714:
44:
727:
2405:
2174:
115:
1187:
1429:
1984:
2034:
1460:
1936:
1338:
1238:
1600:
1523:
1662:, the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis. The
1564:
1034:
485:
1483:
1278:
1258:
1121:
1082:
1054:
1000:
980:
960:
758:
1845:
After
Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications of
2051:
137:
3259:
1659:
2450:
3314:
50:
3254:
342:
3477:
3327:
2541:
2180:
695:
2565:
505:
347:
1864:. Unlike Riemannian manifolds with positive-definite metrics, an indefinite signature allows tangent vectors to be classified into
2760:
2237:
751:
460:
2191:. On the other hand, there are many theorems in Riemannian geometry that do not hold in the generalized case. For example, it is
3507:
3482:
1760:. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.
373:
207:
3369:
2630:
2409:
2359:
412:
1129:
2856:
2214:
provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the
17:
2203:
does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any
2909:
2437:
744:
3193:
124:
1990:
is the local model of a
Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (
3342:
3307:
2379:
2341:
2316:
150:
2958:
713:
430:
43:
2941:
2550:
800:
337:
1344:
3590:
3605:
3595:
3153:
2560:
811:
424:
132:
470:
3600:
3404:
3300:
3138:
2861:
2635:
2232:
792:
490:
2195:
true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain
2179:
Some theorems of
Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the
1948:
3183:
731:
222:
142:
2004:
3188:
3158:
2866:
2822:
2570:
2514:
1437:
635:
1912:
1683:
of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has
3610:
3585:
3554:
2725:
2590:
1286:
1195:
435:
247:
3487:
3347:
3110:
2975:
2667:
2509:
2188:
670:
660:
510:
327:
2204:
3533:
2807:
2777:
2701:
2691:
2647:
2477:
2430:
1740:
1569:
1492:
1103:
to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by
914:
861:
851:
784:
655:
389:
3564:
3497:
3492:
3430:
3389:
3148:
2767:
2662:
2575:
2482:
2215:
2211:
2184:
1999:
625:
610:
455:
217:
3559:
2369:
1528:
2797:
2792:
857:
600:
168:
1009:
3394:
3128:
3066:
2914:
2618:
2608:
2580:
2555:
2465:
2227:
772:
420:
891:-dimensional Euclidean space. In a manifold it may only be possible to define coordinates
8:
3451:
3420:
3408:
3379:
3362:
3323:
3266:
2948:
2826:
2811:
2740:
2499:
2242:
1939:
796:
615:
585:
580:
332:
3239:
595:
565:
3549:
3446:
3415:
3208:
3163:
3060:
2931:
2735:
2423:
1846:
1468:
1263:
1243:
1106:
1067:
1039:
985:
965:
945:
818:
665:
550:
475:
322:
227:
212:
173:
35:
3456:
2745:
1783:
are non-negative. The non-degeneracy condition together with continuity implies that
3461:
3143:
3123:
3118:
3025:
2936:
2750:
2730:
2585:
2524:
2375:
2355:
2337:
2330:
2325:
2312:
1061:
495:
270:
545:
3528:
3523:
3399:
3352:
3281:
3075:
3030:
2953:
2924:
2782:
2715:
2710:
2705:
2695:
2487:
2470:
1987:
1894:
1804:
1664:
1651:
1611:
920:
896:
830:
620:
575:
555:
500:
640:
3357:
3292:
3224:
3133:
2963:
2919:
2685:
2169:{\displaystyle g=dx_{1}^{2}+\cdots +dx_{p}^{2}-dx_{p+1}^{2}-\cdots -dx_{p+q}^{2}}
1943:
1907:
1834:
1825:
865:
675:
650:
535:
530:
394:
275:
237:
445:
3090:
3015:
2985:
2883:
2876:
2816:
2787:
2657:
2652:
2613:
1625:
1486:
1092:
718:
685:
680:
368:
232:
3579:
3384:
3276:
3100:
3095:
3080:
3070:
3020:
2997:
2871:
2831:
2772:
2720:
2519:
1747:
1088:
1003:
937:
933:
807:
788:
645:
560:
540:
465:
363:
242:
2183:
is true of all pseudo-Riemannian manifolds. This allows one to speak of the
1892:, the manifold is also locally (and possibly globally) time-orientable (see
605:
3203:
3198:
3040:
3007:
2980:
2888:
2529:
1096:
1057:
590:
570:
1803:
is an important special case of a pseudo-Riemannian manifold in which the
3046:
3035:
2992:
2893:
2494:
2200:
1100:
480:
450:
3502:
3271:
3229:
3055:
2968:
2600:
2504:
2415:
877:
690:
252:
178:
110:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
1746:
that is equipped with an everywhere non-degenerate, smooth, symmetric
3085:
3050:
2755:
2642:
2247:
1853:
1791:
remain unchanged throughout the manifold (assuming it is connected).
826:
261:
2309:
An introduction to
Spinors and Geometry with Applications in Physics
3374:
3249:
3244:
3234:
2625:
2446:
2196:
1856:
can be modeled as a 4-dimensional
Lorentzian manifold of signature
908:
847:
630:
440:
280:
1650:
associated with the metric tensor applied to each vector of any
2841:
2404:
2374:, Pure and Applied Mathematics, vol. 103, Academic Press,
2207:
2272:
1901:
887:-dimensional differentiable manifold is a generalisation of
2392:, Bucarest: Editura Academiei Republicii Socialiste RomĂąnia
872:-dimensional Euclidean space any point can be specified by
2187:
on a pseudo-Riemannian manifold along with the associated
2390:
Introduction to
Relativity and Pseudo-Riemannian Geometry
2284:
2371:
Semi-Riemannian
Geometry With Applications to Relativity
2352:
Pseudo-Riemannian
Geometry, -invariants and Applications
2260:
29:
Differentiable manifold with nondegenerate metric tensor
1182:{\displaystyle g:T_{p}M\times T_{p}M\to \mathbb {R} .}
2054:
2007:
1951:
1915:
1572:
1531:
1495:
1471:
1440:
1347:
1289:
1266:
1246:
1198:
1132:
1109:
1070:
1042:
1012:
988:
968:
948:
53:
2336:(First Dover 1980 ed.), The Macmillan Company,
927:
3322:
2329:
2168:
2028:
1978:
1930:
1852:A principal premise of general relativity is that
1594:
1558:
1517:
1477:
1454:
1423:
1332:
1272:
1252:
1232:
1181:
1115:
1076:
1048:
1028:
994:
974:
954:
899:: subsets of the manifold that can be mapped into
109:
3577:
1763:The signature of a pseudo-Riemannian metric is
2387:
2324:
2311:(First published 1987 ed.), Adam Hilger,
2278:
3308:
2431:
829:, where tangent vectors can be classified as
752:
2367:
2290:
1833:. They are named after the Dutch physicist
3478:Fundamental theorem of Riemannian geometry
3315:
3301:
2438:
2424:
2306:
2266:
2181:fundamental theorem of Riemannian geometry
1938:can be thought of as the local model of a
1840:
1424:{\displaystyle \,g(aX+Y,Z)=ag(X,Z)+g(Y,Z)}
759:
745:
2010:
1954:
1918:
1902:Properties of pseudo-Riemannian manifolds
1448:
1348:
1290:
1172:
2445:
2238:Hyperbolic partial differential equation
1192:The map is symmetric and bilinear so if
864:is a space that is locally similar to a
14:
3578:
1794:
3296:
2419:
982:-dimensional differentiable manifold
810:of a pseudo-Riemannian manifold is a
2349:
2218:disallows for Riemannian manifolds.
2036:, for which there exist coordinates
1979:{\displaystyle \mathbb {R} ^{n-1,1}}
1605:
1064:of curves passing through the point
1060:whose elements can be thought of as
876:real numbers. These are called the
24:
2029:{\displaystyle \mathbb {R} ^{p,q}}
70:
25:
3622:
2397:
2388:VrÄnceanu, G.; RoĆca, R. (1976),
2307:Benn, I.M.; Tucker, R.W. (1987),
1690:and the signature may be denoted
1455:{\displaystyle a\in \mathbb {R} }
928:Tangent spaces and metric tensors
795:. This is a generalization of a
2403:
1931:{\displaystyle \mathbb {R} ^{n}}
1624:-dimensional real manifold, the
726:
725:
712:
42:
1333:{\displaystyle \,g(X,Y)=g(Y,X)}
1240:are tangent vectors at a point
1233:{\displaystyle X,Y,Z\in T_{p}M}
895:. This is achieved by defining
836:
2478:Differentiable/Smooth manifold
2354:, World Scientific Publisher,
2328:; Goldberg, Samuel I. (1968),
1547:
1535:
1418:
1406:
1397:
1385:
1373:
1352:
1327:
1315:
1306:
1294:
1168:
903:-dimensional Euclidean space.
13:
1:
2300:
2199:obstructions. Furthermore, a
1719:
831:timelike, null, and spacelike
812:pseudo-Euclidean vector space
3405:Raising and lowering indices
2332:Tensor Analysis on Manifolds
2233:Globally hyperbolic manifold
841:
799:in which the requirement of
7:
3184:Classification of manifolds
2221:
1829:). Such metrics are called
1595:{\displaystyle Y\in T_{p}M}
1518:{\displaystyle X\in T_{p}M}
1489:means there is no non-zero
942:Associated with each point
223:Gravitational time dilation
10:
3627:
3426:Pseudo-Riemannian manifold
2279:Bishop & Goldberg 1968
1756:Such a metric is called a
1726:pseudo-Riemannian manifold
1660:Sylvester's law of inertia
1609:
931:
845:
777:pseudo-Riemannian manifold
343:MathissonâPapapetrouâDixon
184:Pseudo-Riemannian manifold
3555:Geometrization conjecture
3542:
3516:
3470:
3439:
3335:
3260:over commutative algebras
3217:
3176:
3109:
3006:
2902:
2849:
2840:
2676:
2599:
2538:
2458:
2368:O'Neill, Barrett (1983),
2976:Riemann curvature tensor
2253:
1758:pseudo-Riemannian metric
1559:{\displaystyle g(X,Y)=0}
1123:we can express this as
781:semi-Riemannian manifold
348:HamiltonâJacobiâEinstein
328:Einstein field equations
151:Mathematical formulation
2350:Chen, Bang-Yen (2011),
1876:. With a signature of
1841:Applications in physics
1805:signature of the metric
1741:differentiable manifold
915:Differentiable manifold
862:differentiable manifold
852:Differentiable manifold
817:A special case used in
785:differentiable manifold
3565:Uniformization theorem
3498:Nash embedding theorem
3431:Riemannian volume form
3390:Levi-Civita connection
2768:Manifold with boundary
2483:Differential structure
2267:Benn & Tucker 1987
2185:Levi-Civita connection
2170:
2030:
2000:pseudo-Euclidean space
1980:
1932:
1616:Given a metric tensor
1596:
1560:
1519:
1479:
1456:
1425:
1334:
1274:
1254:
1234:
1183:
1117:
1078:
1050:
1030:
1029:{\displaystyle T_{p}M}
996:
976:
956:
821:is a four-dimensional
218:Gravitational redshift
111:
3591:Differential geometry
2171:
2031:
1981:
1933:
1597:
1561:
1520:
1480:
1457:
1426:
1335:
1275:
1255:
1235:
1184:
1118:
1095:, smooth, symmetric,
1079:
1051:
1031:
997:
977:
957:
858:differential geometry
801:positive-definiteness
506:WeylâLewisâPapapetrou
461:KerrâNewmanâde Sitter
281:EinsteinâRosen bridge
213:Gravitational lensing
169:Equivalence principle
112:
3606:Riemannian manifolds
3596:Lorentzian manifolds
3488:GaussâBonnet theorem
3395:Covariant derivative
2915:Covariant derivative
2466:Topological manifold
2412:at Wikimedia Commons
2410:Lorentzian manifolds
2228:Causality conditions
2052:
2005:
1949:
1913:
1570:
1529:
1493:
1469:
1438:
1434:for any real number
1345:
1287:
1264:
1244:
1196:
1130:
1107:
1068:
1040:
1010:
986:
966:
946:
773:mathematical physics
436:EinsteinâRosen waves
162:Fundamental concepts
51:
18:Lorentzian manifolds
3601:Riemannian geometry
3560:Poincaré conjecture
3421:Riemannian manifold
3409:Musical isomorphism
3324:Riemannian geometry
2949:Exterior derivative
2551:AtiyahâSinger index
2500:Riemannian manifold
2243:Orientable manifold
2165:
2132:
2105:
2078:
1940:Riemannian manifold
1801:Lorentzian manifold
1795:Lorentzian manifold
1062:equivalence classes
823:Lorentzian manifold
797:Riemannian manifold
791:that is everywhere
390:KaluzaâKlein theory
276:Minkowski spacetime
228:Gravitational waves
3550:General relativity
3493:HopfâRinow theorem
3440:Types of manifolds
3416:Parallel transport
3255:Secondary calculus
3209:Singularity theory
3164:Parallel transport
2932:De Rham cohomology
2571:Generalized Stokes
2326:Bishop, Richard L.
2216:HopfâRinow theorem
2212:CliftonâPohl torus
2166:
2145:
2112:
2091:
2064:
2026:
1976:
1928:
1860:or, equivalently,
1847:general relativity
1831:Lorentzian metrics
1592:
1556:
1515:
1475:
1452:
1421:
1330:
1270:
1250:
1230:
1179:
1113:
1074:
1046:
1026:
992:
972:
952:
924:for more details.
897:coordinate patches
819:general relativity
719:Physics portal
491:OppenheimerâSnyder
431:ReissnerâNordström
323:Linearized gravity
271:Spacetime diagrams
174:Special relativity
107:
36:General relativity
3573:
3572:
3290:
3289:
3172:
3171:
2937:Differential form
2591:Whitney embedding
2525:Differential form
2408:Media related to
2361:978-981-4329-63-7
1606:Metric signatures
1478:{\displaystyle g}
1273:{\displaystyle M}
1253:{\displaystyle p}
1116:{\displaystyle g}
1077:{\displaystyle p}
1049:{\displaystyle n}
995:{\displaystyle M}
975:{\displaystyle n}
955:{\displaystyle p}
769:
768:
402:
401:
288:
287:
16:(Redirected from
3618:
3611:Smooth manifolds
3586:Bernhard Riemann
3317:
3310:
3303:
3294:
3293:
3282:Stratified space
3240:Fréchet manifold
2954:Interior product
2847:
2846:
2544:
2440:
2433:
2426:
2417:
2416:
2407:
2393:
2384:
2364:
2346:
2335:
2321:
2294:
2288:
2282:
2276:
2270:
2264:
2189:curvature tensor
2175:
2173:
2172:
2167:
2164:
2159:
2131:
2126:
2104:
2099:
2077:
2072:
2035:
2033:
2032:
2027:
2025:
2024:
2013:
1988:Minkowski metric
1985:
1983:
1982:
1977:
1975:
1974:
1957:
1937:
1935:
1934:
1929:
1927:
1926:
1921:
1895:Causal structure
1891:
1883:
1863:
1859:
1822:
1814:
1774:
1738:
1715:
1701:
1689:
1682:
1658:real values. By
1652:orthogonal basis
1649:
1612:Metric signature
1601:
1599:
1598:
1593:
1588:
1587:
1565:
1563:
1562:
1557:
1524:
1522:
1521:
1516:
1511:
1510:
1484:
1482:
1481:
1476:
1461:
1459:
1458:
1453:
1451:
1430:
1428:
1427:
1422:
1339:
1337:
1336:
1331:
1279:
1277:
1276:
1271:
1260:to the manifold
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1226:
1225:
1188:
1186:
1185:
1180:
1175:
1164:
1163:
1148:
1147:
1122:
1120:
1119:
1114:
1083:
1081:
1080:
1075:
1055:
1053:
1052:
1047:
1035:
1033:
1032:
1027:
1022:
1021:
1001:
999:
998:
993:
981:
979:
978:
973:
961:
959:
958:
953:
921:Coordinate patch
779:, also called a
761:
754:
747:
734:
729:
728:
721:
717:
716:
501:van Stockum dust
486:RobertsonâWalker
312:
311:
202:
201:
116:
114:
113:
108:
106:
105:
93:
85:
84:
66:
65:
46:
32:
31:
21:
3626:
3625:
3621:
3620:
3619:
3617:
3616:
3615:
3576:
3575:
3574:
3569:
3538:
3517:Generalizations
3512:
3466:
3435:
3370:Exponential map
3331:
3321:
3291:
3286:
3225:Banach manifold
3218:Generalizations
3213:
3168:
3105:
3002:
2964:Ricci curvature
2920:Cotangent space
2898:
2836:
2678:
2672:
2631:Exponential map
2595:
2540:
2534:
2454:
2444:
2400:
2382:
2362:
2344:
2319:
2303:
2298:
2297:
2289:
2285:
2277:
2273:
2265:
2261:
2256:
2224:
2160:
2149:
2127:
2116:
2100:
2095:
2073:
2068:
2053:
2050:
2049:
2044:
2014:
2009:
2008:
2006:
2003:
2002:
1958:
1953:
1952:
1950:
1947:
1946:
1944:Minkowski space
1922:
1917:
1916:
1914:
1911:
1910:
1908:Euclidean space
1904:
1885:
1877:
1861:
1857:
1843:
1835:Hendrik Lorentz
1826:Sign convention
1816:
1815:(equivalently,
1808:
1797:
1764:
1728:
1722:
1703:
1691:
1684:
1668:
1628:
1614:
1608:
1583:
1579:
1571:
1568:
1567:
1530:
1527:
1526:
1506:
1502:
1494:
1491:
1490:
1470:
1467:
1466:
1447:
1439:
1436:
1435:
1346:
1343:
1342:
1288:
1285:
1284:
1265:
1262:
1261:
1245:
1242:
1241:
1221:
1217:
1197:
1194:
1193:
1171:
1159:
1155:
1143:
1139:
1131:
1128:
1127:
1108:
1105:
1104:
1099:that assigns a
1069:
1066:
1065:
1041:
1038:
1037:
1017:
1013:
1011:
1008:
1007:
987:
984:
983:
967:
964:
963:
947:
944:
943:
940:
932:Main articles:
930:
866:Euclidean space
854:
846:Main articles:
844:
839:
765:
724:
711:
710:
703:
702:
526:
525:
516:
515:
471:LemaĂźtreâTolman
416:
415:
404:
403:
395:Quantum gravity
382:Advanced theory
309:
308:
307:
290:
289:
238:Geodetic effect
199:
198:
189:
188:
164:
163:
147:
117:
98:
94:
89:
77:
73:
58:
54:
52:
49:
48:
30:
23:
22:
15:
12:
11:
5:
3624:
3614:
3613:
3608:
3603:
3598:
3593:
3588:
3571:
3570:
3568:
3567:
3562:
3557:
3552:
3546:
3544:
3540:
3539:
3537:
3536:
3534:Sub-Riemannian
3531:
3526:
3520:
3518:
3514:
3513:
3511:
3510:
3505:
3500:
3495:
3490:
3485:
3480:
3474:
3472:
3468:
3467:
3465:
3464:
3459:
3454:
3449:
3443:
3441:
3437:
3436:
3434:
3433:
3428:
3423:
3418:
3413:
3412:
3411:
3402:
3397:
3392:
3382:
3377:
3372:
3367:
3366:
3365:
3360:
3355:
3350:
3339:
3337:
3336:Basic concepts
3333:
3332:
3320:
3319:
3312:
3305:
3297:
3288:
3287:
3285:
3284:
3279:
3274:
3269:
3264:
3263:
3262:
3252:
3247:
3242:
3237:
3232:
3227:
3221:
3219:
3215:
3214:
3212:
3211:
3206:
3201:
3196:
3191:
3186:
3180:
3178:
3174:
3173:
3170:
3169:
3167:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3115:
3113:
3107:
3106:
3104:
3103:
3098:
3093:
3088:
3083:
3078:
3073:
3063:
3058:
3053:
3043:
3038:
3033:
3028:
3023:
3018:
3012:
3010:
3004:
3003:
3001:
3000:
2995:
2990:
2989:
2988:
2978:
2973:
2972:
2971:
2961:
2956:
2951:
2946:
2945:
2944:
2934:
2929:
2928:
2927:
2917:
2912:
2906:
2904:
2900:
2899:
2897:
2896:
2891:
2886:
2881:
2880:
2879:
2869:
2864:
2859:
2853:
2851:
2844:
2838:
2837:
2835:
2834:
2829:
2819:
2814:
2800:
2795:
2790:
2785:
2780:
2778:Parallelizable
2775:
2770:
2765:
2764:
2763:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2698:
2688:
2682:
2680:
2674:
2673:
2671:
2670:
2665:
2660:
2658:Lie derivative
2655:
2653:Integral curve
2650:
2645:
2640:
2639:
2638:
2628:
2623:
2622:
2621:
2614:Diffeomorphism
2611:
2605:
2603:
2597:
2596:
2594:
2593:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2553:
2547:
2545:
2536:
2535:
2533:
2532:
2527:
2522:
2517:
2512:
2507:
2502:
2497:
2492:
2491:
2490:
2485:
2475:
2474:
2473:
2462:
2460:
2459:Basic concepts
2456:
2455:
2443:
2442:
2435:
2428:
2420:
2414:
2413:
2399:
2398:External links
2396:
2395:
2394:
2385:
2380:
2365:
2360:
2347:
2342:
2322:
2317:
2302:
2299:
2296:
2295:
2283:
2271:
2258:
2257:
2255:
2252:
2251:
2250:
2245:
2240:
2235:
2230:
2223:
2220:
2177:
2176:
2163:
2158:
2155:
2152:
2148:
2144:
2141:
2138:
2135:
2130:
2125:
2122:
2119:
2115:
2111:
2108:
2103:
2098:
2094:
2090:
2087:
2084:
2081:
2076:
2071:
2067:
2063:
2060:
2057:
2040:
2023:
2020:
2017:
2012:
1986:with the flat
1973:
1970:
1967:
1964:
1961:
1956:
1925:
1920:
1903:
1900:
1842:
1839:
1796:
1793:
1721:
1718:
1626:quadratic form
1610:Main article:
1607:
1604:
1591:
1586:
1582:
1578:
1575:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1514:
1509:
1505:
1501:
1498:
1487:non-degenerate
1474:
1450:
1446:
1443:
1432:
1431:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1340:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1269:
1249:
1229:
1224:
1220:
1216:
1213:
1210:
1207:
1204:
1201:
1190:
1189:
1178:
1174:
1170:
1167:
1162:
1158:
1154:
1151:
1146:
1142:
1138:
1135:
1112:
1093:non-degenerate
1073:
1045:
1036:). This is an
1025:
1020:
1016:
991:
971:
951:
929:
926:
880:of the point.
843:
840:
838:
835:
767:
766:
764:
763:
756:
749:
741:
738:
737:
736:
735:
722:
705:
704:
701:
700:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
527:
523:
522:
521:
518:
517:
514:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
417:
411:
410:
409:
406:
405:
400:
399:
398:
397:
392:
384:
383:
379:
378:
377:
376:
374:Post-Newtonian
371:
366:
358:
357:
353:
352:
351:
350:
345:
340:
335:
330:
325:
317:
316:
310:
306:
305:
302:
298:
297:
296:
295:
292:
291:
286:
285:
284:
283:
278:
273:
265:
264:
258:
257:
256:
255:
250:
245:
240:
235:
233:Frame-dragging
230:
225:
220:
215:
210:
208:Kepler problem
200:
196:
195:
194:
191:
190:
187:
186:
181:
176:
171:
165:
161:
160:
159:
156:
155:
154:
153:
148:
146:
145:
140:
135:
129:
127:
119:
118:
104:
101:
97:
92:
88:
83:
80:
76:
72:
69:
64:
61:
57:
47:
39:
38:
28:
9:
6:
4:
3:
2:
3623:
3612:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3583:
3581:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3545:
3541:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3519:
3515:
3509:
3508:Schur's lemma
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3483:Gauss's lemma
3481:
3479:
3476:
3475:
3473:
3469:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3444:
3442:
3438:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3410:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3386:
3385:Metric tensor
3383:
3381:
3380:Inner product
3378:
3376:
3373:
3371:
3368:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3345:
3344:
3341:
3340:
3338:
3334:
3329:
3325:
3318:
3313:
3311:
3306:
3304:
3299:
3298:
3295:
3283:
3280:
3278:
3277:Supermanifold
3275:
3273:
3270:
3268:
3265:
3261:
3258:
3257:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3222:
3220:
3216:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3181:
3179:
3175:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3116:
3114:
3112:
3108:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3068:
3064:
3062:
3059:
3057:
3054:
3052:
3048:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3013:
3011:
3009:
3005:
2999:
2998:Wedge product
2996:
2994:
2991:
2987:
2984:
2983:
2982:
2979:
2977:
2974:
2970:
2967:
2966:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2943:
2942:Vector-valued
2940:
2939:
2938:
2935:
2933:
2930:
2926:
2923:
2922:
2921:
2918:
2916:
2913:
2911:
2908:
2907:
2905:
2901:
2895:
2892:
2890:
2887:
2885:
2882:
2878:
2875:
2874:
2873:
2872:Tangent space
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2854:
2852:
2848:
2845:
2843:
2839:
2833:
2830:
2828:
2824:
2820:
2818:
2815:
2813:
2809:
2805:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2762:
2759:
2758:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2703:
2699:
2697:
2693:
2689:
2687:
2684:
2683:
2681:
2675:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2637:
2636:in Lie theory
2634:
2633:
2632:
2629:
2627:
2624:
2620:
2617:
2616:
2615:
2612:
2610:
2607:
2606:
2604:
2602:
2598:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2548:
2546:
2543:
2539:Main results
2537:
2531:
2528:
2526:
2523:
2521:
2520:Tangent space
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2489:
2486:
2484:
2481:
2480:
2479:
2476:
2472:
2469:
2468:
2467:
2464:
2463:
2461:
2457:
2452:
2448:
2441:
2436:
2434:
2429:
2427:
2422:
2421:
2418:
2411:
2406:
2402:
2401:
2391:
2386:
2383:
2381:9780080570570
2377:
2373:
2372:
2366:
2363:
2357:
2353:
2348:
2345:
2343:0-486-64039-6
2339:
2334:
2333:
2327:
2323:
2320:
2318:0-85274-169-3
2314:
2310:
2305:
2304:
2293:, p. 193
2292:
2287:
2281:, p. 208
2280:
2275:
2269:, p. 172
2268:
2263:
2259:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2225:
2219:
2217:
2213:
2209:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2161:
2156:
2153:
2150:
2146:
2142:
2139:
2136:
2133:
2128:
2123:
2120:
2117:
2113:
2109:
2106:
2101:
2096:
2092:
2088:
2085:
2082:
2079:
2074:
2069:
2065:
2061:
2058:
2055:
2048:
2047:
2046:
2043:
2039:
2021:
2018:
2015:
2001:
1997:
1993:
1989:
1971:
1968:
1965:
1962:
1959:
1945:
1941:
1923:
1909:
1899:
1897:
1896:
1889:
1881:
1875:
1871:
1867:
1855:
1850:
1848:
1838:
1836:
1832:
1828:
1827:
1820:
1812:
1806:
1802:
1792:
1790:
1786:
1782:
1778:
1775:, where both
1772:
1768:
1761:
1759:
1754:
1752:
1749:
1748:metric tensor
1745:
1742:
1736:
1732:
1727:
1717:
1714:
1710:
1706:
1699:
1695:
1687:
1680:
1676:
1672:
1667:
1666:
1661:
1657:
1653:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1613:
1603:
1589:
1584:
1580:
1576:
1573:
1553:
1550:
1544:
1541:
1538:
1532:
1512:
1507:
1503:
1499:
1496:
1488:
1472:
1463:
1444:
1441:
1415:
1412:
1409:
1403:
1400:
1394:
1391:
1388:
1382:
1379:
1376:
1370:
1367:
1364:
1361:
1358:
1355:
1349:
1341:
1324:
1321:
1318:
1312:
1309:
1303:
1300:
1297:
1291:
1283:
1282:
1281:
1280:then we have
1267:
1247:
1227:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1199:
1176:
1165:
1160:
1156:
1152:
1149:
1144:
1140:
1136:
1133:
1126:
1125:
1124:
1110:
1102:
1098:
1094:
1090:
1089:metric tensor
1085:
1071:
1063:
1059:
1056:-dimensional
1043:
1023:
1018:
1014:
1005:
1004:tangent space
989:
969:
949:
939:
938:Metric tensor
935:
934:Tangent space
925:
923:
922:
917:
916:
911:
910:
904:
902:
898:
894:
890:
886:
881:
879:
875:
871:
867:
863:
859:
853:
849:
834:
832:
828:
825:for modeling
824:
820:
815:
813:
809:
808:tangent space
804:
803:is relaxed.
802:
798:
794:
793:nondegenerate
790:
789:metric tensor
786:
782:
778:
774:
762:
757:
755:
750:
748:
743:
742:
740:
739:
733:
723:
720:
715:
709:
708:
707:
706:
699:
698:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
636:Chandrasekhar
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
551:Schwarzschild
549:
547:
544:
542:
539:
537:
534:
532:
529:
528:
520:
519:
512:
511:HartleâThorne
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
426:
422:
421:Schwarzschild
419:
418:
414:
408:
407:
396:
393:
391:
388:
387:
386:
385:
381:
380:
375:
372:
370:
367:
365:
362:
361:
360:
359:
355:
354:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
320:
319:
318:
314:
313:
303:
300:
299:
294:
293:
282:
279:
277:
274:
272:
269:
268:
267:
266:
263:
260:
259:
254:
251:
249:
246:
244:
243:Event horizon
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
205:
204:
203:
193:
192:
185:
182:
180:
177:
175:
172:
170:
167:
166:
158:
157:
152:
149:
144:
141:
139:
136:
134:
131:
130:
128:
126:
123:
122:
121:
120:
102:
99:
95:
90:
86:
81:
78:
74:
67:
62:
59:
55:
45:
41:
40:
37:
34:
33:
27:
19:
3543:Applications
3471:Main results
3425:
3204:Moving frame
3199:Morse theory
3189:Gauge theory
2981:Tensor field
2910:Closed/Exact
2889:Vector field
2857:Distribution
2803:
2798:Hypercomplex
2793:Quaternionic
2530:Vector field
2488:Smooth atlas
2389:
2370:
2351:
2331:
2308:
2291:O'Neill 1983
2286:
2274:
2262:
2192:
2178:
2041:
2037:
1995:
1991:
1905:
1893:
1887:
1879:
1873:
1869:
1865:
1851:
1844:
1830:
1824:
1818:
1810:
1800:
1798:
1788:
1784:
1780:
1776:
1770:
1766:
1762:
1757:
1755:
1750:
1743:
1734:
1730:
1725:
1723:
1712:
1708:
1704:
1697:
1693:
1685:
1678:
1674:
1670:
1663:
1655:
1645:
1641:
1637:
1633:
1629:
1621:
1617:
1615:
1464:
1433:
1191:
1097:bilinear map
1086:
1058:vector space
941:
919:
913:
907:
905:
900:
892:
888:
884:
882:
873:
869:
855:
837:Introduction
822:
816:
805:
780:
776:
770:
696:
656:Raychaudhuri
183:
125:Introduction
26:
3149:Levi-Civita
3139:Generalized
3111:Connections
3061:Lie algebra
2993:Volume form
2894:Vector flow
2867:Pushforward
2862:Lie bracket
2761:Lie algebra
2726:G-structure
2515:Pushforward
2495:Submanifold
2201:submanifold
2197:topological
1101:real number
878:coordinates
671:van Stockum
601:Oppenheimer
456:KerrâNewman
248:Singularity
3580:Categories
3503:Ricci flow
3452:Hyperbolic
3272:Stratifold
3230:Diffeology
3026:Associated
2827:Symplectic
2812:Riemannian
2741:Hyperbolic
2668:Submersion
2576:HopfâRinow
2510:Submersion
2505:Smooth map
2301:references
2205:light-like
2045:such that
1720:Definition
1525:such that
524:Scientists
356:Formalisms
304:Formalisms
253:Black hole
179:World line
3447:Hermitian
3400:Signature
3363:Sectional
3343:Curvature
3154:Principal
3129:Ehresmann
3086:Subbundle
3076:Principal
3051:Fibration
3031:Cotangent
2903:Covectors
2756:Lie group
2736:Hermitian
2679:manifolds
2648:Immersion
2643:Foliation
2581:Noether's
2566:Frobenius
2561:De Rham's
2556:Darboux's
2447:Manifolds
2248:Spacetime
2140:−
2137:⋯
2134:−
2107:−
2083:⋯
1963:−
1874:spacelike
1854:spacetime
1665:signature
1654:produces
1577:∈
1500:∈
1445:∈
1215:∈
1169:→
1153:×
1006:(denoted
842:Manifolds
827:spacetime
616:Robertson
581:Friedmann
576:Eddington
566:Nordström
556:de Sitter
413:Solutions
338:Geodesics
333:Friedmann
315:Equations
301:Equations
262:Spacetime
197:Phenomena
103:ν
100:μ
91:κ
82:ν
79:μ
71:Λ
63:ν
60:μ
3462:Kenmotsu
3375:Geodesic
3328:Glossary
3250:Orbifold
3245:K-theory
3235:Diffiety
2959:Pullback
2773:Oriented
2751:Kenmotsu
2731:Hadamard
2677:Types of
2626:Geodesic
2451:Glossary
2222:See also
1906:Just as
1866:timelike
1702:, where
1566:for all
909:Manifold
868:. In an
848:Manifold
732:Category
596:LemaĂźtre
561:Reissner
546:Poincaré
531:Einstein
476:TaubâNUT
441:Wormhole
425:interior
138:Timeline
3529:Hilbert
3524:Finsler
3194:History
3177:Related
3091:Tangent
3069:)
3049:)
3016:Adjoint
3008:Bundles
2986:density
2884:Torsion
2850:Vectors
2842:Tensors
2825:)
2810:)
2806:,
2804:Pseudoâ
2783:Poisson
2716:Finsler
2711:Fibered
2706:Contact
2704:)
2696:Complex
2694:)
2663:Section
893:locally
787:with a
783:, is a
651:Hawking
646:Penrose
621:Bardeen
611:Wheeler
541:Hilbert
536:Lorentz
496:pp-wave
133:History
3457:KĂ€hler
3353:Scalar
3348:tensor
3159:Vector
3144:Koszul
3124:Cartan
3119:Affine
3101:Vector
3096:Tensor
3081:Spinor
3071:Normal
3067:Stable
3021:Affine
2925:bundle
2877:bundle
2823:Almost
2746:KĂ€hler
2702:Almost
2692:Almost
2686:Closed
2586:Sard's
2542:(list)
2378:
2358:
2340:
2315:
2210:. The
1862:(1,â3)
1858:(3,â1)
1823:; see
1821:â1,â1)
1620:on an
962:in an
806:Every
730:
697:others
691:Thorne
681:Newman
661:Taylor
641:Ehlers
626:Walker
591:Zwicky
466:Kasner
3358:Ricci
3267:Sheaf
3041:Fiber
2817:Rizza
2788:Prime
2619:Local
2609:Curve
2471:Atlas
2254:Notes
2208:curve
1998:) is
1739:is a
1465:That
1091:is a
1002:is a
666:Hulse
606:Gödel
586:Milne
481:Milne
446:Gödel
143:Tests
3134:Form
3036:Dual
2969:flow
2832:Tame
2808:Subâ
2721:Flat
2601:Maps
2376:ISBN
2356:ISBN
2338:ISBN
2313:ISBN
1886:(1,
1882:,â1)
1870:null
1809:(1,
1787:and
1779:and
1636:) =
936:and
906:See
860:, a
850:and
775:, a
676:Taub
631:Kerr
571:Weyl
451:Kerr
369:BSSN
3056:Jet
2193:not
1898:).
1884:or
1872:or
1813:â1)
1807:is
1688:= 0
1485:is
883:An
856:In
771:In
686:Yau
364:ADM
3582::
3047:Co
1994:,
1942:,
1868:,
1849:.
1837:.
1799:A
1769:,
1753:.
1733:,
1724:A
1716:.
1711:=
1707:+
1696:,
1677:,
1673:,
1644:,
1602:.
1462:.
1087:A
1084:.
918:,
912:,
833:.
814:.
3407:/
3330:)
3326:(
3316:e
3309:t
3302:v
3065:(
3045:(
2821:(
2802:(
2700:(
2690:(
2453:)
2449:(
2439:e
2432:t
2425:v
2162:2
2157:q
2154:+
2151:p
2147:x
2143:d
2129:2
2124:1
2121:+
2118:p
2114:x
2110:d
2102:2
2097:p
2093:x
2089:d
2086:+
2080:+
2075:2
2070:1
2066:x
2062:d
2059:=
2056:g
2042:i
2038:x
2022:q
2019:,
2016:p
2011:R
1996:q
1992:p
1972:1
1969:,
1966:1
1960:n
1955:R
1924:n
1919:R
1890:)
1888:q
1880:p
1878:(
1819:n
1817:(
1811:n
1789:q
1785:p
1781:q
1777:p
1773:)
1771:q
1767:p
1765:(
1751:g
1744:M
1737:)
1735:g
1731:M
1729:(
1713:n
1709:q
1705:p
1700:)
1698:q
1694:p
1692:(
1686:r
1681:)
1679:r
1675:q
1671:p
1669:(
1656:n
1648:)
1646:x
1642:x
1640:(
1638:g
1634:x
1632:(
1630:q
1622:n
1618:g
1590:M
1585:p
1581:T
1574:Y
1554:0
1551:=
1548:)
1545:Y
1542:,
1539:X
1536:(
1533:g
1513:M
1508:p
1504:T
1497:X
1473:g
1449:R
1442:a
1419:)
1416:Z
1413:,
1410:Y
1407:(
1404:g
1401:+
1398:)
1395:Z
1392:,
1389:X
1386:(
1383:g
1380:a
1377:=
1374:)
1371:Z
1368:,
1365:Y
1362:+
1359:X
1356:a
1353:(
1350:g
1328:)
1325:X
1322:,
1319:Y
1316:(
1313:g
1310:=
1307:)
1304:Y
1301:,
1298:X
1295:(
1292:g
1268:M
1248:p
1228:M
1223:p
1219:T
1212:Z
1209:,
1206:Y
1203:,
1200:X
1177:.
1173:R
1166:M
1161:p
1157:T
1150:M
1145:p
1141:T
1137::
1134:g
1111:g
1072:p
1044:n
1024:M
1019:p
1015:T
990:M
970:n
950:p
901:n
889:n
885:n
874:n
870:n
760:e
753:t
746:v
427:)
423:(
96:T
87:=
75:g
68:+
56:G
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
