2466:, similar to the electromagnetic radiation produced by oscillating electric or magnetic dipoles and higher multipoles. However, only quadrupole and higher moments can radiate gravitationally. The mass monopole represents the total mass-energy in a system, which is conserved—thus it gives off no radiation. Similarly, the mass dipole corresponds to the center of mass of a system and its first derivative represents momentum which is also a conserved quantity so the mass dipole also emits no radiation. The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation.
2180:
2172:
1246:
2111:
2473:. Since the dipole moment is constant, we can for convenience place the coordinate origin right between the two points. Then the dipole moment will be zero, and if we also scale the coordinates so that the points are at unit distance from the center, in opposite direction, the system's quadrupole moment will then simply be
1449:
2621:-vector will rotate, which means that it will have a non-zero first, and also a non-zero second time derivative (this is of course true regardless the choice of the coordinate system). Therefore, the system will radiate gravitational waves. Energy lost in this way was first observed in the changing period of the
1702:
1875:
925:
of two opposite-sign, same-strength point charges, which has no monopole moment, can have a nonzero quadrupole moment if the origin is shifted away from the center of the configuration exactly between the two charges; or the quadrupole moment can be reduced to zero with the origin at the center. In
2403:
For example, because the Earth is rotating, it is oblate (flattened at the poles). This gives it a nonzero quadrupole moment. While the contribution to the Earth's gravitational field from this quadrupole is extremely important for artificial satellites close to Earth, it is less important for the
2201:
All known magnetic sources give dipole fields. However, it is possible to make a magnetic quadrupole by placing four identical bar magnets perpendicular to each other such that the north pole of one is next to the south of the other. Such a configuration cancels the dipole moment and gives a
1126:
2244:
The mass quadrupole is analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density and a negative sign is added because the masses are always positive and the force is attractive. The gravitational potential is then expressed as:
904:
471:
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1273:
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1257:
A simple example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square. The monopole moment (just the total charge) of this arrangement is zero. Similarly, the
326:
2581:
926:
contrast, if the monopole and dipole moments vanish, but the quadrupole moment does not, e.g. four same-strength charges, arranged in a square, with alternating signs, then the quadrupole moment is coordinate independent.
1262:
is zero, regardless of the coordinate origin that has been chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin. The
975:
710:
773:
2628:
Just as electric charge and current multipoles contribute to the electromagnetic field, mass and mass-current multipoles contribute to the gravitational field in general relativity, causing the so-called
2035:
338:
2633:
effects. Changing mass-current multipoles can also give off gravitational radiation. However, contributions from the current multipoles will typically be much smaller than that of the mass quadrupole.
1957:
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The simplest and most important example of a radiating system is a pair of mass points with equal masses orbiting each other on a circular orbit, an approximation to e.g. special case of binary
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1177:
1444:{\displaystyle V_{\text{q}}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ ,}
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239:
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2064:, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g.,
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601:
535:
181:
1697:{\displaystyle Q_{ij}=\int \,\rho (\mathbf {r} )\left({\frac {3}{2}}r_{i}r_{j}-{\frac {1}{2}}\left\|\mathbf {r} \right\|^{2}\delta _{ij}\right)\,d^{3}\mathbf {r} }
955:
500:
2729:
Applequist, J. (1989). "Traceless cartesian tensor forms for spherical harmonic functions: New theorems and applications to electrostatics of dielectric media".
1235:
621:
1870:{\displaystyle V_{\text{q}}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ ,}
244:
2479:
2225:. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a large
1121:{\displaystyle V_{\text{q}}(\mathbf {R} )={\frac {k}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ ,}
899:{\displaystyle Q_{ij}=\int \,\rho (\mathbf {r} )\left(3r_{i}r_{j}-\left\|\mathbf {r} \right\|^{2}\delta _{ij}\right)\,d^{3}\mathbf {r} }
636:
2811:
2158:
466:{\displaystyle Q_{ij}=\sum _{\ell }q_{\ell }\left(3r_{i\ell }r_{j\ell }-\left\|\mathbf {r} _{\ell }\right\|^{2}\delta _{ij}\right).}
1981:
2693:
2393:{\displaystyle V_{\text{q}}(\mathbf {R} )=-{\frac {G}{2|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\ .}
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156:). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and
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are the
Cartesian components of the unit vector pointing from the origin to the field point. Here,
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quadrupole moment, and its field will decrease at large distances faster than that of a dipole.
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in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the
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An example of a magnetic quadrupole, involving permanent magnets, is depicted on the right.
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719:. Conversion between these two forms can be easily achieved using a detracing operator.
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are components of the (unit) position vector of one of the points. As they orbit, this
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1978:. The "octopole moment" of this arrangement would correspond, in the "octopole limit"
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to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order
321:{\displaystyle \mathbf {r} _{\ell }=\left(r_{x\ell },r_{y\ell },r_{z\ell }\right)}
2110:
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2576:{\displaystyle Q_{ij}=M\left(3x_{i}x_{j}-|\mathbf {x} |^{2}\delta _{ij}\right)}
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624:
84:—3×3 matrix. There are several definitions, but it is normally stated in the
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85:
1237:
is a constant that depends on the type of field, and the units being used.
17:
1967:
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In the non-traceless form, the quadrupole moment is sometimes stated as:
1970:") would be: Eight alternating point charges at the eight corners of a
763:, the components of Q are defined by integral over the Cartesian space
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914:
81:
65:
of a more complex structure reflecting various orders of complexity.
1518:
Alternatively, other sources include the factor of one half in the
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2170:
1244:
715:
with this form seeing some usage in the literature regarding the
705:{\displaystyle Q_{ij}=\sum _{\ell }q_{\ell }r_{i\ell }r_{j\ell }}
328:
relative to the coordinate system origin, the components of the
2625:, a pulsar in orbit with another neutron star of similar mass.
61:
that can exist in ideal form, but it is usually just part of a
2104:
722:
For a continuous system with charge density, or mass density,
2030:{\textstyle \lim _{a\to 0}{a^{3}\cdot Q}\to {\text{const. }}}
2229:
that flows in the coils of tubing wrapped around the poles.
160:
property, in this form only five of these are independent.
603:
must be equal, up to sign, to distances from the point to
1952:{\textstyle P_{2}(x)={\frac {3}{2}}x^{2}-{\frac {1}{2}}.}
27:
Arrangement that creates a quadrupole field of some sort
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which result from the multipole expansion, namely here
909:
As with any multipole moment, if a lower-order moment,
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is one of a sequence of configurations of things like
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is a vector with origin in the system of charges and
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2844:"Multipole Expansions of Gravitational Radiation"
2458:The mass quadrupole moment is also important in
1986:
2462:because, if it changes in time, it can produce
2232:A changing magnetic quadrupole moment produces
2731:Journal of Physics A: Mathematical and General
1267:of an electric charge quadrupole is given by
184:
8:
2221:and beam transport lines, a method known as
1880:which makes more explicit the connection to
2139:. Unsourced material may be challenged and
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2159:Learn how and when to remove this message
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183:point charges or masses in the case of a
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2448:{\displaystyle {1}/{|\mathbf {R} |^{3}}}
2829:Introduction to Electrodynamics, 4th ed
2721:
1139:is the unit vector in the direction of
149:{\displaystyle Q_{xx}+Q_{yy}+Q_{zz}=0}
2209:of similar conceptual design (called
7:
2137:adding citations to reliable sources
627:for the Kronecker delta to equal 1.
1974:, e.g., of a cube with edge length
929:If each charge is the source of a "
2175:Coils producing a quadrupole field
1966:An extreme generalization ("point
965:, the contribution to the field's
25:
2770:Eric Weisstein's World of Physics
1962:Generalization: higher multipoles
2539:
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1474:{\displaystyle \varepsilon _{0}}
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2694:Multipolar exchange interaction
2590:is the mass of each point, and
1253:of an electric quadrupole field
969:from the quadrupole moment is:
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1515:follows the definition above.
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1172:{\displaystyle {\hat {R}}_{i}}
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1:
2842:Thorne, Kip S. (April 1980).
2664:Cylindrical multipole moments
2213:) are commonly used to focus
2827:Griffiths, David J. (2013).
2794:Jackson, John David (1975).
2766:"Electric Quadrupole Moment"
756:{\displaystyle \rho (x,y,z)}
560:{\displaystyle \delta _{ij}}
34:, which is sometimes called
2831:. Pearson. p. 153,165.
2751:10.1088/0305-4470/22/20/011
2669:Spherical multipole moments
2090:{\displaystyle 2^{\ell -1}}
957:potential" field, like the
2929:
2215:beams of charged particles
2194:
1548:tensor itself, such that:
29:
2871:10.1103/RevModPhys.52.299
2851:Reviews of Modern Physics
2798:Classical Electrodynamics
2234:electromagnetic radiation
2057:{\displaystyle 2^{\ell }}
234:{\displaystyle m_{\ell }}
207:{\displaystyle q_{\ell }}
163:For a discrete system of
2689:Quadrupole mass analyzer
2240:Gravitational quadrupole
185:gravitational quadrupole
75:quadrupole moment tensor
30:Not to be confused with
2659:Axial multipole moments
2464:gravitational radiation
1210:{\displaystyle i=x,y,z}
623:mutually perpendicular
332:matrix are defined by:
69:Mathematical definition
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2449:
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2176:
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1541:{\displaystyle Q_{ij}}
1509:
1508:{\displaystyle Q_{ij}}
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1251:equipotential surfaces
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2804:John Wiley & Sons
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2610:{\displaystyle x_{i}}
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1483:electric permittivity
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717:fast multipole method
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596:{\displaystyle x,y,z}
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530:{\displaystyle x,y,z}
504:Cartesian coordinates
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178:
176:{\displaystyle \ell }
151:
2679:Legendre polynomials
2594:
2480:
2455:term falls quickly.
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2133:improve this section
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2041:
1982:
1888:
1882:Legendre polynomials
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1249:Contour plot of the
1241:Electric quadrupole
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2888:Multipole expansion
2863:1980RvMP...52..299T
2743:1989JPhA...22.4303A
2684:Quadrupole ion trap
2644:Multipole expansion
2623:Hulse–Taylor binary
2101:Magnetic quadrupole
963:gravitational field
950:{\displaystyle 1/r}
571:. This means that
495:{\displaystyle i,j}
187:, each with charge
63:multipole expansion
2709:Quadrupole formula
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2460:general relativity
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1143:. That is to say,
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921:. For example, a
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59:gravitational mass
2764:Weisstein, Eric.
2737:(20): 4303–4330.
2674:Laplace expansion
2649:Multipole moments
2404:Moon because the
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2197:Quadrupole magnet
2185:quadrupole magnet
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1230:{\displaystyle k}
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919:coordinate origin
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616:{\displaystyle n}
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16:(Redirected from
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2913:Moment (physics)
2903:Electromagnetism
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107:
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32:two-port network
21:
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2699:Star quad cable
2654:Solid harmonics
2639:
2631:gravitomagnetic
2597:
2592:
2591:
2555:
2543:
2520:
2510:
2506:
2502:
2483:
2478:
2477:
2432:
2406:
2405:
2365:
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2301:
2287:
2255:
2250:
2249:
2242:
2223:strong focusing
2199:
2165:
2154:
2148:
2145:
2130:
2114:
2103:
2071:
2066:
2065:
2044:
2039:
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1980:
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1964:
1923:
1891:
1886:
1885:
1842:
1823:
1809:
1778:
1767:
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829:
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777:
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771:
724:
723:
689:
676:
666:
640:
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573:
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569:Kronecker delta
544:
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538:
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506:
478:
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442:
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241:, and position
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127:
111:
95:
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71:
51:electric charge
39:
28:
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12:
11:
5:
2926:
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2882:External links
2880:
2877:
2876:
2857:(2): 299–339.
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2207:Electromagnets
2167:
2166:
2117:
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2108:
2102:
2099:
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2081:
2078:
2074:
2051:
2047:
2021:
2017:
2014:
2009:
2005:
1998:
1995:
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1988:
1972:parallelepiped
1963:
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1948:
1943:
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80:is a rank-two
70:
67:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2925:
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2813:0-471-43132-X
2809:
2805:
2800:
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2748:
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2725:
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2715:
2710:
2707:
2705:
2704:Magnetic lens
2702:
2700:
2697:
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2690:
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2134:
2128:
2127:
2123:
2118:This section
2116:
2112:
2107:
2106:
2100:
2098:
2082:
2079:
2076:
2072:
2049:
2045:
2015:
2012:
2007:
2003:
1996:
1990:
1977:
1973:
1969:
1961:
1959:
1946:
1941:
1938:
1933:
1928:
1924:
1918:
1915:
1910:
1904:
1896:
1892:
1883:
1864:
1856:
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1318:
1314:
1310:
1307:
1303:
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1278:
1270:
1269:
1268:
1266:
1261:
1260:dipole moment
1252:
1247:
1240:
1238:
1224:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1164:
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924:
920:
916:
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876:
870:
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863:
857:
839:
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797:
793:
790:
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782:
778:
770:
769:
768:
766:
747:
744:
741:
738:
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729:
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718:
697:
694:
690:
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681:
677:
671:
667:
661:
657:
653:
648:
645:
641:
633:
632:
631:
628:
626:
610:
590:
587:
584:
581:
578:
570:
552:
549:
545:
524:
521:
518:
515:
512:
505:
502:run over the
489:
486:
483:
460:
456:
450:
447:
443:
437:
427:
412:
407:
404:
400:
394:
391:
387:
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335:
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276:
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226:
222:
199:
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186:
170:
161:
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143:
140:
135:
132:
128:
124:
119:
116:
112:
108:
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100:
96:
87:
83:
79:
76:
68:
66:
64:
60:
56:
52:
48:
44:
37:
33:
19:
2854:
2850:
2837:
2828:
2822:
2797:
2789:
2777:. Retrieved
2769:
2759:
2734:
2730:
2724:
2627:
2618:
2587:
2585:
2468:
2457:
2402:
2243:
2231:
2204:
2200:
2188:
2155:
2146:
2131:Please help
2119:
2024:const.
1975:
1965:
1879:
1517:
1453:
1256:
1140:
1136:
1132:
1130:
928:
908:
764:
721:
714:
629:
476:The indices
475:
329:
162:
77:
74:
72:
46:
42:
40:
35:
2471:black holes
2149:August 2024
625:hyperplanes
88:form (i.e.
2897:Categories
2716:References
2195:See also:
2183:Schematic
214:, or mass
158:zero-trace
47:quadrapole
43:quadrupole
36:quadripole
18:Quadropole
2557:δ
2531:−
2373:^
2354:^
2318:∑
2280:−
2189:four-pole
2120:does not
2080:−
2077:ℓ
2050:ℓ
2020:→
2013:⋅
1994:→
1934:−
1850:^
1831:^
1795:∑
1751:ε
1747:π
1662:δ
1628:−
1579:ρ
1575:∫
1463:ε
1424:^
1405:^
1359:∑
1315:ε
1311:π
1158:^
1101:^
1082:^
1036:∑
967:potential
864:δ
840:−
798:ρ
794:∫
730:ρ
698:ℓ
685:ℓ
672:ℓ
662:ℓ
658:∑
546:δ
444:δ
428:ℓ
413:−
408:ℓ
395:ℓ
374:ℓ
364:ℓ
360:∑
309:ℓ
293:ℓ
277:ℓ
256:ℓ
227:ℓ
200:ℓ
171:ℓ
86:traceless
2637:See also
1968:octopole
1651:‖
1643:‖
959:electric
911:monopole
853:‖
845:‖
433:‖
418:‖
2908:Gravity
2859:Bibcode
2739:Bibcode
2141:removed
2126:sources
1481:is the
567:is the
55:current
2810:
2779:May 8,
2586:where
2385:
1862:
1485:, and
1454:where
1436:
1131:where
1113:
923:dipole
915:dipole
82:tensor
2847:(PDF)
1704:, and
57:, or
2808:ISBN
2781:2012
2124:any
2122:cite
1179:for
537:and
73:The
2867:doi
2747:doi
2217:in
2135:by
1987:lim
961:or
913:or
53:or
45:or
2899::
2865:.
2855:52
2853:.
2849:.
2806:.
2802:.
2772:.
2768:.
2745:.
2735:22
2733:.
2236:.
2191:")
2187:("
2097:.
1137:RĚ‚
767::
41:A
2873:.
2869::
2861::
2816:.
2783:.
2753:.
2749::
2741::
2619:x
2603:i
2599:x
2588:M
2570:)
2564:j
2561:i
2551:2
2546:|
2540:x
2535:|
2526:j
2522:x
2516:i
2512:x
2508:3
2504:(
2500:M
2497:=
2492:j
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2325:,
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2309:3
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2270:R
2266:(
2261:q
2257:V
2162:)
2156:(
2151:)
2147:(
2143:.
2129:.
2083:1
2073:2
2046:2
2016:Q
2008:3
2004:a
1997:0
1991:a
1976:a
1947:.
1942:2
1939:1
1929:2
1925:x
1919:2
1916:3
1911:=
1908:)
1905:x
1902:(
1897:2
1893:P
1865:,
1857:j
1847:R
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748:z
745:,
742:y
739:,
736:x
733:(
695:j
691:r
682:i
678:r
668:q
654:=
649:j
646:i
642:Q
611:n
591:z
588:,
585:y
582:,
579:x
553:j
550:i
525:z
522:,
519:y
516:,
513:x
490:j
487:,
484:i
461:.
457:)
451:j
448:i
438:2
423:r
405:j
401:r
392:i
388:r
384:3
380:(
370:q
356:=
351:j
348:i
344:Q
330:Q
315:)
306:z
302:r
298:,
290:y
286:r
282:,
274:x
270:r
265:(
261:=
251:r
223:m
196:q
144:0
141:=
136:z
133:z
129:Q
125:+
120:y
117:y
113:Q
109:+
104:x
101:x
97:Q
78:Q
38:.
20:)
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