Angular
Momentum| Angular
Momentum |
In Classical Theory > s.a. light [orbital
angular momentum]; rotation; spin
and spinors.
* Particles in 3D flat space: L:= r × p,
or Li(x, p):=
xj pk – xk pj,
with {Li, Lj}
= 
ijk Lk.
* Particles in axially symmetric
spacetime: L = gab 
a ub,
with a:=
(
/
)a, ua =
particle 4-velocity.
* Question: How could
you make the Earth spin faster?
@ II, III: Cowley 94.
@ In field theory: Stewart EJP(05)
[electromagnetic waves, and plane wave paradox].
Gravitational, In the ADM Framework > ADM
canonical gravity.
$ Def: Given a spacelike
surface 
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 wrt a rotational Killing vector
field 
of
the flat metric
is
QN := (1/8
)
limr to infty 
(Kab –
K hab) a dsb = (1/8)
Kab a 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];
Chrusciel CQG(87),
and GR11-05:05.
Gravitational, In the Spi Framework
$ Def: If we impose that the magnetic part of the Weyl tensor Bab
= 0 on the hyperboloid 
of
unit spacelike vectors at i0,
we can define
Mab Fab :=
(1/8) 
ab 
a dsb ,
for any skew tensor Fab at
i0,
where a:=
Fab Xb is
the Lorentz Killing vector field in the tangent space at i0,
and ab is
the piece of the Weyl curvature of order 
–4 in
the physical spacetime.
@ Gravitational radiation: Nesterov PLA(98)gq/04;
Randono & Sloan a0905 [interal spin angular momentum].
Gravitational, Other Definitions and Systems > s.a. kerr
metric; teleparallel
gravity.
$ Komar integral: For
an axisymmetric asymptotically flat spacetime with axial Killing vector field a (tangent
to a hypersurface ),
J = (1/16) 
S abcd 
cd =
– Sigma Tab na b 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).
@ At null infinity: Rizzi PRL(98), PRD(01);
Moreschi gq/03 [no
supertranslation
ambiguity]; Chrusciel & Tod a0706 [inequality];
Helfer GRG(07)-a0709 [twistorial
approach], a0903-in [for non-specialists]; > 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]; Korzynski CQG(07)-a0707 [from
conformal decomposition of
the metric]; > s.a. quasilocal
general relativity.
@ Other systems: Friedman et al PRD(78)
[particles in axisymmetric spacetimes];
Bondi PRS(94)
[cylindrical]; > s.a. critical collapse [at
black hole threshold], topology in physics.
In Quantum Theory > s.a. 6j symbols; clebsch-gordan; Racah
Formula.
* 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.
@ Reason for l (l + 1): in Lee AJP(90)oct;
Milonni AJP(90)oct;
McGervey AJP(91)apr;
Gómez a0803.
@ Quantum gravity: Bojowald gq/00 [loop
representation].
@ Recoupling theory: Aquilanti et al PS(08)-a0901 [general framework]
@ Related topics: Lévy-Leblond AJP(67)may, AJP(76)aug;
Gambini & Setaro
PRL(90)
[fractional]; 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]; Iliev in(04)ht/02 [definitions];
Bakker et al PRD(04)hp [sum
rules for nucleons]; Gatland AJP(06)mar
[integer vs half-integer]; Feng et al qp/07/PRL
[experiment on conservation]; Benavides & Reyes-Lega a0806 [particle
on S2 and projective plane].
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send feedback and suggestions to bombelli at olemiss.edu – modified
1 jun 2009