709:
39:
4064:
2362:
3809:
1183:
5546:
722:
823:
2037:
3132:
844:
2734:
1921:
2357:{\displaystyle -c^{2}{d\tau }^{2}=-{\frac {1}{4}}\left(3{\sqrt {1-{\frac {r_{g}^{2}}{{\mathcal {R}}^{2}}}}}-{\sqrt {1-{\frac {r^{2}}{{\mathcal {R}}^{2}}}}}\right)^{2}c^{2}dt^{2}+\left(1-{\frac {r^{2}}{{\mathcal {R}}^{2}}}\right)^{-1}dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}
2890:
1178:{\displaystyle c^{2}{d\tau }^{2}=-{\frac {1}{4}}\left(3{\sqrt {1-{\frac {r_{s}}{r_{g}}}}}-{\sqrt {1-{\frac {r^{2}r_{s}}{r_{g}^{3}}}}}\right)^{2}c^{2}dt^{2}+\left(1-{\frac {r^{2}r_{s}}{r_{g}^{3}}}\right)^{-1}dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right),}
4156:-coordinate is the angle measured from the centre of the cap, that is, from "above" the slice. The proper radius of the sphere â intuitively, the length of a measuring rod spanning from its centre to a point on its surface â is half the length of the circular arc, or
2504:
1691:
3239:
2496:
3506:
3127:{\displaystyle V=\int _{0}^{r_{g}}A{\sqrt {g_{rr}}}\,{\rm {d}}r=2\pi \left({\frac {r_{g}^{9/2}\arcsin {\sqrt {\frac {r_{s}}{r_{g}}}}}{r_{s}^{3/2}}}-{\frac {r_{g}^{4}{\sqrt {1-{\frac {r_{s}}{r_{g}}}}}}{r_{s}}}\right)}
110:
4054:
2729:{\displaystyle c^{2}{d\tau }^{2}=\left({\frac {3\cos \eta _{g}-\cos \eta }{2}}\right)^{2}c^{2}dt^{2}-{\frac {dr^{2}}{\cos ^{2}\eta }}-r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}
2406:
2842:
2029:
5249:
3722:
480:
3646:
3322:, which seems to disregard that this is less than the proper radius, and that space inside the body is curved so that the volume formula for a "flat" sphere shouldn't hold at all. However,
1916:{\displaystyle -c^{2}{d\tau }^{2}=\left(1-{\frac {r_{s}}{r}}\right)c^{2}dt^{2}-\left(1-{\frac {r_{s}}{r}}\right)^{-1}dr^{2}-r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right),}
4134:
3606:
4188:
3888:
3772:
3289:
3556:
753:
4205:
Here are the relevant parameters for some astronomical objects, disregarding rotation and inhomogeneities such as deviation from the spherical shape and variation in density.
1573:
4291:
3920:
3150:
2411:
3385:
2881:
1683:
1390:
4319:
4193:
This is a purely geometric visualization and does not imply a physical "fourth spatial dimension" into which space would be curved. (Intrinsic curvature does not imply
3947:
1462:
132:
1957:
1432:
3798:
1648:-coordinate at the body's surface. (This is less than its proper (measurable interior) radius, although for the Earth the difference is only about 1.4 millimetres.)
1357:
4266:
4238:
4154:
3320:
1626:
1495:
1209:
5568:
3405:
4340:
3970:
3340:
1646:
1593:
1519:
1410:
1331:
1311:
1291:
1269:
1243:
337:
5296:
5531:
5320:
5066:
4458:
3346:
mass"), which in general relativity is not necessarily equal to the proper mass. This mass difference exactly cancels out the difference of the volumes.
690:
500:
342:
455:
746:
5102:
5056:
368:
202:
708:
5240:
787:
407:
45:
4724:
3829:
of the interior
Schwarzschild metric can be visualized by taking a slice (1) with constant time and (2) through the sphere's equator, i.e.
1392:(as by the usual formulas), but the warping of space means the proper distance from each shell to the center of the body is greater than
3975:
1685:. For a complete metric of the sphere's gravitational field, the interior Schwarzschild metric has to be matched with the exterior one,
739:
5348:
4977:
3511:
As expected, the pressure is zero at the surface of the sphere and increases towards the centre. It becomes infinite at the centre if
119:
4982:
4564:
3826:
145:
5302:
425:
38:
5015:
4486:
3292:
332:
3815:
of a
Schwarzschild metric's slice in three-dimensional Euclidean space. The interior solution is the darker cap at the bottom.
5118:
2370:
5196:
5259:
465:
5549:
5126:
4462:
2752:
811:
127:
4581:
485:
5113:
4671:
1970:
4873:
5154:
4888:
3673:
831:
726:
217:
137:
3611:
3342:
is the mass measured from the outside, for example by observing a test particle orbiting the gravitating body (the "
5441:
4883:
4836:
630:
178:
430:
5471:
5097:
4778:
4077:
242:
505:
5496:
5046:
4806:
4717:
4518:
3561:
665:
655:
322:
5225:
5147:
384:
4159:
3832:
4767:
4758:
650:
5314:
5220:
3730:
806:
at the surface. This is a static solution, meaning that it does not change over time. It was discovered by
450:
5451:
5178:
4852:
4816:
4343:
3664:
3649:
3247:
1596:
620:
605:
212:
5456:
5061:
5000:
3514:
3295:. It may be counterintuitive that the density is the mass divided by the volume of a sphere with radius
595:
163:
4057:
1271:
is the time coordinate (measured by a stationary clock located infinitely far from the spherical body).
4519:"Ăber das Gravitationsfeld einer Kugel aus inkompressibler FlĂŒssigkeit nach der Einsteinschen Theorie"
3234:{\displaystyle \rho ={\frac {M}{{\frac {4\pi }{3}}r_{g}^{3}}}={\frac {3}{\kappa {\mathcal {R}}^{2}}},}
2491:{\displaystyle \eta _{g}=\arcsin {\frac {r_{g}}{\mathcal {R}}}=\arcsin {\sqrt {\frac {r_{s}}{r_{g}}}}}
5361:
5287:
5131:
4733:
4628:
4526:
4494:
1524:
1498:
795:
415:
4521:[On the gravitational field of a ball of incompressible fluid following Einstein's theory].
4272:
3901:
5436:
5431:
5426:
5421:
5253:
5202:
5020:
5005:
4831:
4762:
4710:
4194:
791:
610:
590:
580:
575:
560:
327:
5325:
3360:
2847:
470:
5501:
5401:
5183:
5164:
5158:
5109:
5051:
4960:
4878:
4796:
4773:
4741:
4652:
4644:
3950:
1655:
807:
771:
660:
545:
317:
222:
207:
168:
30:
1926:
at the surface. It can easily be seen that the two have the same value at the surface, i.e., at
1362:
4297:
3925:
5511:
5381:
5343:
5335:
4936:
4893:
4560:
4063:
1447:
540:
490:
265:
1929:
1417:
5476:
5416:
5406:
5353:
5330:
4908:
4636:
3777:
3656:. As this is a time dependent process, the Schwarzschild solution does not hold any longer.
3501:{\displaystyle p=\rho c^{2}{\frac {\cos \eta -\cos \eta _{g}}{3\cos \eta _{g}-\cos \eta }}.}
1336:
635:
615:
570:
550:
495:
4244:
4216:
4139:
3298:
1604:
1473:
1194:
5491:
5466:
5391:
5386:
5269:
5230:
5192:
5136:
5010:
4946:
3388:
3355:
3343:
767:
670:
645:
530:
525:
389:
270:
232:
5274:
440:
4632:
4621:
Proceedings of the Royal
Society of London. Series A, Mathematical and Physical Sciences
4530:
4498:
5516:
5188:
5172:
5168:
5071:
5041:
4898:
4801:
4461:
solution that was found. It was published on 24 February 1916, only three months after
4325:
3955:
3808:
3325:
1631:
1578:
1504:
1395:
1316:
1296:
1276:
1254:
1246:
1228:
713:
680:
675:
363:
227:
3395:, and has equal values for the three spatial diagonal components, meaning pressure is
5562:
5506:
5486:
5481:
5396:
5264:
5092:
5036:
4868:
4821:
4656:
4070:
In this graphic, the blue circular arc represents the interior metric, and the black
3895:
1220:
640:
600:
555:
535:
460:
358:
237:
5526:
5446:
5411:
4941:
4903:
4428:
4365:
3818:
3392:
835:
585:
565:
17:
4559:] (in German) (4th ed.). Spektrum Akademischer Verlag. pp. 231â241.
4489:[On the gravitational field of a point mass following Einstein's theory].
3648:, which is true for a body that is extremely dense or large. Such a body suffers
5310:
5279:
4826:
4405:
1212:
475:
445:
1599:. (For ordinary stars and planets, this is much less than their proper radius.)
5521:
5087:
4931:
4926:
4592:
4487:"Ăber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie"
4371:
3653:
1435:
1216:
685:
247:
173:
4682:
4918:
3891:
3812:
3667:
for radiation from the sphere's surface (for example, light from a star) is
1465:
834:
with the body's centre located at the origin, plus the time coordinate. Its
256:
4640:
822:
5461:
4811:
4071:
3396:
3354:
The pressure of the incompressible fluid can be found by calculating the
803:
625:
435:
275:
4523:
Sitzungsberichte der Königlich-Preussischen
Akademie der Wissenschaften
4491:
Sitzungsberichte der Königlich-Preussischen
Akademie der Wissenschaften
799:
105:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
4062:
3817:
This embedding should not be confused with the unrelated concept of a
794:
in the interior of a non-rotating spherical body which consists of an
5141:
4648:
3894:
in a three-dimensional
Euclidean space and then takes the shape of a
1439:
1333:
has the geometry of a sphere with measurable (proper) circumference
4056:. As the exterior metric can be embedded in the same way (yielding
4351:
4049:{\displaystyle {\mathcal {R}}^{-2}=r_{s}/r_{g}^{3}=\rho \kappa /3}
3807:
821:
3391:(i.e., all off-diagonal elements are zero), meaning there are no
1293:
5122:
4672:"New Embedding of Schwarzschild Geometry. II. Interior Solution"
4619:
P. S. Florides (1974). "A New
Interior Schwarzschild Solution".
3136:
which is larger than the volume of a euclidean reference shell.
4706:
3144:
The fluid has a constant density by definition. It is given by
4381:
4702:
4278:
4175:
4060:), a slice of the complete solution can be drawn like this:
3982:
3907:
3214:
2447:
2392:
2242:
2163:
2122:
1977:
4136:
represent the exterior metric, or Flamm's paraboloid. The
4465:
2401:{\displaystyle \eta =\arcsin {\frac {r}{\mathcal {R}}}}
4328:
4300:
4275:
4247:
4219:
4162:
4142:
4080:
3978:
3958:
3928:
3904:
3835:
3780:
3733:
3676:
3614:
3564:
3517:
3408:
3363:
3328:
3301:
3250:
3153:
2893:
2850:
2755:
2507:
2414:
2373:
2040:
1973:
1932:
1694:
1658:
1634:
1607:
1581:
1527:
1507:
1476:
1450:
1420:
1398:
1365:
1339:
1319:
1299:
1279:
1257:
1231:
1197:
847:
48:
2837:{\displaystyle g_{rr}=(1-r_{s}r^{2}/r_{g}^{3})^{-1}}
2367:
We can also define an alternative radial coordinate
5374:
5239:
5211:
5080:
5029:
4991:
4970:
4959:
4917:
4861:
4845:
4787:
4751:
4740:
4480:
4478:
4457:The interior Schwarzschild solution was the first
4334:
4313:
4285:
4260:
4232:
4182:
4148:
4128:
4048:
3972:is proportional to the fluid's density and equals
3964:
3941:
3914:
3882:
3792:
3766:
3716:
3640:
3600:
3550:
3500:
3379:
3334:
3314:
3283:
3233:
3126:
2875:
2836:
2728:
2490:
2400:
2356:
2024:{\displaystyle {\mathcal {R}}^{2}=r_{g}^{3}/r_{s}}
2023:
1951:
1915:
1677:
1640:
1620:
1587:
1567:
1513:
1489:
1456:
1426:
1404:
1384:
1351:
1325:
1305:
1285:
1263:
1237:
1203:
1177:
104:
830:The interior Schwarzschild metric is framed in a
3717:{\displaystyle z={\frac {1}{\cos \eta _{g}}}-1.}
1215:(time measured by a clock moving along the same
4582:"Interior Schwarzschild Solution and Free Fall"
4512:
4510:
4508:
3641:{\displaystyle \eta _{g}\approx 70.5^{\circ }}
802:is constant throughout the body) and has zero
4718:
747:
8:
4129:{\displaystyle w=2{\sqrt {r_{s}(r-r_{s})}}}
4967:
4748:
4725:
4711:
4703:
4614:
4612:
4459:static spherically symmetric perfect fluid
1501:of the body, which is related to its mass
754:
740:
306:
196:
26:
4546:
4544:
4542:
4540:
4327:
4305:
4299:
4277:
4276:
4274:
4252:
4246:
4224:
4218:
4174:
4173:
4167:
4161:
4141:
4115:
4096:
4090:
4079:
4038:
4023:
4018:
4009:
4003:
3987:
3981:
3980:
3977:
3957:
3933:
3927:
3906:
3905:
3903:
3872:
3834:
3779:
3756:
3744:
3732:
3699:
3683:
3675:
3632:
3619:
3613:
3601:{\displaystyle r_{s}={\frac {8}{9}}r_{g}}
3592:
3578:
3569:
3563:
3540:
3528:
3516:
3474:
3453:
3428:
3422:
3407:
3368:
3362:
3327:
3306:
3300:
3275:
3266:
3249:
3219:
3213:
3212:
3202:
3190:
3185:
3166:
3160:
3152:
3111:
3096:
3086:
3080:
3072:
3066:
3061:
3054:
3039:
3035:
3030:
3016:
3006:
2999:
2983:
2979:
2974:
2967:
2944:
2943:
2942:
2931:
2925:
2914:
2909:
2904:
2892:
2867:
2849:
2825:
2815:
2810:
2801:
2795:
2785:
2760:
2754:
2712:
2704:
2692:
2679:
2661:
2639:
2627:
2617:
2608:
2595:
2585:
2557:
2541:
2527:
2519:
2512:
2506:
2479:
2469:
2462:
2446:
2440:
2434:
2419:
2413:
2391:
2386:
2372:
2340:
2332:
2320:
2307:
2289:
2276:
2260:
2247:
2241:
2240:
2233:
2227:
2206:
2193:
2183:
2168:
2162:
2161:
2154:
2148:
2140:
2127:
2121:
2120:
2113:
2108:
2102:
2094:
2075:
2063:
2055:
2048:
2039:
2015:
2006:
2000:
1995:
1982:
1976:
1975:
1972:
1943:
1931:
1899:
1891:
1879:
1866:
1848:
1835:
1819:
1803:
1797:
1776:
1763:
1743:
1737:
1717:
1709:
1702:
1693:
1669:
1657:
1633:
1612:
1606:
1580:
1559:
1550:
1532:
1526:
1506:
1481:
1475:
1449:
1419:
1397:
1376:
1364:
1338:
1318:
1298:
1278:
1256:
1230:
1196:
1161:
1153:
1141:
1128:
1110:
1097:
1081:
1068:
1063:
1052:
1042:
1035:
1014:
1001:
991:
976:
971:
960:
950:
943:
935:
922:
912:
906:
898:
879:
867:
859:
852:
846:
93:
84:
72:
53:
47:
4207:
3387:from the metric. The Einstein tensor is
4474:
4183:{\displaystyle \eta _{g}{\mathcal {R}}}
3883:{\displaystyle t=const.,\theta =\pi /2}
376:
350:
309:
255:
29:
2885:the integral for the proper volume is
5569:Exact solutions in general relativity
3767:{\displaystyle \cos \eta _{g}>1/3}
7:
3890:. This two-dimensional slice can be
3284:{\displaystyle \kappa =8\pi G/c^{2}}
5297:TolmanâOppenheimerâVolkoff equation
5250:FriedmannâLemaĂźtreâRobertsonâWalker
810:in 1916, who earlier had found the
3551:{\displaystyle \cos \eta _{g}=1/3}
2945:
65:
25:
5067:HamiltonâJacobiâEinstein equation
5545:
5544:
721:
720:
707:
37:
4679:Austrian Reports on Gravitation
4589:Austrian Reports on Gravitation
3293:Einstein gravitational constant
1568:{\displaystyle r_{s}=2GM/c^{2}}
1438:(angle from north, in units of
780:interior Schwarzschild solution
4874:Massâenergy equivalence (E=mc)
4553:Allgemeine RelativitÀtstheorie
4525:(in German). Berlin: 424â434.
4493:(in German). Berlin: 189â196.
4286:{\displaystyle {\mathcal {R}}}
4121:
4102:
3915:{\displaystyle {\mathcal {R}}}
2822:
2772:
2408:and a corresponding parameter
1:
3727:From the stability condition
812:exterior Schwarzschild metric
776:interior Schwarzschild metric
4557:General Theory of Relativity
3380:{\displaystyle G_{\mu \nu }}
2876:{\displaystyle A=4\pi r^{2}}
784:Schwarzschild fluid solution
4889:Relativistic Doppler effect
4517:Karl Schwarzschild (1916).
4485:Karl Schwarzschild (1916).
1678:{\displaystyle r\leq r_{g}}
1652:This solution is valid for
832:spherical coordinate system
218:Gravitational time dilation
5585:
5360:In computational physics:
4884:Relativity of simultaneity
4551:Torsten FlieĂbach (2003).
4463:Einstein's field equations
1385:{\displaystyle 4\pi r^{2}}
338:MathissonâPapapetrouâDixon
179:Pseudo-Riemannian manifold
5542:
5197:LenseâThirring precession
4779:Doubly special relativity
4314:{\displaystyle \eta _{g}}
3942:{\displaystyle \eta _{g}}
5057:Post-Newtonian formalism
5047:Einstein field equations
4983:Mathematical formulation
4807:Hyperbolic orthogonality
1457:{\displaystyle \varphi }
343:HamiltonâJacobiâEinstein
323:Einstein field equations
146:Mathematical formulation
4768:Galilean transformation
4759:Principle of relativity
4074:arcs with the equation
3922:and half opening angle
3558:, which corresponds to
1952:{\displaystyle r=r_{g}}
1427:{\displaystyle \theta }
4853:Lorentz transformation
4641:10.1098/rspa.1974.0065
4394:19 light-minutes
4336:
4315:
4287:
4262:
4234:
4184:
4150:
4130:
4067:
4050:
3966:
3943:
3916:
3884:
3822:
3794:
3793:{\displaystyle z<2}
3768:
3718:
3665:Gravitational redshift
3650:gravitational collapse
3642:
3602:
3552:
3502:
3381:
3350:Pressure and stability
3336:
3316:
3285:
3235:
3128:
2877:
2838:
2730:
2492:
2402:
2358:
2025:
1953:
1917:
1679:
1642:
1622:
1597:gravitational constant
1589:
1569:
1515:
1491:
1458:
1428:
1406:
1386:
1353:
1352:{\displaystyle 2\pi r}
1327:
1307:
1287:
1265:
1239:
1205:
1179:
827:
213:Gravitational redshift
106:
5321:WeylâLewisâPapapetrou
5062:Raychaudhuri equation
5001:Equivalence principle
4670:R. Burghardt (2009).
4580:R. Burghardt (2009).
4337:
4316:
4288:
4263:
4261:{\displaystyle r_{s}}
4235:
4233:{\displaystyle r_{g}}
4185:
4151:
4149:{\displaystyle \eta }
4131:
4066:
4051:
3967:
3944:
3917:
3885:
3811:
3795:
3769:
3719:
3643:
3603:
3553:
3503:
3382:
3337:
3317:
3315:{\displaystyle r_{g}}
3286:
3236:
3129:
2878:
2839:
2731:
2493:
2403:
2359:
2026:
1967:Defining a parameter
1954:
1918:
1680:
1643:
1623:
1621:{\displaystyle r_{g}}
1590:
1570:
1516:
1492:
1490:{\displaystyle r_{s}}
1459:
1429:
1407:
1387:
1354:
1328:
1308:
1288:
1266:
1240:
1206:
1204:{\displaystyle \tau }
1180:
826:Spherical coordinates
825:
501:WeylâLewisâPapapetrou
456:KerrâNewmanâde Sitter
276:EinsteinâRosen bridge
208:Gravitational lensing
164:Equivalence principle
107:
5362:Numerical relativity
5203:pulsar timing arrays
4431:with 2 solar masses
4326:
4298:
4273:
4245:
4217:
4160:
4140:
4078:
3976:
3956:
3926:
3902:
3833:
3778:
3731:
3674:
3612:
3562:
3515:
3406:
3361:
3326:
3299:
3248:
3151:
2891:
2848:
2753:
2505:
2412:
2371:
2038:
1971:
1930:
1692:
1656:
1632:
1628:is the value of the
1605:
1579:
1525:
1505:
1499:Schwarzschild radius
1474:
1448:
1418:
1396:
1363:
1337:
1317:
1297:
1277:
1255:
1229:
1195:
845:
796:incompressible fluid
431:EinsteinâRosen waves
157:Fundamental concepts
46:
5254:Friedmann equations
5148:HulseâTaylor binary
5110:Gravitational waves
5006:Riemannian geometry
4832:Proper acceleration
4817:Maxwell's equations
4763:Galilean relativity
4633:1974RSPSA.337..529F
4531:1916skpa.conf..424S
4499:1916SPAW.......189S
4392:338,000,000 km
4362:170,000,000 km
4195:extrinsic curvature
4028:
3195:
3071:
3048:
2992:
2921:
2820:
2118:
2005:
1073:
981:
792:gravitational field
385:KaluzaâKlein theory
271:Minkowski spacetime
223:Gravitational waves
18:Schwarzschild fluid
5303:ReissnerâNordström
5221:BransâDicke theory
5052:Linearized gravity
4879:Length contraction
4797:Frame of reference
4774:Special relativity
4408:with 1 solar mass
4332:
4311:
4283:
4258:
4230:
4180:
4146:
4126:
4068:
4058:Flamm's paraboloid
4046:
4014:
3962:
3951:Gaussian curvature
3939:
3912:
3880:
3823:
3790:
3764:
3714:
3638:
3598:
3548:
3498:
3377:
3332:
3312:
3281:
3231:
3181:
3124:
3057:
3026:
2970:
2900:
2873:
2834:
2806:
2726:
2488:
2398:
2354:
2104:
2021:
1991:
1963:Other formulations
1949:
1913:
1675:
1638:
1618:
1585:
1565:
1511:
1487:
1468:(also in radians).
1454:
1424:
1402:
1382:
1349:
1323:
1303:
1283:
1261:
1235:
1201:
1175:
1059:
967:
828:
808:Karl Schwarzschild
772:general relativity
714:Physics portal
486:OppenheimerâSnyder
426:ReissnerâNordström
318:Linearized gravity
266:Spacetime diagrams
169:Special relativity
102:
31:General relativity
5556:
5555:
5370:
5369:
5349:OzsvĂĄthâSchĂŒcking
4955:
4954:
4937:Minkowski diagram
4894:Thomas precession
4837:Relativistic mass
4627:(1611): 529â535.
4450:
4449:
4335:{\displaystyle z}
4124:
3965:{\displaystyle K}
3706:
3586:
3493:
3335:{\displaystyle M}
3226:
3197:
3179:
3117:
3104:
3102:
3049:
3023:
3022:
2940:
2652:
2579:
2486:
2485:
2451:
2396:
2253:
2176:
2174:
2135:
2133:
2083:
1812:
1752:
1641:{\displaystyle r}
1588:{\displaystyle G}
1514:{\displaystyle M}
1405:{\displaystyle r}
1326:{\displaystyle r}
1306:{\displaystyle t}
1286:{\displaystyle r}
1264:{\displaystyle t}
1238:{\displaystyle c}
1074:
984:
982:
930:
928:
887:
764:
763:
397:
396:
283:
282:
16:(Redirected from
5576:
5548:
5547:
5331:van Stockum dust
5103:Two-body problem
5021:Mach's principle
4968:
4909:Terrell rotation
4749:
4727:
4720:
4713:
4704:
4697:
4696:
4694:
4693:
4687:
4681:. Archived from
4676:
4667:
4661:
4660:
4616:
4607:
4606:
4604:
4603:
4597:
4591:. Archived from
4586:
4577:
4571:
4570:
4548:
4535:
4534:
4514:
4503:
4502:
4482:
4417:200,000 km
4386:696,000 km
4341:
4339:
4338:
4333:
4320:
4318:
4317:
4312:
4310:
4309:
4292:
4290:
4289:
4284:
4282:
4281:
4267:
4265:
4264:
4259:
4257:
4256:
4239:
4237:
4236:
4231:
4229:
4228:
4208:
4189:
4187:
4186:
4181:
4179:
4178:
4172:
4171:
4155:
4153:
4152:
4147:
4135:
4133:
4132:
4127:
4125:
4120:
4119:
4101:
4100:
4091:
4055:
4053:
4052:
4047:
4042:
4027:
4022:
4013:
4008:
4007:
3995:
3994:
3986:
3985:
3971:
3969:
3968:
3963:
3948:
3946:
3945:
3940:
3938:
3937:
3921:
3919:
3918:
3913:
3911:
3910:
3889:
3887:
3886:
3881:
3876:
3799:
3797:
3796:
3791:
3773:
3771:
3770:
3765:
3760:
3749:
3748:
3723:
3721:
3720:
3715:
3707:
3705:
3704:
3703:
3684:
3647:
3645:
3644:
3639:
3637:
3636:
3624:
3623:
3607:
3605:
3604:
3599:
3597:
3596:
3587:
3579:
3574:
3573:
3557:
3555:
3554:
3549:
3544:
3533:
3532:
3507:
3505:
3504:
3499:
3494:
3492:
3479:
3478:
3459:
3458:
3457:
3429:
3427:
3426:
3386:
3384:
3383:
3378:
3376:
3375:
3341:
3339:
3338:
3333:
3321:
3319:
3318:
3313:
3311:
3310:
3290:
3288:
3287:
3282:
3280:
3279:
3270:
3240:
3238:
3237:
3232:
3227:
3225:
3224:
3223:
3218:
3217:
3203:
3198:
3196:
3194:
3189:
3180:
3175:
3167:
3161:
3133:
3131:
3130:
3125:
3123:
3119:
3118:
3116:
3115:
3106:
3105:
3103:
3101:
3100:
3091:
3090:
3081:
3073:
3070:
3065:
3055:
3050:
3047:
3043:
3034:
3025:
3024:
3021:
3020:
3011:
3010:
3001:
3000:
2991:
2987:
2978:
2968:
2949:
2948:
2941:
2939:
2938:
2926:
2920:
2919:
2918:
2908:
2882:
2880:
2879:
2874:
2872:
2871:
2843:
2841:
2840:
2835:
2833:
2832:
2819:
2814:
2805:
2800:
2799:
2790:
2789:
2768:
2767:
2735:
2733:
2732:
2727:
2722:
2718:
2717:
2716:
2697:
2696:
2684:
2683:
2666:
2665:
2653:
2651:
2644:
2643:
2633:
2632:
2631:
2618:
2613:
2612:
2600:
2599:
2590:
2589:
2584:
2580:
2575:
2562:
2561:
2542:
2532:
2531:
2526:
2517:
2516:
2497:
2495:
2494:
2489:
2487:
2484:
2483:
2474:
2473:
2464:
2463:
2452:
2450:
2445:
2444:
2435:
2424:
2423:
2407:
2405:
2404:
2399:
2397:
2395:
2387:
2363:
2361:
2360:
2355:
2350:
2346:
2345:
2344:
2325:
2324:
2312:
2311:
2294:
2293:
2281:
2280:
2268:
2267:
2259:
2255:
2254:
2252:
2251:
2246:
2245:
2238:
2237:
2228:
2211:
2210:
2198:
2197:
2188:
2187:
2182:
2178:
2177:
2175:
2173:
2172:
2167:
2166:
2159:
2158:
2149:
2141:
2136:
2134:
2132:
2131:
2126:
2125:
2117:
2112:
2103:
2095:
2084:
2076:
2068:
2067:
2062:
2053:
2052:
2030:
2028:
2027:
2022:
2020:
2019:
2010:
2004:
1999:
1987:
1986:
1981:
1980:
1958:
1956:
1955:
1950:
1948:
1947:
1922:
1920:
1919:
1914:
1909:
1905:
1904:
1903:
1884:
1883:
1871:
1870:
1853:
1852:
1840:
1839:
1827:
1826:
1818:
1814:
1813:
1808:
1807:
1798:
1781:
1780:
1768:
1767:
1758:
1754:
1753:
1748:
1747:
1738:
1722:
1721:
1716:
1707:
1706:
1684:
1682:
1681:
1676:
1674:
1673:
1647:
1645:
1644:
1639:
1627:
1625:
1624:
1619:
1617:
1616:
1594:
1592:
1591:
1586:
1574:
1572:
1571:
1566:
1564:
1563:
1554:
1537:
1536:
1520:
1518:
1517:
1512:
1496:
1494:
1493:
1488:
1486:
1485:
1463:
1461:
1460:
1455:
1433:
1431:
1430:
1425:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1381:
1380:
1358:
1356:
1355:
1350:
1332:
1330:
1329:
1324:
1312:
1310:
1309:
1304:
1292:
1290:
1289:
1284:
1270:
1268:
1267:
1262:
1244:
1242:
1241:
1236:
1210:
1208:
1207:
1202:
1184:
1182:
1181:
1176:
1171:
1167:
1166:
1165:
1146:
1145:
1133:
1132:
1115:
1114:
1102:
1101:
1089:
1088:
1080:
1076:
1075:
1072:
1067:
1058:
1057:
1056:
1047:
1046:
1036:
1019:
1018:
1006:
1005:
996:
995:
990:
986:
985:
983:
980:
975:
966:
965:
964:
955:
954:
944:
936:
931:
929:
927:
926:
917:
916:
907:
899:
888:
880:
872:
871:
866:
857:
856:
756:
749:
742:
729:
724:
723:
716:
712:
711:
496:van Stockum dust
481:RobertsonâWalker
307:
197:
111:
109:
108:
103:
101:
100:
88:
80:
79:
61:
60:
41:
27:
21:
5584:
5583:
5579:
5578:
5577:
5575:
5574:
5573:
5559:
5558:
5557:
5552:
5538:
5366:
5270:BKL singularity
5260:LemaĂźtreâTolman
5235:
5231:Quantum gravity
5213:
5207:
5193:geodetic effect
5167:(together with
5137:LISA Pathfinder
5076:
5025:
5011:Penrose diagram
4993:
4987:
4962:
4951:
4947:Minkowski space
4913:
4857:
4841:
4789:
4783:
4743:
4736:
4731:
4701:
4700:
4691:
4689:
4685:
4674:
4669:
4668:
4664:
4618:
4617:
4610:
4601:
4599:
4595:
4584:
4579:
4578:
4574:
4567:
4550:
4549:
4538:
4516:
4515:
4506:
4484:
4483:
4476:
4471:
4455:
4393:
4363:
4324:
4323:
4301:
4296:
4295:
4271:
4270:
4248:
4243:
4242:
4220:
4215:
4214:
4203:
4163:
4158:
4157:
4138:
4137:
4111:
4092:
4076:
4075:
3999:
3979:
3974:
3973:
3954:
3953:
3929:
3924:
3923:
3900:
3899:
3831:
3830:
3816:
3806:
3776:
3775:
3740:
3729:
3728:
3695:
3688:
3672:
3671:
3662:
3628:
3615:
3610:
3609:
3588:
3565:
3560:
3559:
3524:
3513:
3512:
3470:
3460:
3449:
3430:
3418:
3404:
3403:
3399:. Its value is
3364:
3359:
3358:
3356:Einstein tensor
3352:
3324:
3323:
3302:
3297:
3296:
3271:
3246:
3245:
3211:
3207:
3168:
3165:
3149:
3148:
3142:
3107:
3092:
3082:
3056:
3012:
3002:
2969:
2966:
2962:
2927:
2910:
2889:
2888:
2863:
2846:
2845:
2821:
2791:
2781:
2756:
2751:
2750:
2747:
2742:
2708:
2688:
2675:
2671:
2667:
2657:
2635:
2634:
2623:
2619:
2604:
2591:
2553:
2543:
2537:
2536:
2518:
2508:
2503:
2502:
2475:
2465:
2436:
2415:
2410:
2409:
2369:
2368:
2336:
2316:
2303:
2299:
2295:
2285:
2272:
2239:
2229:
2220:
2216:
2215:
2202:
2189:
2160:
2150:
2119:
2090:
2086:
2085:
2054:
2044:
2036:
2035:
2011:
1974:
1969:
1968:
1965:
1939:
1928:
1927:
1895:
1875:
1862:
1858:
1854:
1844:
1831:
1799:
1790:
1786:
1785:
1772:
1759:
1739:
1730:
1726:
1708:
1698:
1690:
1689:
1665:
1654:
1653:
1630:
1629:
1608:
1603:
1602:
1577:
1576:
1555:
1528:
1523:
1522:
1503:
1502:
1477:
1472:
1471:
1446:
1445:
1416:
1415:
1394:
1393:
1372:
1361:
1360:
1335:
1334:
1315:
1314:
1295:
1294:
1275:
1274:
1253:
1252:
1227:
1226:
1193:
1192:
1157:
1137:
1124:
1120:
1116:
1106:
1093:
1048:
1038:
1037:
1028:
1024:
1023:
1010:
997:
956:
946:
945:
918:
908:
894:
890:
889:
858:
848:
843:
842:
820:
798:(implying that
760:
719:
706:
705:
698:
697:
521:
520:
511:
510:
466:LemaĂźtreâTolman
411:
410:
399:
398:
390:Quantum gravity
377:Advanced theory
304:
303:
302:
285:
284:
233:Geodetic effect
194:
193:
184:
183:
159:
158:
142:
112:
89:
68:
49:
44:
43:
23:
22:
15:
12:
11:
5:
5582:
5580:
5572:
5571:
5561:
5560:
5554:
5553:
5543:
5540:
5539:
5537:
5536:
5529:
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5457:Choquet-Bruhat
5454:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5378:
5376:
5372:
5371:
5368:
5367:
5365:
5364:
5357:
5356:
5351:
5346:
5339:
5338:
5333:
5328:
5323:
5318:
5309:Axisymmetric:
5306:
5305:
5300:
5294:
5283:
5282:
5277:
5272:
5267:
5262:
5257:
5248:Cosmological:
5245:
5243:
5237:
5236:
5234:
5233:
5228:
5223:
5217:
5215:
5209:
5208:
5206:
5205:
5200:
5189:frame-dragging
5186:
5181:
5176:
5173:Einstein rings
5169:Einstein cross
5162:
5151:
5150:
5145:
5139:
5134:
5129:
5116:
5106:
5105:
5100:
5095:
5090:
5084:
5082:
5078:
5077:
5075:
5074:
5072:Ernst equation
5069:
5064:
5059:
5054:
5049:
5044:
5042:BSSN formalism
5039:
5033:
5031:
5027:
5026:
5024:
5023:
5018:
5013:
5008:
5003:
4997:
4995:
4989:
4988:
4986:
4985:
4980:
4974:
4972:
4965:
4957:
4956:
4953:
4952:
4950:
4949:
4944:
4939:
4934:
4929:
4923:
4921:
4915:
4914:
4912:
4911:
4906:
4901:
4899:Ladder paradox
4896:
4891:
4886:
4881:
4876:
4871:
4865:
4863:
4859:
4858:
4856:
4855:
4849:
4847:
4843:
4842:
4840:
4839:
4834:
4829:
4824:
4819:
4814:
4809:
4804:
4802:Speed of light
4799:
4793:
4791:
4785:
4784:
4782:
4781:
4776:
4771:
4765:
4755:
4753:
4746:
4738:
4737:
4732:
4730:
4729:
4722:
4715:
4707:
4699:
4698:
4662:
4608:
4572:
4565:
4536:
4504:
4473:
4472:
4470:
4467:
4454:
4451:
4448:
4447:
4444:
4441:
4438:
4435:
4432:
4425:
4424:
4421:
4418:
4415:
4412:
4409:
4402:
4401:
4398:
4395:
4390:
4387:
4384:
4378:
4377:
4374:
4368:
4360:
4357:
4356:6,370 km
4354:
4348:
4347:
4331:
4321:
4308:
4304:
4293:
4280:
4268:
4255:
4251:
4240:
4227:
4223:
4212:
4202:
4199:
4177:
4170:
4166:
4145:
4123:
4118:
4114:
4110:
4107:
4104:
4099:
4095:
4089:
4086:
4083:
4045:
4041:
4037:
4034:
4031:
4026:
4021:
4017:
4012:
4006:
4002:
3998:
3993:
3990:
3984:
3961:
3936:
3932:
3909:
3879:
3875:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3805:
3802:
3789:
3786:
3783:
3763:
3759:
3755:
3752:
3747:
3743:
3739:
3736:
3725:
3724:
3713:
3710:
3702:
3698:
3694:
3691:
3687:
3682:
3679:
3661:
3658:
3635:
3631:
3627:
3622:
3618:
3595:
3591:
3585:
3582:
3577:
3572:
3568:
3547:
3543:
3539:
3536:
3531:
3527:
3523:
3520:
3509:
3508:
3497:
3491:
3488:
3485:
3482:
3477:
3473:
3469:
3466:
3463:
3456:
3452:
3448:
3445:
3442:
3439:
3436:
3433:
3425:
3421:
3417:
3414:
3411:
3393:shear stresses
3374:
3371:
3367:
3351:
3348:
3331:
3309:
3305:
3278:
3274:
3269:
3265:
3262:
3259:
3256:
3253:
3242:
3241:
3230:
3222:
3216:
3210:
3206:
3201:
3193:
3188:
3184:
3178:
3174:
3171:
3164:
3159:
3156:
3141:
3138:
3122:
3114:
3110:
3099:
3095:
3089:
3085:
3079:
3076:
3069:
3064:
3060:
3053:
3046:
3042:
3038:
3033:
3029:
3019:
3015:
3009:
3005:
2998:
2995:
2990:
2986:
2982:
2977:
2973:
2965:
2961:
2958:
2955:
2952:
2947:
2937:
2934:
2930:
2924:
2917:
2913:
2907:
2903:
2899:
2896:
2870:
2866:
2862:
2859:
2856:
2853:
2831:
2828:
2824:
2818:
2813:
2809:
2804:
2798:
2794:
2788:
2784:
2780:
2777:
2774:
2771:
2766:
2763:
2759:
2746:
2743:
2741:
2738:
2737:
2736:
2725:
2721:
2715:
2711:
2707:
2703:
2700:
2695:
2691:
2687:
2682:
2678:
2674:
2670:
2664:
2660:
2656:
2650:
2647:
2642:
2638:
2630:
2626:
2622:
2616:
2611:
2607:
2603:
2598:
2594:
2588:
2583:
2578:
2574:
2571:
2568:
2565:
2560:
2556:
2552:
2549:
2546:
2540:
2535:
2530:
2525:
2522:
2515:
2511:
2482:
2478:
2472:
2468:
2461:
2458:
2455:
2449:
2443:
2439:
2433:
2430:
2427:
2422:
2418:
2394:
2390:
2385:
2382:
2379:
2376:
2365:
2364:
2353:
2349:
2343:
2339:
2335:
2331:
2328:
2323:
2319:
2315:
2310:
2306:
2302:
2298:
2292:
2288:
2284:
2279:
2275:
2271:
2266:
2263:
2258:
2250:
2244:
2236:
2232:
2226:
2223:
2219:
2214:
2209:
2205:
2201:
2196:
2192:
2186:
2181:
2171:
2165:
2157:
2153:
2147:
2144:
2139:
2130:
2124:
2116:
2111:
2107:
2101:
2098:
2093:
2089:
2082:
2079:
2074:
2071:
2066:
2061:
2058:
2051:
2047:
2043:
2018:
2014:
2009:
2003:
1998:
1994:
1990:
1985:
1979:
1964:
1961:
1946:
1942:
1938:
1935:
1924:
1923:
1912:
1908:
1902:
1898:
1894:
1890:
1887:
1882:
1878:
1874:
1869:
1865:
1861:
1857:
1851:
1847:
1843:
1838:
1834:
1830:
1825:
1822:
1817:
1811:
1806:
1802:
1796:
1793:
1789:
1784:
1779:
1775:
1771:
1766:
1762:
1757:
1751:
1746:
1742:
1736:
1733:
1729:
1725:
1720:
1715:
1712:
1705:
1701:
1697:
1672:
1668:
1664:
1661:
1650:
1649:
1637:
1615:
1611:
1600:
1584:
1562:
1558:
1553:
1549:
1546:
1543:
1540:
1535:
1531:
1510:
1484:
1480:
1469:
1453:
1443:
1423:
1413:
1401:
1379:
1375:
1371:
1368:
1348:
1345:
1342:
1322:
1302:
1282:
1272:
1260:
1250:
1247:speed of light
1234:
1224:
1200:
1186:
1185:
1174:
1170:
1164:
1160:
1156:
1152:
1149:
1144:
1140:
1136:
1131:
1127:
1123:
1119:
1113:
1109:
1105:
1100:
1096:
1092:
1087:
1084:
1079:
1071:
1066:
1062:
1055:
1051:
1045:
1041:
1034:
1031:
1027:
1022:
1017:
1013:
1009:
1004:
1000:
994:
989:
979:
974:
970:
963:
959:
953:
949:
942:
939:
934:
925:
921:
915:
911:
905:
902:
897:
893:
886:
883:
878:
875:
870:
865:
862:
855:
851:
819:
816:
788:exact solution
762:
761:
759:
758:
751:
744:
736:
733:
732:
731:
730:
717:
700:
699:
696:
695:
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:
528:
522:
518:
517:
516:
513:
512:
509:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
423:
412:
406:
405:
404:
401:
400:
395:
394:
393:
392:
387:
379:
378:
374:
373:
372:
371:
369:Post-Newtonian
366:
361:
353:
352:
348:
347:
346:
345:
340:
335:
330:
325:
320:
312:
311:
305:
301:
300:
297:
293:
292:
291:
290:
287:
286:
281:
280:
279:
278:
273:
268:
260:
259:
253:
252:
251:
250:
245:
240:
235:
230:
228:Frame-dragging
225:
220:
215:
210:
205:
203:Kepler problem
195:
191:
190:
189:
186:
185:
182:
181:
176:
171:
166:
160:
156:
155:
154:
151:
150:
149:
148:
143:
141:
140:
135:
130:
124:
122:
114:
113:
99:
96:
92:
87:
83:
78:
75:
71:
67:
64:
59:
56:
52:
42:
34:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5581:
5570:
5567:
5566:
5564:
5551:
5541:
5535:
5534:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5442:Chandrasekhar
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5402:Schwarzschild
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5379:
5377:
5373:
5363:
5359:
5358:
5355:
5352:
5350:
5347:
5345:
5341:
5340:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5316:
5312:
5308:
5307:
5304:
5301:
5298:
5295:
5293:
5289:
5288:Schwarzschild
5285:
5284:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5255:
5251:
5247:
5246:
5244:
5242:
5238:
5232:
5229:
5227:
5224:
5222:
5219:
5218:
5216:
5210:
5204:
5201:
5198:
5194:
5190:
5187:
5185:
5184:Shapiro delay
5182:
5180:
5177:
5174:
5170:
5166:
5163:
5160:
5156:
5153:
5152:
5149:
5146:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5127:collaboration
5124:
5120:
5117:
5115:
5111:
5108:
5107:
5104:
5101:
5099:
5096:
5094:
5093:Event horizon
5091:
5089:
5086:
5085:
5083:
5079:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5037:ADM formalism
5035:
5034:
5032:
5028:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4990:
4984:
4981:
4979:
4976:
4975:
4973:
4969:
4966:
4964:
4958:
4948:
4945:
4943:
4942:Biquaternions
4940:
4938:
4935:
4933:
4930:
4928:
4925:
4924:
4922:
4920:
4916:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4869:Time dilation
4867:
4866:
4864:
4860:
4854:
4851:
4850:
4848:
4844:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4822:Proper length
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4794:
4792:
4786:
4780:
4777:
4775:
4772:
4769:
4766:
4764:
4760:
4757:
4756:
4754:
4750:
4747:
4745:
4739:
4735:
4728:
4723:
4721:
4716:
4714:
4709:
4708:
4705:
4688:on 2016-05-08
4684:
4680:
4673:
4666:
4663:
4658:
4654:
4650:
4646:
4642:
4638:
4634:
4630:
4626:
4622:
4615:
4613:
4609:
4598:on 2017-03-05
4594:
4590:
4583:
4576:
4573:
4568:
4566:3-8274-1356-7
4562:
4558:
4554:
4547:
4545:
4543:
4541:
4537:
4532:
4528:
4524:
4520:
4513:
4511:
4509:
4505:
4500:
4496:
4492:
4488:
4481:
4479:
4475:
4468:
4466:
4464:
4460:
4452:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4426:
4422:
4419:
4416:
4414:2.95 km
4413:
4411:5000 km
4410:
4407:
4404:
4403:
4399:
4396:
4391:
4389:2.95 km
4388:
4385:
4383:
4380:
4379:
4375:
4373:
4369:
4367:
4366:light-minutes
4361:
4359:8.87 mm
4358:
4355:
4353:
4350:
4349:
4345:
4329:
4322:
4306:
4302:
4294:
4269:
4253:
4249:
4241:
4225:
4221:
4213:
4210:
4209:
4206:
4200:
4198:
4196:
4191:
4168:
4164:
4143:
4116:
4112:
4108:
4105:
4097:
4093:
4087:
4084:
4081:
4073:
4065:
4061:
4059:
4043:
4039:
4035:
4032:
4029:
4024:
4019:
4015:
4010:
4004:
4000:
3996:
3991:
3988:
3959:
3952:
3934:
3930:
3897:
3896:spherical cap
3893:
3877:
3873:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3828:
3820:
3814:
3810:
3804:Visualization
3803:
3801:
3787:
3784:
3781:
3761:
3757:
3753:
3750:
3745:
3741:
3737:
3734:
3711:
3708:
3700:
3696:
3692:
3689:
3685:
3680:
3677:
3670:
3669:
3668:
3666:
3659:
3657:
3655:
3651:
3633:
3629:
3625:
3620:
3616:
3593:
3589:
3583:
3580:
3575:
3570:
3566:
3545:
3541:
3537:
3534:
3529:
3525:
3521:
3518:
3495:
3489:
3486:
3483:
3480:
3475:
3471:
3467:
3464:
3461:
3454:
3450:
3446:
3443:
3440:
3437:
3434:
3431:
3423:
3419:
3415:
3412:
3409:
3402:
3401:
3400:
3398:
3394:
3390:
3372:
3369:
3365:
3357:
3349:
3347:
3345:
3329:
3307:
3303:
3294:
3276:
3272:
3267:
3263:
3260:
3257:
3254:
3251:
3228:
3220:
3208:
3204:
3199:
3191:
3186:
3182:
3176:
3172:
3169:
3162:
3157:
3154:
3147:
3146:
3145:
3139:
3137:
3134:
3120:
3112:
3108:
3097:
3093:
3087:
3083:
3077:
3074:
3067:
3062:
3058:
3051:
3044:
3040:
3036:
3031:
3027:
3017:
3013:
3007:
3003:
2996:
2993:
2988:
2984:
2980:
2975:
2971:
2963:
2959:
2956:
2953:
2950:
2935:
2932:
2928:
2922:
2915:
2911:
2905:
2901:
2897:
2894:
2886:
2883:
2868:
2864:
2860:
2857:
2854:
2851:
2844:and the area
2829:
2826:
2816:
2811:
2807:
2802:
2796:
2792:
2786:
2782:
2778:
2775:
2769:
2764:
2761:
2757:
2744:
2739:
2723:
2719:
2713:
2709:
2705:
2701:
2698:
2693:
2689:
2685:
2680:
2676:
2672:
2668:
2662:
2658:
2654:
2648:
2645:
2640:
2636:
2628:
2624:
2620:
2614:
2609:
2605:
2601:
2596:
2592:
2586:
2581:
2576:
2572:
2569:
2566:
2563:
2558:
2554:
2550:
2547:
2544:
2538:
2533:
2528:
2523:
2520:
2513:
2509:
2501:
2500:
2499:
2480:
2476:
2470:
2466:
2459:
2456:
2453:
2441:
2437:
2431:
2428:
2425:
2420:
2416:
2388:
2383:
2380:
2377:
2374:
2351:
2347:
2341:
2337:
2333:
2329:
2326:
2321:
2317:
2313:
2308:
2304:
2300:
2296:
2290:
2286:
2282:
2277:
2273:
2269:
2264:
2261:
2256:
2248:
2234:
2230:
2224:
2221:
2217:
2212:
2207:
2203:
2199:
2194:
2190:
2184:
2179:
2169:
2155:
2151:
2145:
2142:
2137:
2128:
2114:
2109:
2105:
2099:
2096:
2091:
2087:
2080:
2077:
2072:
2069:
2064:
2059:
2056:
2049:
2045:
2041:
2034:
2033:
2032:
2016:
2012:
2007:
2001:
1996:
1992:
1988:
1983:
1962:
1960:
1944:
1940:
1936:
1933:
1910:
1906:
1900:
1896:
1892:
1888:
1885:
1880:
1876:
1872:
1867:
1863:
1859:
1855:
1849:
1845:
1841:
1836:
1832:
1828:
1823:
1820:
1815:
1809:
1804:
1800:
1794:
1791:
1787:
1782:
1777:
1773:
1769:
1764:
1760:
1755:
1749:
1744:
1740:
1734:
1731:
1727:
1723:
1718:
1713:
1710:
1703:
1699:
1695:
1688:
1687:
1686:
1670:
1666:
1662:
1659:
1635:
1613:
1609:
1601:
1598:
1582:
1560:
1556:
1551:
1547:
1544:
1541:
1538:
1533:
1529:
1508:
1500:
1482:
1478:
1470:
1467:
1451:
1444:
1441:
1437:
1421:
1414:
1399:
1377:
1373:
1369:
1366:
1346:
1343:
1340:
1320:
1300:
1280:
1273:
1258:
1251:
1248:
1232:
1225:
1222:
1221:test particle
1218:
1214:
1198:
1191:
1190:
1189:
1172:
1168:
1162:
1158:
1154:
1150:
1147:
1142:
1138:
1134:
1129:
1125:
1121:
1117:
1111:
1107:
1103:
1098:
1094:
1090:
1085:
1082:
1077:
1069:
1064:
1060:
1053:
1049:
1043:
1039:
1032:
1029:
1025:
1020:
1015:
1011:
1007:
1002:
998:
992:
987:
977:
972:
968:
961:
957:
951:
947:
940:
937:
932:
923:
919:
913:
909:
903:
900:
895:
891:
884:
881:
876:
873:
868:
863:
860:
853:
849:
841:
840:
839:
837:
833:
824:
817:
815:
813:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
770:'s theory of
769:
757:
752:
750:
745:
743:
738:
737:
735:
734:
728:
718:
715:
710:
704:
703:
702:
701:
694:
693:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
631:Chandrasekhar
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:
549:
547:
546:Schwarzschild
544:
542:
539:
537:
534:
532:
529:
527:
524:
523:
515:
514:
507:
506:HartleâThorne
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:
427:
424:
421:
417:
416:Schwarzschild
414:
413:
409:
403:
402:
391:
388:
386:
383:
382:
381:
380:
375:
370:
367:
365:
362:
360:
357:
356:
355:
354:
349:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
315:
314:
313:
308:
298:
295:
294:
289:
288:
277:
274:
272:
269:
267:
264:
263:
262:
261:
258:
254:
249:
246:
244:
241:
239:
238:Event horizon
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
200:
199:
198:
188:
187:
180:
177:
175:
172:
170:
167:
165:
162:
161:
153:
152:
147:
144:
139:
136:
134:
131:
129:
126:
125:
123:
121:
118:
117:
116:
115:
97:
94:
90:
85:
81:
76:
73:
69:
62:
57:
54:
50:
40:
36:
35:
32:
28:
19:
5532:
5291:
5226:KaluzaâKlein
4978:Introduction
4904:Twin paradox
4690:. Retrieved
4683:the original
4678:
4665:
4624:
4620:
4600:. Retrieved
4593:the original
4588:
4575:
4556:
4552:
4522:
4490:
4456:
4429:Neutron star
4204:
4192:
4069:
3898:with radius
3825:The spatial
3824:
3819:gravity well
3726:
3663:
3510:
3353:
3243:
3143:
3135:
2887:
2884:
2748:
2366:
1966:
1925:
1651:
1187:
836:line element
829:
783:
779:
775:
765:
691:
651:Raychaudhuri
419:
120:Introduction
5315:KerrâNewman
5286:Spherical:
5155:Other tests
5098:Singularity
5030:Formulation
4992:Fundamental
4846:Formulation
4827:Proper time
4788:Fundamental
4440:37 km
4434:20 km
4406:White dwarf
2498:, yielding
1213:proper time
818:Mathematics
666:van Stockum
596:Oppenheimer
451:KerrâNewman
243:Singularity
5467:Zel'dovich
5375:Scientists
5354:Alcubierre
5161:of Mercury
5159:precession
5088:Black hole
4971:Background
4963:relativity
4932:World line
4927:Light cone
4752:Background
4744:relativity
4734:Relativity
4692:2016-05-03
4602:2016-05-05
4469:References
4437:6 km
3654:black hole
2740:Properties
1436:colatitude
1217:world line
519:Scientists
351:Formalisms
299:Formalisms
248:Black hole
174:World line
5437:Robertson
5422:Friedmann
5417:Eddington
5407:de Sitter
5241:Solutions
5119:detectors
5114:astronomy
5081:Phenomena
5016:Geodesics
4919:Spacetime
4862:Phenomena
4657:122449954
4364:9.5
4303:η
4165:η
4144:η
4109:−
4072:parabolic
4036:κ
4033:ρ
3989:−
3931:η
3870:π
3864:θ
3827:curvature
3813:Embedding
3742:η
3738:
3709:−
3697:η
3693:
3634:∘
3626:≈
3617:η
3526:η
3522:
3490:η
3487:
3481:−
3472:η
3468:
3451:η
3447:
3441:−
3438:η
3435:
3416:ρ
3397:isotropic
3373:ν
3370:μ
3261:π
3252:κ
3209:κ
3173:π
3155:ρ
3078:−
3052:−
2997:
2960:π
2902:∫
2861:π
2827:−
2779:−
2710:φ
2702:θ
2699:
2677:θ
2655:−
2649:η
2646:
2615:−
2573:η
2570:
2564:−
2555:η
2551:
2524:τ
2460:
2432:
2417:η
2384:
2375:η
2338:φ
2330:θ
2327:
2305:θ
2262:−
2225:−
2146:−
2138:−
2100:−
2073:−
2060:τ
2042:−
2031:, we get
1897:φ
1889:θ
1886:
1864:θ
1842:−
1821:−
1795:−
1783:−
1735:−
1714:τ
1696:−
1663:≤
1466:longitude
1452:φ
1422:θ
1370:π
1359:and area
1344:π
1219:with the
1199:τ
1159:φ
1151:θ
1148:
1126:θ
1083:−
1033:−
941:−
933:−
904:−
877:−
864:τ
611:Robertson
576:Friedmann
571:Eddington
561:Nordström
551:de Sitter
408:Solutions
333:Geodesics
328:Friedmann
310:Equations
296:Equations
257:Spacetime
192:Phenomena
98:ν
95:μ
86:κ
77:ν
74:μ
66:Λ
58:ν
55:μ
5563:Category
5550:Category
5427:LemaĂźtre
5392:Einstein
5382:Poincaré
5342:Others:
5326:TaubâNUT
5292:interior
5214:theories
5212:Advanced
5179:redshift
4994:concepts
4812:Rapidity
4790:concepts
4344:redshift
4201:Examples
3892:embedded
3774:follows
3660:Redshift
3389:diagonal
1575:, where
804:pressure
790:for the
786:) is an
768:Einstein
727:Category
591:LemaĂźtre
556:Reissner
541:Poincaré
526:Einstein
471:TaubâNUT
436:Wormhole
420:interior
133:Timeline
5492:Hawking
5487:Penrose
5472:Novikov
5452:Wheeler
5397:Hilbert
5387:Lorentz
5344:pp-wave
5165:lensing
4961:General
4742:Special
4629:Bibcode
4527:Bibcode
4495:Bibcode
4453:History
4211:Object
3652:into a
3291:is the
3140:Density
1595:is the
1497:is the
1464:is the
1440:radians
1434:is the
1245:is the
1211:is the
800:density
646:Hawking
641:Penrose
616:Bardeen
606:Wheeler
536:Hilbert
531:Lorentz
491:pp-wave
128:History
5533:others
5522:Thorne
5512:Misner
5497:Taylor
5482:Geroch
5477:Ehlers
5447:Zwicky
5265:Kasner
4655:
4647:
4563:
3949:. Its
3344:Kepler
3244:where
2994:arcsin
2745:Volume
2457:arcsin
2429:arcsin
2381:arcsin
1188:where
778:(also
774:, the
725:
692:others
686:Thorne
676:Newman
656:Taylor
636:Ehlers
621:Walker
586:Zwicky
461:Kasner
5527:Weiss
5507:Bondi
5502:Hulse
5432:Milne
5336:discs
5280:Milne
5275:Gödel
5132:Virgo
4686:(PDF)
4675:(PDF)
4653:S2CID
4649:78530
4645:JSTOR
4596:(PDF)
4585:(PDF)
4555:[
4446:0.15
4423:3Ă10
4420:1.4°
4400:2Ă10
4397:7.0âČ
4376:7Ă10
4352:Earth
2749:With
661:Hulse
601:Gödel
581:Milne
476:Milne
441:Gödel
138:Tests
5462:Kerr
5412:Weyl
5311:Kerr
5171:and
5125:and
5123:LIGO
4561:ISBN
4443:30°
3785:<
3751:>
3630:70.5
1313:and
671:Taub
626:Kerr
566:Weyl
446:Kerr
364:BSSN
5517:Yau
5142:GEO
4637:doi
4625:337
4382:Sun
4370:7.7
4197:.)
3735:cos
3690:cos
3608:or
3519:cos
3484:cos
3465:cos
3444:cos
3432:cos
2690:sin
2637:cos
2567:cos
2548:cos
2318:sin
1877:sin
1521:by
1139:sin
838:is
782:or
766:In
681:Yau
359:ADM
5565::
5191:/
5157::
5112::
4677:.
4651:.
4643:.
4635:.
4623:.
4611:^
4587:.
4539:^
4507:^
4477:^
4346:)
4190:.
3800:.
3712:1.
1959:.
1442:).
1223:).
814:.
5317:)
5313:(
5299:)
5290:(
5256:)
5252:(
5199:)
5195:(
5175:)
5144:)
5121:(
4770:)
4761:(
4726:e
4719:t
4712:v
4695:.
4659:.
4639::
4631::
4605:.
4569:.
4533:.
4529::
4501:.
4497::
4372:âł
4342:(
4330:z
4307:g
4279:R
4254:s
4250:r
4226:g
4222:r
4176:R
4169:g
4122:)
4117:s
4113:r
4106:r
4103:(
4098:s
4094:r
4088:2
4085:=
4082:w
4044:3
4040:/
4030:=
4025:3
4020:g
4016:r
4011:/
4005:s
4001:r
3997:=
3992:2
3983:R
3960:K
3935:g
3908:R
3878:2
3874:/
3867:=
3861:,
3858:.
3855:t
3852:s
3849:n
3846:o
3843:c
3840:=
3837:t
3821:.
3788:2
3782:z
3762:3
3758:/
3754:1
3746:g
3701:g
3686:1
3681:=
3678:z
3621:g
3594:g
3590:r
3584:9
3581:8
3576:=
3571:s
3567:r
3546:3
3542:/
3538:1
3535:=
3530:g
3496:.
3476:g
3462:3
3455:g
3424:2
3420:c
3413:=
3410:p
3366:G
3330:M
3308:g
3304:r
3277:2
3273:c
3268:/
3264:G
3258:8
3255:=
3229:,
3221:2
3215:R
3205:3
3200:=
3192:3
3187:g
3183:r
3177:3
3170:4
3163:M
3158:=
3121:)
3113:s
3109:r
3098:g
3094:r
3088:s
3084:r
3075:1
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3063:g
3059:r
3045:2
3041:/
3037:3
3032:s
3028:r
3018:g
3014:r
3008:s
3004:r
2989:2
2985:/
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2976:g
2972:r
2964:(
2957:2
2954:=
2951:r
2946:d
2936:r
2933:r
2929:g
2923:A
2916:g
2912:r
2906:0
2898:=
2895:V
2869:2
2865:r
2858:4
2855:=
2852:A
2830:1
2823:)
2817:3
2812:g
2808:r
2803:/
2797:2
2793:r
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2783:r
2776:1
2773:(
2770:=
2765:r
2762:r
2758:g
2724:.
2720:)
2714:2
2706:d
2694:2
2686:+
2681:2
2673:d
2669:(
2663:2
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2641:2
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2621:d
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2606:t
2602:d
2597:2
2593:c
2587:2
2582:)
2577:2
2559:g
2545:3
2539:(
2534:=
2529:2
2521:d
2514:2
2510:c
2481:g
2477:r
2471:s
2467:r
2454:=
2448:R
2442:g
2438:r
2426:=
2421:g
2393:R
2389:r
2378:=
2352:.
2348:)
2342:2
2334:d
2322:2
2314:+
2309:2
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2297:(
2291:2
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2257:)
2249:2
2243:R
2235:2
2231:r
2222:1
2218:(
2213:+
2208:2
2204:t
2200:d
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2191:c
2185:2
2180:)
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2164:R
2156:2
2152:r
2143:1
2129:2
2123:R
2115:2
2110:g
2106:r
2097:1
2092:3
2088:(
2081:4
2078:1
2070:=
2065:2
2057:d
2050:2
2046:c
2017:s
2013:r
2008:/
2002:3
1997:g
1993:r
1989:=
1984:2
1978:R
1945:g
1941:r
1937:=
1934:r
1911:,
1907:)
1901:2
1893:d
1881:2
1873:+
1868:2
1860:d
1856:(
1850:2
1846:r
1837:2
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1824:1
1816:)
1810:r
1805:s
1801:r
1792:1
1788:(
1778:2
1774:t
1770:d
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1761:c
1756:)
1750:r
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1741:r
1732:1
1728:(
1724:=
1719:2
1711:d
1704:2
1700:c
1671:g
1667:r
1660:r
1636:r
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1610:r
1583:G
1561:2
1557:c
1552:/
1548:M
1545:G
1542:2
1539:=
1534:s
1530:r
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1412:.
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1169:)
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1118:(
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1021:+
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978:3
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952:2
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924:g
920:r
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901:1
896:3
892:(
885:4
882:1
874:=
869:2
861:d
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850:c
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418:(
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