Multipole Moments in Field Theory| Multipole Moments in Field Theory |
In General
* Idea: A set of numbers
characterizing an extended source for a field, in terms of which the field
can be expanded in a series of terms, each depending on one of the multipole
moments, and usually having different radial and angular dependences.
@ References: Ross PRD(12)-a1202
[multipole expansion at the level of the action, for scalar field, electromagnetism and linearized gravity].
Flat Space, Cartesian > s.a. spherical harmonics.
$ Def: For a distribution
ρ(x) (of charge, mass, probability, ...) in Euclidean
space, the monopole moment is just the integral interpreted as total charge,
M = ∫ \({\mathbb R}^3\) d3x ρ(x) ,
the dipole moment is the vector
Di = ∫ \({\mathbb R}^3\) d3x xi ρ(x) ,
and the quadrupole moment is the rank-2 symmetric, traceless tensor
Qij = ∫ \({\mathbb R}^3\) d3x (3 xix j − r2 δij) ρ(x) .
* Properties:
D i = 0
if the origin is the center of mass.
@ References: Vrejoiu JPA(02) [general reduction procedure].
Flat Space, Special Cases > s.a. atomic physics;
laplacian [Maxwell multipoles]; magnetism.
* For an ellipsoid: If
the ellipsoid is given by (x2
+ y2)/a2
+ z2/b2 = 1,
then Q12 = Q13
= Q23 = 0.
@ Electromagnetic field: de Lange & Raab PRS(03) [D and H];
Vrejoiu & Nicmorus JPA(04) [radiation];
Raab & de Lange 05 [r JPA(05)];
> s.a. modified theories of electrodynamics [Podolsky theory].
In General Relativity > s.a. detection
of gravitational radiation; models in numerical relativity.
* Status: 1988, A good definition
has been given for the stationary case only, and it is known that the field outside
the source is uniquely determined by them.
* Applications: Calculations
of energy loss by gravitational-wave emission.
@ General references:
Fodor, Hoenselaers & Perjés JMP(89);
Compère et al JHEP(18)-a1711 [in terms of canonical Noether charges for multipole symmetries].
@ Static: Geroch JMP(70),
JMP(70);
Beig gq/00 [rev];
Bäckdahl & Herberthson CQG(05)gq,
Herberthson CQG(09)-a0906 [metric from multipoles].
@ Stationary: Hansen JMP(74);
Xanthopoulos JPA(79);
Beig & Simon CMP(80),
PRS(81);
Beig APA(81);
Kundu JMP(81),
JMP(81);
Simon & Beig JMP(83);
Gürsel GRG(83);
Simon JMP(84);
Quevedo FdP(90);
Herrera & Manko PLA(93);
Sotiriou & Apostolatos CQG(04) [axisymmetric, electrovac];
Bäckdahl & Herberthson CQG(05)gq [axisymmetric, asymptotically flat],
CQG(06)gq [calculation and bound];
Bäckdahl CQG(07)gq/06 [solutions with prescribed multipole moments].
@ And perihelion precession:
Fernández-Jambrina in(01)-a0906;
Boisseau & Letelier GRG(02)gq.
@ Phenomenology: Sotiriou & Apostolatos AIP(06)gq [and gravitational waves];
Brink PRD(08)-a0807 [reconstructing from gravitational waves emitted by orbiting body];
Iorio CQG(13)-a1302 [and tests of general relativity with the Juno mission];
> s.a. gravitating matter [motion of extended objects];
light propagation.
@ In scalar-tensor theories: Pappas & Sotiriou PRD(15)-a1412;
Pappas & Sotiriou MNRAS(15)-a1505 [and geodesic properties].
@ Related topics:
Nolan PRIA(97)gq/95 [Lorentz-covariant gravity];
Blanchet et al CQG(05)gq/04 [post-Newtonian sources].
Specific Types of Spacetimes
> s.a. solar system [solar multipoles].
@ Extended objects: Kleinwächter et al PLA(95) [rotating disk];
Ohashi PRD(03)gq [gravitational field and motion];
Quevedo GRG(11)-a1003 [metrics with quadrupole moment],
a1201-proc
[arbitrarily rotating electrovac, with infinite sets of multipole moments].
@ Of gravitational radiation:
Thorne RMP(80);
Leonard & Poisson CQG(98)gq/97.
@ Non-spherical pure monopoles: Connes et al NPB(97)gq/96.
@ Black holes and horizons: Ashtekar et al CQG(04)gq [axisymmetric isolated horizons];
Damour & Lecian PRD(09) [polarizability of black holes, Love numbers];
Bena & Mayerson a2006 [4D non-extremal and supersymmetric black holes];
> s.a. black-hole uniqueness.
@ Slowly moving source: Blanchet CQG(98)gq.
@ Other types: Suen PRD(86);
Bondi & Rindler GRG(91);
Chakraborty et al a2105
[asymptotically de Sitter spacetimes].
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