4586:
701:
4572:
31:
1933:
714:
1506:
3939:
in the orthogonal directions; the eigenvalues have the pattern (-2,1,1). For example, a spacecraft orbiting the Earth experiences a tiny tension along a radius from the center of the Earth, and a tiny compression in the orthogonal directions. Just as in
Newtonian gravitation, this tidal field
1074:
In (four-dimensional) Lorentzian spacetimes, there is a six-dimensional space of antisymmetric bivectors at each event. However, the symmetries of the Weyl tensor imply that any eigenbivectors must belong to a four-dimensional subset. Thus, the Weyl tensor (at a given event) can in fact have
1928:{\displaystyle {\begin{aligned}C_{abcd}&=\Psi _{0}U_{ab}U_{cd}\\&\,\,\,+\Psi _{1}(U_{ab}W_{cd}+W_{ab}U_{cd})\\&\,\,\,+\Psi _{2}(V_{ab}U_{cd}+U_{ab}V_{cd}+W_{ab}W_{cd})\\&\,\,\,+\Psi _{3}(V_{ab}W_{cd}+W_{ab}V_{cd})\\&\,\,\,+\Psi _{4}V_{ab}V_{cd}+c.c.\end{aligned}}}
3866:
In fact, for each criterion above, there are equivalent conditions for the Weyl tensor to have that type. These equivalent conditions are stated in terms of the dual and self-dual of the Weyl tensor and certain bivectors and are collected together in Hall (2004).
3761:
3653:
102:
1069:
919:
2377:
2930:
3518:
3195:
2286:
2143:
3426:
3103:
797:, but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation. The classification was found in 1954 by
1511:
1173:
The possible transitions between Petrov types are shown in the figure, which can also be interpreted as stating that some of the Petrov types are "more special" than others. For example, type
472:
2208:
1116:
is somewhat involved (see the citations below), but the resulting classification theorem states that there are precisely six possible types of algebraic symmetry. These are known as the
4545:
is a WAND if and only if it is a principal null direction in the sense defined above. This approach gives a natural higher-dimensional extension of each of the various algebraic types
2065:
3790:
3547:
3224:
1495:
3896:
fields occur as the exterior field of a gravitating object which is completely characterized by its mass and angular momentum. (A more general object might have nonzero higher
2673:
745:
3870:
The Bel criteria find application in general relativity where determining the Petrov type of algebraically special Weyl tensors is accomplished by searching for null vectors.
3841:
1969:
1325:
1388:
2013:
4189:
4128:
4063:
3977:
3287:
2992:
2776:
2736:
2583:
2536:
945:
124:
3339:
2460:
979:
2809:
2616:
4996:
4471:
4543:
4519:
4499:
4445:
4425:
4401:
3997:
3861:
3314:
3248:
3016:
2953:
2696:
2500:
2480:
2430:
2410:
329:
4351:
Some classes of solutions can be invariantly characterized using algebraic symmetries of the Weyl tensor: for example, the class of non-conformally flat null
682:
492:
334:
447:
4599:
3882:, the various algebraically special Petrov types have some interesting physical interpretations, the classification then sometimes being called the
738:
4991:
360:
194:
4585:
4383:
A. Coley, R. Milson, V. Pravda and A. Pravdová (2004) developed a generalization of algebraic classification to arbitrary spacetime dimension
700:
4604:
1975:
and c.c. is the complex conjugate. The six different Petrov types are distinguished by which of the Weyl scalars vanish. The conditions are
399:
37:
987:
842:
2297:
731:
2818:
3658:
2462:, there is a useful set of conditions, found by Lluis (or Louis) Bel and Robert Debever, for determining precisely the Petrov type at
111:
4969:
4807:
4785:
137:
417:
3431:
3108:
30:
3559:
4329:
2219:
2076:
1124:
324:
5001:
4945:
1093:
of the Weyl tensor at the given event. The different types of Weyl tensor (at a given event) can be determined by solving a
3348:
3025:
4223:
regions, are associated with places where the Weyl tensor vanishes identically. In this case, the curvature is said to be
457:
794:
411:
119:
1239:
477:
4317:
4141:
3892:
regions are associated with the gravitational fields of isolated massive objects, such as stars. More precisely, type
2389:
718:
209:
129:
4624:
4268:
4259:
describes the way in which, as one moves farther way from the source of the radiation, the various components of the
4245:
on events in our region. More precisely, if there are any time varying gravitational fields in distant regions, the
2154:
4961:
1086:
622:
170:
4643:
Ortaggio, Marcello (2009). "Bel–Debever criteria for the classification of the Weyl tensor in higher dimensions".
4295:
4230:. In a conformally flat region, any gravitational effects must be due to the immediate presence of matter or the
422:
1094:
234:
4341:
497:
4080:
1242:
is often used in practice for the classification. Consider the following set of bivectors, constructed out of
657:
647:
314:
376:
4777:
4478:
4252:
3928:
642:
4556:
An alternative, but inequivalent, generalization was previously defined by de Smet (2002), based on a
2024:
442:
4352:
4242:
4231:
4079:
effect. This possibility is often neglected, in part because the gravitational radiation which arises in
3908:
3766:
612:
597:
204:
3523:
3200:
1394:
4238:
759:
587:
155:
4571:
2625:
4011:
3799:
4908:
4867:
4830:
4740:
4693:
4609:
4260:
407:
4800:
Symmetries and
Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)
1208:
Different events in a given spacetime can have different Petrov types. A Weyl tensor that has type
1101:. All the above happens similarly to the theory of the eigenvectors of an ordinary linear operator.
4821:
MacCallum, M.A.H. (2000). "Editor's note: Classification of spaces defining gravitational fields".
4279:
In some (more or less) familiar solutions, the Weyl tensor has the same Petrov type at each event:
4246:
1941:
1256:
1113:
783:
763:
602:
582:
572:
567:
552:
319:
1331:
462:
4924:
4846:
4756:
4730:
4709:
4683:
4648:
4577:
4356:
4306:
3879:
1985:
652:
537:
309:
214:
199:
160:
22:
4155:
4094:
4029:
3943:
4965:
4941:
4803:
4781:
4072:
3920:
3897:
3256:
2961:
2745:
2705:
2552:
2505:
930:
532:
482:
257:
4916:
4875:
4838:
4748:
4701:
4658:
4619:
4220:
4007:
4003:
2439:
1243:
1226:
1098:
954:
627:
607:
562:
542:
487:
4359:
solutions admitting an expanding but nontwisting null congruence is precisely the class of
4614:
4256:
3901:
2785:
2592:
833:
790:
662:
637:
522:
517:
381:
262:
224:
4674:
Coley, A.; et al. (2004). "Classification of the Weyl tensor in higher dimensions".
4662:
4450:
432:
4912:
4871:
4834:
4744:
4697:
3319:
4887:
Petrov, A.Z. (1954). "Klassifikacya prostranstv opredelyayushchikh polya tyagoteniya".
4770:
4591:
4528:
4504:
4484:
4430:
4410:
4386:
3982:
3846:
3299:
3233:
3001:
2938:
2681:
2485:
2465:
2415:
2395:
705:
672:
667:
355:
219:
4752:
770:(also known as Petrov–Pirani–Penrose classification) describes the possible algebraic
4985:
4879:
4850:
4705:
4152:
describing the direction of propagation of this radiation. It typically decays like
4076:
632:
592:
547:
527:
452:
350:
229:
4928:
4760:
4713:
3919:
region is very closely analogous to the gravitational fields which are described in
3900:.) The two double principal null directions define "radially" ingoing and outgoing
4226:
948:
802:
798:
577:
557:
4956:
Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C. & Herlt, E. (2003).
4899:
Petrov, A.Z. (2000). "Classification of spaces defined by gravitational fields".
4815:
See sections 7.3, 7.4 for a comprehensive discussion of the Petrov classification
4474:
4284:
4149:
4019:
1972:
1247:
1105:
925:
821:
775:
467:
437:
4255:
emitted from an isolated system will usually not be algebraically special. The
4920:
4842:
4721:
de Smet, P. (2002). "Black holes on cylinders are not algebraically special".
4567:
4015:
677:
239:
165:
1500:
The Weyl tensor can be expressed as a combination of these bivectors through
779:
248:
4144:
gravitational radiation, which is the type astronomers have detected with
4404:
4241:). In a sense, this means that any distant objects are not exerting any
829:
771:
617:
427:
267:
1131:
showing the possible degenerations of the Petrov type of the Weyl tensor
4735:
4560:. However, de Smet's approach is restricted to 5 dimensions only.
4557:
3924:
97:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
4688:
1064:{\displaystyle {\frac {1}{2}}\,{C^{ab}}_{mn}\,X^{mn}=\lambda \,X^{ab}}
4234:
914:{\displaystyle X^{ab}\rightarrow {\frac {1}{2}}\,{C^{ab}}_{mn}X^{mn}}
817:
814:
1089:
and any multiplicities among the eigenbivectors indicates a kind of
4267:
radiation is noticeable at large distances. This is similar to the
4075:
gravitational radiation. In such regions, the tidal forces have a
2372:{\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{2}=\Psi _{3}=\Psi _{4}=0}
16:
Classification used in differential geometry and general relativity
4858:
Penrose, Roger (1960). "A spinor approach to general relativity".
4653:
1123:
1122:
4340:). All algebraically special spacetimes having various types of
2925:{\displaystyle C_{abcd}\,k^{b}k^{d}=0={^{*}C}_{abcd}\,k^{b}k^{d}}
1250:(note that in some notations, symbols l and n are interchanged):
4145:
3756:{\displaystyle {}^{*}C_{abcd}\,k'^{b}k'^{d}=\delta k'_{a}k'_{c}}
4477:
are classified by their transformation properties under local
1085:
The eigenbivectors of the Weyl tensor can occur with various
4407:
approach, that is a frame basis containing two null vectors
4148:. The quadruple principal null direction corresponds to the
3513:{\displaystyle {}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}}
3190:{\displaystyle {}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}}
4006:, in addition to the tidal effects, there will be various
3648:{\displaystyle C_{abcd}\,k'^{b}k'^{d}=\gamma k'_{a}k'_{c}}
2281:{\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{2}=\Psi _{3}=0}
2138:{\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0}
4958:
Exact
Solutions of Einstein's Field Equations (2nd edn.)
2432:
for this metric may be computed. If the Weyl tensor is
924:
Then, it is natural to consider the problem of finding
3421:{\displaystyle C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}}
3098:{\displaystyle C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}}
1145:: one double and two simple principal null directions,
4531:
4507:
4487:
4453:
4433:
4413:
4389:
4158:
4097:
4032:
3985:
3946:
3849:
3802:
3769:
3661:
3562:
3526:
3434:
3351:
3322:
3302:
3259:
3236:
3203:
3111:
3028:
3004:
2964:
2941:
2821:
2788:
2748:
2708:
2684:
2628:
2595:
2555:
2508:
2488:
2468:
2442:
2418:
2398:
2300:
2222:
2157:
2079:
2027:
1988:
1944:
1509:
1397:
1334:
1259:
1157:: one triple and one simple principal null direction,
990:
957:
933:
845:
40:
4026:
vacuum solution, this part of the field decays like
4769:
4537:
4513:
4493:
4473:spacelike vectors. Frame basis components of the
4465:
4439:
4419:
4395:
4201:regions combine the effects noted above for types
4183:
4122:
4057:
3991:
3971:
3904:near the object which is the source of the field.
3855:
3835:
3784:
3755:
3647:
3541:
3512:
3420:
3333:
3308:
3281:
3242:
3218:
3189:
3097:
3010:
2986:
2947:
2924:
2803:
2770:
2730:
2690:
2667:
2610:
2577:
2530:
2494:
2474:
2454:
2424:
2404:
2371:
2280:
2202:
2137:
2059:
2007:
1963:
1927:
1489:
1382:
1319:
1063:
973:
939:
913:
96:
4952:, translated by R. F. Kelleher & J. Woodrow.
4249:has not yet reached our conformally flat region.
1104:These eigenbivectors are associated with certain
3250:is necessarily null and unique (up to scaling).
2955:is necessarily null and unique (up to scaling).
2698:is necessarily null and unique (up to scaling).
1108:in the original spacetime, which are called the
2203:{\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{2}=0}
4481:. If particular Weyl components vanish, then
1220:(at that event). In General Relativity, type
951:(which are now referred to as eigenbivectors)
739:
8:
4191:, so the long-range radiation field is type
1958:
1945:
4237:of some nongravitational field (such as an
4022:, which is the best known example of type
3931:. Such a tidal field is characterized by
746:
732:
298:
188:
18:
4734:
4687:
4652:
4530:
4506:
4486:
4452:
4432:
4412:
4388:
4213:, in a rather complicated nonlinear way.
4169:
4157:
4108:
4096:
4043:
4031:
3984:
3957:
3945:
3848:
3818:
3808:
3806:
3804:
3801:
3768:
3744:
3731:
3714:
3699:
3690:
3675:
3665:
3663:
3660:
3636:
3623:
3606:
3591:
3582:
3567:
3561:
3525:
3504:
3494:
3478:
3468:
3463:
3448:
3438:
3436:
3433:
3412:
3402:
3386:
3376:
3371:
3356:
3350:
3321:
3301:
3264:
3258:
3235:
3202:
3181:
3171:
3155:
3145:
3140:
3125:
3115:
3113:
3110:
3089:
3079:
3063:
3053:
3048:
3033:
3027:
3003:
2969:
2963:
2940:
2916:
2906:
2901:
2886:
2876:
2872:
2856:
2846:
2841:
2826:
2820:
2787:
2753:
2747:
2713:
2707:
2683:
2653:
2648:
2633:
2627:
2594:
2560:
2554:
2513:
2507:
2487:
2482:. Denoting the Weyl tensor components at
2467:
2441:
2417:
2397:
2357:
2344:
2331:
2318:
2305:
2299:
2266:
2253:
2240:
2227:
2221:
2188:
2175:
2162:
2156:
2123:
2110:
2097:
2084:
2078:
2045:
2032:
2026:
1993:
1987:
1952:
1943:
1897:
1884:
1874:
1866:
1865:
1864:
1845:
1832:
1816:
1803:
1790:
1782:
1781:
1780:
1761:
1748:
1732:
1719:
1703:
1690:
1677:
1669:
1668:
1667:
1648:
1635:
1619:
1606:
1593:
1585:
1584:
1583:
1567:
1554:
1544:
1518:
1510:
1508:
1475:
1462:
1443:
1432:
1431:
1421:
1402:
1396:
1371:
1358:
1339:
1333:
1308:
1297:
1296:
1286:
1264:
1258:
1163:: one quadruple principal null direction,
1052:
1047:
1032:
1027:
1018:
1008:
1003:
1001:
991:
989:
962:
956:
932:
902:
889:
879:
874:
872:
862:
850:
844:
85:
76:
64:
45:
39:
4600:Classification of electromagnetic fields
1139:: four simple principal null directions,
4802:. Singapore: World Scientific Pub. Co.
4635:
3711:
3696:
3603:
3588:
1151:: two double principal null directions,
368:
342:
301:
247:
21:
4071:regions are associated with a kind of
3884:classification of gravitational fields
4997:Exact solutions in general relativity
4605:Exact solutions in general relativity
4344:are known, for example, all the type
4002:If the object is rotating about some
2998:if and only if there exists a vector
2782:if and only if there exists a vector
2589:if and only if there exists a vector
2538:(assumed non-zero, i.e., not of type
2060:{\displaystyle \Psi _{0}=\Psi _{1}=0}
1079:linearly independent eigenbivectors.
789:It is most often applied in studying
7:
4263:"peel" off, until finally only type
3785:{\displaystyle \gamma \delta \neq 0}
4379:Generalization to higher dimensions
3542:{\displaystyle \alpha \beta \neq 0}
3219:{\displaystyle \alpha \beta \neq 0}
1490:{\displaystyle W_{ab}=2m_{}-2n_{}.}
4901:General Relativity and Gravitation
4823:General Relativity and Gravitation
3843:is the dual of the Weyl tensor at
2354:
2341:
2328:
2315:
2302:
2263:
2250:
2237:
2224:
2185:
2172:
2159:
2120:
2107:
2094:
2081:
2042:
2029:
1990:
1949:
1871:
1787:
1674:
1590:
1541:
57:
14:
4772:Introducing Einstein's Relativity
3999:is the distance from the object.
2668:{\displaystyle C_{abcd}\,k^{d}=0}
1112:(at a given event). The relevant
4584:
4570:
4018:carried by an observer. In the
3836:{\displaystyle {{}^{*}C}_{abcd}}
3295:two linearly independent vectors
713:
712:
699:
29:
4330:spherically symmetric spacetime
4269:electromagnetic peeling theorem
4663:10.1088/0264-9381/26/19/195015
4525:(WANDs). In four dimensions,
4403:. Their approach uses a null
4178:
4162:
4117:
4101:
4052:
4036:
3966:
3950:
2798:
2792:
2605:
2599:
1854:
1796:
1770:
1683:
1657:
1599:
1479:
1463:
1447:
1437:
1422:
1375:
1359:
1312:
1302:
1287:
859:
1:
4992:Tensors in general relativity
4889:Uch. Zapiski Kazan. Gos. Univ
4723:Classical and Quantum Gravity
4676:Classical and Quantum Gravity
4645:Classical and Quantum Gravity
4361:Robinson/Trautmann spacetimes
1964:{\displaystyle \{\Psi _{j}\}}
1320:{\displaystyle U_{ab}=-2l_{}}
1177:, the most general type, can
4880:10.1016/0003-4916(60)90021-X
4523:Weyl-Aligned Null Directions
4140:regions are associated with
4130:, which is faster than type
3293:if and only if there exists
1383:{\displaystyle V_{ab}=2n_{}}
828:, as acting on the space of
4753:10.1088/0264-9381/19/19/307
4087:, and in part because type
2008:{\displaystyle \Psi _{0}=0}
1169:: the Weyl tensor vanishes.
210:Gravitational time dilation
5018:
4962:Cambridge University Press
4706:10.1088/0264-9381/21/7/L01
4363:. These are usually type
1216:; otherwise, it is called
1212:(at some event) is called
836:acting on a vector space:
795:Einstein's field equations
330:Mathisson–Papapetrou–Dixon
171:Pseudo-Riemannian manifold
4296:Robinson/Trautman vacuums
4184:{\displaystyle O(r^{-1})}
4123:{\displaystyle O(r^{-2})}
4058:{\displaystyle O(r^{-4})}
3972:{\displaystyle O(r^{-3})}
3341:satisfying the conditions
2392:on a Lorentzian manifold
1110:principal null directions
813:We can think of a fourth
3282:{\displaystyle C_{abcd}}
2987:{\displaystyle C_{abcd}}
2771:{\displaystyle C_{abcd}}
2731:{\displaystyle C_{abcd}}
2578:{\displaystyle C_{abcd}}
2531:{\displaystyle C_{abcd}}
1240:Newman–Penrose formalism
1234:Newman–Penrose formalism
1193:can degenerate to types
940:{\displaystyle \lambda }
335:Hamilton–Jacobi–Einstein
315:Einstein field equations
138:Mathematical formulation
4921:10.1023/A:1001910908054
4843:10.1023/A:1001958823984
4793:See sections 21.7, 21.8
4778:Oxford University Press
4768:d'Inverno, Ray (1992).
4253:Gravitational radiation
3929:gravitational potential
3874:Physical interpretation
1095:characteristic equation
826:evaluated at some event
4625:Goldberg–Sachs theorem
4539:
4515:
4495:
4467:
4441:
4421:
4397:
4185:
4124:
4091:radiation decays like
4059:
3993:
3973:
3940:typically decays like
3909:electrogravitic tensor
3857:
3837:
3786:
3757:
3649:
3543:
3514:
3422:
3335:
3310:
3283:
3244:
3220:
3191:
3099:
3012:
2988:
2949:
2926:
2805:
2772:
2732:
2692:
2669:
2612:
2579:
2532:
2496:
2476:
2456:
2455:{\displaystyle p\in M}
2426:
2406:
2373:
2282:
2204:
2139:
2061:
2009:
1965:
1929:
1491:
1384:
1321:
1132:
1065:
975:
974:{\displaystyle X^{ab}}
941:
915:
809:Classification theorem
205:Gravitational redshift
98:
5002:Differential geometry
4936:Petrov, A.Z. (1969).
4798:Hall, Graham (2004).
4540:
4516:
4496:
4468:
4442:
4422:
4398:
4239:electromagnetic field
4186:
4125:
4060:
3994:
3974:
3935:in one direction and
3858:
3838:
3787:
3758:
3650:
3544:
3515:
3423:
3336:
3311:
3284:
3245:
3221:
3192:
3100:
3013:
2989:
2950:
2927:
2806:
2773:
2733:
2693:
2670:
2613:
2580:
2533:
2497:
2477:
2457:
2434:algebraically special
2427:
2407:
2374:
2283:
2205:
2140:
2062:
2010:
1966:
1930:
1492:
1385:
1322:
1218:algebraically special
1214:algebraically general
1126:
1066:
976:
942:
916:
832:at that event like a
801:and independently by
768:Petrov classification
760:differential geometry
493:Weyl−Lewis−Papapetrou
448:Kerr–Newman–de Sitter
268:Einstein–Rosen bridge
200:Gravitational lensing
156:Equivalence principle
99:
4940:. Oxford: Pergamon.
4897:English translation
4610:Segre classification
4553:etc. defined above.
4529:
4505:
4485:
4451:
4431:
4411:
4387:
4342:stress–energy tensor
4328:More generally, any
4320:are everywhere type
4309:are everywhere type
4298:are everywhere type
4243:long range influence
4156:
4095:
4030:
3983:
3944:
3847:
3800:
3767:
3659:
3560:
3524:
3432:
3349:
3320:
3300:
3257:
3234:
3201:
3109:
3026:
3002:
2962:
2939:
2819:
2804:{\displaystyle k(p)}
2786:
2746:
2706:
2682:
2626:
2611:{\displaystyle k(p)}
2593:
2553:
2506:
2486:
2466:
2440:
2416:
2396:
2298:
2220:
2155:
2077:
2025:
1986:
1942:
1507:
1395:
1332:
1257:
988:
955:
931:
843:
423:Einstein–Rosen waves
149:Fundamental concepts
38:
4913:2000GReGr..32.1665P
4872:1960AnPhy..10..171P
4835:2000GReGr..32.1661P
4745:2002CQGra..19.4877D
4698:2004CQGra..21L..35C
4466:{\displaystyle d-2}
4367:, but include type
4287:is everywhere type
3752:
3739:
3644:
3631:
1114:multilinear algebra
784:Lorentzian manifold
764:theoretical physics
377:Kaluza–Klein theory
263:Minkowski spacetime
215:Gravitational waves
4977:See chapters 4, 26
4578:Mathematics portal
4558:spinorial approach
4535:
4511:
4491:
4463:
4437:
4417:
4393:
4348:vacuum solutions.
4307:pp-wave spacetimes
4181:
4120:
4055:
3989:
3969:
3880:general relativity
3853:
3833:
3782:
3753:
3740:
3727:
3645:
3632:
3619:
3539:
3510:
3418:
3334:{\displaystyle k'}
3331:
3306:
3279:
3240:
3216:
3187:
3095:
3008:
2984:
2945:
2922:
2801:
2768:
2728:
2688:
2665:
2608:
2575:
2546:may be stated as:
2528:
2492:
2472:
2452:
2422:
2412:, the Weyl tensor
2402:
2369:
2278:
2200:
2135:
2057:
2005:
1961:
1925:
1923:
1487:
1380:
1317:
1133:
1091:algebraic symmetry
1061:
971:
937:
911:
706:Physics portal
478:Oppenheimer–Snyder
418:Reissner–Nordström
310:Linearized gravity
258:Spacetime diagrams
161:Special relativity
94:
23:General relativity
4860:Annals of Physics
4729:(19): 4877–4896.
4538:{\displaystyle l}
4514:{\displaystyle n}
4494:{\displaystyle l}
4440:{\displaystyle n}
4420:{\displaystyle l}
4396:{\displaystyle d}
4081:weak-field theory
4010:effects, such as
3992:{\displaystyle r}
3921:Newtonian gravity
3898:multipole moments
3856:{\displaystyle p}
3309:{\displaystyle k}
3243:{\displaystyle k}
3011:{\displaystyle k}
2948:{\displaystyle k}
2691:{\displaystyle k}
2495:{\displaystyle p}
2475:{\displaystyle p}
2425:{\displaystyle C}
2405:{\displaystyle M}
1440:
1305:
1097:, in this case a
999:
870:
756:
755:
389:
388:
275:
274:
5009:
4975:
4951:
4932:
4907:(8): 1665–1685.
4896:
4883:
4854:
4829:(8): 1661–1663.
4813:
4791:
4775:
4764:
4738:
4717:
4691:
4667:
4666:
4656:
4640:
4620:Plebanski tensor
4594:
4589:
4588:
4580:
4575:
4574:
4544:
4542:
4541:
4536:
4520:
4518:
4517:
4512:
4500:
4498:
4497:
4492:
4472:
4470:
4469:
4464:
4446:
4444:
4443:
4438:
4426:
4424:
4423:
4418:
4402:
4400:
4399:
4394:
4332:must be of type
4221:conformally flat
4190:
4188:
4187:
4182:
4177:
4176:
4129:
4127:
4126:
4121:
4116:
4115:
4064:
4062:
4061:
4056:
4051:
4050:
4012:spin-spin forces
3998:
3996:
3995:
3990:
3978:
3976:
3975:
3970:
3965:
3964:
3902:null congruences
3862:
3860:
3859:
3854:
3842:
3840:
3839:
3834:
3832:
3831:
3817:
3813:
3812:
3807:
3791:
3789:
3788:
3783:
3762:
3760:
3759:
3754:
3748:
3735:
3720:
3719:
3718:
3705:
3704:
3703:
3689:
3688:
3670:
3669:
3664:
3654:
3652:
3651:
3646:
3640:
3627:
3612:
3611:
3610:
3597:
3596:
3595:
3581:
3580:
3548:
3546:
3545:
3540:
3519:
3517:
3516:
3511:
3509:
3508:
3499:
3498:
3483:
3482:
3473:
3472:
3462:
3461:
3443:
3442:
3437:
3427:
3425:
3424:
3419:
3417:
3416:
3407:
3406:
3391:
3390:
3381:
3380:
3370:
3369:
3340:
3338:
3337:
3332:
3330:
3315:
3313:
3312:
3307:
3288:
3286:
3285:
3280:
3278:
3277:
3249:
3247:
3246:
3241:
3225:
3223:
3222:
3217:
3196:
3194:
3193:
3188:
3186:
3185:
3176:
3175:
3160:
3159:
3150:
3149:
3139:
3138:
3120:
3119:
3114:
3104:
3102:
3101:
3096:
3094:
3093:
3084:
3083:
3068:
3067:
3058:
3057:
3047:
3046:
3017:
3015:
3014:
3009:
2993:
2991:
2990:
2985:
2983:
2982:
2954:
2952:
2951:
2946:
2931:
2929:
2928:
2923:
2921:
2920:
2911:
2910:
2900:
2899:
2885:
2881:
2880:
2861:
2860:
2851:
2850:
2840:
2839:
2810:
2808:
2807:
2802:
2777:
2775:
2774:
2769:
2767:
2766:
2737:
2735:
2734:
2729:
2727:
2726:
2697:
2695:
2694:
2689:
2674:
2672:
2671:
2666:
2658:
2657:
2647:
2646:
2617:
2615:
2614:
2609:
2584:
2582:
2581:
2576:
2574:
2573:
2537:
2535:
2534:
2529:
2527:
2526:
2501:
2499:
2498:
2493:
2481:
2479:
2478:
2473:
2461:
2459:
2458:
2453:
2431:
2429:
2428:
2423:
2411:
2409:
2408:
2403:
2378:
2376:
2375:
2370:
2362:
2361:
2349:
2348:
2336:
2335:
2323:
2322:
2310:
2309:
2287:
2285:
2284:
2279:
2271:
2270:
2258:
2257:
2245:
2244:
2232:
2231:
2209:
2207:
2206:
2201:
2193:
2192:
2180:
2179:
2167:
2166:
2144:
2142:
2141:
2136:
2128:
2127:
2115:
2114:
2102:
2101:
2089:
2088:
2066:
2064:
2063:
2058:
2050:
2049:
2037:
2036:
2014:
2012:
2011:
2006:
1998:
1997:
1970:
1968:
1967:
1962:
1957:
1956:
1934:
1932:
1931:
1926:
1924:
1905:
1904:
1892:
1891:
1879:
1878:
1860:
1853:
1852:
1840:
1839:
1824:
1823:
1811:
1810:
1795:
1794:
1776:
1769:
1768:
1756:
1755:
1740:
1739:
1727:
1726:
1711:
1710:
1698:
1697:
1682:
1681:
1663:
1656:
1655:
1643:
1642:
1627:
1626:
1614:
1613:
1598:
1597:
1579:
1575:
1574:
1562:
1561:
1549:
1548:
1532:
1531:
1496:
1494:
1493:
1488:
1483:
1482:
1470:
1469:
1451:
1450:
1442:
1441:
1433:
1429:
1428:
1410:
1409:
1389:
1387:
1386:
1381:
1379:
1378:
1366:
1365:
1347:
1346:
1326:
1324:
1323:
1318:
1316:
1315:
1307:
1306:
1298:
1294:
1293:
1272:
1271:
1227:conformally flat
1099:quartic equation
1070:
1068:
1067:
1062:
1060:
1059:
1040:
1039:
1026:
1025:
1017:
1016:
1015:
1000:
992:
980:
978:
977:
972:
970:
969:
946:
944:
943:
938:
920:
918:
917:
912:
910:
909:
897:
896:
888:
887:
886:
871:
863:
858:
857:
748:
741:
734:
721:
716:
715:
708:
704:
703:
488:van Stockum dust
473:Robertson–Walker
299:
189:
103:
101:
100:
95:
93:
92:
80:
72:
71:
53:
52:
33:
19:
5017:
5016:
5012:
5011:
5010:
5008:
5007:
5006:
4982:
4981:
4972:
4955:
4948:
4938:Einstein Spaces
4935:
4898:
4886:
4857:
4820:
4810:
4797:
4788:
4767:
4720:
4673:
4670:
4642:
4641:
4637:
4633:
4615:Peeling theorem
4590:
4583:
4576:
4569:
4566:
4527:
4526:
4521:are said to be
4503:
4502:
4483:
4482:
4449:
4448:
4429:
4428:
4409:
4408:
4385:
4384:
4381:
4277:
4261:radiation field
4257:peeling theorem
4165:
4154:
4153:
4104:
4093:
4092:
4039:
4028:
4027:
4008:gravitomagnetic
3981:
3980:
3953:
3942:
3941:
3876:
3845:
3844:
3805:
3803:
3798:
3797:
3765:
3764:
3710:
3706:
3695:
3691:
3671:
3662:
3657:
3656:
3602:
3598:
3587:
3583:
3563:
3558:
3557:
3522:
3521:
3500:
3490:
3474:
3464:
3444:
3435:
3430:
3429:
3408:
3398:
3382:
3372:
3352:
3347:
3346:
3323:
3318:
3317:
3298:
3297:
3260:
3255:
3254:
3232:
3231:
3199:
3198:
3177:
3167:
3151:
3141:
3121:
3112:
3107:
3106:
3085:
3075:
3059:
3049:
3029:
3024:
3023:
3000:
2999:
2965:
2960:
2959:
2937:
2936:
2912:
2902:
2873:
2871:
2852:
2842:
2822:
2817:
2816:
2784:
2783:
2749:
2744:
2743:
2709:
2704:
2703:
2680:
2679:
2649:
2629:
2624:
2623:
2591:
2590:
2556:
2551:
2550:
2509:
2504:
2503:
2484:
2483:
2464:
2463:
2438:
2437:
2414:
2413:
2394:
2393:
2386:
2353:
2340:
2327:
2314:
2301:
2296:
2295:
2262:
2249:
2236:
2223:
2218:
2217:
2184:
2171:
2158:
2153:
2152:
2119:
2106:
2093:
2080:
2075:
2074:
2041:
2028:
2023:
2022:
1989:
1984:
1983:
1948:
1940:
1939:
1922:
1921:
1893:
1880:
1870:
1858:
1857:
1841:
1828:
1812:
1799:
1786:
1774:
1773:
1757:
1744:
1728:
1715:
1699:
1686:
1673:
1661:
1660:
1644:
1631:
1615:
1602:
1589:
1577:
1576:
1563:
1550:
1540:
1533:
1514:
1505:
1504:
1471:
1458:
1430:
1417:
1398:
1393:
1392:
1367:
1354:
1335:
1330:
1329:
1295:
1282:
1260:
1255:
1254:
1236:
1224:spacetimes are
1129:Penrose diagram
1082:
1048:
1028:
1004:
1002:
986:
985:
958:
953:
952:
929:
928:
898:
875:
873:
846:
841:
840:
834:linear operator
811:
791:exact solutions
752:
711:
698:
697:
690:
689:
513:
512:
503:
502:
458:Lemaître–Tolman
403:
402:
391:
390:
382:Quantum gravity
369:Advanced theory
296:
295:
294:
277:
276:
225:Geodetic effect
186:
185:
176:
175:
151:
150:
134:
104:
81:
60:
41:
36:
35:
17:
12:
11:
5:
5015:
5013:
5005:
5004:
4999:
4994:
4984:
4983:
4980:
4979:
4970:
4953:
4946:
4933:
4884:
4866:(2): 171–201.
4855:
4818:
4808:
4795:
4786:
4765:
4736:hep-th/0206106
4718:
4682:(7): L35–L42.
4669:
4668:
4634:
4632:
4629:
4628:
4627:
4622:
4617:
4612:
4607:
4602:
4596:
4595:
4592:Physics portal
4581:
4565:
4562:
4534:
4510:
4490:
4479:Lorentz boosts
4462:
4459:
4456:
4436:
4416:
4392:
4380:
4377:
4326:
4325:
4314:
4303:
4292:
4276:
4273:
4180:
4175:
4172:
4168:
4164:
4161:
4119:
4114:
4111:
4107:
4103:
4100:
4054:
4049:
4046:
4042:
4038:
4035:
3988:
3968:
3963:
3960:
3956:
3952:
3949:
3875:
3872:
3852:
3830:
3827:
3824:
3821:
3816:
3811:
3794:
3793:
3781:
3778:
3775:
3772:
3751:
3747:
3743:
3738:
3734:
3730:
3726:
3723:
3717:
3713:
3709:
3702:
3698:
3694:
3687:
3684:
3681:
3678:
3674:
3668:
3643:
3639:
3635:
3630:
3626:
3622:
3618:
3615:
3609:
3605:
3601:
3594:
3590:
3586:
3579:
3576:
3573:
3570:
3566:
3551:
3550:
3538:
3535:
3532:
3529:
3507:
3503:
3497:
3493:
3489:
3486:
3481:
3477:
3471:
3467:
3460:
3457:
3454:
3451:
3447:
3441:
3415:
3411:
3405:
3401:
3397:
3394:
3389:
3385:
3379:
3375:
3368:
3365:
3362:
3359:
3355:
3343:
3342:
3329:
3326:
3305:
3276:
3273:
3270:
3267:
3263:
3239:
3228:
3227:
3215:
3212:
3209:
3206:
3184:
3180:
3174:
3170:
3166:
3163:
3158:
3154:
3148:
3144:
3137:
3134:
3131:
3128:
3124:
3118:
3092:
3088:
3082:
3078:
3074:
3071:
3066:
3062:
3056:
3052:
3045:
3042:
3039:
3036:
3032:
3020:
3019:
3007:
2981:
2978:
2975:
2972:
2968:
2944:
2933:
2932:
2919:
2915:
2909:
2905:
2898:
2895:
2892:
2889:
2884:
2879:
2875:
2870:
2867:
2864:
2859:
2855:
2849:
2845:
2838:
2835:
2832:
2829:
2825:
2813:
2812:
2800:
2797:
2794:
2791:
2765:
2762:
2759:
2756:
2752:
2725:
2722:
2719:
2716:
2712:
2687:
2676:
2675:
2664:
2661:
2656:
2652:
2645:
2642:
2639:
2636:
2632:
2620:
2619:
2607:
2604:
2601:
2598:
2572:
2569:
2566:
2563:
2559:
2525:
2522:
2519:
2516:
2512:
2491:
2471:
2451:
2448:
2445:
2421:
2401:
2385:
2382:
2381:
2380:
2368:
2365:
2360:
2356:
2352:
2347:
2343:
2339:
2334:
2330:
2326:
2321:
2317:
2313:
2308:
2304:
2289:
2277:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2243:
2239:
2235:
2230:
2226:
2211:
2199:
2196:
2191:
2187:
2183:
2178:
2174:
2170:
2165:
2161:
2146:
2134:
2131:
2126:
2122:
2118:
2113:
2109:
2105:
2100:
2096:
2092:
2087:
2083:
2068:
2056:
2053:
2048:
2044:
2040:
2035:
2031:
2016:
2004:
2001:
1996:
1992:
1960:
1955:
1951:
1947:
1936:
1935:
1920:
1917:
1914:
1911:
1908:
1903:
1900:
1896:
1890:
1887:
1883:
1877:
1873:
1869:
1863:
1861:
1859:
1856:
1851:
1848:
1844:
1838:
1835:
1831:
1827:
1822:
1819:
1815:
1809:
1806:
1802:
1798:
1793:
1789:
1785:
1779:
1777:
1775:
1772:
1767:
1764:
1760:
1754:
1751:
1747:
1743:
1738:
1735:
1731:
1725:
1722:
1718:
1714:
1709:
1706:
1702:
1696:
1693:
1689:
1685:
1680:
1676:
1672:
1666:
1664:
1662:
1659:
1654:
1651:
1647:
1641:
1638:
1634:
1630:
1625:
1622:
1618:
1612:
1609:
1605:
1601:
1596:
1592:
1588:
1582:
1580:
1578:
1573:
1570:
1566:
1560:
1557:
1553:
1547:
1543:
1539:
1536:
1534:
1530:
1527:
1524:
1521:
1517:
1513:
1512:
1498:
1497:
1486:
1481:
1478:
1474:
1468:
1465:
1461:
1457:
1454:
1449:
1446:
1439:
1436:
1427:
1424:
1420:
1416:
1413:
1408:
1405:
1401:
1390:
1377:
1374:
1370:
1364:
1361:
1357:
1353:
1350:
1345:
1342:
1338:
1327:
1314:
1311:
1304:
1301:
1292:
1289:
1285:
1281:
1278:
1275:
1270:
1267:
1263:
1235:
1232:
1171:
1170:
1164:
1158:
1152:
1146:
1140:
1087:multiplicities
1072:
1071:
1058:
1055:
1051:
1046:
1043:
1038:
1035:
1031:
1024:
1021:
1014:
1011:
1007:
998:
995:
968:
965:
961:
936:
922:
921:
908:
905:
901:
895:
892:
885:
882:
878:
869:
866:
861:
856:
853:
849:
810:
807:
754:
753:
751:
750:
743:
736:
728:
725:
724:
723:
722:
709:
692:
691:
688:
687:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
514:
510:
509:
508:
505:
504:
501:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
404:
398:
397:
396:
393:
392:
387:
386:
385:
384:
379:
371:
370:
366:
365:
364:
363:
361:Post-Newtonian
358:
353:
345:
344:
340:
339:
338:
337:
332:
327:
322:
317:
312:
304:
303:
297:
293:
292:
289:
285:
284:
283:
282:
279:
278:
273:
272:
271:
270:
265:
260:
252:
251:
245:
244:
243:
242:
237:
232:
227:
222:
220:Frame-dragging
217:
212:
207:
202:
197:
195:Kepler problem
187:
183:
182:
181:
178:
177:
174:
173:
168:
163:
158:
152:
148:
147:
146:
143:
142:
141:
140:
135:
133:
132:
127:
122:
116:
114:
106:
105:
91:
88:
84:
79:
75:
70:
67:
63:
59:
56:
51:
48:
44:
34:
26:
25:
15:
13:
10:
9:
6:
4:
3:
2:
5014:
5003:
5000:
4998:
4995:
4993:
4990:
4989:
4987:
4978:
4973:
4971:0-521-46136-7
4967:
4963:
4960:. Cambridge:
4959:
4954:
4949:
4943:
4939:
4934:
4930:
4926:
4922:
4918:
4914:
4910:
4906:
4902:
4894:
4890:
4885:
4881:
4877:
4873:
4869:
4865:
4861:
4856:
4852:
4848:
4844:
4840:
4836:
4832:
4828:
4824:
4819:
4816:
4811:
4809:981-02-1051-5
4805:
4801:
4796:
4794:
4789:
4787:0-19-859686-3
4783:
4779:
4774:
4773:
4766:
4762:
4758:
4754:
4750:
4746:
4742:
4737:
4732:
4728:
4724:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4690:
4689:gr-qc/0401008
4685:
4681:
4677:
4672:
4671:
4664:
4660:
4655:
4650:
4646:
4639:
4636:
4630:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4597:
4593:
4587:
4582:
4579:
4573:
4568:
4563:
4561:
4559:
4554:
4552:
4548:
4532:
4524:
4508:
4488:
4480:
4476:
4460:
4457:
4454:
4447:, along with
4434:
4414:
4406:
4390:
4378:
4376:
4374:
4370:
4366:
4362:
4358:
4354:
4353:electrovacuum
4349:
4347:
4343:
4339:
4335:
4331:
4323:
4319:
4315:
4312:
4308:
4304:
4301:
4297:
4293:
4290:
4286:
4282:
4281:
4280:
4274:
4272:
4270:
4266:
4262:
4258:
4254:
4250:
4248:
4244:
4240:
4236:
4233:
4229:
4228:
4222:
4218:
4214:
4212:
4208:
4204:
4200:
4196:
4194:
4173:
4170:
4166:
4159:
4151:
4147:
4143:
4139:
4135:
4133:
4112:
4109:
4105:
4098:
4090:
4086:
4082:
4078:
4074:
4070:
4066:
4047:
4044:
4040:
4033:
4025:
4021:
4017:
4013:
4009:
4005:
4000:
3986:
3961:
3958:
3954:
3947:
3938:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3905:
3903:
3899:
3895:
3891:
3887:
3885:
3881:
3878:According to
3873:
3871:
3868:
3864:
3850:
3828:
3825:
3822:
3819:
3814:
3809:
3779:
3776:
3773:
3770:
3749:
3745:
3741:
3736:
3732:
3728:
3724:
3721:
3715:
3707:
3700:
3692:
3685:
3682:
3679:
3676:
3672:
3666:
3641:
3637:
3633:
3628:
3624:
3620:
3616:
3613:
3607:
3599:
3592:
3584:
3577:
3574:
3571:
3568:
3564:
3556:
3555:
3554:
3536:
3533:
3530:
3527:
3505:
3501:
3495:
3491:
3487:
3484:
3479:
3475:
3469:
3465:
3458:
3455:
3452:
3449:
3445:
3439:
3413:
3409:
3403:
3399:
3395:
3392:
3387:
3383:
3377:
3373:
3366:
3363:
3360:
3357:
3353:
3345:
3344:
3327:
3324:
3303:
3296:
3292:
3274:
3271:
3268:
3265:
3261:
3253:
3252:
3251:
3237:
3213:
3210:
3207:
3204:
3182:
3178:
3172:
3168:
3164:
3161:
3156:
3152:
3146:
3142:
3135:
3132:
3129:
3126:
3122:
3116:
3090:
3086:
3080:
3076:
3072:
3069:
3064:
3060:
3054:
3050:
3043:
3040:
3037:
3034:
3030:
3022:
3021:
3005:
2997:
2979:
2976:
2973:
2970:
2966:
2958:
2957:
2956:
2942:
2917:
2913:
2907:
2903:
2896:
2893:
2890:
2887:
2882:
2877:
2874:
2868:
2865:
2862:
2857:
2853:
2847:
2843:
2836:
2833:
2830:
2827:
2823:
2815:
2814:
2795:
2789:
2781:
2763:
2760:
2757:
2754:
2750:
2741:
2723:
2720:
2717:
2714:
2710:
2701:
2700:
2699:
2685:
2662:
2659:
2654:
2650:
2643:
2640:
2637:
2634:
2630:
2622:
2621:
2602:
2596:
2588:
2570:
2567:
2564:
2561:
2557:
2549:
2548:
2547:
2545:
2541:
2523:
2520:
2517:
2514:
2510:
2489:
2469:
2449:
2446:
2443:
2435:
2419:
2399:
2391:
2383:
2366:
2363:
2358:
2350:
2345:
2337:
2332:
2324:
2319:
2311:
2306:
2293:
2290:
2275:
2272:
2267:
2259:
2254:
2246:
2241:
2233:
2228:
2215:
2212:
2197:
2194:
2189:
2181:
2176:
2168:
2163:
2150:
2147:
2132:
2129:
2124:
2116:
2111:
2103:
2098:
2090:
2085:
2072:
2069:
2054:
2051:
2046:
2038:
2033:
2020:
2017:
2002:
1999:
1994:
1981:
1978:
1977:
1976:
1974:
1953:
1918:
1915:
1912:
1909:
1906:
1901:
1898:
1894:
1888:
1885:
1881:
1875:
1867:
1862:
1849:
1846:
1842:
1836:
1833:
1829:
1825:
1820:
1817:
1813:
1807:
1804:
1800:
1791:
1783:
1778:
1765:
1762:
1758:
1752:
1749:
1745:
1741:
1736:
1733:
1729:
1723:
1720:
1716:
1712:
1707:
1704:
1700:
1694:
1691:
1687:
1678:
1670:
1665:
1652:
1649:
1645:
1639:
1636:
1632:
1628:
1623:
1620:
1616:
1610:
1607:
1603:
1594:
1586:
1581:
1571:
1568:
1564:
1558:
1555:
1551:
1545:
1537:
1535:
1528:
1525:
1522:
1519:
1515:
1503:
1502:
1501:
1484:
1476:
1472:
1466:
1459:
1455:
1452:
1444:
1434:
1425:
1418:
1414:
1411:
1406:
1403:
1399:
1391:
1372:
1368:
1362:
1355:
1351:
1348:
1343:
1340:
1336:
1328:
1309:
1299:
1290:
1283:
1279:
1276:
1273:
1268:
1265:
1261:
1253:
1252:
1251:
1249:
1245:
1241:
1233:
1231:
1229:
1228:
1223:
1219:
1215:
1211:
1206:
1204:
1200:
1196:
1192:
1189:, while type
1188:
1184:
1180:
1176:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1134:
1130:
1125:
1121:
1119:
1115:
1111:
1107:
1102:
1100:
1096:
1092:
1088:
1083:
1080:
1078:
1056:
1053:
1049:
1044:
1041:
1036:
1033:
1029:
1022:
1019:
1012:
1009:
1005:
996:
993:
984:
983:
982:
966:
963:
959:
950:
934:
927:
906:
903:
899:
893:
890:
883:
880:
876:
867:
864:
854:
851:
847:
839:
838:
837:
835:
831:
827:
823:
819:
816:
808:
806:
804:
800:
796:
792:
787:
785:
781:
777:
773:
769:
765:
761:
749:
744:
742:
737:
735:
730:
729:
727:
726:
720:
710:
707:
702:
696:
695:
694:
693:
686:
685:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
623:Chandrasekhar
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
538:Schwarzschild
536:
534:
531:
529:
526:
524:
521:
519:
516:
515:
507:
506:
499:
498:Hartle–Thorne
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
413:
409:
408:Schwarzschild
406:
405:
401:
395:
394:
383:
380:
378:
375:
374:
373:
372:
367:
362:
359:
357:
354:
352:
349:
348:
347:
346:
341:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
307:
306:
305:
300:
290:
287:
286:
281:
280:
269:
266:
264:
261:
259:
256:
255:
254:
253:
250:
246:
241:
238:
236:
233:
231:
230:Event horizon
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
192:
191:
190:
180:
179:
172:
169:
167:
164:
162:
159:
157:
154:
153:
145:
144:
139:
136:
131:
128:
126:
123:
121:
118:
117:
115:
113:
110:
109:
108:
107:
89:
86:
82:
77:
73:
68:
65:
61:
54:
49:
46:
42:
32:
28:
27:
24:
20:
4976:
4957:
4937:
4904:
4900:
4892:
4888:
4863:
4859:
4826:
4822:
4814:
4799:
4792:
4771:
4726:
4722:
4679:
4675:
4644:
4638:
4555:
4550:
4546:
4522:
4382:
4372:
4368:
4364:
4360:
4350:
4345:
4337:
4333:
4327:
4321:
4310:
4299:
4288:
4278:
4264:
4251:
4224:
4219:regions, or
4216:
4215:
4210:
4206:
4202:
4198:
4197:
4192:
4137:
4136:
4131:
4088:
4084:
4073:longitudinal
4068:
4067:
4023:
4001:
3936:
3932:
3916:
3915:) in a type
3913:tidal tensor
3912:
3906:
3893:
3889:
3888:
3883:
3877:
3869:
3865:
3795:
3552:
3294:
3290:
3229:
2995:
2934:
2779:
2739:
2677:
2586:
2544:Bel criteria
2543:
2539:
2433:
2387:
2384:Bel criteria
2291:
2213:
2148:
2070:
2018:
1979:
1973:Weyl scalars
1937:
1499:
1248:null vectors
1237:
1225:
1221:
1217:
1213:
1209:
1207:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1172:
1166:
1160:
1154:
1148:
1142:
1136:
1128:
1118:Petrov types
1117:
1109:
1106:null vectors
1103:
1090:
1084:
1081:
1077:at most four
1076:
1073:
949:eigenvectors
923:
825:
820:such as the
812:
803:Felix Pirani
799:A. Z. Petrov
788:
767:
757:
683:
643:Raychaudhuri
112:Introduction
4895:(8): 55–69.
4475:Weyl tensor
4405:frame basis
4318:FLRW models
4285:Kerr vacuum
4150:wave vector
4134:radiation.
4020:Kerr vacuum
3937:compression
3289:is of type
2994:is of type
2778:is of type
926:eigenvalues
822:Weyl tensor
776:Weyl tensor
658:van Stockum
588:Oppenheimer
443:Kerr–Newman
235:Singularity
4986:Categories
4947:0080123155
4776:. Oxford:
4631:References
4375:examples.
4142:transverse
4016:gyroscopes
3018:satisfying
2811:satisfying
2740:not type N
2618:satisfying
1938:where the
1179:degenerate
981:such that
772:symmetries
511:Scientists
343:Formalisms
291:Formalisms
240:Black hole
166:World line
4851:116370483
4654:0906.3818
4458:−
4371:and type
4357:null dust
4171:−
4110:−
4045:−
3959:−
3810:∗
3777:≠
3774:δ
3771:γ
3725:δ
3667:∗
3617:γ
3534:≠
3531:β
3528:α
3488:β
3440:∗
3396:α
3211:≠
3208:β
3205:α
3165:β
3117:∗
3073:α
2878:∗
2447:∈
2355:Ψ
2342:Ψ
2329:Ψ
2316:Ψ
2303:Ψ
2264:Ψ
2251:Ψ
2238:Ψ
2225:Ψ
2186:Ψ
2173:Ψ
2160:Ψ
2121:Ψ
2108:Ψ
2095:Ψ
2082:Ψ
2043:Ψ
2030:Ψ
1991:Ψ
1950:Ψ
1872:Ψ
1788:Ψ
1675:Ψ
1591:Ψ
1542:Ψ
1453:−
1438:¯
1303:¯
1277:−
1181:to types
1045:λ
935:λ
860:→
830:bivectors
805:in 1957.
603:Robertson
568:Friedmann
563:Eddington
553:Nordström
543:de Sitter
400:Solutions
325:Geodesics
320:Friedmann
302:Equations
288:Equations
249:Spacetime
184:Phenomena
90:ν
87:μ
78:κ
69:ν
66:μ
58:Λ
50:ν
47:μ
4929:73540912
4761:15772816
4714:31859828
4564:See also
4294:certain
4275:Examples
4083:is type
4077:shearing
4069:Type III
3979:, where
3750:′
3737:′
3712:′
3697:′
3642:′
3629:′
3604:′
3589:′
3328:′
2585:is type
2436:at some
2388:Given a
2294: :
2216: :
2151: :
2149:Type III
2073: :
2021: :
1982: :
1971:are the
1155:Type III
778:at each
719:Category
583:Lemaître
548:Reissner
533:Poincaré
518:Einstein
463:Taub–NUT
428:Wormhole
412:interior
125:Timeline
4909:Bibcode
4868:Bibcode
4831:Bibcode
4741:Bibcode
4694:Bibcode
4501:and/or
4199:Type II
3933:tension
3925:Coulomb
2742:, then
2542:), the
2019:Type II
1244:tetrads
1143:Type II
774:of the
638:Hawking
633:Penrose
608:Bardeen
598:Wheeler
528:Hilbert
523:Lorentz
483:pp-wave
120:History
4968:
4944:
4927:
4849:
4806:
4784:
4759:
4712:
4235:energy
4217:Type O
4209:, and
4138:Type N
3890:Type D
3796:where
3230:where
2935:where
2678:where
2390:metric
2292:Type O
2214:Type N
2071:Type D
1980:Type I
1167:Type O
1161:Type N
1149:Type D
1137:Type I
818:tensor
766:, the
717:
684:others
678:Thorne
668:Newman
648:Taylor
628:Ehlers
613:Walker
578:Zwicky
453:Kasner
4925:S2CID
4847:S2CID
4757:S2CID
4731:arXiv
4710:S2CID
4684:arXiv
4649:arXiv
4232:field
4227:Ricci
4225:pure
3927:type
3923:by a
1201:, or
782:in a
780:event
653:Hulse
593:Gödel
573:Milne
468:Milne
433:Gödel
130:Tests
4966:ISBN
4942:ISBN
4804:ISBN
4782:ISBN
4427:and
4336:(or
4316:the
4305:the
4283:the
4247:news
4146:LIGO
4004:axis
3911:(or
3907:The
3553:and
3105:and
1238:The
1127:The
947:and
815:rank
762:and
663:Taub
618:Kerr
558:Weyl
438:Kerr
356:BSSN
4917:doi
4893:114
4876:doi
4839:doi
4749:doi
4702:doi
4659:doi
4369:III
4355:or
4300:III
4207:III
4089:III
4014:on
2780:III
2738:is
2702:If
2502:by
1246:of
1195:III
1185:or
793:of
758:In
673:Yau
351:ADM
4988::
4964:.
4923:.
4915:.
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4891:.
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4862:.
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4725:.
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4700:.
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4680:21
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4657:.
4647:.
4547:II
4365:II
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4195:.
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2996:II
1230:.
1205:.
1197:,
1191:II
1183:II
1120::
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786:.
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4882:.
4878::
4870::
4853:.
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4790:.
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4743::
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4704::
4696::
4686::
4665:.
4661::
4651::
4551:D
4549:,
4533:l
4509:n
4489:l
4461:2
4455:d
4435:n
4415:l
4391:d
4373:N
4346:D
4338:O
4334:D
4324:.
4322:O
4313:,
4311:N
4302:,
4291:,
4289:D
4265:N
4211:N
4203:D
4193:N
4179:)
4174:1
4167:r
4163:(
4160:O
4132:N
4118:)
4113:2
4106:r
4102:(
4099:O
4085:N
4053:)
4048:4
4041:r
4037:(
4034:O
4024:D
3987:r
3967:)
3962:3
3955:r
3951:(
3948:O
3917:D
3894:D
3851:p
3829:d
3826:c
3823:b
3820:a
3815:C
3780:0
3763:(
3746:c
3742:k
3733:a
3729:k
3722:=
3716:d
3708:k
3701:b
3693:k
3686:d
3683:c
3680:b
3677:a
3673:C
3638:c
3634:k
3625:a
3621:k
3614:=
3608:d
3600:k
3593:b
3585:k
3578:d
3575:c
3572:b
3569:a
3565:C
3549:)
3537:0
3520:(
3506:c
3502:k
3496:a
3492:k
3485:=
3480:d
3476:k
3470:b
3466:k
3459:d
3456:c
3453:b
3450:a
3446:C
3414:c
3410:k
3404:a
3400:k
3393:=
3388:d
3384:k
3378:b
3374:k
3367:d
3364:c
3361:b
3358:a
3354:C
3325:k
3304:k
3291:D
3275:d
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3238:k
3226:)
3214:0
3197:(
3183:c
3179:k
3173:a
3169:k
3162:=
3157:d
3153:k
3147:b
3143:k
3136:d
3133:c
3130:b
3127:a
3123:C
3091:c
3087:k
3081:a
3077:k
3070:=
3065:d
3061:k
3055:b
3051:k
3044:d
3041:c
3038:b
3035:a
3031:C
3006:k
2980:d
2977:c
2974:b
2971:a
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2943:k
2918:d
2914:k
2908:b
2904:k
2897:d
2894:c
2891:b
2888:a
2883:C
2869:=
2866:0
2863:=
2858:d
2854:k
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2844:k
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2796:p
2793:(
2790:k
2764:d
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2755:a
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2724:d
2721:c
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2715:a
2711:C
2686:k
2663:0
2660:=
2655:d
2651:k
2644:d
2641:c
2638:b
2635:a
2631:C
2606:)
2603:p
2600:(
2597:k
2587:N
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2568:c
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2540:O
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2490:p
2470:p
2450:M
2444:p
2420:C
2400:M
2379:.
2367:0
2364:=
2359:4
2351:=
2346:3
2338:=
2333:2
2325:=
2320:1
2312:=
2307:0
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2273:=
2268:3
2260:=
2255:2
2247:=
2242:1
2234:=
2229:0
2210:,
2198:0
2195:=
2190:2
2182:=
2177:1
2169:=
2164:0
2145:,
2133:0
2130:=
2125:4
2117:=
2112:3
2104:=
2099:1
2091:=
2086:0
2067:,
2055:0
2052:=
2047:1
2039:=
2034:0
2015:,
2003:0
2000:=
1995:0
1959:}
1954:j
1946:{
1919:.
1916:c
1913:.
1910:c
1907:+
1902:d
1899:c
1895:V
1889:b
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1882:V
1876:4
1868:+
1855:)
1850:d
1847:c
1843:V
1837:b
1834:a
1830:W
1826:+
1821:d
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1814:W
1808:b
1805:a
1801:V
1797:(
1792:3
1784:+
1771:)
1766:d
1763:c
1759:W
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1742:+
1737:d
1734:c
1730:V
1724:b
1721:a
1717:U
1713:+
1708:d
1705:c
1701:U
1695:b
1692:a
1688:V
1684:(
1679:2
1671:+
1658:)
1653:d
1650:c
1646:U
1640:b
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1633:W
1629:+
1624:d
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1608:a
1604:U
1600:(
1595:1
1587:+
1572:d
1569:c
1565:U
1559:b
1556:a
1552:U
1546:0
1538:=
1529:d
1526:c
1523:b
1520:a
1516:C
1485:.
1480:]
1477:b
1473:l
1467:a
1464:[
1460:n
1456:2
1448:]
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1426:a
1423:[
1419:m
1415:2
1412:=
1407:b
1404:a
1400:W
1376:]
1373:b
1369:m
1363:a
1360:[
1356:n
1352:2
1349:=
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1341:a
1337:V
1313:]
1310:b
1300:m
1291:a
1288:[
1284:l
1280:2
1274:=
1269:b
1266:a
1262:U
1222:O
1210:I
1203:D
1199:N
1187:D
1175:I
1057:b
1054:a
1050:X
1042:=
1037:n
1034:m
1030:X
1023:n
1020:m
1013:b
1010:a
1006:C
997:2
994:1
967:b
964:a
960:X
907:n
904:m
900:X
894:n
891:m
884:b
881:a
877:C
868:2
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855:b
852:a
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747:e
740:t
733:v
414:)
410:(
83:T
74:=
62:g
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