71:, the formal notion of a Cauchy surface can be understood in familiar terms. Suppose that humans can travel at a maximum speed of 20 miles per hour. This places constraints, for any given person, upon where they can reach by a certain time. For instance, it is impossible for a person who is in Mexico at 3 o'clock to arrive in Libya by 4 o'clock; however it is
100:
this time; furthermore, no traveler can be at multiple locations at this time. By contrast, there cannot be any Cauchy surface for this causal structure that contains both the pair (Manhattan, 1 o'clock) and (Brooklyn, 2 o'clock) since there are hypothetical travelers that could have been in
Manhattan at 1 o'clock and Brooklyn at 2 o'clock.
1224:. The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve.
288:
define causal structures which are schematically of the above type ("a traveler either can or cannot reach a certain spacetime point from a certain other spacetime point"), with the exception that locations and times are not cleanly separable from one another. Hence one can speak of Cauchy surfaces
1227:
When there are closed timelike curves, or even when there are closed non-spacelike curves, a Cauchy surface still determines the future, but the future includes the surface itself. This means that the initial conditions obey a constraint, and the Cauchy surface is not of the same character as when
1562:
Since a black hole Cauchy horizon only forms in a region where the geodesics are outgoing, in radial coordinates, in a region where the central singularity is repulsive, it is hard to imagine exactly how it forms. For this reason, Kerr and others suggest that a Cauchy horizon never forms, instead
99:
There are a number of uninteresting Cauchy surfaces. For instance, one Cauchy surface for this causal structure is given by considering the pairing of every location with the time of 1 o'clock (on a certain specified day), since any hypothetical traveler must have been at one specific location at
1559:, beyond which information cannot escape, but where the future is still determined from the conditions outside. Inside the inner horizon, the Cauchy horizon, the singularity is visible and to predict the future requires additional data about what comes out of the singularity.
95:
for this causal structure is a collection of pairs of locations and times such that, for any hypothetical traveler whatsoever, there is exactly one location and time pair in the collection for which the traveler was at the indicated location at the indicated time.
75:
for a person who is in
Manhattan at 1 o'clock to reach Brooklyn by 2 o'clock, since the locations are ten miles apart. So as to speak semi-formally, ignore time zones and travel difficulties, and suppose that travelers are immortal beings who have lived forever.
1485:
1339:
186:
641:
103:
There are, also, some more interesting Cauchy surfaces which are harder to describe verbally. One could define a function τ from the collection of all locations into the collection of all times, such that the
1963:
Conference Board of the
Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1972. viii+72 pp.
1529:
650:
makes clear that, even for the "simplest" Lorentzian manifolds, Cauchy surfaces may fail to be differentiable everywhere (in this case, at the origin), and that the homeomorphism
1553:
1367:
1253:
1222:
1198:
1404:
1258:
1174:
1147:
731:
Furthermore, at the cost of not being able to consider arbitrary Cauchy surface, it is always possible to find smooth Cauchy surfaces (Bernal & Sánchez 2003):
114:
1713:
Hamilton, Andrew J.S.; Avelino, Pedro P. (2010), "The physics of the relativistic counter-streaming instability that drives mass inflation inside black holes",
1387:
527:
1564:
32:, a Cauchy surface is usually interpreted as defining an "instant of time". In the mathematics of general relativity, Cauchy surfaces provide
1555:. A clear physical example of a Cauchy horizon is the second horizon inside a charged or rotating black hole. The outermost horizon is an
1176:
are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of
517:. By considering the restriction of the inverse to another Cauchy surface, one sees that any two Cauchy surfaces are homeomorphic.
1991:
1914:
Second edition. Monographs and
Textbooks in Pure and Applied Mathematics, 202. Marcel Dekker, Inc., New York, 1996. xiv+635 pp.
1563:
that the inner horizon is in fact a spacelike or timelike singularity. The inner horizon corresponds to the instability due to
108:
of τ is everywhere less than 1/20 hours per mile. Then another example of a Cauchy surface is given by the collection of pairs
1933:
1975:
1954:
1919:
1942:
Cambridge
Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp.
1822:
Di
Filippo, Francesco; Carballo-Rubio, Raúl; Liberati, Stefano; Pacilio, Costantino; Visser, Matt (28 Mar 2022).
1494:
1996:
204:
17:
1480:{\displaystyle D^{+}({\mathcal {S}})\cup {\mathcal {S}}\cup D^{-}({\mathcal {S}})\not ={\mathcal {M}}}
277:, which must be larger than "20 miles per hour" due to the gradient condition on τ: a contradiction.
1534:
1348:
1234:
1203:
1179:
1774:
1334:{\displaystyle D^{+}({\mathcal {S}})\cup {\mathcal {S}}\cup D^{-}({\mathcal {S}})={\mathcal {M}}}
52:
181:{\displaystyle {\Big \{}{\big (}p,\tau (p){\big )}:p{\text{ a location on Earth}}{\Big \}}.}
1845:
1788:
1732:
1571:
1390:
1152:
1125:
497:
is topologically closed and is an embedded continuous (and even
Lipschitz) submanifold of
8:
1898:
Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes.
1849:
1792:
1736:
1949:
Pure and
Applied Mathematics, 103. Academic Press, Inc. , New York, 1983. xiii+468 pp.
1835:
1756:
1722:
1372:
668:-diffeomorphism. However, the same argument as for a general Cauchy surface shows that
285:
281:
240:
68:
41:
33:
29:
521:
It is hard to say more about the nature of Cauchy surfaces in general. The example of
28:
of a
Lorentzian manifold. In the application of Lorentzian geometry to the physics of
1971:
1950:
1929:
1915:
1804:
1779:
1760:
1748:
1700:
1928:
Oxford
Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp.
1853:
1796:
1740:
1583:
37:
1744:
1057:
by the same criteria, replacing "past-inextensible" with "future-inextensible".
636:{\displaystyle {\Big \{}(t,x,y,z):t^{2}={\frac {x^{2}+y^{2}+z^{2}}{2}}{\Big \}}}
1488:
1114:
is the union of the future Cauchy development and the past Cauchy development.
56:
191:
The point is that, for any hypothetical traveler, there must be some location
1985:
1800:
1752:
1556:
45:
1858:
1823:
1808:
505:. The flow of any continuous timelike vector field defines a homeomorphism
83:"A person in (location 1) at (time 1) can reach (location 2) by (time 2)"
25:
1531:
and regions of the manifold not completely determined by information on
1821:
862:
is either future-directed timelike or future-directed null for each
1840:
1342:
105:
1727:
983:
is any past-inextensible differentiable causal curve such that
292:
1891:
On smooth Cauchy hypersurfaces and Geroch's splitting theorem.
1824:"On the inner horizon instability of non-singular black holes"
79:
The system of all possible ways to fill in the four blanks in
1970:
University of Chicago Press, Chicago, IL, 1984. xiii+491 pp.
207:. Furthermore, it is impossible that there are two locations
805:
be a time-oriented Lorentzian manifold. One says that a map
690:, then the flow of a smooth timelike vector field defines a
1877:
Global aspects of the Cauchy problem in general relativity.
1947:
Semi-Riemannian geometry. With applications to relativity.
751:
which has a Cauchy surface, there exists a Cauchy surface
757:
which is an embedded and spacelike smooth submanifold of
477:
The following is automatically true of a Cauchy surface
434:
if every inextensible differentiable timelike curve in
1537:
1497:
1407:
1375:
1351:
1261:
1237:
1206:
1182:
1155:
1128:
530:
243:
they would at some point have had to travel at speed
117:
1231:If there are no closed timelike curves, then given
1961:Techniques of differential topology in relativity.
1910:Beem, John K.; Ehrlich, Paul E.; Easley, Kevin L.
1570:A homogeneous space-time with a Cauchy horizon is
1547:
1523:
1479:
1381:
1361:
1333:
1247:
1216:
1192:
1168:
1141:
708:, and that any two Cauchy surfaces which are both
635:
180:
628:
533:
170:
120:
1983:
1712:
903:future-inextensible differentiable causal curve
1926:General relativity and the Einstein equations.
1369:is a Cauchy surface. Any surface of constant
313:be a Lorentzian manifold. One says that a map
1893:Comm. Math. Phys. 243 (2003), no. 3, 461–470.
1772:
827:past-inextensible differentiable causal curve
152:
127:
293:Mathematical definition and basic properties
1900:Comm. Math. Phys. 257 (2005), no. 1, 43–50.
450:has exactly one point of intersection with
67:Although it is usually phrased in terms of
55:(1789-1857) due to their relevance for the
1122:When there are no closed timelike curves,
905:by the same criteria, with the phrase "as
335:inextensible differentiable timelike curve
215:and that there is some traveler who is at
1886:J. Mathematical Phys. 11 (1970), 437–449.
1857:
1839:
1726:
1940:The large scale structure of space-time.
1875:Choquet-Bruhat, Yvonne; Geroch, Robert.
1524:{\displaystyle D^{\pm }({\mathcal {S}})}
646:as a Cauchy surface for Minkowski space
62:
51:They are named for French mathematician
44:can be solved (using, for example, the
1984:
1773:Poisson, Eric; Israel, Werner (1990).
1691:. This definition makes sense even if
1228:the future and the past are disjoint.
784:
454:; if there exists such a subset, then
1896:Bernal, Antonio N.; Sánchez, Miguel.
1889:Bernal, Antonio N.; Sánchez, Miguel.
1879:Comm. Math. Phys. 14 (1969), 329–335.
1601:One is requiring that for all points
735:Given any smooth Lorentzian manifold
289:for these causal structures as well.
1613:, there exists an open neighborhood
1088:such that any observer leaving from
951:is defined to consist of all points
1775:"Internal structure of black holes"
1072:such that any observer arriving at
195:which the traveler was at, at time
13:
1540:
1513:
1472:
1459:
1436:
1423:
1354:
1326:
1313:
1290:
1277:
1240:
1209:
1185:
14:
2008:
1396:
1080:; the past Cauchy development of
1064:The future Cauchy development of
1255:a partial Cauchy surface and if
1992:Partial differential equations
1815:
1766:
1706:
1595:
1548:{\displaystyle {\mathcal {S}}}
1518:
1508:
1464:
1454:
1428:
1418:
1362:{\displaystyle {\mathcal {S}}}
1318:
1308:
1282:
1272:
1248:{\displaystyle {\mathcal {S}}}
1217:{\displaystyle {\mathcal {S}}}
1200:are the same and both include
1193:{\displaystyle {\mathcal {S}}}
562:
538:
147:
141:
1:
1938:Hawking, S.W.; Ellis, G.F.R.
1745:10.1016/j.physrep.2010.06.002
1589:
1117:
890:does not approach a limit as
772:is smoothly diffeomorphic to
398:does not approach a limit as
16:In the mathematical field of
1699:only has the structure of a
7:
1912:Global Lorentzian geometry.
1577:
10:
2013:
1092:will have to pass through
205:intermediate value theorem
1076:must have passed through
1013:, then there exists some
935:future Cauchy development
280:The physical theories of
165: a location on Earth
1924:Choquet-Bruhat, Yvonne.
1801:10.1103/PhysRevD.41.1796
203:; this follows from the
87:defines the notion of a
1859:10.3390/universe8040204
1084:consists of all points
1068:consists of all points
1045:past Cauchy development
59:of general relativity.
1549:
1525:
1481:
1383:
1363:
1335:
1249:
1218:
1194:
1170:
1143:
1098:
782:
662:may fail to be even a
637:
519:
182:
85:
1884:Domain of dependence.
1683:are not contained in
1550:
1526:
1482:
1393:is a Cauchy surface.
1384:
1364:
1336:
1250:
1219:
1195:
1171:
1169:{\displaystyle D^{-}}
1144:
1142:{\displaystyle D^{+}}
1062:
733:
638:
483:
370:is timelike for each
183:
81:
63:Informal introduction
53:Augustin-Louis Cauchy
24:is a certain kind of
1997:Lorentzian manifolds
1572:anti-de Sitter space
1535:
1495:
1487:then there exists a
1405:
1391:Minkowski space-time
1373:
1349:
1259:
1235:
1204:
1180:
1153:
1126:
849:it is differentiable
528:
357:it is differentiable
115:
1968:General relativity.
1850:2022Univ....8..204D
1793:1990PhRvD..41.1796P
1737:2010PhR...495....1H
1632:which increases to
785:Cauchy developments
472:globally hyperbolic
34:boundary conditions
18:Lorentzian geometry
1945:O'Neill, Barrett.
1545:
1521:
1477:
1379:
1359:
1331:
1245:
1214:
1190:
1166:
1139:
1102:Cauchy development
1043:. One defines the
921:". Given a subset
913:" replaced by "as
633:
286:general relativity
282:special relativity
241:mean value theorem
178:
69:general relativity
42:Einstein equations
30:general relativity
1934:978-0-19-923072-3
1870:Research articles
1780:Physical Review D
1701:topological space
1382:{\displaystyle t}
714:-submanifolds of
672:a Cauchy surface
624:
166:
2004:
1966:Wald, Robert M.
1959:Penrose, Roger.
1882:Geroch, Robert.
1864:
1863:
1861:
1843:
1819:
1813:
1812:
1787:(6): 1796–1809.
1770:
1764:
1763:
1730:
1710:
1704:
1698:
1697:
1690:
1686:
1682:
1666:
1650:
1646:
1635:
1631:
1620:
1616:
1612:
1611:
1604:
1599:
1584:Causal structure
1554:
1552:
1551:
1546:
1544:
1543:
1530:
1528:
1527:
1522:
1517:
1516:
1507:
1506:
1486:
1484:
1483:
1478:
1476:
1475:
1463:
1462:
1453:
1452:
1440:
1439:
1427:
1426:
1417:
1416:
1388:
1386:
1385:
1380:
1368:
1366:
1365:
1360:
1358:
1357:
1340:
1338:
1337:
1332:
1330:
1329:
1317:
1316:
1307:
1306:
1294:
1293:
1281:
1280:
1271:
1270:
1254:
1252:
1251:
1246:
1244:
1243:
1223:
1221:
1220:
1215:
1213:
1212:
1199:
1197:
1196:
1191:
1189:
1188:
1175:
1173:
1172:
1167:
1165:
1164:
1148:
1146:
1145:
1140:
1138:
1137:
1113:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1056:
1042:
1028:
1016:
1012:
1000:
996:
982:
981:
962:
961:
954:
950:
946:
932:
931:
924:
920:
916:
912:
908:
897:
893:
889:
877:
866:in the interval
865:
861:
844:
842:
836:
824:
823:
804:
802:
796:
779:
778:
771:
764:
763:
756:
750:
748:
742:
728:-diffeomorphic.
727:
721:
720:
713:
707:
706:
696:-diffeomorphism
695:
689:
688:
682:-submanifold of
681:
675:
667:
661:
660:
649:
642:
640:
639:
634:
632:
631:
625:
620:
619:
618:
606:
605:
593:
592:
582:
577:
576:
537:
536:
516:
515:
504:
503:
496:
495:
480:
469:
467:
461:
453:
449:
447:
441:
429:
428:
421:
413:
409:
405:
401:
397:
385:
374:in the interval
373:
369:
352:
350:
344:
332:
331:
312:
310:
304:
276:
275:
273:
272:
261:
258:
238:
230:
226:
218:
214:
210:
202:
194:
187:
185:
184:
179:
174:
173:
167:
164:
156:
155:
131:
130:
124:
123:
89:causal structure
38:causal structure
2012:
2011:
2007:
2006:
2005:
2003:
2002:
2001:
1982:
1981:
1978:; 0-226-87033-2
1867:
1820:
1816:
1771:
1767:
1715:Physics Reports
1711:
1707:
1693:
1692:
1688:
1684:
1680:
1668:
1664:
1652:
1648:
1645:
1637:
1636:and a sequence
1633:
1630:
1622:
1621:and a sequence
1618:
1614:
1607:
1606:
1602:
1600:
1596:
1592:
1580:
1539:
1538:
1536:
1533:
1532:
1512:
1511:
1502:
1498:
1496:
1493:
1492:
1471:
1470:
1458:
1457:
1448:
1444:
1435:
1434:
1422:
1421:
1412:
1408:
1406:
1403:
1402:
1399:
1374:
1371:
1370:
1353:
1352:
1350:
1347:
1346:
1325:
1324:
1312:
1311:
1302:
1298:
1289:
1288:
1276:
1275:
1266:
1262:
1260:
1257:
1256:
1239:
1238:
1236:
1233:
1232:
1208:
1207:
1205:
1202:
1201:
1184:
1183:
1181:
1178:
1177:
1160:
1156:
1154:
1151:
1150:
1133:
1129:
1127:
1124:
1123:
1120:
1104:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1047:
1030:
1018:
1014:
1002:
998:
984:
977:
964:
957:
956:
952:
948:
937:
927:
926:
922:
918:
914:
910:
906:
895:
891:
880:
867:
863:
852:
838:
832:
830:
819:
806:
798:
792:
790:
787:
774:
773:
766:
759:
758:
752:
744:
738:
736:
723:
716:
715:
709:
702:
697:
691:
684:
683:
677:
673:
663:
656:
651:
647:
627:
626:
614:
610:
601:
597:
588:
584:
583:
581:
572:
568:
532:
531:
529:
526:
525:
511:
506:
499:
498:
491:
486:
478:
463:
457:
455:
451:
443:
437:
435:
424:
423:
419:
411:
407:
403:
399:
388:
375:
371:
360:
346:
340:
338:
327:
314:
306:
300:
298:
295:
262:
259:
248:
247:
245:
244:
239:, since by the
232:
228:
220:
216:
212:
208:
196:
192:
169:
168:
163:
151:
150:
126:
125:
119:
118:
116:
113:
112:
65:
12:
11:
5:
2010:
2000:
1999:
1994:
1980:
1979:
1964:
1957:
1943:
1936:
1922:
1902:
1901:
1894:
1887:
1880:
1866:
1865:
1814:
1765:
1705:
1676:
1660:
1647:decreasing to
1641:
1626:
1593:
1591:
1588:
1587:
1586:
1579:
1576:
1565:mass inflation
1542:
1520:
1515:
1510:
1505:
1501:
1489:Cauchy horizon
1474:
1469:
1466:
1461:
1456:
1451:
1447:
1443:
1438:
1433:
1430:
1425:
1420:
1415:
1411:
1398:
1397:Cauchy horizon
1395:
1378:
1356:
1328:
1323:
1320:
1315:
1310:
1305:
1301:
1297:
1292:
1287:
1284:
1279:
1274:
1269:
1265:
1242:
1211:
1187:
1163:
1159:
1136:
1132:
1119:
1116:
901:One defines a
899:
898:
878:
850:
786:
783:
765:and such that
644:
643:
630:
623:
617:
613:
609:
604:
600:
596:
591:
587:
580:
575:
571:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
535:
432:Cauchy surface
416:
415:
386:
358:
294:
291:
189:
188:
177:
172:
162:
159:
154:
149:
146:
143:
140:
137:
134:
129:
122:
93:Cauchy surface
64:
61:
57:Cauchy problem
22:Cauchy surface
9:
6:
4:
3:
2:
2009:
1998:
1995:
1993:
1990:
1989:
1987:
1977:
1976:0-226-87032-4
1973:
1969:
1965:
1962:
1958:
1956:
1955:0-12-526740-1
1952:
1948:
1944:
1941:
1937:
1935:
1931:
1927:
1923:
1921:
1920:0-8247-9324-2
1917:
1913:
1909:
1908:
1907:
1906:
1899:
1895:
1892:
1888:
1885:
1881:
1878:
1874:
1873:
1872:
1871:
1860:
1855:
1851:
1847:
1842:
1837:
1833:
1829:
1825:
1818:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1781:
1776:
1769:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1729:
1724:
1720:
1716:
1709:
1702:
1696:
1679:
1675:
1671:
1663:
1659:
1655:
1644:
1640:
1629:
1625:
1610:
1598:
1594:
1585:
1582:
1581:
1575:
1573:
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847:
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781:
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726:
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518:
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475:
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440:
433:
427:
410:decreases to
402:increases to
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356:
355:
354:
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278:
270:
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252:
242:
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101:
97:
94:
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49:
47:
46:ADM formalism
43:
40:in which the
39:
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1911:
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512:
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492:
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431:
430:is called a
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485:The subset
26:submanifold
1986:Categories
1841:2203.14516
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1590:References
1118:Discussion
470:is called
1905:Textbooks
1761:118546967
1753:0370-1573
1728:0811.1926
1504:±
1450:−
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1162:−
997:for some
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810: : (
418:A subset
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1687:for any
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1491:between
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231:at time
219:at time
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73:possible
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1789:Bibcode
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274:
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