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
Qij =
R^3
d3x (3 xix j – r2
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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send feedback and suggestions to bombelli at olemiss.edu – modified 18
oct 2009