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.

Flat Space, Cartesian > s.a. spherical harmonics.
$ Def: For a distribution (x) (of charge, mass, probability, ...) in Eucliden space, the monopole moment is just the integral interpreted as total charge,

M = _R^3 d³x (x) ,

the dipole moment is the vector

Dⁱ = _R^3 d³x xⁱ (x) ,

and the quadrupole moment is the rank-2 symmetric, traceless tensor

Q^ij = _R^3 d³x (3 xⁱx ^j – r^{2 ij}) (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)].

In General Relativity > s.a. detection of gravitational radiation.
* 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) [reconstructing from gravitational waves emitted by orbiting body]; > s.a. gravitating matter [motion of extended objects]; light propagation.
@ Related topics: Nolan PRIA(97)gq/95 [Lorentz-covariant gravity]; Blanchet et al CQG(05)gq/04 [post-Newtonian sources].

Specific Types of Spacetimes
@ Extended objects: Kleinwächter et al PLA(95) [rotating disk]; Ohashi PRD(03)gq [gravitational field and motion].
@ 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].
@ Slowly moving source: Blanchet CQG(98)gq.
@ Other types: Suen PRD(86); Bondi & Rindler GRG(91).

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