in(88); Helfer PLA(90); Szabados CQG(99)gq, CQG(01)gq; Moreschi CQG(04)gq/02 [intrinsic angular momentum of sources]; Korzyński CQG(07)-a0707 [from conformal decomposition of the metric]; Chen et al a1312; > s.a. quasilocal general relativity.
@ Other systems: Friedman et al PRD(78) [particles in axisymmetric spacetimes]; Bondi PRS(94) [cylindrical]; Elhashash & Nichols a2101 [compact-binary mergers]; > s.a. black-hole binaries; critical collapse [at black hole threshold]; topology in physics.

In Quantum Theory > s.a. 6j symbols; clebsch-gordan coefficients; Racah Formula; uncertainty relations; Wigner-Eckart Theorem.
* Recoupling theory: The problem of determining all states of n coupled spins that give a total angular momentum j; Applies to spin networks.
@ Texts: Edmonds 74; Biederharn & Louck 81; Feenberg & Pake 99.
@ General references: Lévy-Leblond AJP(67)may, AJP(76)aug; Iliev in(04)ht/02 [definitions]; Mostowski & Pietraszewicz EJP(20)-a1907 [vs classical].
@ Reason for l (l + 1): in Lee AJP(90)oct; Milonni AJP(90)oct; McGervey AJP(91)apr; Gómez a0803.
@ Quantum field theory: Bojowald gq/00 [quantum gravity, loop representation]; Stewart JMO(11)-a1011 [electromagnetic field].
@ Recoupling theory: Aquilanti et al PS(08)-a0901 [general framework].
@ Fractional angular momentum: Gambini & Setaro PRL(90); Jing et al a1805 [cold neutral atom with a permanent magnetic dipole moment].
@ Model systems: Benavides & Reyes-Lega in(10)-a0806 [particle on S2 and projective plane]; Sławianowski et al JGSP(11)-a1007 [quasiclassical systems with underlying SU(2) and SO(3) groups]; Le Floch & Pelayo a1607 [semitoric system of two coupled angular momenta].
@ Related topics: Salasnich & Sattin MPLB(97)qp [from supersymmetric semiclassical quantum mechanics], JPA(97)qp [WKB series]; Campos & Pimentel NCB(01) [finite-dimensional representation]; Bandyopadhyay & Rai qp/00 [coherence and squeezing]; Bakker et al PRD(04)hp [sum rules for nucleons]; Gatland AJP(06)mar [integer vs half-integer]; Feng et al SPIE(07)qp [experiment on conservation]; Puniani a1005 [solutions to commutation relations]; Nauts & Gatti AJP(10)dec [unusual commutation relations]; Luis & Rivas PRA(11) [non-classicality by breaking classical bounds on statistics]; Friedmann & Hagen JMP(12)-a1211 [spectrum of L2 in arbitrary dimensions]; Sehrawat a1810 [allowed region for expectation value, uncertainty relations].
> Specific particles: see photon.

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 26 may 2021