1998:
28:
1134:
1717:
1405:
906:
1603:
821:
1294:
227:
983:
1761:
956:
1608:
1299:
515:
256:
172:
1497:
1451:
556:
1198:
347:
315:
119:
1820:
976:
707:
687:
663:
643:
616:
596:
576:
470:
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407:
387:
367:
280:
143:
99:
79:
826:
2039:
1506:
746:
1186:
provides a coordinate system for the sphere in which conformal flatness is explicit, as the metric is proportional to the flat one.
17:
1219:
145:
the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means there exists a function
2032:
2073:
2025:
1211:
1916:
Garat, Alcides; Price, Richard H. (2000-05-18). "Nonexistence of conformally flat slices of the Kerr spacetime".
177:
1831:
1129:{\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,={\frac {4}{(1+r^{2})^{2}}}(dx^{2}+dy^{2})}
2063:
736:
259:
2058:
1725:
1712:{\displaystyle g_{ik}={\begin{bmatrix}1-{\frac {2GM}{r}}&0\\0&-1+{\frac {2GM}{r}}\end{bmatrix}}}
1400:{\displaystyle g_{ik}={\begin{bmatrix}0&1-{\frac {2GM}{r}}\\1-{\frac {2GM}{r}}&0\end{bmatrix}}}
915:
36:
1773:
2013:
1183:
909:
522:
52:
475:
232:
148:
1879:
Garecki, Janusz (2008). "On Energy of the
Friedman Universes in Conformally Flat Coordinates".
1456:
1410:
528:
1935:
1898:
320:
288:
104:
8:
2005:
722:
40:
1939:
1902:
2068:
1959:
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961:
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412:
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265:
128:
84:
64:
1821:
Spherical coordinate system - Integration and differentiation in spherical coordinates
31:
The upper manifold is flat. The lower one is not, but it is conformal to the first one
1963:
1951:
1943:
1853:
1168:
1141:
A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the
901:{\displaystyle g_{ik}={\begin{bmatrix}1&0\\0&sin^{2}\theta \end{bmatrix}}}
430:
2009:
1947:
1783:
1202:
2052:
1955:
1201:. However it was also shown that there are no conformally flat slices of the
1171:, conformally Euclidean Riemannian manifold is conformally equivalent to the
1164:
1142:
518:
59:
48:
1997:
732:
1157:
1865:
735:
of the two dimensional spherical coordinates, like the one used in the
1930:
1197:
conformally flat manifolds can often be used, for example to describe
1598:{\displaystyle ds^{2}=\left(1-{\frac {2GM}{r}}\right)(dt^{2}-dx^{2})}
1857:
728:
Every 2-dimensional pseudo-Riemannian manifold is conformally flat.
1172:
122:
1893:
816:{\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,}
1844:
Kuiper, N. H. (1949). "On conformally flat spaces in the large".
27:
1289:{\displaystyle ds^{2}=\left(1-{\frac {2GM}{r}}\right)dv\,du}
978:
is the distance from the origin of the flat space, obtaining
912:
can be mapped to a flat space using the conformal factor
47:
if each point has a neighborhood that can be mapped to
1633:
1324:
851:
1728:
1611:
1509:
1459:
1413:
1302:
1222:
986:
964:
919:
829:
749:
695:
675:
651:
631:
604:
584:
564:
531:
478:
458:
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375:
355:
323:
291:
268:
235:
180:
151:
131:
107:
87:
67:
1722:
which is the flat metric times the conformal factor
669:
for the case in which the relation is valid for all
1755:
1711:
1597:
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1445:
1399:
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1128:
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381:
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309:
274:
250:
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166:
137:
113:
93:
73:
1407:and so is not flat. But with the transformations
2050:
2033:
1976:
1805:
1977:Ray D'Inverno. "17.2 The Kruskal solution".
1152:-dimensional pseudo-Riemannian manifold for
1156:≥ 4 is conformally flat if and only if the
222:{\displaystyle g(x)=\lambda ^{2}(x)\,\eta }
2040:
2026:
1915:
1199:Friedmann–Lemaître–Robertson–Walker metric
1929:
1892:
1279:
1049:
1035:
812:
798:
215:
26:
1878:
1806:Ray D'Inverno. "6.13 The Weyl tensor".
101:has to be conformal to the flat metric
14:
2051:
349:is conformally flat if for each point
317:be a pseudo-Riemannian manifold. Then
1832:Stereographic projection - Properties
1992:
1801:
1799:
908: and is not flat but with the
621:Some authors use the definition of
24:
1756:{\displaystyle 1-{\frac {2GM}{r}}}
25:
2085:
1979:Introducing Einstein's Relativity
1808:Introducing Einstein's Relativity
1796:
951:{\displaystyle 2 \over (1+r^{2})}
625:when referred to just some point
1996:
1970:
1909:
1872:
1837:
1825:
1814:
1592:
1560:
1478:
1466:
1432:
1420:
1123:
1091:
1079:
1059:
944:
925:
665:and reserve the definition of
598:need not be defined on all of
504:
479:
389:, there exists a neighborhood
336:
324:
304:
292:
245:
239:
212:
206:
190:
184:
161:
155:
13:
1:
1789:
2012:. You can help Knowledge by
1212:Kruskal-Szekeres coordinates
737:geographic coordinate system
282:is a point on the manifold.
7:
1843:
1767:
712:
510:{\displaystyle (U,e^{2f}g)}
251:{\displaystyle \lambda (x)}
167:{\displaystyle \lambda (x)}
10:
2090:
1991:
1948:10.1103/PhysRevD.61.124011
125:maintain in all points of
2074:Riemannian geometry stubs
1492:{\displaystyle x=(v-u)/2}
1446:{\displaystyle t=(v+u)/2}
1184:stereographic projection
910:stereographic projection
623:locally conformally flat
53:conformal transformation
18:Locally conformally flat
1881:Acta Physica Polonica B
1834:. The Riemann's formula
551:{\displaystyle e^{2f}g}
2008:-related article is a
1757:
1713:
1599:
1493:
1447:
1401:
1290:
1130:
972:
952:
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817:
703:
683:
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639:
612:
592:
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552:
511:
466:
446:
423:
403:
383:
363:
343:
311:
276:
252:
223:
168:
139:
115:
95:
75:
32:
1846:Annals of Mathematics
1774:Weyl–Schouten theorem
1758:
1714:
1600:
1494:
1448:
1402:
1291:
1131:
973:
953:
903:
818:
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684:
660:
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613:
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553:
512:
467:
447:
424:
404:
384:
364:
344:
342:{\displaystyle (M,g)}
312:
310:{\displaystyle (M,g)}
277:
253:
224:
169:
140:
116:
114:{\displaystyle \eta }
96:
76:
30:
1726:
1609:
1507:
1457:
1411:
1300:
1220:
984:
962:
916:
827:
823:, has metric tensor
747:
725:is conformally flat.
718:Every manifold with
693:
673:
649:
629:
602:
582:
562:
529:
476:
456:
436:
413:
393:
373:
353:
321:
289:
266:
233:
178:
149:
129:
105:
85:
65:
2064:Riemannian geometry
2006:Riemannian geometry
1981:. pp. 230–231.
1940:2000PhRvD..61l4011G
1903:2008AcPPB..39..781G
1605:with metric tensor
1296:with metric tensor
723:sectional curvature
285:More formally, let
41:Riemannian manifold
2059:Conformal geometry
1779:conformal geometry
1753:
1709:
1703:
1595:
1489:
1443:
1397:
1391:
1286:
1195:general relativity
1126:
968:
923:
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813:
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248:
219:
164:
135:
111:
91:
71:
33:
2021:
2020:
1918:Physical Review D
1810:. pp. 88–89.
1751:
1699:
1658:
1553:
1382:
1354:
1266:
1214:have line element
1210:For example, the
1089:
971:{\displaystyle r}
948:
702:{\displaystyle M}
682:{\displaystyle x}
658:{\displaystyle M}
638:{\displaystyle x}
611:{\displaystyle M}
591:{\displaystyle f}
571:{\displaystyle U}
465:{\displaystyle U}
445:{\displaystyle f}
422:{\displaystyle x}
402:{\displaystyle U}
382:{\displaystyle M}
362:{\displaystyle x}
275:{\displaystyle x}
138:{\displaystyle M}
94:{\displaystyle M}
74:{\displaystyle g}
58:In practice, the
16:(Redirected from
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2042:
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1234:
1169:simply connected
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708:
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667:conformally flat
664:
662:
661:
656:
644:
642:
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617:
615:
614:
609:
597:
595:
594:
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578:). The function
577:
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569:
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348:
346:
345:
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316:
314:
313:
308:
281:
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273:
260:conformal factor
258:is known as the
257:
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228:
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173:
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165:
144:
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120:
118:
117:
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97:
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81:of the manifold
80:
78:
77:
72:
45:conformally flat
21:
2089:
2088:
2084:
2083:
2082:
2080:
2079:
2078:
2049:
2048:
2047:
2046:
1989:
1987:
1986:
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1971:
1914:
1910:
1877:
1873:
1858:10.2307/1969587
1842:
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1804:
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530:
527:
526:
492:
488:
477:
474:
473:
457:
454:
453:
437:
434:
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431:smooth function
414:
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410:
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23:
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15:
12:
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5:
2087:
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2037:
2030:
2022:
2019:
2018:
2001:
1985:
1984:
1969:
1924:(12): 124011.
1908:
1887:(4): 781–797.
1871:
1852:(4): 916–924.
1836:
1824:
1813:
1794:
1793:
1791:
1788:
1787:
1786:
1784:Yamabe problem
1781:
1776:
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1233:
1229:
1225:
1215:
1207:
1206:
1203:Kerr spacetime
1190:
1189:
1188:
1187:
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1176:
1161:
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1125:
1120:
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90:
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9:
6:
4:
3:
2:
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2024:
2023:
2017:
2015:
2011:
2007:
2002:
1999:
1995:
1994:
1990:
1980:
1973:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1932:
1931:gr-qc/0002013
1927:
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1231:
1227:
1223:
1216:
1213:
1209:
1208:
1204:
1200:
1196:
1192:
1191:
1185:
1181:
1180:
1179:
1178:
1174:
1170:
1166:
1162:
1159:
1155:
1151:
1147:
1144:
1143:Cotton tensor
1140:
1118:
1114:
1110:
1107:
1102:
1098:
1094:
1083:
1073:
1069:
1065:
1062:
1055:
1050:
1044:
1040:
1036:
1032:
1029:
1024:
1020:
1016:
1011:
1007:
1003:
1000:
995:
991:
987:
980:
965:
939:
935:
931:
928:
920:
911:
893:
887:
882:
878:
874:
871:
866:
859:
854:
848:
843:
838:
835:
831:
807:
803:
799:
795:
792:
787:
783:
779:
774:
770:
766:
763:
758:
754:
750:
743:
742:
738:
734:
730:
729:
727:
724:
721:
717:
716:
710:
696:
676:
668:
652:
632:
624:
619:
605:
585:
565:
545:
540:
537:
533:
524:
520:
501:
496:
493:
489:
485:
482:
459:
439:
432:
416:
396:
376:
356:
333:
330:
327:
301:
298:
295:
283:
269:
261:
242:
236:
216:
209:
201:
197:
193:
187:
181:
158:
152:
132:
124:
108:
88:
68:
61:
56:
54:
50:
46:
42:
38:
29:
19:
2014:expanding it
2003:
1988:
1978:
1972:
1921:
1917:
1911:
1884:
1880:
1874:
1849:
1845:
1839:
1827:
1816:
1807:
1173:round sphere
1153:
1149:
733:line element
666:
622:
620:
558:vanishes on
284:
121:, i.e., the
57:
44:
34:
1158:Weyl tensor
452:defined on
2053:Categories
1790:References
521:(i.e. the
472:such that
174:such that
49:flat space
2069:Manifolds
1964:119452751
1956:0556-2821
1894:0708.2783
1733:−
1675:−
1640:−
1577:−
1535:−
1473:−
1364:−
1336:−
1248:−
1160:vanishes.
1145:vanishes.
1041:ϕ
1033:θ
1030:
1008:θ
888:θ
804:ϕ
796:θ
793:
771:θ
523:curvature
237:λ
217:η
198:λ
153:λ
123:geodesics
109:η
1768:See also
958:, where
720:constant
713:Examples
229:, where
1936:Bibcode
1899:Bibcode
1866:1969587
1501:becomes
1165:compact
1962:
1954:
1864:
1163:Every
429:and a
60:metric
37:pseudo
2004:This
1960:S2CID
1926:arXiv
1889:arXiv
1862:JSTOR
51:by a
2010:stub
1952:ISSN
1453:and
1182:The
731:The
519:flat
262:and
1944:doi
1854:doi
1193:In
1148:An
1021:sin
784:sin
689:on
645:on
525:of
517:is
409:of
369:in
43:is
35:A (
2055::
1958:.
1950:.
1942:.
1934:.
1922:61
1920:.
1897:.
1885:39
1883:.
1860:.
1850:50
1848:.
1798:^
1167:,
709:.
618:.
55:.
39:-)
2041:e
2034:t
2027:v
2016:.
1966:.
1946::
1938::
1928::
1905:.
1901::
1891::
1868:.
1856::
1763:.
1749:r
1745:M
1742:G
1739:2
1730:1
1719:,
1705:]
1697:r
1693:M
1690:G
1687:2
1681:+
1678:1
1670:0
1663:0
1656:r
1652:M
1649:G
1646:2
1637:1
1631:[
1626:=
1621:k
1618:i
1614:g
1593:)
1588:2
1584:x
1580:d
1572:2
1568:t
1564:d
1561:(
1557:)
1551:r
1547:M
1544:G
1541:2
1532:1
1528:(
1524:=
1519:2
1515:s
1511:d
1487:2
1483:/
1479:)
1476:u
1470:v
1467:(
1464:=
1461:x
1441:2
1437:/
1433:)
1430:u
1427:+
1424:v
1421:(
1418:=
1415:t
1393:]
1387:0
1380:r
1376:M
1373:G
1370:2
1361:1
1352:r
1348:M
1345:G
1342:2
1333:1
1328:0
1322:[
1317:=
1312:k
1309:i
1305:g
1284:u
1281:d
1277:v
1274:d
1270:)
1264:r
1260:M
1257:G
1254:2
1245:1
1241:(
1237:=
1232:2
1228:s
1224:d
1205:.
1175:.
1154:n
1150:n
1136:.
1124:)
1119:2
1115:y
1111:d
1108:+
1103:2
1099:x
1095:d
1092:(
1084:2
1080:)
1074:2
1070:r
1066:+
1063:1
1060:(
1056:4
1051:=
1045:2
1037:d
1025:2
1017:+
1012:2
1004:d
1001:=
996:2
992:s
988:d
966:r
945:)
940:2
936:r
932:+
929:1
926:(
921:2
894:]
883:2
879:n
875:i
872:s
867:0
860:0
855:1
849:[
844:=
839:k
836:i
832:g
808:2
800:d
788:2
780:+
775:2
767:d
764:=
759:2
755:s
751:d
739:,
697:M
677:x
653:M
633:x
606:M
586:f
566:U
546:g
541:f
538:2
534:e
505:)
502:g
497:f
494:2
490:e
486:,
483:U
480:(
460:U
440:f
417:x
397:U
377:M
357:x
337:)
334:g
331:,
328:M
325:(
305:)
302:g
299:,
296:M
293:(
270:x
246:)
243:x
240:(
213:)
210:x
207:(
202:2
194:=
191:)
188:x
185:(
182:g
162:)
159:x
156:(
133:M
89:M
69:g
20:)
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