25:
1674:
arises for rotating sources due to the rotational kinetic energy which, because of mass–energy equivalence, can also act as the source of a gravitational field. The situation is analogous to a static electromagnetic field where one has two sets of potentials, electric and magnetic. In general
338:
684:
1555:
1291:
2154:
1884:
1615:
578:
1364:
756:. The twist vector measures the extent to which the Killing vector fails to be orthogonal to a family of 3-surfaces. A non-zero twist indicates the presence of rotation in the spacetime geometry.
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985:
2059:
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is a 3-vector, called the twist vector, which vanishes when the
Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector
1135:
2195:
144:
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711:
605:
445:
194:
1707:
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1387:
1025:
522:
1407:
1314:
201:
1783:
1082:
610:
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835:
803:. By identifying the spacetime points that lie on a particular trajectory (also called orbit) one gets a 3-dimensional space (the manifold of Killing trajectories)
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and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case
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527:
146:, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form
1322:
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the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
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39:
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33:
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it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials,
119:
1647:
plays the role of the
Newtonian gravitational potential. A nontrivial angular momentum potential
1030:
759:
The coordinate representation described above has an interesting geometrical interpretation. The
689:
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423:
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is the metric tensor of 3-dimensional space. In this coordinate system the
Killing vector field
333:{\displaystyle ds^{2}=\lambda (dt-\omega _{i}\,dy^{i})^{2}-\lambda ^{-1}h_{ij}\,dy^{i}\,dy^{j},}
1685:
1650:
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679:{\displaystyle \omega _{\mu }=e_{\mu \nu \rho \sigma }\xi ^{\nu }\nabla ^{\rho }\xi ^{\sigma }}
50:
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1785:. In terms of these quantities the Einstein vacuum field equations can be put in the form
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102:
1550:{\displaystyle \Phi _{M}={\frac {1}{4}}\lambda ^{-1}(\lambda ^{2}+\omega ^{2}-1),}
1682:
A stationary vacuum metric is thus expressible in terms of the Hansen potentials
98:
1286:{\displaystyle \nabla _{\mu }\omega _{\nu }-\nabla _{\nu }\omega _{\mu }=0,\,}
2354:
2149:{\displaystyle \Phi ^{2}=\Phi _{A}\Phi _{A}=(\Phi _{M}^{2}+\Phi _{J}^{2})}
686:(see, for example, p. 163) which is orthogonal to the Killing vector
1146:
524:
is a positive scalar representing the norm of the
Killing vector, i.e.,
16:
Spacetime that admits a
Killing vector that is asymptotically timelike
1879:{\displaystyle (h^{ij}\nabla _{i}\nabla _{j}-2R^{(3)})\Phi _{A}=0,\,}
90:
1157:
In a stationary spacetime satisfying the vacuum
Einstein equations
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105:
1145:
is stationary, but the converse is not generally true, as the
1610:{\displaystyle \Phi _{J}={\frac {1}{2}}\lambda ^{-1}\omega .}
2324:
Wald, R.M., (1984). General
Relativity, (U. Chicago Press)
573:{\displaystyle \lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }}
763:
Killing vector generates a one-parameter group of motion
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In a stationary spacetime, the metric tensor components,
1359:{\displaystyle \omega _{\mu }=\nabla _{\mu }\omega .\,}
877:. This identification, called a canonical projection,
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1296:and is therefore locally the gradient of a scalar
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1153:Use as starting point for vacuum field equations
2197:is the Ricci tensor of the spatial metric and
2345:Hansen, R.O. (1974). J. Math. Phys. 15, 46.
909:is a mapping that sends each trajectory in
749:{\displaystyle \omega _{\mu }\xi ^{\mu }=0}
111:
2333:Geroch, R., (1971). J. Math. Phys. 12, 918
2263:{\displaystyle R^{(3)}=h^{ij}R_{ij}^{(3)}}
857:represents a trajectory in the spacetime
1875:
1620:In general relativity the mass potential
1355:
1282:
313:
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246:
69:Learn how and when to remove this message
2341:
2339:
2305:General Relativity: A Geometric Approach
1193:outside the sources, the twist 4-vector
32:This article includes a list of general
1675:relativity, rotating sources produce a
2353:
390:are the three spatial coordinates and
2336:
837:, the quotient space. Each point of
497:{\displaystyle \xi ^{\mu }=(1,0,0,0)}
18:
902:{\displaystyle \pi :M\rightarrow V}
13:
2307:, Cambridge University Press, 1999
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38:it lacks sufficient corresponding
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980:{\displaystyle h=-\lambda \pi *g}
23:
2285:Spherically symmetric spacetime
2054:{\displaystyle R_{ij}^{(3)}=2,}
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1679:that has no Newtonian analog.
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1213:{\displaystyle \omega _{\mu }}
1186:{\displaystyle R_{\mu \nu }=0}
893:
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467:
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1:
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1130:{\displaystyle \omega _{i}=0}
1007:via pullback. The quantities
2190:{\displaystyle R_{ij}^{(3)}}
1137:the spacetime is said to be
139:{\displaystyle g_{\mu \nu }}
7:
2273:
1316:(called the twist scalar):
1149:provides a counterexample.
1047:{\displaystyle \omega _{i}}
706:{\displaystyle \xi ^{\mu }}
600:{\displaystyle \omega _{i}}
440:{\displaystyle \xi ^{\mu }}
189:{\displaystyle (i,j=1,2,3)}
10:
2377:
1702:{\displaystyle \Phi _{A}}
1667:{\displaystyle \Phi _{J}}
1640:{\displaystyle \Phi _{M}}
1456:{\displaystyle \Phi _{J}}
1429:{\displaystyle \Phi _{M}}
1382:{\displaystyle \lambda }
1020:{\displaystyle \lambda }
517:{\displaystyle \lambda }
363:is the time coordinate,
112:Description and analysis
87:Einstein field equations
1402:{\displaystyle \omega }
1369:Instead of the scalars
1309:{\displaystyle \omega }
1141:. By definition, every
53:more precise citations.
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1778:{\displaystyle h_{ij}}
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1077:{\displaystyle h_{ij}}
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413:{\displaystyle h_{ij}}
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85:, specifically in the
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1677:gravitomagnetic field
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949:and induces a metric
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830:{\displaystyle V=M/G}
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383:{\displaystyle y^{i}}
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2361:Lorentzian manifolds
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1728:{\displaystyle A=M}
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1084:are all fields on
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83:general relativity
1748:{\displaystyle J}
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1097:{\displaystyle V}
1000:{\displaystyle V}
942:{\displaystyle V}
922:{\displaystyle M}
870:{\displaystyle M}
850:{\displaystyle V}
796:{\displaystyle M}
783:in the spacetime
776:{\displaystyle G}
356:{\displaystyle t}
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2280:Static spacetime
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1143:static spacetime
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761:time translation
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1528:
1524:
1519:
1515:
1511:
1506:
1503:
1499:
1493:
1490:
1485:
1480:
1476:
1450:
1446:
1423:
1419:
1398:
1378:
1367:
1366:
1354:
1351:
1346:
1342:
1338:
1333:
1329:
1305:
1294:
1293:
1281:
1278:
1275:
1270:
1266:
1260:
1256:
1252:
1247:
1243:
1237:
1233:
1220:is curl-free,
1207:
1203:
1182:
1179:
1174:
1171:
1167:
1154:
1151:
1126:
1123:
1118:
1114:
1093:
1071:
1068:
1064:
1041:
1037:
1016:
996:
976:
973:
970:
967:
964:
961:
958:
938:
918:
898:
895:
892:
889:
886:
866:
846:
826:
822:
818:
815:
812:
792:
772:
745:
742:
737:
733:
727:
723:
700:
696:
673:
669:
663:
659:
653:
649:
643:
640:
637:
634:
630:
626:
621:
617:
594:
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547:
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540:
536:
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493:
490:
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472:
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400:
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320:
316:
310:
306:
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296:
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289:
283:
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276:
272:
267:
263:
257:
253:
249:
243:
239:
235:
232:
229:
226:
223:
220:
215:
211:
207:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
133:
130:
126:
113:
110:
103:asymptotically
99:Killing vector
93:is said to be
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2373:
2362:
2359:
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2308:
2306:
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2241:
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2164:
2138:
2133:
2125:
2120:
2115:
2104:
2099:
2089:
2081:
2076:
2048:
2040:
2030:
2020:
2010:
2000:
1997:
1987:
1979:
1976:
1973:
1967:
1962:
1952:
1942:
1932:
1921:
1918:
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1902:
1899:
1895:
1887:
1872:
1869:
1866:
1861:
1845:
1838:
1834:
1831:
1826:
1816:
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1788:
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1722:
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1680:
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1513:
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1491:
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1483:
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1465:
1464:
1463:, defined as
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538:
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473:
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1147:Kerr metric
51:introducing
2314:052163976X
2291:References
95:stationary
59:April 2021
34:references
2130:Φ
2112:Φ
2096:Φ
2086:Φ
2073:Φ
2037:Φ
2027:∇
2017:Φ
2007:∇
1998:−
1984:Φ
1968:−
1959:Φ
1949:∇
1939:Φ
1929:∇
1858:Φ
1832:−
1823:∇
1813:∇
1691:Φ
1656:Φ
1629:Φ
1602:ω
1594:−
1590:λ
1567:Φ
1536:−
1527:ω
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1502:−
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1475:Φ
1445:Φ
1418:Φ
1397:ω
1377:λ
1350:ω
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1341:∇
1332:μ
1328:ω
1304:ω
1269:μ
1265:ω
1259:ν
1255:∇
1251:−
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1242:ω
1236:μ
1232:∇
1206:μ
1202:ω
1173:ν
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1113:ω
1036:ω
1015:λ
972:∗
969:π
966:λ
963:−
894:→
885:π
736:μ
732:ξ
726:μ
722:ω
699:μ
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672:σ
668:ξ
662:ρ
658:∇
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642:σ
639:ρ
636:ν
633:μ
620:μ
616:ω
589:ω
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543:μ
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512:λ
460:μ
456:ξ
433:μ
429:ξ
279:−
275:λ
271:−
238:ω
234:−
222:λ
132:ν
129:μ
91:spacetime
2355:Category
2274:See also
106:timelike
101:that is
47:improve
2312:
2156:, and
2064:where
1139:static
580:, and
343:where
36:, but
2310:ISBN
1436:and
1389:and
1054:and
89:, a
987:on
81:In
2357::
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