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 jr2 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.
@ 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].
@ Computation: Fodor, Hoenselaers & Perjés JMP(89).
@ 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]; > s.a. black-hole uniqueness.
@ Slowly moving source: Blanchet CQG(98)gq.
@ Other types: Suen PRD(86); Bondi & Rindler GRG(91).


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