712:
3485:
2968:
42:
770:
3480:{\displaystyle {\begin{aligned}\partial _{t}\pi ^{ij}=&-N{\sqrt {g}}\left(R^{ij}-{\tfrac {1}{2}}Rg^{ij}\right)+{\frac {N}{2{\sqrt {g}}}}g^{ij}\left(\pi ^{mn}\pi _{mn}-{\tfrac {1}{2}}\pi ^{2}\right)-{\frac {2N}{\sqrt {g}}}\left(\pi ^{in}{\pi _{n}}^{j}-{\tfrac {1}{2}}\pi \pi ^{ij}\right)\\&+{\sqrt {g}}\left(\nabla ^{i}\nabla ^{j}N-g^{ij}\nabla ^{n}\nabla _{n}N\right)+\nabla _{n}\left(\pi ^{ij}N^{n}\right)-{N^{i}}_{;n}\pi ^{nj}-{N^{j}}_{;n}\pi ^{ni}\end{aligned}}}
5469:
725:
2499:
1455:
denotes the separation of the spacetime metric into three spatial components and one temporal component (foliation). It separates the spacetime metric into its spatial and temporal parts, which facilitates the study of the evolution of gravitational fields. The basic idea is to express the spacetime
3929:
In
Hamiltonian formulations, the basic point is replacement of set of second order equations by another first order set of equations. We may get this second set of equations by Hamiltonian formulation in an easy way. Of course this is very useful for numerical physics, because reducing the order of
1217:
Most references adopt notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime indices taking values (0, 1, 2, 3) and Latin indices are spatial indices taking values (1, 2, 3). In the derivation here, a superscript (4) is prepended to
1698:
is the emergent 3D spatial metric on each hypersurface. This decomposition allows for a separation of the spacetime evolution equations into constraints (which relate the initial data on a spatial hypersurface) and evolution equations (which describe how the geometry of spacetime changes from one
4021:
2085:
2957:
3921:
are used to find approximate solutions to the equations. In order to construct such solutions numerically, most researchers start with a formulation of the
Einstein equations closely related to the ADM formulation. The most common approaches start with an
3882:
2308:
3964:. The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form. In other words, the ADM energy is computed as the strength of the gravitational field at infinity.
3975:
then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds â for example, it is completely violated in
1616:
113:
3761:
1789:
1896:
2228:
4027:
This is done in M. Montesinos and J. Romero, Linking the ADM formulation to other
Hamiltonian formulations of general relativity, Phys. Rev. D 107, 044052 (2023). DOI 10.1103/PhysRevD.107.044052
2973:
2810:
1205:
of the foliation of spacetime are welded together. The equations of motion for these variables can be freely specified; this freedom corresponds to the freedom to specify how to lay out the
2711:
2140:
1877:
991:
2294:
1441:
1293:
2641:
1813:
The desired outcome of the derivation is to define an embedding of three-dimensional spatial slices in the four-dimensional spacetime. The metric of the three-dimensional slices
3656:
1046:
1334:
1087:
1365:
1203:
888:
2802:
1117:
3570:
3767:
2772:
1696:
1246:
5172:
2494:{\displaystyle {\mathcal {L}}=-g_{ij}\partial _{t}\pi ^{ij}-NH-N_{i}P^{i}-2\partial _{i}\left(\pi ^{ij}N_{j}-{\frac {1}{2}}\pi N^{i}+\nabla ^{i}N{\sqrt {g}}\right)}
2302:
Having identified the quantities for the formulation, the next step is to rewrite the
Lagrangian in terms of these variables. The new expression for the Lagrangian
1666:
1176:
935:
483:
3528:
1639:
1392:
1145:
908:
1148:
756:
5491:
3575:
and the lapse and shift themselves can be freely specified, reflecting the fact that coordinate systems can be freely specified in both space and time.
1464:
that represents spatial coordinate changes between these hypersurfaces) along with a 3D spatial metric. Mathematically, this separation is written as:
1124:
135:
1470:
1218:
quantities that typically have both a three-dimensional and a four-dimensional version, such as the metric tensor for three-dimensional slices
48:
340:
5219:
5454:
4005:
693:
3664:
5243:
4989:
4036:
503:
345:
2080:{\displaystyle \pi ^{ij}={\sqrt {-^{(4)}g}}\left({^{(4)}}\Gamma _{pq}^{0}-g_{pq}{^{(4)}}\Gamma _{rs}^{0}g^{rs}\right)g^{ip}g^{jq},}
749:
458:
3967:
If the required asymptotic form is time-independent (such as the
Minkowski space itself), then it respects the time-translational
1721:
5025:
4979:
371:
205:
5163:
4616:
410:
4647:
2952:{\displaystyle \partial _{t}g_{ij}={\frac {2N}{\sqrt {g}}}\left(\pi _{ij}-{\tfrac {1}{2}}\pi g_{ij}\right)+N_{i;j}+N_{j;i}}
2152:
4547:; Sasaki, Misao; Sendouda, Yuuiti; Yamauchi, Daisuke (2010). "Hamiltonian formulation of f(Riemann) theories of gravity".
1394:(with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as
742:
4900:
837:
122:
5271:
4905:
148:
5225:
711:
428:
41:
4938:
335:
5496:
5041:
3888:
3495:
2652:
835:
The comprehensive review of the formalism that the authors published in 1962 has been reprinted in the journal
2093:
5472:
5214:
5119:
5049:
4065:
422:
130:
5182:
468:
5036:
1819:
944:
488:
2239:
1397:
1251:
5077:
4811:
4796:
3892:
3589:
2510:
1807:
729:
220:
140:
3609:
999:
5364:
4806:
4759:
3939:
633:
181:
5394:
5020:
4701:
1713:
825:
433:
245:
4289:
4152:
Arnowitt, R.; Deser, S.; Misner, C. (2008). "Republication of: The dynamics of general relativity".
5419:
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4729:
4640:
4457:
4325:
4245:
4094:
3910:
1312:
668:
658:
508:
325:
4501:
4413:
4369:
4209:
1062:
5148:
5070:
4690:
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4608:
1883:
1343:
1181:
866:
653:
387:
3599:
5374:
5237:
5143:
5101:
4775:
4739:
3877:{\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\mapsto -i{\frac {\delta }{\delta g_{ij}(t,x^{k})}}.}
2777:
2717:
1092:
1053:
623:
608:
453:
215:
3539:
2720:
and the momentum constraint respectively. The lapse and the shift appear in the
Lagrangian as
5379:
4984:
4923:
4031:
3923:
1049:
994:
817:
598:
166:
2747:
1671:
1221:
5284:
5210:
5054:
4656:
4566:
4516:
4472:
4428:
4384:
4340:
4304:
4260:
4224:
4171:
4109:
3972:
3930:
differential equations is often convenient if we want to prepare equations for a computer.
3914:
3904:
2741:
2721:
1644:
1337:
1154:
1120:
913:
829:
418:
8:
5359:
5354:
5344:
5176:
5125:
4943:
4928:
4754:
4685:
4633:
3507:
2143:
613:
583:
578:
330:
5349:
4570:
4520:
4476:
4432:
4388:
4344:
4308:
4264:
4228:
4175:
4113:
593:
563:
5424:
5324:
5248:
5106:
5087:
5081:
5032:
4974:
4883:
4801:
4719:
4696:
4664:
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4444:
4400:
4356:
4276:
4187:
4161:
3989:
3977:
3949:
1624:
1377:
1306:
1206:
1130:
893:
821:
663:
548:
473:
320:
225:
210:
171:
33:
5434:
5266:
5258:
4859:
4816:
4612:
4586:
4544:
4492:
4448:
4404:
4360:
4280:
4001:
3603:
1887:
1299:
813:
493:
268:
20:
5304:
4191:
2744:
to derive "equations of motion" that describe the time evolution of both the metric
1668:
is the shift vector, encoding how spatial coordinates change between hypersurfaces.
543:
5339:
5329:
5276:
5253:
4831:
4574:
4524:
4480:
4436:
4392:
4348:
4312:
4268:
4232:
4179:
4117:
4022:
Linking the ADM formulation to other
Hamiltonian formulations of general relativity
4009:
3981:
3887:
More precisely, the replacing of classical variables by operators is restricted by
3501:
Taking variations with respect to the lapse and shift provide constraint equations
1803:
805:
773:
618:
573:
553:
498:
5399:
1052:, and thereby write the equations of motion for general relativity in the form of
638:
5414:
5389:
5314:
5309:
5192:
5153:
5115:
5059:
4933:
4869:
3961:
3595:
843:
673:
648:
533:
528:
392:
273:
235:
5197:
443:
5439:
5111:
5095:
5091:
4994:
4964:
4821:
4724:
4046:
4000:
By using the ADM decomposition and introducing extra auxiliary fields, in 2009
3918:
3913:. In order to find other solutions, there is an active field of study known as
3898:
781:
716:
683:
678:
366:
230:
4370:"Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem"
4316:
4183:
3984:
in particular is able to produce energy (and mass) from "nothing" because the
788:
conference held in
November 2009 to honor the 50th anniversary of their paper.
5485:
5429:
5409:
5404:
5319:
5187:
5015:
4791:
4744:
4484:
4352:
4272:
4121:
3985:
3957:
3658:
and the spatial metric functions by linear functional differential operators
2733:
2146:
associated with the metric of the full four-dimensional spacetime. The lapse
1799:
938:
809:
777:
769:
643:
558:
538:
463:
240:
4528:
4440:
4396:
4236:
4012:
theories "whose
Lagrangian is an arbitrary function of the Riemann tensor".
603:
5449:
5369:
5334:
4864:
4826:
4135:
4041:
588:
568:
4210:"Quantum Theory of Gravitation: General Formulation and Linearized Theory"
1611:{\displaystyle ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)}
5233:
5202:
4749:
1795:
1371:
478:
448:
4578:
5444:
5010:
4854:
4849:
3491:
688:
250:
176:
108:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
4166:
4841:
4414:"Interior Schwarzschild Solutions and Interpretation of Source Terms"
3960:
at infinity â for example a spacetime that asymptotically approaches
3953:
2737:
860:
856:
259:
4502:"Coordinate Invariance and Energy Expressions in General Relativity"
4095:"Dynamical Structure and Definition of Energy in General Relativity"
3891:. The hats represent operators in quantum theory. This leads to the
3594:
Using the ADM formulation, it is possible to attempt to construct a
5384:
4734:
3968:
628:
438:
278:
4561:
4069:
3756:{\displaystyle {\hat {g}}_{ij}(t,x^{k})\mapsto g_{ij}(t,x^{k}),}
2740:, it is possible and desirable to use the usual procedures from
1460:
that represents the time evolution between hypersurfaces, and a
5064:
3945:
4543:
1374:
of the matrix of metric tensor coefficients is represented by
1248:
and the metric tensor for the full four-dimensional spacetime
4625:
4326:"Energy and the Criteria for Radiation in General Relativity"
2736:
on three-dimensional spaces embedded in the four-dimensional
3988:
density is roughly constant, but the volume of the
Universe
3899:
Application to numerical solutions of the Einstein equations
937:. The dynamic variables of this theory are taken to be the
841:, while the original papers can be found in the archives of
5045:
2504:
is conveniently written in terms of the two new quantities
1784:{\displaystyle {\mathcal {L}}={^{(4)}R}{\sqrt {-^{(4)}g}},}
1641:
is the lapse function encoding the proper time evolution,
3952:, which is only applicable to some special geometries of
24:
2732:
Although the variables in the Lagrangian represent the
2090:
using standard techniques and definitions. The symbols
3909:
There are relatively few known exact solutions to the
3230:
3143:
3041:
2879:
2299:
are the remaining elements of the four-metric tensor.
3770:
3667:
3612:
3542:
3510:
2971:
2813:
2780:
2750:
2655:
2513:
2311:
2242:
2223:{\displaystyle N=\left(-{^{(4)}g^{00}}\right)^{-1/2}}
2155:
2096:
1899:
1822:
1724:
1674:
1647:
1627:
1473:
1400:
1380:
1346:
1315:
1302:
in which summation over repeated indices is assumed.
1254:
1224:
1184:
1157:
1133:
1095:
1065:
1002:
947:
916:
896:
869:
51:
1048:. Using these variables it is possible to define a
4499:
4455:
4411:
4367:
4323:
4287:
4243:
4151:
4092:
3995:
3876:
3755:
3650:
3564:
3522:
3479:
2951:
2796:
2766:
2705:
2635:
2493:
2288:
2222:
2134:
2079:
1871:
1783:
1712:The starting point for the ADM formulation is the
1690:
1660:
1633:
1610:
1435:
1386:
1359:
1328:
1287:
1240:
1197:
1170:
1139:
1111:
1081:
1040:
985:
929:
902:
882:
107:
4290:"Finite Self-Energy of Classical Point Particles"
3583:
5483:
4140:Gravitation: An introduction to current research
4066:"ADM-50: A Celebration of Current GR Innovation"
1702:
910:, and with coordinates on each slice given by
786:ADM-50: A Celebration of Current GR Innovation
4641:
4602:
4207:
1794:which is a product of the square root of the
750:
16:Hamiltonian formulation of general relativity
4500:Arnowitt, R.; Deser, S.; Misner, C. (1961).
4456:Arnowitt, R.; Deser, S.; Misner, C. (1961).
4412:Arnowitt, R.; Deser, S.; Misner, C. (1960).
4368:Arnowitt, R.; Deser, S.; Misner, C. (1960).
4324:Arnowitt, R.; Deser, S.; Misner, C. (1960).
4288:Arnowitt, R.; Deser, S.; Misner, C. (1960).
4246:"Canonical Variables for General Relativity"
4244:Arnowitt, R.; Deser, S.; Misner, C. (1960).
4093:Arnowitt, R.; Deser, S.; Misner, C. (1959).
3956:that asymptotically approach a well-defined
1178:. These describe how each of the "leaves"
5492:Mathematical methods in general relativity
4648:
4634:
3944:ADM energy is a special way to define the
1367:or by subscripts preceded by a semicolon.
757:
743:
4560:
4165:
3606:. That is, replace the canonical momenta
2706:{\displaystyle P^{i}=-2\pi ^{ij}{}_{;j},}
1707:
3602:corresponding to a given Hamiltonian in
3598:in the same way that one constructs the
2135:{\displaystyle {^{(4)}}\Gamma _{ij}^{0}}
768:
1336:or by subscripts preceded by a comma.
5484:
3933:
2727:
4629:
4088:
4086:
1872:{\displaystyle g_{ij}={^{(4)}}g_{ij}}
986:{\displaystyle \gamma _{ij}(t,x^{k})}
2289:{\displaystyle N_{i}={^{(4)}g_{0i}}}
1436:{\displaystyle \pi =g^{ij}\pi _{ij}}
1305:Two types of derivatives are used:
1288:{\displaystyle {^{(4)}}g_{\mu \nu }}
1059:In addition to the twelve variables
941:of three-dimensional spatial slices
863:into a family of spacelike surfaces
5220:TolmanâOppenheimerâVolkoff equation
5173:FriedmannâLemaĂźtreâRobertsonâWalker
2636:{\displaystyle H=-{\sqrt {g}}\left}
1886:for a Hamiltonian formulation. The
1340:are denoted either by the operator
1309:are denoted either by the operator
890:, labeled by their time coordinate
13:
4154:General Relativity and Gravitation
4083:
4006:GibbonsâHawkingâYork boundary term
4004:et al. found a method to find the
3651:{\displaystyle \pi ^{ij}(t,x^{k})}
3355:
3334:
3324:
3295:
3285:
2977:
2815:
2467:
2400:
2339:
2314:
2115:
2013:
1962:
1806:. This is the Lagrangian from the
1727:
1348:
1317:
1186:
1041:{\displaystyle \pi ^{ij}(t,x^{k})}
871:
838:General Relativity and Gravitation
832:. It was first published in 1959.
68:
14:
5508:
4990:HamiltonâJacobiâEinstein equation
4458:"Wave Zone in General Relativity"
4037:HamiltonâJacobiâEinstein equation
5468:
5467:
4208:Arnowitt, R.; Deser, S. (1959).
824:that plays an important role in
724:
723:
710:
40:
4549:Progress of Theoretical Physics
3996:Application to modified gravity
3578:
1802:for the full spacetime and its
4797:Massâenergy equivalence (E=mc)
4655:
4537:
4198:
4145:
4128:
4058:
3865:
3846:
3815:
3812:
3793:
3778:
3747:
3728:
3712:
3709:
3690:
3675:
3645:
3626:
3584:Application to quantum gravity
3496:partial differential equations
2545:
2539:
2267:
2261:
2182:
2176:
2108:
2102:
2006:
2000:
1955:
1949:
1929:
1923:
1850:
1844:
1768:
1762:
1746:
1740:
1605:
1570:
1567:
1532:
1266:
1260:
1035:
1016:
980:
961:
1:
4596:
1329:{\displaystyle \partial _{i}}
3926:based on the ADM formalism.
1446:
1082:{\displaystyle \gamma _{ij}}
855:The formalism supposes that
7:
4812:Relativistic Doppler effect
4134:Chapter 7 (pp. 227â265) of
4015:
2774:and its conjugate momentum
1703:Derivation of ADM formalism
1360:{\displaystyle \nabla _{i}}
1212:
1198:{\displaystyle \Sigma _{t}}
883:{\displaystyle \Sigma _{t}}
850:
221:Gravitational time dilation
10:
5513:
5283:In computational physics:
4807:Relativity of simultaneity
3940:Mass in general relativity
3937:
3902:
3587:
1699:hypersurface to another).
1370:The absolute value of the
341:MathissonâPapapetrouâDixon
182:Pseudo-Riemannian manifold
18:
5465:
5297:
5162:
5134:
5120:LenseâThirring precession
5003:
4952:
4914:
4893:
4882:
4840:
4784:
4768:
4710:
4702:Doubly special relativity
4674:
4663:
4317:10.1103/PhysRevLett.4.375
4184:10.1007/s10714-008-0661-1
3596:quantum theory of gravity
2797:{\displaystyle \pi ^{ij}}
1112:{\displaystyle \pi ^{ij}}
826:canonical quantum gravity
4980:Post-Newtonian formalism
4970:Einstein field equations
4906:Mathematical formulation
4730:Hyperbolic orthogonality
4485:10.1103/PhysRev.121.1556
4353:10.1103/PhysRev.118.1100
4273:10.1103/PhysRev.117.1595
4142:, Wiley: New York, 1962.
4122:10.1103/PhysRev.116.1322
4052:
3911:Einstein field equations
3565:{\displaystyle P^{i}=0,}
1890:can then be computed as
1798:of the four-dimensional
346:HamiltonâJacobiâEinstein
326:Einstein field equations
149:Mathematical formulation
19:Not to be confused with
4691:Galilean transformation
4682:Principle of relativity
4609:Oxford University Press
4529:10.1103/PhysRev.122.997
4441:10.1103/PhysRev.120.321
4397:10.1103/PhysRev.120.313
4297:Physical Review Letters
4237:10.1103/PhysRev.113.745
3893:WheelerâDeWitt equation
3590:WheelerâDeWitt equation
2716:which are known as the
1884:generalized coordinates
1808:EinsteinâHilbert action
804:(named for its authors
4776:Lorentz transformation
4603:Kiefer, Claus (2007).
3878:
3757:
3652:
3566:
3524:
3481:
2953:
2798:
2768:
2767:{\displaystyle g_{ij}}
2718:Hamiltonian constraint
2707:
2637:
2495:
2290:
2224:
2136:
2081:
1873:
1785:
1708:Lagrangian formulation
1692:
1691:{\displaystyle g_{ij}}
1662:
1635:
1612:
1437:
1388:
1361:
1330:
1289:
1242:
1241:{\displaystyle g_{ij}}
1199:
1172:
1141:
1113:
1083:
1042:
987:
931:
904:
884:
789:
216:Gravitational redshift
109:
5497:Formalism (deductive)
5244:WeylâLewisâPapapetrou
4985:Raychaudhuri equation
4924:Equivalence principle
4032:Canonical coordinates
3924:initial value problem
3889:commutation relations
3879:
3758:
3653:
3567:
3525:
3482:
2954:
2799:
2769:
2708:
2638:
2496:
2291:
2233:and the shift vector
2225:
2137:
2082:
1874:
1786:
1693:
1663:
1661:{\displaystyle N_{i}}
1636:
1613:
1456:metric in terms of a
1438:
1389:
1362:
1338:Covariant derivatives
1331:
1290:
1243:
1200:
1173:
1171:{\displaystyle N_{i}}
1142:
1114:
1084:
1043:
988:
932:
930:{\displaystyle x^{i}}
905:
885:
794:ArnowittâDeserâMisner
772:
504:WeylâLewisâPapapetrou
459:KerrâNewmanâde Sitter
279:EinsteinâRosen bridge
211:Gravitational lensing
167:Equivalence principle
110:
5285:Numerical relativity
5126:pulsar timing arrays
4607:. Oxford, New York:
3915:numerical relativity
3905:Numerical relativity
3768:
3665:
3610:
3600:Schrödinger equation
3540:
3508:
2969:
2811:
2778:
2748:
2742:Lagrangian mechanics
2722:Lagrange multipliers
2653:
2511:
2309:
2240:
2153:
2094:
1897:
1820:
1722:
1672:
1645:
1625:
1471:
1398:
1378:
1344:
1313:
1252:
1222:
1182:
1155:
1147:, and components of
1131:
1121:Lagrange multipliers
1093:
1063:
1054:Hamilton's equations
1000:
945:
914:
894:
867:
830:numerical relativity
434:EinsteinâRosen waves
160:Fundamental concepts
49:
5177:Friedmann equations
5071:HulseâTaylor binary
5033:Gravitational waves
4929:Riemannian geometry
4755:Proper acceleration
4740:Maxwell's equations
4686:Galilean relativity
4579:10.1143/PTP.123.169
4571:2010PThPh.123..169D
4521:1961PhRv..122..997A
4477:1961PhRv..121.1556A
4433:1960PhRv..120..321A
4389:1960PhRv..120..313A
4345:1960PhRv..118.1100A
4309:1960PhRvL...4..375A
4265:1960PhRv..117.1595A
4229:1959PhRv..113..745A
4176:2008GReGr..40.1997A
4114:1959PhRv..116.1322A
3990:grows exponentially
3934:ADM energy and mass
3523:{\displaystyle H=0}
2728:Equations of motion
2144:Christoffel symbols
2131:
2029:
1978:
1307:Partial derivatives
1298:The text here uses
1209:in space and time.
388:KaluzaâKlein theory
274:Minkowski spacetime
226:Gravitational waves
5226:ReissnerâNordström
5144:BransâDicke theory
4975:Linearized gravity
4802:Length contraction
4720:Frame of reference
4697:Special relativity
4545:Deruelle, Nathalie
3978:physical cosmology
3950:general relativity
3874:
3753:
3648:
3562:
3520:
3477:
3475:
3239:
3152:
3050:
2949:
2888:
2794:
2764:
2703:
2633:
2491:
2286:
2220:
2132:
2114:
2077:
2012:
1961:
1869:
1781:
1688:
1658:
1631:
1608:
1433:
1384:
1357:
1326:
1285:
1238:
1195:
1168:
1149:shift vector field
1137:
1109:
1079:
1038:
983:
927:
900:
880:
822:general relativity
790:
717:Physics portal
489:OppenheimerâSnyder
429:ReissnerâNordström
321:Linearized gravity
269:Spacetime diagrams
172:Special relativity
105:
34:General relativity
5479:
5478:
5293:
5292:
5272:OzsvĂĄthâSchĂŒcking
4878:
4877:
4860:Minkowski diagram
4817:Thomas precession
4760:Relativistic mass
4618:978-0-19-921252-1
3973:Noether's theorem
3869:
3781:
3678:
3604:quantum mechanics
3277:
3238:
3187:
3186:
3151:
3093:
3090:
3049:
3017:
2887:
2855:
2854:
2582:
2528:
2484:
2448:
1937:
1888:conjugate momenta
1776:
1634:{\displaystyle N}
1387:{\displaystyle g}
1300:Einstein notation
1207:coordinate system
1140:{\displaystyle N}
1119:, there are four
995:conjugate momenta
903:{\displaystyle t}
814:Charles W. Misner
767:
766:
400:
399:
286:
285:
21:ADHM construction
5504:
5471:
5470:
5254:van Stockum dust
5026:Two-body problem
4944:Mach's principle
4891:
4890:
4832:Terrell rotation
4672:
4671:
4650:
4643:
4636:
4627:
4626:
4622:
4591:
4590:
4564:
4541:
4535:
4532:
4506:
4496:
4471:(5): 1556â1566.
4462:
4452:
4418:
4408:
4374:
4364:
4339:(4): 1100â1104.
4330:
4320:
4294:
4284:
4259:(6): 1595â1602.
4250:
4240:
4214:
4204:The papers are:
4202:
4196:
4195:
4169:
4160:(9): 1997â2027.
4149:
4143:
4132:
4126:
4125:
4108:(5): 1322â1330.
4099:
4090:
4081:
4080:
4078:
4077:
4068:. Archived from
4062:
4010:modified gravity
3982:Cosmic inflation
3883:
3881:
3880:
3875:
3870:
3868:
3864:
3863:
3845:
3844:
3825:
3811:
3810:
3792:
3791:
3783:
3782:
3774:
3762:
3760:
3759:
3754:
3746:
3745:
3727:
3726:
3708:
3707:
3689:
3688:
3680:
3679:
3671:
3657:
3655:
3654:
3649:
3644:
3643:
3625:
3624:
3571:
3569:
3568:
3563:
3552:
3551:
3529:
3527:
3526:
3521:
3486:
3484:
3483:
3478:
3476:
3472:
3471:
3459:
3458:
3450:
3449:
3448:
3434:
3433:
3421:
3420:
3412:
3411:
3410:
3396:
3392:
3391:
3390:
3381:
3380:
3363:
3362:
3350:
3346:
3342:
3341:
3332:
3331:
3322:
3321:
3303:
3302:
3293:
3292:
3278:
3273:
3265:
3261:
3257:
3256:
3255:
3240:
3231:
3225:
3224:
3219:
3218:
3217:
3206:
3205:
3188:
3182:
3181:
3173:
3168:
3164:
3163:
3162:
3153:
3144:
3138:
3137:
3125:
3124:
3107:
3106:
3094:
3092:
3091:
3086:
3077:
3072:
3068:
3067:
3066:
3051:
3042:
3036:
3035:
3018:
3013:
2998:
2997:
2985:
2984:
2958:
2956:
2955:
2950:
2948:
2947:
2929:
2928:
2910:
2906:
2905:
2904:
2889:
2880:
2874:
2873:
2856:
2850:
2849:
2841:
2836:
2835:
2823:
2822:
2803:
2801:
2800:
2795:
2793:
2792:
2773:
2771:
2770:
2765:
2763:
2762:
2712:
2710:
2709:
2704:
2699:
2698:
2690:
2687:
2686:
2665:
2664:
2642:
2640:
2639:
2634:
2632:
2628:
2627:
2623:
2622:
2621:
2609:
2608:
2593:
2592:
2583:
2575:
2568:
2567:
2549:
2548:
2529:
2524:
2500:
2498:
2497:
2492:
2490:
2486:
2485:
2480:
2475:
2474:
2462:
2461:
2449:
2441:
2436:
2435:
2426:
2425:
2408:
2407:
2392:
2391:
2382:
2381:
2360:
2359:
2347:
2346:
2337:
2336:
2318:
2317:
2295:
2293:
2292:
2287:
2285:
2284:
2283:
2271:
2270:
2252:
2251:
2229:
2227:
2226:
2221:
2219:
2218:
2214:
2202:
2198:
2197:
2196:
2195:
2186:
2185:
2141:
2139:
2138:
2133:
2130:
2125:
2113:
2112:
2111:
2086:
2084:
2083:
2078:
2073:
2072:
2060:
2059:
2047:
2043:
2042:
2041:
2028:
2023:
2011:
2010:
2009:
1994:
1993:
1977:
1972:
1960:
1959:
1958:
1938:
1933:
1932:
1917:
1912:
1911:
1878:
1876:
1875:
1870:
1868:
1867:
1855:
1854:
1853:
1835:
1834:
1790:
1788:
1787:
1782:
1777:
1772:
1771:
1756:
1754:
1750:
1749:
1731:
1730:
1697:
1695:
1694:
1689:
1687:
1686:
1667:
1665:
1664:
1659:
1657:
1656:
1640:
1638:
1637:
1632:
1617:
1615:
1614:
1609:
1598:
1597:
1585:
1584:
1560:
1559:
1547:
1546:
1531:
1530:
1515:
1514:
1502:
1501:
1486:
1485:
1442:
1440:
1439:
1434:
1432:
1431:
1419:
1418:
1393:
1391:
1390:
1385:
1366:
1364:
1363:
1358:
1356:
1355:
1335:
1333:
1332:
1327:
1325:
1324:
1294:
1292:
1291:
1286:
1284:
1283:
1271:
1270:
1269:
1247:
1245:
1244:
1239:
1237:
1236:
1204:
1202:
1201:
1196:
1194:
1193:
1177:
1175:
1174:
1169:
1167:
1166:
1146:
1144:
1143:
1138:
1118:
1116:
1115:
1110:
1108:
1107:
1088:
1086:
1085:
1080:
1078:
1077:
1047:
1045:
1044:
1039:
1034:
1033:
1015:
1014:
992:
990:
989:
984:
979:
978:
960:
959:
936:
934:
933:
928:
926:
925:
909:
907:
906:
901:
889:
887:
886:
881:
879:
878:
806:Richard Arnowitt
774:Richard Arnowitt
759:
752:
745:
732:
727:
726:
719:
715:
714:
499:van Stockum dust
484:RobertsonâWalker
310:
309:
200:
199:
114:
112:
111:
106:
104:
103:
91:
83:
82:
64:
63:
44:
30:
29:
5512:
5511:
5507:
5506:
5505:
5503:
5502:
5501:
5482:
5481:
5480:
5475:
5461:
5289:
5193:BKL singularity
5183:LemaĂźtreâTolman
5158:
5154:Quantum gravity
5136:
5130:
5116:geodetic effect
5090:(together with
5060:LISA Pathfinder
4999:
4948:
4934:Penrose diagram
4916:
4910:
4885:
4874:
4870:Minkowski space
4836:
4780:
4764:
4712:
4706:
4666:
4659:
4654:
4619:
4605:Quantum Gravity
4599:
4594:
4542:
4538:
4515:(3): 997â1006.
4509:Physical Review
4504:
4465:Physical Review
4460:
4421:Physical Review
4416:
4377:Physical Review
4372:
4333:Physical Review
4328:
4292:
4253:Physical Review
4248:
4217:Physical Review
4212:
4203:
4199:
4150:
4146:
4133:
4129:
4102:Physical Review
4097:
4091:
4084:
4075:
4073:
4064:
4063:
4059:
4055:
4018:
3998:
3962:Minkowski space
3942:
3936:
3907:
3901:
3859:
3855:
3837:
3833:
3829:
3824:
3806:
3802:
3784:
3773:
3772:
3771:
3769:
3766:
3765:
3741:
3737:
3719:
3715:
3703:
3699:
3681:
3670:
3669:
3668:
3666:
3663:
3662:
3639:
3635:
3617:
3613:
3611:
3608:
3607:
3592:
3586:
3581:
3547:
3543:
3541:
3538:
3537:
3509:
3506:
3505:
3474:
3473:
3464:
3460:
3451:
3444:
3440:
3439:
3438:
3426:
3422:
3413:
3406:
3402:
3401:
3400:
3386:
3382:
3373:
3369:
3368:
3364:
3358:
3354:
3337:
3333:
3327:
3323:
3314:
3310:
3298:
3294:
3288:
3284:
3283:
3279:
3272:
3263:
3262:
3248:
3244:
3229:
3220:
3213:
3209:
3208:
3207:
3198:
3194:
3193:
3189:
3174:
3172:
3158:
3154:
3142:
3130:
3126:
3117:
3113:
3112:
3108:
3099:
3095:
3085:
3081:
3076:
3059:
3055:
3040:
3028:
3024:
3023:
3019:
3012:
3002:
2990:
2986:
2980:
2976:
2972:
2970:
2967:
2966:
2937:
2933:
2918:
2914:
2897:
2893:
2878:
2866:
2862:
2861:
2857:
2842:
2840:
2828:
2824:
2818:
2814:
2812:
2809:
2808:
2785:
2781:
2779:
2776:
2775:
2755:
2751:
2749:
2746:
2745:
2730:
2691:
2689:
2688:
2679:
2675:
2660:
2656:
2654:
2651:
2650:
2614:
2610:
2601:
2597:
2588:
2584:
2574:
2573:
2569:
2560:
2556:
2538:
2535:
2534:
2530:
2523:
2512:
2509:
2508:
2479:
2470:
2466:
2457:
2453:
2440:
2431:
2427:
2418:
2414:
2413:
2409:
2403:
2399:
2387:
2383:
2377:
2373:
2352:
2348:
2342:
2338:
2329:
2325:
2313:
2312:
2310:
2307:
2306:
2276:
2272:
2260:
2257:
2256:
2247:
2243:
2241:
2238:
2237:
2210:
2203:
2191:
2187:
2175:
2172:
2171:
2167:
2163:
2162:
2154:
2151:
2150:
2126:
2118:
2101:
2098:
2097:
2095:
2092:
2091:
2065:
2061:
2052:
2048:
2034:
2030:
2024:
2016:
1999:
1996:
1995:
1986:
1982:
1973:
1965:
1948:
1945:
1944:
1943:
1939:
1922:
1918:
1916:
1904:
1900:
1898:
1895:
1894:
1860:
1856:
1843:
1840:
1839:
1827:
1823:
1821:
1818:
1817:
1761:
1757:
1755:
1739:
1736:
1735:
1726:
1725:
1723:
1720:
1719:
1710:
1705:
1679:
1675:
1673:
1670:
1669:
1652:
1648:
1646:
1643:
1642:
1626:
1623:
1622:
1593:
1589:
1580:
1576:
1555:
1551:
1542:
1538:
1523:
1519:
1510:
1506:
1497:
1493:
1481:
1477:
1472:
1469:
1468:
1449:
1424:
1420:
1411:
1407:
1399:
1396:
1395:
1379:
1376:
1375:
1351:
1347:
1345:
1342:
1341:
1320:
1316:
1314:
1311:
1310:
1276:
1272:
1259:
1256:
1255:
1253:
1250:
1249:
1229:
1225:
1223:
1220:
1219:
1215:
1189:
1185:
1183:
1180:
1179:
1162:
1158:
1156:
1153:
1152:
1132:
1129:
1128:
1100:
1096:
1094:
1091:
1090:
1070:
1066:
1064:
1061:
1060:
1029:
1025:
1007:
1003:
1001:
998:
997:
974:
970:
952:
948:
946:
943:
942:
921:
917:
915:
912:
911:
895:
892:
891:
874:
870:
868:
865:
864:
853:
844:Physical Review
820:formulation of
763:
722:
709:
708:
701:
700:
524:
523:
514:
513:
469:LemaĂźtreâTolman
414:
413:
402:
401:
393:Quantum gravity
380:Advanced theory
307:
306:
305:
288:
287:
236:Geodetic effect
197:
196:
187:
186:
162:
161:
145:
115:
96:
92:
87:
75:
71:
56:
52:
50:
47:
46:
28:
17:
12:
11:
5:
5510:
5500:
5499:
5494:
5477:
5476:
5466:
5463:
5462:
5460:
5459:
5452:
5447:
5442:
5437:
5432:
5427:
5422:
5417:
5412:
5407:
5402:
5397:
5392:
5387:
5382:
5380:Choquet-Bruhat
5377:
5372:
5367:
5362:
5357:
5352:
5347:
5342:
5337:
5332:
5327:
5322:
5317:
5312:
5307:
5301:
5299:
5295:
5294:
5291:
5290:
5288:
5287:
5280:
5279:
5274:
5269:
5262:
5261:
5256:
5251:
5246:
5241:
5232:Axisymmetric:
5229:
5228:
5223:
5217:
5206:
5205:
5200:
5195:
5190:
5185:
5180:
5171:Cosmological:
5168:
5166:
5160:
5159:
5157:
5156:
5151:
5146:
5140:
5138:
5132:
5131:
5129:
5128:
5123:
5112:frame-dragging
5109:
5104:
5099:
5096:Einstein rings
5092:Einstein cross
5085:
5074:
5073:
5068:
5062:
5057:
5052:
5039:
5029:
5028:
5023:
5018:
5013:
5007:
5005:
5001:
5000:
4998:
4997:
4995:Ernst equation
4992:
4987:
4982:
4977:
4972:
4967:
4965:BSSN formalism
4962:
4956:
4954:
4950:
4949:
4947:
4946:
4941:
4936:
4931:
4926:
4920:
4918:
4912:
4911:
4909:
4908:
4903:
4897:
4895:
4888:
4880:
4879:
4876:
4875:
4873:
4872:
4867:
4862:
4857:
4852:
4846:
4844:
4838:
4837:
4835:
4834:
4829:
4824:
4822:Ladder paradox
4819:
4814:
4809:
4804:
4799:
4794:
4788:
4786:
4782:
4781:
4779:
4778:
4772:
4770:
4766:
4765:
4763:
4762:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4725:Speed of light
4722:
4716:
4714:
4708:
4707:
4705:
4704:
4699:
4694:
4688:
4678:
4676:
4669:
4661:
4660:
4653:
4652:
4645:
4638:
4630:
4624:
4623:
4617:
4598:
4595:
4593:
4592:
4555:(1): 169â185.
4536:
4534:
4533:
4497:
4453:
4427:(1): 321â324.
4409:
4383:(1): 313â320.
4365:
4321:
4303:(7): 375â377.
4285:
4241:
4223:(2): 745â750.
4197:
4144:
4127:
4082:
4056:
4054:
4051:
4050:
4049:
4047:Shape dynamics
4044:
4039:
4034:
4025:
4024:
4017:
4014:
3997:
3994:
3935:
3932:
3919:supercomputers
3903:Main article:
3900:
3897:
3885:
3884:
3873:
3867:
3862:
3858:
3854:
3851:
3848:
3843:
3840:
3836:
3832:
3828:
3823:
3820:
3817:
3814:
3809:
3805:
3801:
3798:
3795:
3790:
3787:
3780:
3777:
3763:
3752:
3749:
3744:
3740:
3736:
3733:
3730:
3725:
3722:
3718:
3714:
3711:
3706:
3702:
3698:
3695:
3692:
3687:
3684:
3677:
3674:
3647:
3642:
3638:
3634:
3631:
3628:
3623:
3620:
3616:
3588:Main article:
3585:
3582:
3580:
3577:
3573:
3572:
3561:
3558:
3555:
3550:
3546:
3531:
3530:
3519:
3516:
3513:
3488:
3487:
3470:
3467:
3463:
3457:
3454:
3447:
3443:
3437:
3432:
3429:
3425:
3419:
3416:
3409:
3405:
3399:
3395:
3389:
3385:
3379:
3376:
3372:
3367:
3361:
3357:
3353:
3349:
3345:
3340:
3336:
3330:
3326:
3320:
3317:
3313:
3309:
3306:
3301:
3297:
3291:
3287:
3282:
3276:
3271:
3268:
3266:
3264:
3260:
3254:
3251:
3247:
3243:
3237:
3234:
3228:
3223:
3216:
3212:
3204:
3201:
3197:
3192:
3185:
3180:
3177:
3171:
3167:
3161:
3157:
3150:
3147:
3141:
3136:
3133:
3129:
3123:
3120:
3116:
3111:
3105:
3102:
3098:
3089:
3084:
3080:
3075:
3071:
3065:
3062:
3058:
3054:
3048:
3045:
3039:
3034:
3031:
3027:
3022:
3016:
3011:
3008:
3005:
3003:
3001:
2996:
2993:
2989:
2983:
2979:
2975:
2974:
2960:
2959:
2946:
2943:
2940:
2936:
2932:
2927:
2924:
2921:
2917:
2913:
2909:
2903:
2900:
2896:
2892:
2886:
2883:
2877:
2872:
2869:
2865:
2860:
2853:
2848:
2845:
2839:
2834:
2831:
2827:
2821:
2817:
2804:. The result
2791:
2788:
2784:
2761:
2758:
2754:
2729:
2726:
2714:
2713:
2702:
2697:
2694:
2685:
2682:
2678:
2674:
2671:
2668:
2663:
2659:
2644:
2643:
2631:
2626:
2620:
2617:
2613:
2607:
2604:
2600:
2596:
2591:
2587:
2581:
2578:
2572:
2566:
2563:
2559:
2555:
2552:
2547:
2544:
2541:
2537:
2533:
2527:
2522:
2519:
2516:
2502:
2501:
2489:
2483:
2478:
2473:
2469:
2465:
2460:
2456:
2452:
2447:
2444:
2439:
2434:
2430:
2424:
2421:
2417:
2412:
2406:
2402:
2398:
2395:
2390:
2386:
2380:
2376:
2372:
2369:
2366:
2363:
2358:
2355:
2351:
2345:
2341:
2335:
2332:
2328:
2324:
2321:
2316:
2297:
2296:
2282:
2279:
2275:
2269:
2266:
2263:
2259:
2255:
2250:
2246:
2231:
2230:
2217:
2213:
2209:
2206:
2201:
2194:
2190:
2184:
2181:
2178:
2174:
2170:
2166:
2161:
2158:
2129:
2124:
2121:
2117:
2110:
2107:
2104:
2100:
2088:
2087:
2076:
2071:
2068:
2064:
2058:
2055:
2051:
2046:
2040:
2037:
2033:
2027:
2022:
2019:
2015:
2008:
2005:
2002:
1998:
1992:
1989:
1985:
1981:
1976:
1971:
1968:
1964:
1957:
1954:
1951:
1947:
1942:
1936:
1931:
1928:
1925:
1921:
1915:
1910:
1907:
1903:
1880:
1879:
1866:
1863:
1859:
1852:
1849:
1846:
1842:
1838:
1833:
1830:
1826:
1792:
1791:
1780:
1775:
1770:
1767:
1764:
1760:
1753:
1748:
1745:
1742:
1738:
1734:
1729:
1709:
1706:
1704:
1701:
1685:
1682:
1678:
1655:
1651:
1630:
1619:
1618:
1607:
1604:
1601:
1596:
1592:
1588:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1558:
1554:
1550:
1545:
1541:
1537:
1534:
1529:
1526:
1522:
1518:
1513:
1509:
1505:
1500:
1496:
1492:
1489:
1484:
1480:
1476:
1458:lapse function
1448:
1445:
1430:
1427:
1423:
1417:
1414:
1410:
1406:
1403:
1383:
1354:
1350:
1323:
1319:
1282:
1279:
1275:
1268:
1265:
1262:
1258:
1235:
1232:
1228:
1214:
1211:
1192:
1188:
1165:
1161:
1136:
1125:lapse function
1106:
1103:
1099:
1076:
1073:
1069:
1037:
1032:
1028:
1024:
1021:
1018:
1013:
1010:
1006:
982:
977:
973:
969:
966:
963:
958:
955:
951:
924:
920:
899:
877:
873:
852:
849:
782:Charles Misner
765:
764:
762:
761:
754:
747:
739:
736:
735:
734:
733:
720:
703:
702:
699:
698:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
525:
521:
520:
519:
516:
515:
512:
511:
506:
501:
496:
491:
486:
481:
476:
471:
466:
461:
456:
451:
446:
441:
436:
431:
426:
415:
409:
408:
407:
404:
403:
398:
397:
396:
395:
390:
382:
381:
377:
376:
375:
374:
372:Post-Newtonian
369:
364:
356:
355:
351:
350:
349:
348:
343:
338:
333:
328:
323:
315:
314:
308:
304:
303:
300:
296:
295:
294:
293:
290:
289:
284:
283:
282:
281:
276:
271:
263:
262:
256:
255:
254:
253:
248:
243:
238:
233:
231:Frame-dragging
228:
223:
218:
213:
208:
206:Kepler problem
198:
194:
193:
192:
189:
188:
185:
184:
179:
174:
169:
163:
159:
158:
157:
154:
153:
152:
151:
146:
144:
143:
138:
133:
127:
125:
117:
116:
102:
99:
95:
90:
86:
81:
78:
74:
70:
67:
62:
59:
55:
45:
37:
36:
15:
9:
6:
4:
3:
2:
5509:
5498:
5495:
5493:
5490:
5489:
5487:
5474:
5464:
5458:
5457:
5453:
5451:
5448:
5446:
5443:
5441:
5438:
5436:
5433:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5365:Chandrasekhar
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5325:Schwarzschild
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5302:
5300:
5296:
5286:
5282:
5281:
5278:
5275:
5273:
5270:
5268:
5264:
5263:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5242:
5239:
5235:
5231:
5230:
5227:
5224:
5221:
5218:
5216:
5212:
5211:Schwarzschild
5208:
5207:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5178:
5174:
5170:
5169:
5167:
5165:
5161:
5155:
5152:
5150:
5147:
5145:
5142:
5141:
5139:
5133:
5127:
5124:
5121:
5117:
5113:
5110:
5108:
5107:Shapiro delay
5105:
5103:
5100:
5097:
5093:
5089:
5086:
5083:
5079:
5076:
5075:
5072:
5069:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5050:collaboration
5047:
5043:
5040:
5038:
5034:
5031:
5030:
5027:
5024:
5022:
5019:
5017:
5016:Event horizon
5014:
5012:
5009:
5008:
5006:
5002:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4960:ADM formalism
4958:
4957:
4955:
4951:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4921:
4919:
4913:
4907:
4904:
4902:
4899:
4898:
4896:
4892:
4889:
4887:
4881:
4871:
4868:
4866:
4865:Biquaternions
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4847:
4845:
4843:
4839:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4792:Time dilation
4790:
4789:
4787:
4783:
4777:
4774:
4773:
4771:
4767:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4745:Proper length
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4717:
4715:
4709:
4703:
4700:
4698:
4695:
4692:
4689:
4687:
4683:
4680:
4679:
4677:
4673:
4670:
4668:
4662:
4658:
4651:
4646:
4644:
4639:
4637:
4632:
4631:
4628:
4620:
4614:
4610:
4606:
4601:
4600:
4588:
4584:
4580:
4576:
4572:
4568:
4563:
4558:
4554:
4550:
4546:
4540:
4530:
4526:
4522:
4518:
4514:
4510:
4503:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4470:
4466:
4459:
4454:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4422:
4415:
4410:
4406:
4402:
4398:
4394:
4390:
4386:
4382:
4378:
4371:
4366:
4362:
4358:
4354:
4350:
4346:
4342:
4338:
4334:
4327:
4322:
4318:
4314:
4310:
4306:
4302:
4298:
4291:
4286:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4254:
4247:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4211:
4206:
4205:
4201:
4193:
4189:
4185:
4181:
4177:
4173:
4168:
4167:gr-qc/0405109
4163:
4159:
4155:
4148:
4141:
4137:
4131:
4123:
4119:
4115:
4111:
4107:
4103:
4096:
4089:
4087:
4072:on 2011-07-20
4071:
4067:
4061:
4057:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4029:
4028:
4023:
4020:
4019:
4013:
4011:
4007:
4003:
3993:
3991:
3987:
3986:vacuum energy
3983:
3979:
3974:
3970:
3965:
3963:
3959:
3958:metric tensor
3955:
3951:
3947:
3941:
3931:
3927:
3925:
3920:
3916:
3912:
3906:
3896:
3894:
3890:
3871:
3860:
3856:
3852:
3849:
3841:
3838:
3834:
3830:
3826:
3821:
3818:
3807:
3803:
3799:
3796:
3788:
3785:
3775:
3764:
3750:
3742:
3738:
3734:
3731:
3723:
3720:
3716:
3704:
3700:
3696:
3693:
3685:
3682:
3672:
3661:
3660:
3659:
3640:
3636:
3632:
3629:
3621:
3618:
3614:
3605:
3601:
3597:
3591:
3576:
3559:
3556:
3553:
3548:
3544:
3536:
3535:
3534:
3517:
3514:
3511:
3504:
3503:
3502:
3499:
3497:
3493:
3468:
3465:
3461:
3455:
3452:
3445:
3441:
3435:
3430:
3427:
3423:
3417:
3414:
3407:
3403:
3397:
3393:
3387:
3383:
3377:
3374:
3370:
3365:
3359:
3351:
3347:
3343:
3338:
3328:
3318:
3315:
3311:
3307:
3304:
3299:
3289:
3280:
3274:
3269:
3267:
3258:
3252:
3249:
3245:
3241:
3235:
3232:
3226:
3221:
3214:
3210:
3202:
3199:
3195:
3190:
3183:
3178:
3175:
3169:
3165:
3159:
3155:
3148:
3145:
3139:
3134:
3131:
3127:
3121:
3118:
3114:
3109:
3103:
3100:
3096:
3087:
3082:
3078:
3073:
3069:
3063:
3060:
3056:
3052:
3046:
3043:
3037:
3032:
3029:
3025:
3020:
3014:
3009:
3006:
3004:
2999:
2994:
2991:
2987:
2981:
2965:
2964:
2963:
2944:
2941:
2938:
2934:
2930:
2925:
2922:
2919:
2915:
2911:
2907:
2901:
2898:
2894:
2890:
2884:
2881:
2875:
2870:
2867:
2863:
2858:
2851:
2846:
2843:
2837:
2832:
2829:
2825:
2819:
2807:
2806:
2805:
2789:
2786:
2782:
2759:
2756:
2752:
2743:
2739:
2735:
2734:metric tensor
2725:
2723:
2719:
2700:
2695:
2692:
2683:
2680:
2676:
2672:
2669:
2666:
2661:
2657:
2649:
2648:
2647:
2629:
2624:
2618:
2615:
2611:
2605:
2602:
2598:
2594:
2589:
2585:
2579:
2576:
2570:
2564:
2561:
2557:
2553:
2550:
2542:
2536:
2531:
2525:
2520:
2517:
2514:
2507:
2506:
2505:
2487:
2481:
2476:
2471:
2463:
2458:
2454:
2450:
2445:
2442:
2437:
2432:
2428:
2422:
2419:
2415:
2410:
2404:
2396:
2393:
2388:
2384:
2378:
2374:
2370:
2367:
2364:
2361:
2356:
2353:
2349:
2343:
2333:
2330:
2326:
2322:
2319:
2305:
2304:
2303:
2300:
2280:
2277:
2273:
2264:
2258:
2253:
2248:
2244:
2236:
2235:
2234:
2215:
2211:
2207:
2204:
2199:
2192:
2188:
2179:
2173:
2168:
2164:
2159:
2156:
2149:
2148:
2147:
2145:
2127:
2122:
2119:
2105:
2099:
2074:
2069:
2066:
2062:
2056:
2053:
2049:
2044:
2038:
2035:
2031:
2025:
2020:
2017:
2003:
1997:
1990:
1987:
1983:
1979:
1974:
1969:
1966:
1952:
1946:
1940:
1934:
1926:
1919:
1913:
1908:
1905:
1901:
1893:
1892:
1891:
1889:
1885:
1864:
1861:
1857:
1847:
1841:
1836:
1831:
1828:
1824:
1816:
1815:
1814:
1811:
1809:
1805:
1801:
1800:metric tensor
1797:
1778:
1773:
1765:
1758:
1751:
1743:
1737:
1732:
1718:
1717:
1716:
1715:
1700:
1683:
1680:
1676:
1653:
1649:
1628:
1602:
1599:
1594:
1590:
1586:
1581:
1577:
1573:
1564:
1561:
1556:
1552:
1548:
1543:
1539:
1535:
1527:
1524:
1520:
1516:
1511:
1507:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1467:
1466:
1465:
1463:
1459:
1454:
1444:
1428:
1425:
1421:
1415:
1412:
1408:
1404:
1401:
1381:
1373:
1368:
1352:
1339:
1321:
1308:
1303:
1301:
1296:
1280:
1277:
1273:
1263:
1257:
1233:
1230:
1226:
1210:
1208:
1190:
1163:
1159:
1150:
1134:
1126:
1122:
1104:
1101:
1097:
1074:
1071:
1067:
1057:
1055:
1051:
1030:
1026:
1022:
1019:
1011:
1008:
1004:
996:
975:
971:
967:
964:
956:
953:
949:
940:
939:metric tensor
922:
918:
897:
875:
862:
858:
848:
846:
845:
840:
839:
833:
831:
827:
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549:Schwarzschild
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419:Schwarzschild
417:
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241:Event horizon
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4959:
4901:Introduction
4827:Twin paradox
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4070:the original
4060:
4042:Peres metric
4026:
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3928:
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3886:
3593:
3579:Applications
3574:
3532:
3500:
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2961:
2731:
2715:
2645:
2503:
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2089:
1882:will be the
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1462:shift vector
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654:Raychaudhuri
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123:Introduction
5238:KerrâNewman
5209:Spherical:
5078:Other tests
5021:Singularity
4953:Formulation
4915:Fundamental
4769:Formulation
4750:Proper time
4711:Fundamental
1796:determinant
1372:determinant
1050:Hamiltonian
818:Hamiltonian
669:van Stockum
599:Oppenheimer
454:KerrâNewman
246:Singularity
5486:Categories
5390:Zel'dovich
5298:Scientists
5277:Alcubierre
5084:of Mercury
5082:precession
5011:Black hole
4894:Background
4886:relativity
4855:World line
4850:Light cone
4675:Background
4667:relativity
4657:Relativity
4597:References
4076:2021-03-25
3938:See also:
3492:non-linear
1714:Lagrangian
993:and their
522:Scientists
354:Formalisms
302:Formalisms
251:Black hole
177:World line
5360:Robertson
5345:Friedmann
5340:Eddington
5330:de Sitter
5164:Solutions
5042:detectors
5037:astronomy
5004:Phenomena
4939:Geodesics
4842:Spacetime
4785:Phenomena
4587:118570242
4562:0908.0679
4493:123381338
4449:121420368
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4281:120715041
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2014:Γ
1980:−
1963:Γ
1920:−
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1447:ADM Split
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1068:γ
1005:π
950:γ
872:Σ
857:spacetime
802:formalism
614:Robertson
579:Friedmann
574:Eddington
564:Nordström
554:de Sitter
411:Solutions
336:Geodesics
331:Friedmann
313:Equations
299:Equations
260:Spacetime
195:Phenomena
101:ν
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5473:Category
5350:LemaĂźtre
5315:Einstein
5305:Poincaré
5265:Others:
5249:TaubâNUT
5215:interior
5137:theories
5135:Advanced
5102:redshift
4917:concepts
4735:Rapidity
4713:concepts
4192:14054267
4016:See also
4002:Deruelle
3969:symmetry
1213:Notation
861:foliated
851:Overview
730:Category
594:LemaĂźtre
559:Reissner
544:Poincaré
529:Einstein
474:TaubâNUT
439:Wormhole
423:interior
136:Timeline
5415:Hawking
5410:Penrose
5395:Novikov
5375:Wheeler
5320:Hilbert
5310:Lorentz
5267:pp-wave
5088:lensing
4884:General
4665:Special
4567:Bibcode
4517:Bibcode
4473:Bibcode
4429:Bibcode
4385:Bibcode
4341:Bibcode
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4261:Bibcode
4225:Bibcode
4172:Bibcode
4138:(ed.),
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644:Penrose
619:Bardeen
609:Wheeler
539:Hilbert
534:Lorentz
494:pp-wave
131:History
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679:Newman
659:Taylor
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624:Walker
589:Zwicky
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5430:Bondi
5425:Hulse
5355:Milne
5259:discs
5203:Milne
5198:Gödel
5055:Virgo
4583:S2CID
4557:arXiv
4505:(PDF)
4489:S2CID
4461:(PDF)
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4329:(PDF)
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4162:arXiv
4098:(PDF)
4053:Notes
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479:Milne
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141:Tests
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