6850:(except in the trivial case of a locally flat spacetime); rather, the light cones appear (radially flattened) or (radially elongated). This is of course just another way of saying that Schwarzschild charts correctly represent distances within each nested round sphere, but the radial coordinate does not faithfully represent radial proper distance.
3941:
1289:
4873:
416:
4707:
6514:
This is all valid for any
Lorentzian manifold, but we note that in general relativity, the electrogravitic tensor controls tidal stresses on small objects, as measured by the observers corresponding to our frame, and the magnetogravitic tensor controls any spin-spin forces on spinning objects, as
7450:
6248:
1533:. In particular, the three spatial Killing vector fields have exactly the same form as the three nontranslational Killing vector fields in a spherically symmetric chart on E; that is, they exhibit the notion of arbitrary Euclidean rotation about the origin or spherical symmetry.
2992:
3759:
6509:
2815:
2520:
7087:
3250:
The point is that the defining characteristic of a
Schwarzschild chart in terms of the geometric interpretation of the radial coordinate is just what we need to carry out (in principle) this kind of spherically symmetric embedding of the spatial hyperslices.
7197:
526:
630:
and other properties expected of reasonable perfect fluid), then, with appropriate tensor fields representing physical quantities such as matter and momentum densities, we have a piece of a possibly larger spacetime; a piece which can be considered a
3666:
1087:
4048:
4718:
2294:(they are of course all isometric to one another) in a flat Euclidean space. People who find it difficult to visualize four-dimensional Euclidean space will be glad to observe that we can take advantage of the spherical symmetry to
858:
787:
159:
5122:
5008:
5225:
4551:
4531:(In this example, only four of the six are nonvanishing.) We can collect these one-forms into a matrix of one-forms, or even better an SO(1,3)-valued one-form. Note that the resulting matrix of one-forms will not quite be
3533:. (The fact that our spacetime admits a frame having this particular trigonometric form is yet another equivalent expression of the notion of a Schwarzschild chart in a static, spherically symmetric Lorentzian manifold).
6799:
7212:
5323:
6017:
2668:
1688:
Similarly, we can regard each sphere as the locus of a spherical cloud of idealized observers, who must (in general) use rocket engines to accelerate radially outward in order to maintain their position. These are
4467:
7602:
is notation for a vector field pointing in the timelike direction. It is written so as to resemble the differential operator with respect to t, because derivatives can be taken along this direction. The notation
6727:
6666:
6594:
3936:{\displaystyle d\sigma ^{3}=\sin \theta \,dr\wedge d\phi +r\,\cos \theta \,d\theta \wedge d\phi =-\left({\frac {\sin \theta \,d\phi }{b(r)}}\wedge \sigma ^{1}+\cos \theta \,d\phi \wedge \sigma ^{2}\right)}
4233:
5398:
2826:
106:
is not static inside the horizon, and the family of (spacelike) nested spheres cannot be extended inside the horizon, so the
Schwarzschild chart for this solution necessarily breaks down at the horizon.
5929:(eight linearly independent components, in general), which we think of as representing a linear operator on the six-dimensional vector space of two forms (at each event). From this we can read off the
4388:
2038:. Just as for an ordinary polar spherical chart on E, for topological reasons we cannot obtain continuous coordinates on the entire sphere; we must choose some longitude (a great circle) to act as the
1762:
1888:
1683:
4526:
6877:
3095:
2679:
6296:
3484:
2032:
55:
to these nested round spheres. The defining characteristic of
Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and
4160:
3753:
2350:
146:
and verify that the same tensor is defined on the overlaps of the domains of the charts. In this article, we will only attempt to define the metric tensor in the domain of a single chart.
1962:
6913:
6029:
4107:
1010:
2253:
excise the region outside some ball, or inside some ball, from the domain of our chart. This happens whenever f or g blow up at some value of the
Schwarzschild radial coordinate r.
903:
3434:
1468:
567:
7093:
3343:
422:
7659:
3393:
4319:
4276:
3237:
2342:
6840:
3160:
2154:
1075:
716:
3704:
2247:
1373:
7628:
7600:
1565:
1423:
683:
615:). Alternatively, we can plug in specific functions (possibly depending on some parameters) to obtain a Schwarzschild coordinate chart on a specific Lorentzian spacetime.
5920:
1319:
594:
7539:
6288:
2214:
1284:{\displaystyle g|_{t=t_{0},r=r_{0}}=r_{0}^{2}g_{\Omega }=r_{0}^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right),\;0<\theta <\pi ,\;-\pi <\phi <\pi }
4868:{\displaystyle {\Omega ^{\hat {m}}}_{\hat {n}}={R^{\hat {m}}}_{{\hat {n}}|{\hat {\imath }}{\hat {\jmath }}|}\,\sigma ^{\hat {\imath }}\wedge \sigma ^{\hat {\jmath }}}
2292:
2098:
2065:
1488:
955:
3956:
1812:
1607:
2174:
1512:
411:{\displaystyle g=-a(r)^{2}\,dt^{2}+b(r)^{2}\,dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)=-a(r)^{2}\,dt^{2}+b(r)^{2}\,dr^{2}+r^{2}g_{\Omega }}
7547:, a chart covering the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity,
3511:
3024:
2570:
3542:
2124:
793:
722:
3531:
136:
6883:
the Janis-Newman-Winacour solution (which models the exterior of a static spherically symmetric object endowed with a massless minimally coupled scalar field),
4702:{\displaystyle {\Omega ^{\hat {m}}}_{\hat {n}}=d{\omega ^{\hat {m}}}_{\hat {n}}-{\omega ^{\hat {m}}}_{\hat {\ell }}\wedge {\omega ^{\hat {\ell }}}_{\hat {n}}}
4536:
1081:
in polar spherical fashion), and from its form, we see that the
Schwarzschild metric restricted to any of these surfaces is positive definite and given by
5014:
6904:
5128:
7445:{\displaystyle g=-a(r)^{2}\,dt^{2}+b(r)^{2}\,dr^{2}+{\frac {dx^{2}+dy^{2}}{(1+x^{2}+y^{2})^{2}}},\;-\infty <t,x,y<\infty ,r_{1}<r<r_{2}}
7681:
6733:
5231:
5936:
2582:
4394:
4057:. (The hats are just a notational device for reminding us that the indices refer to our cobasis one-forms, not to the coordinate one-forms
4908:
6672:
7473:
6887:
4879:
6600:
6528:
2987:{\displaystyle d\rho ^{2}=\left(1+f^{\prime }(r)^{2}\right)\,dr^{2}+r^{2}\,d\phi ^{2},\;r_{1}<r<r_{2},\,-\pi <\phi <\pi }
1525:
on our
Lorentzian manifold, give the truest expression of both the symmetries of a static spherically symmetric spacetime, while the
3243:
embeddings of annular rings (for regions of positive or negative
Gaussian curvature). In general, we should not expect to obtain a
611:
as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the
5331:
2261:
To better understand the significance of the
Schwarzschild radial coordinate, it may help to embed one of the spatial hyperslices
4325:
1696:
917:. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a
7491:
6903:
It is natural to consider nonstatic but spherically symmetric spacetimes, with a generalized
Schwarzschild chart in which the
4168:
1820:
1615:
7550:
4473:
3287:
2810:{\displaystyle \partial _{r}=(f^{\prime }(r),\,\cos \phi ,\,\sin \phi ),\;\;\partial _{\phi }=(0,-r\sin \phi ,r\cos \phi )}
6870:
6504:{\displaystyle L_{11}={\frac {1}{r^{2}}}{\frac {1-b^{2}}{b^{2}}}(r),\;L_{22}=L_{33}={\frac {1}{r}}{\frac {-b'}{b^{3}}}(r)}
7202:
Generalizing in another direction, we can use other coordinate systems on our round two-spheres, to obtain for example a
3032:
597:
7497:
6519:
4899:
3440:
2515:{\displaystyle g|_{t=0,\theta =\pi /2}=b(r)^{2}dr^{2}+r^{2}d\phi ^{2},\;\;r_{1}<r<r_{2},\,-\pi <\phi <\pi }
79:
1971:
7082:{\displaystyle g=-a(t,r)^{2}\,dt^{2}+b(t,r)^{2}\,dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right),}
4115:
3710:
7544:
6243:{\displaystyle E_{11}={\frac {a''\,b-a'\,b'}{a\,b^{3}}}(r),\;E_{22}=E_{33}={\frac {1}{r}}{\frac {a'}{a\,b^{2}}}(r)}
1901:
7509:
7467:
962:
910:
45:
21:
4060:
2067:
and cut this out of the chart. The result is that we cut out a closed half plane from each spatial hyperslice
975:
7192:{\displaystyle -\infty <t<\infty ,\,r_{0}<r<r_{1},\,0<\theta <\pi ,\,-\pi <\phi <\pi }
866:
521:{\displaystyle -\infty <t<\infty ,\,r_{0}<r<r_{1},\,0<\theta <\pi ,\,-\pi <\phi <\pi }
7686:
3399:
1435:
534:
4112:
If we recall which pairs of indices are symmetric (space-time) and which are antisymmetric (space-space) in
3296:
3276:
63:
7633:
3349:
4282:
4239:
3180:
2820:
The induced metric inherited when we restrict the Euclidean metric on E to our parameterized surface is
2301:
7485:
6807:
3103:
2132:
1027:
694:
6290:
is irrotational) we can determine the three-dimensional Riemann tensor of the spatial hyperslices, is
3672:
2537:
is defined on a parameterized surface, which will inherit the desired metric from the embedding space,
2219:
1334:
1540:, i.e. distances taken along the spacelike geodesic congruence which arise as the integral curves of
7606:
7578:
2180:
coordinate, and indeed thinking of our three spacelike Killing vectors as acting on round spheres.
2156:
is a Killing vector field, we omitted the pedantic but important qualifier that we are thinking of
1543:
1382:
661:
619:
5406:
1297:
623:
612:
572:
87:
7556:
7515:
6264:
2186:
6254:
6020:
4535:
as for an SO(4)-valued one-form; we need to use instead a notion of transpose arising from the
4043:{\displaystyle d\sigma ^{\hat {m}}=-{\omega ^{\hat {m}}}_{\hat {n}}\,\wedge \sigma ^{\hat {n}}}
7479:
6894:
to an exterior region, which is locally isometric to part of the Schwarzschild vacuum region.
2526:
2264:
2070:
2044:
1473:
927:
37:
3661:{\displaystyle d\sigma ^{0}=-a'(r)\,dr\wedge dt={\frac {a'(r)}{b(r)}}\,dt\wedge \sigma ^{1}}
1782:
1577:
6863:
2344:. Now we have a two-dimensional Riemannian manifold with a local radial coordinate chart,
2159:
1497:
1013:
972:
Note the last two fields are rotations of one-another, under the coordinate transformation
853:{\displaystyle \cos \phi \,\partial _{\theta }-\cot \theta \,\sin \phi \,\partial _{\phi }}
782:{\displaystyle \sin \phi \,\partial _{\theta }+\cot \theta \,\cos \phi \,\partial _{\phi }}
648:
95:
3492:
3165:
This works for surfaces in which true distances between two radially separated points are
3000:
2546:
8:
6258:
4054:
3174:
2103:
1078:
139:
17:
4542:
Third, we compute the exterior derivatives of the connection one-forms and use Cartan's
3516:
3282:
As an illustration, we will indicate how to compute the connection and curvature using
1377:
121:
75:
67:
56:
59:
of each sphere. However, radial distances and angles are not accurately represented.
7503:
5930:
1898:
Looking back at the coordinate ranges above, note that the coordinate singularity at
627:
5117:{\displaystyle {R^{0}}_{202}={\frac {1}{r}}{\frac {-a'}{a\,b^{2}}}(r)={R^{0}}_{303}}
7461:
6843:
5925:
where E,L are symmetric (six linearly independent components, in general) and B is
3283:
3275:
in deriving static spherically symmetric solutions in general relativity (or other
918:
906:
143:
71:
41:
29:
1518:. This is essentially the defining geometric feature of the Schwarzschild chart.
3169:
than the difference between their radial coordinates. If the true distances are
2530:
1529:
which they take in our chart is the truest expression of the meaning of the term
5220:{\displaystyle {R^{1}}_{212}={\frac {1}{r}}{\frac {b'}{b^{3}}}(r)={R^{1}}_{313}}
7662:
4895:
2997:
To identify this with the metric of our hyperslice, we should evidently choose
7494:, a simple chart that's valid inside the event horizon of a static black hole.
7488:, a less common alternative chart for static spherically symmetric spacetimes,
1321:
is the standard Riemannian metric on the unit radius 2-sphere. That is, these
1016:
provides a detailed derivation and discussion of the three space-like fields.
142:. The simplest way to define this tensor is to define it in compatible local
7675:
1568:
116:
99:
91:
6858:
Some examples of exact solutions which can be obtained in this way include:
6794:{\displaystyle {\vec {e}}_{3}={\frac {1}{r\sin \theta }}\,\partial _{\phi }}
149:
In a Schwarzschild chart (on a static spherically symmetric spacetime), the
6880:
electrolambdavacuum, which includes the previous example as a special case,
5318:{\displaystyle {R^{2}}_{323}={\frac {1}{r^{2}}}{\frac {b^{2}-1}{b^{2}}}(r)}
1521:
It may help to add that the four Killing fields given above, considered as
1329:
150:
6012:{\displaystyle {\vec {X}}={\vec {e}}_{0}={\frac {1}{a(r)}}\,\partial _{t}}
3267:
regarded as undetermined functions of the Schwarzschild radial coordinate
2663:{\displaystyle (z,r,\phi )\rightarrow (f(r),\,r\cos \phi ,\,r\sin \phi )}
1772:
966:
644:
3536:
Second, we compute the exterior derivatives of these cobasis one-forms:
969:, where the timelike coordinate vector is not hypersurface orthogonal.)
626:(say, for a static spherically symmetric perfect fluid obeying suitable
4462:{\displaystyle {\omega ^{1}}_{3}=-{\frac {\sin \theta \,d\phi }{b(r)}}}
1776:
103:
6853:
5926:
48:
7482:, another popular chart for static spherically symmetric spacetimes,
1536:
However, note well: in general, the Schwarzschild radial coordinate
7661:
is frequently and generically used to denote a vector field in the
7553:, an alternative chart for static spherically symmetric spacetimes,
6891:
5003:{\displaystyle {R^{0}}_{101}={\frac {-a''\,b+a'\,b'}{a\,b^{3}}}(r)}
1572:
6886:
stellar models obtained by matching an interior region which is a
6722:{\displaystyle {\vec {e}}_{2}={\frac {1}{r}}\,\partial _{\theta }}
3247:
embedding in any one flat space (with vanishing Riemann tensor).
3177:
as a spacelike surface in E instead. For example, we might have
6661:{\displaystyle {\vec {e}}_{1}={\frac {1}{b(r)}}\,\partial _{r}}
6589:{\displaystyle {\vec {e}}_{0}={\frac {1}{a(r)}}\,\partial _{t}}
4712:
to compute the curvature two forms. Fourth, using the formula
3272:
4894:), we can read off the linearly independent components of the
86:
spherically symmetric vacuum or electrovacuum solution of the
5393:{\displaystyle R_{{\hat {m}}{\hat {n}}{\hat {i}}{\hat {j}}}}
4383:{\displaystyle {\omega ^{1}}_{2}=-{\frac {d\theta }{b(r)}}}
1757:{\displaystyle r=r_{0},\theta =\theta _{0},\phi =\phi _{0}}
1470:
are exactly the usual polar spherical angular coordinates:
7559:, an earliest chart which is regular at the event horizon.
28:. In such a spacetime, a particularly important kind of
6873:, which includes the previous example as a special case,
5328:
Fifth, we can lower indices and organize the components
4228:{\displaystyle {\omega ^{0}}_{1}={\frac {a'}{b}}(r)\,dt}
1883:{\displaystyle \Delta \tau =\int _{t_{1}}^{t_{2}}a(r)dt}
1678:{\displaystyle \Delta \rho =\int _{r_{1}}^{r_{2}}b(r)dr}
4521:{\displaystyle {\omega ^{2}}_{3}=-\cos \theta \,d\phi }
4162:, we can confirm that the six connection one-forms are
6515:
measured by the observers corresponding to our frame.
5874:
5415:
603:
Depending on context, it may be appropriate to regard
7636:
7609:
7581:
7518:
7215:
7096:
6916:
6810:
6736:
6675:
6603:
6531:
6299:
6267:
6032:
5939:
5409:
5334:
5234:
5131:
5017:
4911:
4721:
4554:
4476:
4397:
4328:
4285:
4242:
4171:
4118:
4063:
3959:
3762:
3713:
3675:
3545:
3519:
3495:
3443:
3402:
3352:
3299:
3183:
3106:
3035:
3003:
2829:
2682:
2585:
2549:
2353:
2304:
2267:
2256:
2222:
2189:
2162:
2135:
2106:
2073:
2047:
1974:
1904:
1823:
1785:
1699:
1618:
1580:
1546:
1500:
1476:
1438:
1385:
1337:
1300:
1090:
1030:
978:
930:
869:
796:
725:
697:
664:
575:
537:
425:
162:
124:
1019:
6854:
Some exact solutions admitting Schwarzschild charts
6842:only multiplies the first of the three orthonormal
3090:{\displaystyle f^{\prime }(r)={\sqrt {1-b(r)^{2}}}}
600:for a more detailed derivation of this expression.
7653:
7622:
7594:
7533:
7444:
7191:
7081:
6834:
6793:
6721:
6660:
6588:
6503:
6282:
6242:
6011:
5914:
5392:
5317:
5219:
5116:
5002:
4867:
4701:
4520:
4461:
4382:
4313:
4270:
4227:
4154:
4101:
4042:
3935:
3747:
3698:
3660:
3525:
3505:
3478:
3428:
3387:
3337:
3231:
3154:
3089:
3018:
2986:
2809:
2662:
2564:
2514:
2336:
2286:
2241:
2208:
2168:
2148:
2118:
2092:
2059:
2026:
1956:
1882:
1806:
1756:
1677:
1601:
1559:
1506:
1482:
1462:
1417:
1367:
1313:
1283:
1069:
1004:
949:
897:
852:
781:
710:
677:
588:
561:
520:
410:
130:
74:spherically symmetric spacetimes. In the case of
7500:, for more about frame fields and coframe fields,
3479:{\displaystyle \sigma ^{3}=r\sin \theta \,d\phi }
2673:The coordinate vector fields on this surface are
596:is the standard metric on the unit 2-sphere. See
7673:
6846:fields here means that Schwarzschild charts are
3100:To take a somewhat silly example, we might have
2298:. This may be conveniently achieved by setting
2027:{\displaystyle t=t_{0},\,r=r_{0},\,\theta =\pi }
6890:solution across a spherical locus of vanishing
5933:with respect to the timelike unit vector field
4155:{\displaystyle {\omega ^{\hat {m}}}_{\hat {n}}}
1571:' between two of our nested spheres, we should
4882:indicate that we should sum only over the six
3748:{\displaystyle d\sigma ^{2}=dr\wedge d\theta }
1538:does not accurately represent radial distances
94:. The extension of the exterior region of the
90:is static, but this is certainly not true for
1957:{\displaystyle t=t_{0},\,r=r_{0},\,\theta =0}
1779:of one of these observers, we must integrate
1325:do in fact represent geometric spheres with
913:vanishes; thus, this Killing vector field is
643:With respect to the Schwarzschild chart, the
3513:are as yet undetermined smooth functions of
622:such that the resulting model satisfies the
7474:static spherically symmetric perfect fluids
2576:To wit, consider the parameterized surface
2126:and a half plane extending from that axis.
1968:of one of our static nested spheres, while
1893:
1609:along some coordinate ray from the origin:
7379:
6888:static spherically symmetric perfect fluid
6392:
6130:
2932:
2751:
2750:
2460:
2459:
1262:
1243:
688:and three spacelike Killing vector fields
7280:
7244:
7170:
7151:
7118:
7057:
6993:
6951:
6780:
6708:
6647:
6575:
6217:
6104:
6090:
6075:
5998:
5069:
4977:
4963:
4948:
4898:with respect to our coframe and its dual
4827:
4511:
4435:
4218:
4102:{\displaystyle dt,\,dr,\,d\theta ,d\phi }
4083:
4073:
4019:
3950:(or rather its integrability condition),
3908:
3859:
3820:
3810:
3788:
3695:
3638:
3582:
3499:
3469:
3425:
3378:
3328:
3286:. First, we read off the line element a
2965:
2915:
2888:
2734:
2721:
2644:
2628:
2493:
2014:
1994:
1944:
1924:
1567:. Rather, to find a suitable notion of '
1221:
1050:
1024:In the Schwarzschild chart, the surfaces
1005:{\displaystyle \phi \mapsto \phi +\pi /2}
921:. One immediate consequence is that the
839:
829:
806:
768:
758:
735:
569:is the standard spherical coordinate and
499:
480:
447:
371:
335:
291:
227:
191:
961:. (This is not true for example in the
898:{\displaystyle {\vec {X}}=\partial _{t}}
638:
7682:Coordinate charts in general relativity
3429:{\displaystyle \sigma ^{2}=rd\theta \,}
3239:. Sometimes we might need two or more
1814:along the appropriate coordinate line:
1463:{\displaystyle \Omega =(\theta ,\phi )}
562:{\displaystyle \Omega =(\theta ,\phi )}
62:These charts have many applications in
7674:
1077:appear as round spheres (when we plot
7512:, for more about congruences such as
3338:{\displaystyle \sigma ^{0}=-a(r)\,dt}
2533:, we adopt a frame field in E which
1432:. Moreover, the angular coordinates
7654:{\displaystyle \partial /\partial x}
3388:{\displaystyle \sigma ^{1}=b(r)\,dr}
1693:, and they have world lines of form
4314:{\displaystyle {\omega ^{0}}_{3}=0}
4271:{\displaystyle {\omega ^{0}}_{2}=0}
3259:The line element given above, with
1775:interval between two events on the
1764:, which of course have the form of
598:Deriving the Schwarzschild solution
13:
7645:
7637:
7611:
7583:
7498:frame fields in general relativity
7407:
7383:
7112:
7100:
6898:
6782:
6710:
6649:
6577:
6261:, from which (using the fact that
6000:
4725:
4558:
3254:
3232:{\displaystyle b(r)=f(r)=\sinh(r)}
3041:
2862:
2753:
2704:
2684:
2543:features an undetermined function
2337:{\displaystyle t=0,\theta =\pi /2}
2257:Visualizing the static hyperslices
2236:
2137:
1824:
1619:
1548:
1439:
1306:
1161:
886:
841:
808:
770:
737:
699:
666:
581:
538:
441:
429:
403:
14:
7698:
7551:Eddington–Finkelstein coordinates
7204:stereographic Schwarzschild chart
6835:{\displaystyle {\frac {1}{b(r)}}}
3284:Cartan's exterior calculus method
3155:{\displaystyle b(r)=f(r)=\sin(r)}
2149:{\displaystyle \partial _{\phi }}
1070:{\displaystyle t=t_{0},\,r=r_{0}}
1020:A family of static nested spheres
923:constant time coordinate surfaces
711:{\displaystyle \partial _{\phi }}
138:is part of the definition of any
6871:Reissner–Nordström electrovacuum
3699:{\displaystyle d\sigma ^{1}=0\,}
2525:To embed this surface (or at an
2242:{\displaystyle r_{2}<\infty }
1368:{\displaystyle A=4\pi r_{0}^{2}}
635:of the Einstein field equation.
22:spherically symmetric spacetimes
7510:congruence (general relativity)
7492:Gullstrand–Painlevé coordinates
7468:spherically symmetric spacetime
2540:is adapted to our radial chart,
965:for the exterior region of the
905:is irrotational means that the
70:. They are most often used in
7570:
7525:
7364:
7331:
7271:
7264:
7235:
7228:
6984:
6971:
6942:
6929:
6826:
6820:
6744:
6683:
6641:
6635:
6611:
6569:
6563:
6539:
6498:
6492:
6443:
6436:
6427:
6412:
6405:
6396:
6386:
6380:
6319:
6312:
6303:
6274:
6257:vanishes identically, and the
6237:
6231:
6181:
6174:
6165:
6150:
6143:
6134:
6124:
6118:
6052:
6045:
6036:
5992:
5986:
5962:
5946:
5382:
5370:
5358:
5346:
5312:
5306:
5192:
5186:
5089:
5083:
4997:
4991:
4858:
4838:
4821:
4814:
4802:
4792:
4785:
4770:
4748:
4734:
4692:
4678:
4656:
4642:
4620:
4606:
4581:
4567:
4453:
4447:
4374:
4368:
4215:
4209:
4145:
4131:
4033:
4012:
3998:
3973:
3877:
3871:
3632:
3626:
3618:
3612:
3579:
3573:
3375:
3369:
3325:
3319:
3277:metric theories of gravitation
3226:
3220:
3208:
3202:
3193:
3187:
3149:
3143:
3131:
3125:
3116:
3110:
3076:
3069:
3052:
3046:
3013:
3007:
2874:
2867:
2804:
2765:
2744:
2715:
2709:
2696:
2657:
2622:
2616:
2610:
2607:
2604:
2586:
2559:
2553:
2408:
2401:
2359:
1871:
1865:
1795:
1789:
1666:
1660:
1590:
1584:
1457:
1445:
1096:
982:
876:
556:
544:
362:
355:
326:
319:
218:
211:
182:
175:
64:metric theories of gravitation
1:
7623:{\displaystyle \partial _{x}}
7595:{\displaystyle \partial _{t}}
4053:we guess expressions for the
1560:{\displaystyle \partial _{r}}
1527:particular trigonometric form
1418:{\displaystyle K=1/r_{0}^{2}}
957:form a family of (isometric)
678:{\displaystyle \partial _{t}}
651:is generated by the timelike
618:If this turns out to admit a
110:
7545:Kruskal–Szekeres coordinates
6878:Reissner–Nordström–de Sitter
3271:, is often used as a metric
1768:in the Schwarzschild chart.
7:
7455:
7206:which is sometimes useful:
6862:the exterior region of the
5915:{\displaystyle \left=\left}
1314:{\displaystyle g_{\Omega }}
589:{\displaystyle g_{\Omega }}
102:of a spherically symmetric
10:
7703:
7534:{\displaystyle {\vec {X}}}
7486:Gaussian polar coordinates
6283:{\displaystyle {\vec {X}}}
4544:second structural equation
2209:{\displaystyle r_{1}>0}
2034:marks the location of the
1964:marks the location of the
38:polar spherical coordinate
6804:The fact that the factor
3948:first structural equation
1766:vertical coordinate lines
1323:nested coordinate spheres
7563:
6522:of our coframe field is
3946:Comparing with Cartan's
2249:, in which case we must
2129:When we said above that
1894:Coordinate singularities
1771:In order to compute the
1490:is sometimes called the
1428:In particular, they are
6848:not spatially isotropic
2296:suppress one coordinate
2287:{\displaystyle t=t_{0}}
2093:{\displaystyle t=t_{0}}
2060:{\displaystyle \phi =0}
1483:{\displaystyle \theta }
1430:geometric round spheres
950:{\displaystyle t=t_{0}}
915:hypersurface orthogonal
624:Einstein field equation
613:Einstein field equation
88:Einstein field equation
7655:
7624:
7596:
7535:
7506:of the Riemann tensor,
7446:
7193:
7083:
6836:
6795:
6723:
6662:
6590:
6505:
6284:
6255:magnetogravitic tensor
6244:
6021:electrogravitic tensor
6013:
5916:
5394:
5319:
5221:
5118:
5004:
4869:
4703:
4522:
4463:
4384:
4315:
4272:
4229:
4156:
4103:
4044:
3937:
3749:
3700:
3662:
3527:
3507:
3480:
3430:
3389:
3339:
3233:
3173:, we should embed our
3156:
3091:
3020:
2988:
2811:
2664:
2566:
2516:
2338:
2288:
2243:
2210:
2170:
2150:
2120:
2094:
2061:
2028:
1958:
1884:
1808:
1807:{\displaystyle a(r)dt}
1758:
1679:
1603:
1602:{\displaystyle b(r)dr}
1561:
1523:abstract vector fields
1514:is usually called the
1508:
1484:
1464:
1419:
1369:
1315:
1285:
1071:
1006:
951:
899:
854:
783:
712:
679:
590:
563:
522:
412:
132:
7656:
7625:
7597:
7536:
7480:isotropic coordinates
7447:
7194:
7084:
6837:
6796:
6724:
6663:
6591:
6506:
6285:
6245:
6014:
5917:
5395:
5320:
5222:
5119:
5005:
4870:
4704:
4523:
4464:
4385:
4316:
4273:
4230:
4157:
4104:
4045:
3938:
3750:
3701:
3663:
3528:
3508:
3481:
3431:
3390:
3340:
3234:
3157:
3092:
3021:
2989:
2812:
2665:
2567:
2517:
2339:
2289:
2244:
2211:
2183:Possibly, of course,
2171:
2169:{\displaystyle \phi }
2151:
2121:
2095:
2062:
2029:
1959:
1885:
1809:
1759:
1680:
1604:
1562:
1509:
1507:{\displaystyle \phi }
1485:
1465:
1420:
1370:
1316:
1286:
1072:
1014:Killing vector fields
1007:
963:Boyer–Lindquist chart
952:
909:of the corresponding
900:
855:
784:
713:
680:
655:Killing vector field
649:Killing vector fields
639:Killing vector fields
591:
564:
523:
413:
133:
46:spherically symmetric
7687:Lorentzian manifolds
7634:
7607:
7579:
7557:Lemaître coordinates
7516:
7213:
7094:
6914:
6864:Schwarzschild vacuum
6808:
6734:
6673:
6601:
6529:
6297:
6265:
6030:
5937:
5407:
5332:
5232:
5129:
5015:
4909:
4719:
4552:
4474:
4395:
4326:
4283:
4240:
4169:
4116:
4061:
4055:connection one-forms
3957:
3760:
3711:
3673:
3543:
3517:
3506:{\displaystyle a\,b}
3493:
3441:
3400:
3350:
3297:
3181:
3104:
3033:
3019:{\displaystyle f(r)}
3001:
2827:
2680:
2583:
2565:{\displaystyle f(r)}
2547:
2351:
2302:
2265:
2220:
2187:
2160:
2133:
2104:
2071:
2045:
1972:
1902:
1821:
1783:
1697:
1616:
1578:
1544:
1498:
1474:
1436:
1383:
1335:
1298:
1088:
1028:
976:
928:
867:
794:
723:
695:
662:
620:stress–energy tensor
573:
535:
423:
160:
122:
98:solution inside the
96:Schwarzschild vacuum
26:nested round spheres
18:Lorentzian manifolds
6259:topogravitic tensor
3175:Riemannian manifold
2119:{\displaystyle r=0}
2100:including the axis
1861:
1656:
1531:Schwarzschild chart
1414:
1364:
1183:
1155:
959:spatial hyperslices
911:timelike congruence
140:Lorentzian manifold
34:Schwarzschild chart
7651:
7620:
7592:
7531:
7442:
7189:
7079:
6832:
6791:
6719:
6658:
6586:
6501:
6280:
6240:
6009:
5912:
5906:
5860:
5390:
5315:
5217:
5114:
5000:
4865:
4699:
4537:Lorentzian adjoint
4518:
4459:
4380:
4311:
4268:
4225:
4152:
4099:
4040:
3933:
3745:
3696:
3658:
3523:
3503:
3476:
3426:
3385:
3335:
3229:
3152:
3087:
3016:
2984:
2807:
2660:
2562:
2512:
2334:
2284:
2239:
2206:
2166:
2146:
2116:
2090:
2057:
2024:
1954:
1880:
1833:
1804:
1754:
1675:
1628:
1599:
1557:
1504:
1480:
1460:
1415:
1400:
1378:Gaussian curvature
1365:
1350:
1311:
1281:
1169:
1141:
1067:
1012:. The article on
1002:
947:
895:
863:Here, saying that
850:
779:
708:
675:
586:
559:
518:
408:
128:
82:states that every
80:Birkhoff's theorem
76:general relativity
68:general relativity
57:Gaussian curvature
24:admit a family of
7528:
7504:Bel decomposition
7374:
6830:
6778:
6747:
6706:
6686:
6645:
6614:
6573:
6542:
6490:
6463:
6439:
6408:
6378:
6346:
6315:
6277:
6229:
6201:
6177:
6146:
6116:
6048:
5996:
5965:
5949:
5931:Bel decomposition
5385:
5373:
5361:
5349:
5304:
5272:
5184:
5162:
5081:
5048:
4989:
4861:
4841:
4817:
4805:
4788:
4773:
4751:
4737:
4695:
4681:
4659:
4645:
4623:
4609:
4584:
4570:
4457:
4378:
4207:
4148:
4134:
4036:
4015:
4001:
3976:
3881:
3636:
3526:{\displaystyle r}
3085:
879:
628:energy conditions
144:coordinate charts
131:{\displaystyle g}
16:In the theory of
7694:
7666:
7660:
7658:
7657:
7652:
7644:
7629:
7627:
7626:
7621:
7619:
7618:
7601:
7599:
7598:
7593:
7591:
7590:
7574:
7540:
7538:
7537:
7532:
7530:
7529:
7521:
7462:static spacetime
7451:
7449:
7448:
7443:
7441:
7440:
7422:
7421:
7375:
7373:
7372:
7371:
7362:
7361:
7349:
7348:
7329:
7328:
7327:
7312:
7311:
7298:
7293:
7292:
7279:
7278:
7257:
7256:
7243:
7242:
7198:
7196:
7195:
7190:
7147:
7146:
7128:
7127:
7088:
7086:
7085:
7080:
7075:
7071:
7070:
7069:
7050:
7049:
7037:
7036:
7019:
7018:
7006:
7005:
6992:
6991:
6964:
6963:
6950:
6949:
6844:spacelike vector
6841:
6839:
6838:
6833:
6831:
6829:
6812:
6800:
6798:
6797:
6792:
6790:
6789:
6779:
6777:
6760:
6755:
6754:
6749:
6748:
6740:
6728:
6726:
6725:
6720:
6718:
6717:
6707:
6699:
6694:
6693:
6688:
6687:
6679:
6667:
6665:
6664:
6659:
6657:
6656:
6646:
6644:
6627:
6622:
6621:
6616:
6615:
6607:
6595:
6593:
6592:
6587:
6585:
6584:
6574:
6572:
6555:
6550:
6549:
6544:
6543:
6535:
6510:
6508:
6507:
6502:
6491:
6489:
6488:
6479:
6478:
6466:
6464:
6456:
6451:
6450:
6441:
6440:
6432:
6420:
6419:
6410:
6409:
6401:
6379:
6377:
6376:
6367:
6366:
6365:
6349:
6347:
6345:
6344:
6332:
6327:
6326:
6317:
6316:
6308:
6289:
6287:
6286:
6281:
6279:
6278:
6270:
6249:
6247:
6246:
6241:
6230:
6228:
6227:
6226:
6212:
6204:
6202:
6194:
6189:
6188:
6179:
6178:
6170:
6158:
6157:
6148:
6147:
6139:
6117:
6115:
6114:
6113:
6099:
6098:
6089:
6074:
6065:
6060:
6059:
6050:
6049:
6041:
6018:
6016:
6015:
6010:
6008:
6007:
5997:
5995:
5978:
5973:
5972:
5967:
5966:
5958:
5951:
5950:
5942:
5921:
5919:
5918:
5913:
5911:
5907:
5898:
5897:
5865:
5861:
5857:
5856:
5845:
5844:
5833:
5832:
5821:
5820:
5809:
5808:
5797:
5796:
5783:
5782:
5771:
5770:
5759:
5758:
5747:
5746:
5735:
5734:
5723:
5722:
5709:
5708:
5697:
5696:
5685:
5684:
5673:
5672:
5661:
5660:
5649:
5648:
5635:
5634:
5623:
5622:
5611:
5610:
5599:
5598:
5587:
5586:
5575:
5574:
5561:
5560:
5549:
5548:
5537:
5536:
5525:
5524:
5513:
5512:
5501:
5500:
5487:
5486:
5475:
5474:
5463:
5462:
5451:
5450:
5439:
5438:
5427:
5426:
5399:
5397:
5396:
5391:
5389:
5388:
5387:
5386:
5378:
5375:
5374:
5366:
5363:
5362:
5354:
5351:
5350:
5342:
5324:
5322:
5321:
5316:
5305:
5303:
5302:
5293:
5286:
5285:
5275:
5273:
5271:
5270:
5258:
5253:
5252:
5247:
5246:
5245:
5226:
5224:
5223:
5218:
5216:
5215:
5210:
5209:
5208:
5185:
5183:
5182:
5173:
5165:
5163:
5155:
5150:
5149:
5144:
5143:
5142:
5123:
5121:
5120:
5115:
5113:
5112:
5107:
5106:
5105:
5082:
5080:
5079:
5078:
5064:
5063:
5051:
5049:
5041:
5036:
5035:
5030:
5029:
5028:
5009:
5007:
5006:
5001:
4990:
4988:
4987:
4986:
4972:
4971:
4962:
4947:
4935:
4930:
4929:
4924:
4923:
4922:
4884:increasing pairs
4874:
4872:
4871:
4866:
4864:
4863:
4862:
4854:
4844:
4843:
4842:
4834:
4826:
4825:
4824:
4819:
4818:
4810:
4807:
4806:
4798:
4795:
4790:
4789:
4781:
4777:
4776:
4775:
4774:
4766:
4754:
4753:
4752:
4744:
4741:
4740:
4739:
4738:
4730:
4708:
4706:
4705:
4700:
4698:
4697:
4696:
4688:
4685:
4684:
4683:
4682:
4674:
4662:
4661:
4660:
4652:
4649:
4648:
4647:
4646:
4638:
4626:
4625:
4624:
4616:
4613:
4612:
4611:
4610:
4602:
4587:
4586:
4585:
4577:
4574:
4573:
4572:
4571:
4563:
4527:
4525:
4524:
4519:
4495:
4494:
4489:
4488:
4487:
4468:
4466:
4465:
4460:
4458:
4456:
4442:
4424:
4416:
4415:
4410:
4409:
4408:
4389:
4387:
4386:
4381:
4379:
4377:
4363:
4355:
4347:
4346:
4341:
4340:
4339:
4320:
4318:
4317:
4312:
4304:
4303:
4298:
4297:
4296:
4277:
4275:
4274:
4269:
4261:
4260:
4255:
4254:
4253:
4234:
4232:
4231:
4226:
4208:
4203:
4195:
4190:
4189:
4184:
4183:
4182:
4161:
4159:
4158:
4153:
4151:
4150:
4149:
4141:
4138:
4137:
4136:
4135:
4127:
4108:
4106:
4105:
4100:
4049:
4047:
4046:
4041:
4039:
4038:
4037:
4029:
4018:
4017:
4016:
4008:
4005:
4004:
4003:
4002:
3994:
3979:
3978:
3977:
3969:
3942:
3940:
3939:
3934:
3932:
3928:
3927:
3926:
3895:
3894:
3882:
3880:
3866:
3848:
3775:
3774:
3754:
3752:
3751:
3746:
3726:
3725:
3705:
3703:
3702:
3697:
3688:
3687:
3667:
3665:
3664:
3659:
3657:
3656:
3637:
3635:
3621:
3611:
3602:
3572:
3558:
3557:
3532:
3530:
3529:
3524:
3512:
3510:
3509:
3504:
3489:where we regard
3485:
3483:
3482:
3477:
3453:
3452:
3435:
3433:
3432:
3427:
3412:
3411:
3394:
3392:
3391:
3386:
3362:
3361:
3344:
3342:
3341:
3336:
3309:
3308:
3238:
3236:
3235:
3230:
3161:
3159:
3158:
3153:
3096:
3094:
3093:
3088:
3086:
3084:
3083:
3059:
3045:
3044:
3025:
3023:
3022:
3017:
2993:
2991:
2990:
2985:
2961:
2960:
2942:
2941:
2928:
2927:
2914:
2913:
2901:
2900:
2887:
2883:
2882:
2881:
2866:
2865:
2842:
2841:
2816:
2814:
2813:
2808:
2761:
2760:
2708:
2707:
2692:
2691:
2669:
2667:
2666:
2661:
2571:
2569:
2568:
2563:
2521:
2519:
2518:
2513:
2489:
2488:
2470:
2469:
2455:
2454:
2442:
2441:
2429:
2428:
2416:
2415:
2394:
2393:
2389:
2362:
2343:
2341:
2340:
2335:
2330:
2293:
2291:
2290:
2285:
2283:
2282:
2248:
2246:
2245:
2240:
2232:
2231:
2215:
2213:
2212:
2207:
2199:
2198:
2175:
2173:
2172:
2167:
2155:
2153:
2152:
2147:
2145:
2144:
2125:
2123:
2122:
2117:
2099:
2097:
2096:
2091:
2089:
2088:
2066:
2064:
2063:
2058:
2033:
2031:
2030:
2025:
2010:
2009:
1990:
1989:
1963:
1961:
1960:
1955:
1940:
1939:
1920:
1919:
1889:
1887:
1886:
1881:
1860:
1859:
1858:
1848:
1847:
1846:
1813:
1811:
1810:
1805:
1763:
1761:
1760:
1755:
1753:
1752:
1734:
1733:
1715:
1714:
1691:static observers
1684:
1682:
1681:
1676:
1655:
1654:
1653:
1643:
1642:
1641:
1608:
1606:
1605:
1600:
1569:spatial distance
1566:
1564:
1563:
1558:
1556:
1555:
1513:
1511:
1510:
1505:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1424:
1422:
1421:
1416:
1413:
1408:
1399:
1374:
1372:
1371:
1366:
1363:
1358:
1320:
1318:
1317:
1312:
1310:
1309:
1290:
1288:
1287:
1282:
1239:
1235:
1234:
1233:
1214:
1213:
1201:
1200:
1182:
1177:
1165:
1164:
1154:
1149:
1137:
1136:
1135:
1134:
1116:
1115:
1099:
1076:
1074:
1073:
1068:
1066:
1065:
1046:
1045:
1011:
1009:
1008:
1003:
998:
956:
954:
953:
948:
946:
945:
919:static spacetime
907:vorticity tensor
904:
902:
901:
896:
894:
893:
881:
880:
872:
859:
857:
856:
851:
849:
848:
816:
815:
788:
786:
785:
780:
778:
777:
745:
744:
717:
715:
714:
709:
707:
706:
684:
682:
681:
676:
674:
673:
595:
593:
592:
587:
585:
584:
568:
566:
565:
560:
527:
525:
524:
519:
476:
475:
457:
456:
417:
415:
414:
409:
407:
406:
397:
396:
384:
383:
370:
369:
348:
347:
334:
333:
309:
305:
304:
303:
284:
283:
271:
270:
253:
252:
240:
239:
226:
225:
204:
203:
190:
189:
137:
135:
134:
129:
30:coordinate chart
7702:
7701:
7697:
7696:
7695:
7693:
7692:
7691:
7672:
7671:
7670:
7669:
7640:
7635:
7632:
7631:
7614:
7610:
7608:
7605:
7604:
7586:
7582:
7580:
7577:
7576:
7575:
7571:
7566:
7520:
7519:
7517:
7514:
7513:
7458:
7436:
7432:
7417:
7413:
7367:
7363:
7357:
7353:
7344:
7340:
7330:
7323:
7319:
7307:
7303:
7299:
7297:
7288:
7284:
7274:
7270:
7252:
7248:
7238:
7234:
7214:
7211:
7210:
7142:
7138:
7123:
7119:
7095:
7092:
7091:
7065:
7061:
7045:
7041:
7032:
7028:
7024:
7020:
7014:
7010:
7001:
6997:
6987:
6983:
6959:
6955:
6945:
6941:
6915:
6912:
6911:
6907:takes the form
6901:
6899:Generalizations
6876:ditto, for the
6869:ditto, for the
6856:
6816:
6811:
6809:
6806:
6805:
6785:
6781:
6764:
6759:
6750:
6739:
6738:
6737:
6735:
6732:
6731:
6713:
6709:
6698:
6689:
6678:
6677:
6676:
6674:
6671:
6670:
6652:
6648:
6631:
6626:
6617:
6606:
6605:
6604:
6602:
6599:
6598:
6580:
6576:
6559:
6554:
6545:
6534:
6533:
6532:
6530:
6527:
6526:
6484:
6480:
6471:
6467:
6465:
6455:
6446:
6442:
6431:
6430:
6415:
6411:
6400:
6399:
6372:
6368:
6361:
6357:
6350:
6348:
6340:
6336:
6331:
6322:
6318:
6307:
6306:
6298:
6295:
6294:
6269:
6268:
6266:
6263:
6262:
6222:
6218:
6213:
6205:
6203:
6193:
6184:
6180:
6169:
6168:
6153:
6149:
6138:
6137:
6109:
6105:
6100:
6091:
6082:
6067:
6066:
6064:
6055:
6051:
6040:
6039:
6031:
6028:
6027:
6003:
5999:
5982:
5977:
5968:
5957:
5956:
5955:
5941:
5940:
5938:
5935:
5934:
5905:
5904:
5899:
5893:
5889:
5886:
5885:
5880:
5873:
5869:
5859:
5858:
5852:
5848:
5846:
5840:
5836:
5834:
5828:
5824:
5822:
5816:
5812:
5810:
5804:
5800:
5798:
5792:
5788:
5785:
5784:
5778:
5774:
5772:
5766:
5762:
5760:
5754:
5750:
5748:
5742:
5738:
5736:
5730:
5726:
5724:
5718:
5714:
5711:
5710:
5704:
5700:
5698:
5692:
5688:
5686:
5680:
5676:
5674:
5668:
5664:
5662:
5656:
5652:
5650:
5644:
5640:
5637:
5636:
5630:
5626:
5624:
5618:
5614:
5612:
5606:
5602:
5600:
5594:
5590:
5588:
5582:
5578:
5576:
5570:
5566:
5563:
5562:
5556:
5552:
5550:
5544:
5540:
5538:
5532:
5528:
5526:
5520:
5516:
5514:
5508:
5504:
5502:
5496:
5492:
5489:
5488:
5482:
5478:
5476:
5470:
5466:
5464:
5458:
5454:
5452:
5446:
5442:
5440:
5434:
5430:
5428:
5422:
5418:
5414:
5410:
5408:
5405:
5404:
5377:
5376:
5365:
5364:
5353:
5352:
5341:
5340:
5339:
5335:
5333:
5330:
5329:
5298:
5294:
5281:
5277:
5276:
5274:
5266:
5262:
5257:
5248:
5241:
5237:
5236:
5235:
5233:
5230:
5229:
5211:
5204:
5200:
5199:
5198:
5178:
5174:
5166:
5164:
5154:
5145:
5138:
5134:
5133:
5132:
5130:
5127:
5126:
5108:
5101:
5097:
5096:
5095:
5074:
5070:
5065:
5056:
5052:
5050:
5040:
5031:
5024:
5020:
5019:
5018:
5016:
5013:
5012:
4982:
4978:
4973:
4964:
4955:
4940:
4936:
4934:
4925:
4918:
4914:
4913:
4912:
4910:
4907:
4906:
4853:
4852:
4848:
4833:
4832:
4828:
4820:
4809:
4808:
4797:
4796:
4791:
4780:
4779:
4778:
4765:
4764:
4760:
4759:
4758:
4743:
4742:
4729:
4728:
4724:
4723:
4722:
4720:
4717:
4716:
4687:
4686:
4673:
4672:
4668:
4667:
4666:
4651:
4650:
4637:
4636:
4632:
4631:
4630:
4615:
4614:
4601:
4600:
4596:
4595:
4594:
4576:
4575:
4562:
4561:
4557:
4556:
4555:
4553:
4550:
4549:
4490:
4483:
4479:
4478:
4477:
4475:
4472:
4471:
4443:
4425:
4423:
4411:
4404:
4400:
4399:
4398:
4396:
4393:
4392:
4364:
4356:
4354:
4342:
4335:
4331:
4330:
4329:
4327:
4324:
4323:
4299:
4292:
4288:
4287:
4286:
4284:
4281:
4280:
4256:
4249:
4245:
4244:
4243:
4241:
4238:
4237:
4196:
4194:
4185:
4178:
4174:
4173:
4172:
4170:
4167:
4166:
4140:
4139:
4126:
4125:
4121:
4120:
4119:
4117:
4114:
4113:
4062:
4059:
4058:
4028:
4027:
4023:
4007:
4006:
3993:
3992:
3988:
3987:
3986:
3968:
3967:
3963:
3958:
3955:
3954:
3922:
3918:
3890:
3886:
3867:
3849:
3847:
3846:
3842:
3770:
3766:
3761:
3758:
3757:
3721:
3717:
3712:
3709:
3708:
3683:
3679:
3674:
3671:
3670:
3652:
3648:
3622:
3604:
3603:
3601:
3565:
3553:
3549:
3544:
3541:
3540:
3518:
3515:
3514:
3494:
3491:
3490:
3448:
3444:
3442:
3439:
3438:
3407:
3403:
3401:
3398:
3397:
3357:
3353:
3351:
3348:
3347:
3304:
3300:
3298:
3295:
3294:
3257:
3255:A metric Ansatz
3182:
3179:
3178:
3105:
3102:
3101:
3079:
3075:
3058:
3040:
3036:
3034:
3031:
3030:
3002:
2999:
2998:
2956:
2952:
2937:
2933:
2923:
2919:
2909:
2905:
2896:
2892:
2877:
2873:
2861:
2857:
2850:
2846:
2837:
2833:
2828:
2825:
2824:
2756:
2752:
2703:
2699:
2687:
2683:
2681:
2678:
2677:
2584:
2581:
2580:
2548:
2545:
2544:
2484:
2480:
2465:
2461:
2450:
2446:
2437:
2433:
2424:
2420:
2411:
2407:
2385:
2363:
2358:
2357:
2352:
2349:
2348:
2326:
2303:
2300:
2299:
2278:
2274:
2266:
2263:
2262:
2259:
2227:
2223:
2221:
2218:
2217:
2194:
2190:
2188:
2185:
2184:
2161:
2158:
2157:
2140:
2136:
2134:
2131:
2130:
2105:
2102:
2101:
2084:
2080:
2072:
2069:
2068:
2046:
2043:
2042:
2005:
2001:
1985:
1981:
1973:
1970:
1969:
1935:
1931:
1915:
1911:
1903:
1900:
1899:
1896:
1854:
1850:
1849:
1842:
1838:
1837:
1822:
1819:
1818:
1784:
1781:
1780:
1748:
1744:
1729:
1725:
1710:
1706:
1698:
1695:
1694:
1649:
1645:
1644:
1637:
1633:
1632:
1617:
1614:
1613:
1579:
1576:
1575:
1551:
1547:
1545:
1542:
1541:
1499:
1496:
1495:
1475:
1472:
1471:
1437:
1434:
1433:
1409:
1404:
1395:
1384:
1381:
1380:
1359:
1354:
1336:
1333:
1332:
1305:
1301:
1299:
1296:
1295:
1229:
1225:
1209:
1205:
1196:
1192:
1188:
1184:
1178:
1173:
1160:
1156:
1150:
1145:
1130:
1126:
1111:
1107:
1100:
1095:
1094:
1089:
1086:
1085:
1061:
1057:
1041:
1037:
1029:
1026:
1025:
1022:
994:
977:
974:
973:
941:
937:
929:
926:
925:
889:
885:
871:
870:
868:
865:
864:
844:
840:
811:
807:
795:
792:
791:
773:
769:
740:
736:
724:
721:
720:
702:
698:
696:
693:
692:
669:
665:
663:
660:
659:
641:
580:
576:
574:
571:
570:
536:
533:
532:
471:
467:
452:
448:
424:
421:
420:
402:
398:
392:
388:
379:
375:
365:
361:
343:
339:
329:
325:
299:
295:
279:
275:
266:
262:
258:
254:
248:
244:
235:
231:
221:
217:
199:
195:
185:
181:
161:
158:
157:
153:takes the form
123:
120:
119:
113:
12:
11:
5:
7700:
7690:
7689:
7684:
7668:
7667:
7663:tangent bundle
7650:
7647:
7643:
7639:
7617:
7613:
7589:
7585:
7568:
7567:
7565:
7562:
7561:
7560:
7554:
7548:
7542:
7527:
7524:
7507:
7501:
7495:
7489:
7483:
7477:
7471:
7465:
7457:
7454:
7453:
7452:
7439:
7435:
7431:
7428:
7425:
7420:
7416:
7412:
7409:
7406:
7403:
7400:
7397:
7394:
7391:
7388:
7385:
7382:
7378:
7370:
7366:
7360:
7356:
7352:
7347:
7343:
7339:
7336:
7333:
7326:
7322:
7318:
7315:
7310:
7306:
7302:
7296:
7291:
7287:
7283:
7277:
7273:
7269:
7266:
7263:
7260:
7255:
7251:
7247:
7241:
7237:
7233:
7230:
7227:
7224:
7221:
7218:
7200:
7199:
7188:
7185:
7182:
7179:
7176:
7173:
7169:
7166:
7163:
7160:
7157:
7154:
7150:
7145:
7141:
7137:
7134:
7131:
7126:
7122:
7117:
7114:
7111:
7108:
7105:
7102:
7099:
7089:
7078:
7074:
7068:
7064:
7060:
7056:
7053:
7048:
7044:
7040:
7035:
7031:
7027:
7023:
7017:
7013:
7009:
7004:
7000:
6996:
6990:
6986:
6982:
6979:
6976:
6973:
6970:
6967:
6962:
6958:
6954:
6948:
6944:
6940:
6937:
6934:
6931:
6928:
6925:
6922:
6919:
6900:
6897:
6896:
6895:
6884:
6881:
6874:
6867:
6855:
6852:
6828:
6825:
6822:
6819:
6815:
6802:
6801:
6788:
6784:
6776:
6773:
6770:
6767:
6763:
6758:
6753:
6746:
6743:
6729:
6716:
6712:
6705:
6702:
6697:
6692:
6685:
6682:
6668:
6655:
6651:
6643:
6640:
6637:
6634:
6630:
6625:
6620:
6613:
6610:
6596:
6583:
6579:
6571:
6568:
6565:
6562:
6558:
6553:
6548:
6541:
6538:
6512:
6511:
6500:
6497:
6494:
6487:
6483:
6477:
6474:
6470:
6462:
6459:
6454:
6449:
6445:
6438:
6435:
6429:
6426:
6423:
6418:
6414:
6407:
6404:
6398:
6395:
6391:
6388:
6385:
6382:
6375:
6371:
6364:
6360:
6356:
6353:
6343:
6339:
6335:
6330:
6325:
6321:
6314:
6311:
6305:
6302:
6276:
6273:
6251:
6250:
6239:
6236:
6233:
6225:
6221:
6216:
6211:
6208:
6200:
6197:
6192:
6187:
6183:
6176:
6173:
6167:
6164:
6161:
6156:
6152:
6145:
6142:
6136:
6133:
6129:
6126:
6123:
6120:
6112:
6108:
6103:
6097:
6094:
6088:
6085:
6081:
6078:
6073:
6070:
6063:
6058:
6054:
6047:
6044:
6038:
6035:
6006:
6002:
5994:
5991:
5988:
5985:
5981:
5976:
5971:
5964:
5961:
5954:
5948:
5945:
5923:
5922:
5910:
5903:
5900:
5896:
5892:
5888:
5887:
5884:
5881:
5879:
5876:
5875:
5872:
5868:
5864:
5855:
5851:
5847:
5843:
5839:
5835:
5831:
5827:
5823:
5819:
5815:
5811:
5807:
5803:
5799:
5795:
5791:
5787:
5786:
5781:
5777:
5773:
5769:
5765:
5761:
5757:
5753:
5749:
5745:
5741:
5737:
5733:
5729:
5725:
5721:
5717:
5713:
5712:
5707:
5703:
5699:
5695:
5691:
5687:
5683:
5679:
5675:
5671:
5667:
5663:
5659:
5655:
5651:
5647:
5643:
5639:
5638:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5605:
5601:
5597:
5593:
5589:
5585:
5581:
5577:
5573:
5569:
5565:
5564:
5559:
5555:
5551:
5547:
5543:
5539:
5535:
5531:
5527:
5523:
5519:
5515:
5511:
5507:
5503:
5499:
5495:
5491:
5490:
5485:
5481:
5477:
5473:
5469:
5465:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5433:
5429:
5425:
5421:
5417:
5416:
5413:
5400:into a matrix
5384:
5381:
5372:
5369:
5360:
5357:
5348:
5345:
5338:
5326:
5325:
5314:
5311:
5308:
5301:
5297:
5292:
5289:
5284:
5280:
5269:
5265:
5261:
5256:
5251:
5244:
5240:
5227:
5214:
5207:
5203:
5197:
5194:
5191:
5188:
5181:
5177:
5172:
5169:
5161:
5158:
5153:
5148:
5141:
5137:
5124:
5111:
5104:
5100:
5094:
5091:
5088:
5085:
5077:
5073:
5068:
5062:
5059:
5055:
5047:
5044:
5039:
5034:
5027:
5023:
5010:
4999:
4996:
4993:
4985:
4981:
4976:
4970:
4967:
4961:
4958:
4954:
4951:
4946:
4943:
4939:
4933:
4928:
4921:
4917:
4902:. We obtain:
4896:Riemann tensor
4876:
4875:
4860:
4857:
4851:
4847:
4840:
4837:
4831:
4823:
4816:
4813:
4804:
4801:
4794:
4787:
4784:
4772:
4769:
4763:
4757:
4750:
4747:
4736:
4733:
4727:
4710:
4709:
4694:
4691:
4680:
4677:
4671:
4665:
4658:
4655:
4644:
4641:
4635:
4629:
4622:
4619:
4608:
4605:
4599:
4593:
4590:
4583:
4580:
4569:
4566:
4560:
4529:
4528:
4517:
4514:
4510:
4507:
4504:
4501:
4498:
4493:
4486:
4482:
4469:
4455:
4452:
4449:
4446:
4441:
4438:
4434:
4431:
4428:
4422:
4419:
4414:
4407:
4403:
4390:
4376:
4373:
4370:
4367:
4362:
4359:
4353:
4350:
4345:
4338:
4334:
4321:
4310:
4307:
4302:
4295:
4291:
4278:
4267:
4264:
4259:
4252:
4248:
4235:
4224:
4221:
4217:
4214:
4211:
4206:
4202:
4199:
4193:
4188:
4181:
4177:
4147:
4144:
4133:
4130:
4124:
4098:
4095:
4092:
4089:
4086:
4082:
4079:
4076:
4072:
4069:
4066:
4051:
4050:
4035:
4032:
4026:
4022:
4014:
4011:
4000:
3997:
3991:
3985:
3982:
3975:
3972:
3966:
3962:
3944:
3943:
3931:
3925:
3921:
3917:
3914:
3911:
3907:
3904:
3901:
3898:
3893:
3889:
3885:
3879:
3876:
3873:
3870:
3865:
3862:
3858:
3855:
3852:
3845:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3819:
3816:
3813:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3787:
3784:
3781:
3778:
3773:
3769:
3765:
3755:
3744:
3741:
3738:
3735:
3732:
3729:
3724:
3720:
3716:
3706:
3694:
3691:
3686:
3682:
3678:
3668:
3655:
3651:
3647:
3644:
3641:
3634:
3631:
3628:
3625:
3620:
3617:
3614:
3610:
3607:
3600:
3597:
3594:
3591:
3588:
3585:
3581:
3578:
3575:
3571:
3568:
3564:
3561:
3556:
3552:
3548:
3522:
3502:
3498:
3487:
3486:
3475:
3472:
3468:
3465:
3462:
3459:
3456:
3451:
3447:
3436:
3424:
3421:
3418:
3415:
3410:
3406:
3395:
3384:
3381:
3377:
3374:
3371:
3368:
3365:
3360:
3356:
3345:
3334:
3331:
3327:
3324:
3321:
3318:
3315:
3312:
3307:
3303:
3256:
3253:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3098:
3097:
3082:
3078:
3074:
3071:
3068:
3065:
3062:
3057:
3054:
3051:
3048:
3043:
3039:
3015:
3012:
3009:
3006:
2995:
2994:
2983:
2980:
2977:
2974:
2971:
2968:
2964:
2959:
2955:
2951:
2948:
2945:
2940:
2936:
2931:
2926:
2922:
2918:
2912:
2908:
2904:
2899:
2895:
2891:
2886:
2880:
2876:
2872:
2869:
2864:
2860:
2856:
2853:
2849:
2845:
2840:
2836:
2832:
2818:
2817:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2759:
2755:
2749:
2746:
2743:
2740:
2737:
2733:
2730:
2727:
2724:
2720:
2717:
2714:
2711:
2706:
2702:
2698:
2695:
2690:
2686:
2671:
2670:
2659:
2656:
2653:
2650:
2647:
2643:
2640:
2637:
2634:
2631:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2574:
2573:
2561:
2558:
2555:
2552:
2541:
2538:
2523:
2522:
2511:
2508:
2505:
2502:
2499:
2496:
2492:
2487:
2483:
2479:
2476:
2473:
2468:
2464:
2458:
2453:
2449:
2445:
2440:
2436:
2432:
2427:
2423:
2419:
2414:
2410:
2406:
2403:
2400:
2397:
2392:
2388:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2361:
2356:
2333:
2329:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2281:
2277:
2273:
2270:
2258:
2255:
2238:
2235:
2230:
2226:
2205:
2202:
2197:
2193:
2165:
2143:
2139:
2115:
2112:
2109:
2087:
2083:
2079:
2076:
2056:
2053:
2050:
2040:prime meridian
2023:
2020:
2017:
2013:
2008:
2004:
2000:
1997:
1993:
1988:
1984:
1980:
1977:
1953:
1950:
1947:
1943:
1938:
1934:
1930:
1927:
1923:
1918:
1914:
1910:
1907:
1895:
1892:
1891:
1890:
1879:
1876:
1873:
1870:
1867:
1864:
1857:
1853:
1845:
1841:
1836:
1832:
1829:
1826:
1803:
1800:
1797:
1794:
1791:
1788:
1751:
1747:
1743:
1740:
1737:
1732:
1728:
1724:
1721:
1718:
1713:
1709:
1705:
1702:
1686:
1685:
1674:
1671:
1668:
1665:
1662:
1659:
1652:
1648:
1640:
1636:
1631:
1627:
1624:
1621:
1598:
1595:
1592:
1589:
1586:
1583:
1554:
1550:
1503:
1479:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1426:
1425:
1412:
1407:
1403:
1398:
1394:
1391:
1388:
1375:
1362:
1357:
1353:
1349:
1346:
1343:
1340:
1308:
1304:
1292:
1291:
1280:
1277:
1274:
1271:
1268:
1265:
1261:
1258:
1255:
1252:
1249:
1246:
1242:
1238:
1232:
1228:
1224:
1220:
1217:
1212:
1208:
1204:
1199:
1195:
1191:
1187:
1181:
1176:
1172:
1168:
1163:
1159:
1153:
1148:
1144:
1140:
1133:
1129:
1125:
1122:
1119:
1114:
1110:
1106:
1103:
1098:
1093:
1064:
1060:
1056:
1053:
1049:
1044:
1040:
1036:
1033:
1021:
1018:
1001:
997:
993:
990:
987:
984:
981:
944:
940:
936:
933:
892:
888:
884:
878:
875:
861:
860:
847:
843:
838:
835:
832:
828:
825:
822:
819:
814:
810:
805:
802:
799:
789:
776:
772:
767:
764:
761:
757:
754:
751:
748:
743:
739:
734:
731:
728:
718:
705:
701:
686:
685:
672:
668:
640:
637:
633:local solution
583:
579:
558:
555:
552:
549:
546:
543:
540:
529:
528:
517:
514:
511:
508:
505:
502:
498:
495:
492:
489:
486:
483:
479:
474:
470:
466:
463:
460:
455:
451:
446:
443:
440:
437:
434:
431:
428:
418:
405:
401:
395:
391:
387:
382:
378:
374:
368:
364:
360:
357:
354:
351:
346:
342:
338:
332:
328:
324:
321:
318:
315:
312:
308:
302:
298:
294:
290:
287:
282:
278:
274:
269:
265:
261:
257:
251:
247:
243:
238:
234:
230:
224:
220:
216:
213:
210:
207:
202:
198:
194:
188:
184:
180:
177:
174:
171:
168:
165:
127:
112:
109:
92:perfect fluids
9:
6:
4:
3:
2:
7699:
7688:
7685:
7683:
7680:
7679:
7677:
7664:
7648:
7641:
7615:
7587:
7573:
7569:
7558:
7555:
7552:
7549:
7546:
7543:
7522:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7460:
7459:
7437:
7433:
7429:
7426:
7423:
7418:
7414:
7410:
7404:
7401:
7398:
7395:
7392:
7389:
7386:
7380:
7376:
7368:
7358:
7354:
7350:
7345:
7341:
7337:
7334:
7324:
7320:
7316:
7313:
7308:
7304:
7300:
7294:
7289:
7285:
7281:
7275:
7267:
7261:
7258:
7253:
7249:
7245:
7239:
7231:
7225:
7222:
7219:
7216:
7209:
7208:
7207:
7205:
7186:
7183:
7180:
7177:
7174:
7171:
7167:
7164:
7161:
7158:
7155:
7152:
7148:
7143:
7139:
7135:
7132:
7129:
7124:
7120:
7115:
7109:
7106:
7103:
7097:
7090:
7076:
7072:
7066:
7062:
7058:
7054:
7051:
7046:
7042:
7038:
7033:
7029:
7025:
7021:
7015:
7011:
7007:
7002:
6998:
6994:
6988:
6980:
6977:
6974:
6968:
6965:
6960:
6956:
6952:
6946:
6938:
6935:
6932:
6926:
6923:
6920:
6917:
6910:
6909:
6908:
6906:
6893:
6889:
6885:
6882:
6879:
6875:
6872:
6868:
6865:
6861:
6860:
6859:
6851:
6849:
6845:
6823:
6817:
6813:
6786:
6774:
6771:
6768:
6765:
6761:
6756:
6751:
6741:
6730:
6714:
6703:
6700:
6695:
6690:
6680:
6669:
6653:
6638:
6632:
6628:
6623:
6618:
6608:
6597:
6581:
6566:
6560:
6556:
6551:
6546:
6536:
6525:
6524:
6523:
6521:
6516:
6495:
6485:
6481:
6475:
6472:
6468:
6460:
6457:
6452:
6447:
6433:
6424:
6421:
6416:
6402:
6393:
6389:
6383:
6373:
6369:
6362:
6358:
6354:
6351:
6341:
6337:
6333:
6328:
6323:
6309:
6300:
6293:
6292:
6291:
6271:
6260:
6256:
6234:
6223:
6219:
6214:
6209:
6206:
6198:
6195:
6190:
6185:
6171:
6162:
6159:
6154:
6140:
6131:
6127:
6121:
6110:
6106:
6101:
6095:
6092:
6086:
6083:
6079:
6076:
6071:
6068:
6061:
6056:
6042:
6033:
6026:
6025:
6024:
6022:
6004:
5989:
5983:
5979:
5974:
5969:
5959:
5952:
5943:
5932:
5928:
5908:
5901:
5894:
5890:
5882:
5877:
5870:
5866:
5862:
5853:
5849:
5841:
5837:
5829:
5825:
5817:
5813:
5805:
5801:
5793:
5789:
5779:
5775:
5767:
5763:
5755:
5751:
5743:
5739:
5731:
5727:
5719:
5715:
5705:
5701:
5693:
5689:
5681:
5677:
5669:
5665:
5657:
5653:
5645:
5641:
5631:
5627:
5619:
5615:
5607:
5603:
5595:
5591:
5583:
5579:
5571:
5567:
5557:
5553:
5545:
5541:
5533:
5529:
5521:
5517:
5509:
5505:
5497:
5493:
5483:
5479:
5471:
5467:
5459:
5455:
5447:
5443:
5435:
5431:
5423:
5419:
5411:
5403:
5402:
5401:
5379:
5367:
5355:
5343:
5336:
5309:
5299:
5295:
5290:
5287:
5282:
5278:
5267:
5263:
5259:
5254:
5249:
5242:
5238:
5228:
5212:
5205:
5201:
5195:
5189:
5179:
5175:
5170:
5167:
5159:
5156:
5151:
5146:
5139:
5135:
5125:
5109:
5102:
5098:
5092:
5086:
5075:
5071:
5066:
5060:
5057:
5053:
5045:
5042:
5037:
5032:
5025:
5021:
5011:
4994:
4983:
4979:
4974:
4968:
4965:
4959:
4956:
4952:
4949:
4944:
4941:
4937:
4931:
4926:
4919:
4915:
4905:
4904:
4903:
4901:
4897:
4893:
4889:
4885:
4881:
4855:
4849:
4845:
4835:
4829:
4811:
4799:
4782:
4767:
4761:
4755:
4745:
4731:
4715:
4714:
4713:
4689:
4675:
4669:
4663:
4653:
4639:
4633:
4627:
4617:
4603:
4597:
4591:
4588:
4578:
4564:
4548:
4547:
4546:
4545:
4540:
4538:
4534:
4533:antisymmetric
4515:
4512:
4508:
4505:
4502:
4499:
4496:
4491:
4484:
4480:
4470:
4450:
4444:
4439:
4436:
4432:
4429:
4426:
4420:
4417:
4412:
4405:
4401:
4391:
4371:
4365:
4360:
4357:
4351:
4348:
4343:
4336:
4332:
4322:
4308:
4305:
4300:
4293:
4289:
4279:
4265:
4262:
4257:
4250:
4246:
4236:
4222:
4219:
4212:
4204:
4200:
4197:
4191:
4186:
4179:
4175:
4165:
4164:
4163:
4142:
4128:
4122:
4110:
4096:
4093:
4090:
4087:
4084:
4080:
4077:
4074:
4070:
4067:
4064:
4056:
4030:
4024:
4020:
4009:
3995:
3989:
3983:
3980:
3970:
3964:
3960:
3953:
3952:
3951:
3949:
3929:
3923:
3919:
3915:
3912:
3909:
3905:
3902:
3899:
3896:
3891:
3887:
3883:
3874:
3868:
3863:
3860:
3856:
3853:
3850:
3843:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3817:
3814:
3811:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3785:
3782:
3779:
3776:
3771:
3767:
3763:
3756:
3742:
3739:
3736:
3733:
3730:
3727:
3722:
3718:
3714:
3707:
3692:
3689:
3684:
3680:
3676:
3669:
3653:
3649:
3645:
3642:
3639:
3629:
3623:
3615:
3608:
3605:
3598:
3595:
3592:
3589:
3586:
3583:
3576:
3569:
3566:
3562:
3559:
3554:
3550:
3546:
3539:
3538:
3537:
3534:
3520:
3500:
3496:
3473:
3470:
3466:
3463:
3460:
3457:
3454:
3449:
3445:
3437:
3422:
3419:
3416:
3413:
3408:
3404:
3396:
3382:
3379:
3372:
3366:
3363:
3358:
3354:
3346:
3332:
3329:
3322:
3316:
3313:
3310:
3305:
3301:
3293:
3292:
3291:
3289:
3288:coframe field
3285:
3280:
3278:
3274:
3270:
3266:
3262:
3252:
3248:
3246:
3242:
3223:
3217:
3214:
3211:
3205:
3199:
3196:
3190:
3184:
3176:
3172:
3168:
3163:
3146:
3140:
3137:
3134:
3128:
3122:
3119:
3113:
3107:
3080:
3072:
3066:
3063:
3060:
3055:
3049:
3037:
3029:
3028:
3027:
3010:
3004:
2981:
2978:
2975:
2972:
2969:
2966:
2962:
2957:
2953:
2949:
2946:
2943:
2938:
2934:
2929:
2924:
2920:
2916:
2910:
2906:
2902:
2897:
2893:
2889:
2884:
2878:
2870:
2858:
2854:
2851:
2847:
2843:
2838:
2834:
2830:
2823:
2822:
2821:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2762:
2757:
2747:
2741:
2738:
2735:
2731:
2728:
2725:
2722:
2718:
2712:
2700:
2693:
2688:
2676:
2675:
2674:
2654:
2651:
2648:
2645:
2641:
2638:
2635:
2632:
2629:
2625:
2619:
2613:
2601:
2598:
2595:
2592:
2589:
2579:
2578:
2577:
2556:
2550:
2542:
2539:
2536:
2535:
2534:
2532:
2528:
2509:
2506:
2503:
2500:
2497:
2494:
2490:
2485:
2481:
2477:
2474:
2471:
2466:
2462:
2456:
2451:
2447:
2443:
2438:
2434:
2430:
2425:
2421:
2417:
2412:
2404:
2398:
2395:
2390:
2386:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2354:
2347:
2346:
2345:
2331:
2327:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2297:
2279:
2275:
2271:
2268:
2254:
2252:
2233:
2228:
2224:
2203:
2200:
2195:
2191:
2181:
2179:
2163:
2141:
2127:
2113:
2110:
2107:
2085:
2081:
2077:
2074:
2054:
2051:
2048:
2041:
2037:
2021:
2018:
2015:
2011:
2006:
2002:
1998:
1995:
1991:
1986:
1982:
1978:
1975:
1967:
1951:
1948:
1945:
1941:
1936:
1932:
1928:
1925:
1921:
1916:
1912:
1908:
1905:
1877:
1874:
1868:
1862:
1855:
1851:
1843:
1839:
1834:
1830:
1827:
1817:
1816:
1815:
1801:
1798:
1792:
1786:
1778:
1774:
1769:
1767:
1749:
1745:
1741:
1738:
1735:
1730:
1726:
1722:
1719:
1716:
1711:
1707:
1703:
1700:
1692:
1672:
1669:
1663:
1657:
1650:
1646:
1638:
1634:
1629:
1625:
1622:
1612:
1611:
1610:
1596:
1593:
1587:
1581:
1574:
1570:
1552:
1539:
1534:
1532:
1528:
1524:
1519:
1517:
1501:
1493:
1477:
1454:
1451:
1448:
1442:
1431:
1410:
1405:
1401:
1396:
1392:
1389:
1386:
1379:
1376:
1360:
1355:
1351:
1347:
1344:
1341:
1338:
1331:
1328:
1327:
1326:
1324:
1302:
1278:
1275:
1272:
1269:
1266:
1263:
1259:
1256:
1253:
1250:
1247:
1244:
1240:
1236:
1230:
1226:
1222:
1218:
1215:
1210:
1206:
1202:
1197:
1193:
1189:
1185:
1179:
1174:
1170:
1166:
1157:
1151:
1146:
1142:
1138:
1131:
1127:
1123:
1120:
1117:
1112:
1108:
1104:
1101:
1091:
1084:
1083:
1082:
1080:
1062:
1058:
1054:
1051:
1047:
1042:
1038:
1034:
1031:
1017:
1015:
999:
995:
991:
988:
985:
979:
970:
968:
964:
960:
942:
938:
934:
931:
924:
920:
916:
912:
908:
890:
882:
873:
845:
836:
833:
830:
826:
823:
820:
817:
812:
803:
800:
797:
790:
774:
765:
762:
759:
755:
752:
749:
746:
741:
732:
729:
726:
719:
703:
691:
690:
689:
670:
658:
657:
656:
654:
650:
646:
636:
634:
629:
625:
621:
616:
614:
610:
606:
601:
599:
577:
553:
550:
547:
541:
515:
512:
509:
506:
503:
500:
496:
493:
490:
487:
484:
481:
477:
472:
468:
464:
461:
458:
453:
449:
444:
438:
435:
432:
426:
419:
399:
393:
389:
385:
380:
376:
372:
366:
358:
352:
349:
344:
340:
336:
330:
322:
316:
313:
310:
306:
300:
296:
292:
288:
285:
280:
276:
272:
267:
263:
259:
255:
249:
245:
241:
236:
232:
228:
222:
214:
208:
205:
200:
196:
192:
186:
178:
172:
169:
166:
163:
156:
155:
154:
152:
147:
145:
141:
125:
118:
117:metric tensor
115:Specifying a
108:
105:
101:
100:event horizon
97:
93:
89:
85:
81:
77:
73:
69:
65:
60:
58:
54:
50:
47:
43:
39:
35:
31:
27:
23:
19:
7572:
7203:
7201:
6902:
6857:
6847:
6803:
6517:
6513:
6252:
5924:
5327:
4891:
4887:
4886:of indices (
4883:
4877:
4711:
4543:
4541:
4532:
4530:
4111:
4052:
3947:
3945:
3535:
3488:
3281:
3268:
3264:
3260:
3258:
3249:
3244:
3240:
3170:
3166:
3164:
3099:
2996:
2819:
2672:
2575:
2524:
2295:
2260:
2250:
2182:
2177:
2128:
2039:
2035:
1965:
1897:
1770:
1765:
1690:
1687:
1537:
1535:
1530:
1526:
1522:
1520:
1515:
1491:
1429:
1427:
1330:surface area
1322:
1293:
1023:
971:
958:
922:
914:
862:
687:
653:irrotational
652:
642:
632:
617:
608:
604:
602:
530:
151:line element
148:
114:
83:
61:
52:
36:, a kind of
33:
25:
15:
6520:frame field
4900:frame field
1773:proper time
967:Kerr vacuum
645:Lie algebra
51:, which is
40:chart on a
7676:Categories
4878:where the
3026:such that
2036:South pole
1966:North pole
1777:world line
1492:colatitude
111:Definition
104:black hole
7646:∂
7638:∂
7612:∂
7584:∂
7526:→
7408:∞
7384:∞
7381:−
7223:−
7187:π
7181:ϕ
7175:π
7172:−
7165:π
7159:θ
7113:∞
7101:∞
7098:−
7063:ϕ
7055:θ
7052:
7030:θ
6924:−
6787:ϕ
6783:∂
6775:θ
6772:
6745:→
6715:θ
6711:∂
6684:→
6650:∂
6612:→
6578:∂
6540:→
6518:The dual
6469:−
6437:→
6406:→
6355:−
6313:→
6275:→
6175:→
6144:→
6080:−
6046:→
6001:∂
5963:→
5947:→
5927:traceless
5383:^
5371:^
5359:^
5347:^
5288:−
5054:−
4938:−
4880:Bach bars
4859:^
4856:ȷ
4850:σ
4846:∧
4839:^
4836:ı
4830:σ
4815:^
4812:ȷ
4803:^
4800:ı
4786:^
4771:^
4749:^
4735:^
4726:Ω
4693:^
4679:^
4676:ℓ
4670:ω
4664:∧
4657:^
4654:ℓ
4643:^
4634:ω
4628:−
4621:^
4607:^
4598:ω
4582:^
4568:^
4559:Ω
4516:ϕ
4509:θ
4506:
4500:−
4481:ω
4440:ϕ
4433:θ
4430:
4421:−
4402:ω
4361:θ
4352:−
4333:ω
4290:ω
4247:ω
4176:ω
4146:^
4132:^
4123:ω
4097:ϕ
4088:θ
4034:^
4025:σ
4021:∧
4013:^
3999:^
3990:ω
3984:−
3974:^
3965:σ
3920:σ
3916:∧
3913:ϕ
3906:θ
3903:
3888:σ
3884:∧
3864:ϕ
3857:θ
3854:
3840:−
3834:ϕ
3828:∧
3825:θ
3818:θ
3815:
3802:ϕ
3796:∧
3786:θ
3783:
3768:σ
3743:θ
3737:∧
3719:σ
3681:σ
3650:σ
3646:∧
3590:∧
3563:−
3551:σ
3474:ϕ
3467:θ
3464:
3446:σ
3423:θ
3405:σ
3355:σ
3314:−
3302:σ
3218:
3141:
3064:−
3042:′
2982:π
2976:ϕ
2970:π
2967:−
2921:ϕ
2863:′
2835:ρ
2802:ϕ
2799:
2787:ϕ
2784:
2775:−
2758:ϕ
2754:∂
2742:ϕ
2739:
2729:ϕ
2726:
2705:′
2685:∂
2655:ϕ
2652:
2639:ϕ
2636:
2608:→
2602:ϕ
2529:ring) in
2510:π
2504:ϕ
2498:π
2495:−
2448:ϕ
2383:π
2377:θ
2324:π
2318:θ
2237:∞
2164:ϕ
2142:ϕ
2138:∂
2049:ϕ
2022:π
2016:θ
1946:θ
1835:∫
1828:τ
1825:Δ
1746:ϕ
1739:ϕ
1727:θ
1720:θ
1630:∫
1623:ρ
1620:Δ
1573:integrate
1549:∂
1516:longitude
1502:ϕ
1478:θ
1455:ϕ
1449:θ
1440:Ω
1348:π
1307:Ω
1279:π
1273:ϕ
1267:π
1264:−
1257:π
1251:θ
1227:ϕ
1219:θ
1216:
1194:θ
1162:Ω
992:π
986:ϕ
983:↦
980:ϕ
887:∂
877:→
846:ϕ
842:∂
837:ϕ
834:
827:θ
824:
818:−
813:θ
809:∂
804:ϕ
801:
775:ϕ
771:∂
766:ϕ
763:
756:θ
753:
742:θ
738:∂
733:ϕ
730:
704:ϕ
700:∂
667:∂
582:Ω
554:ϕ
548:θ
539:Ω
516:π
510:ϕ
504:π
501:−
494:π
488:θ
442:∞
430:∞
427:−
404:Ω
314:−
297:ϕ
289:θ
286:
264:θ
170:−
49:spacetime
7456:See also
6892:pressure
6476:′
6210:′
6096:′
6087:′
6072:″
5171:′
5061:′
4969:′
4960:′
4945:″
4201:′
3609:′
3570:′
84:isolated
66:such as
6019:. The
3171:smaller
2527:annular
53:adapted
32:is the
7541:above,
6905:metric
3273:ansatz
3245:global
3167:larger
2178:cyclic
1294:Where
531:Where
72:static
42:static
7564:Notes
3241:local
2176:as a
7430:<
7424:<
7405:<
7387:<
7184:<
7178:<
7162:<
7156:<
7136:<
7130:<
7110:<
7104:<
6253:The
5854:1212
5842:1231
5830:1223
5818:1203
5806:1202
5794:1201
5780:3112
5768:3131
5756:3123
5744:3103
5732:3102
5720:3101
5706:2312
5694:2331
5682:2323
5670:2303
5658:2302
5646:2301
5632:0312
5620:0331
5608:0323
5596:0303
5584:0302
5572:0301
5558:0212
5546:0231
5534:0223
5522:0203
5510:0202
5498:0201
5484:0112
5472:0131
5460:0123
5448:0103
5436:0102
5424:0101
3215:sinh
2979:<
2973:<
2950:<
2944:<
2507:<
2501:<
2478:<
2472:<
2251:also
2234:<
2201:>
1494:and
1276:<
1270:<
1254:<
1248:<
1079:loci
607:and
513:<
507:<
491:<
485:<
465:<
459:<
439:<
433:<
44:and
7043:sin
6769:sin
6023:is
5250:323
5213:313
5147:212
5110:303
5033:202
4927:101
4503:cos
4427:sin
4109:.)
3900:cos
3851:sin
3812:cos
3780:sin
3461:sin
3279:).
3138:sin
2796:cos
2781:sin
2736:sin
2723:cos
2649:sin
2633:cos
2216:or
1207:sin
831:sin
821:cot
798:cos
760:cos
750:cot
727:sin
647:of
277:sin
7678::
7630:=
6448:33
6417:22
6324:11
6186:33
6155:22
6057:11
4539:.
3290:,
3162:.
78:,
20:,
7665:.
7649:x
7642:/
7616:x
7588:t
7523:X
7476:,
7470:,
7464:,
7438:2
7434:r
7427:r
7419:1
7415:r
7411:,
7402:y
7399:,
7396:x
7393:,
7390:t
7377:,
7369:2
7365:)
7359:2
7355:y
7351:+
7346:2
7342:x
7338:+
7335:1
7332:(
7325:2
7321:y
7317:d
7314:+
7309:2
7305:x
7301:d
7295:+
7290:2
7286:r
7282:d
7276:2
7272:)
7268:r
7265:(
7262:b
7259:+
7254:2
7250:t
7246:d
7240:2
7236:)
7232:r
7229:(
7226:a
7220:=
7217:g
7168:,
7153:0
7149:,
7144:1
7140:r
7133:r
7125:0
7121:r
7116:,
7107:t
7077:,
7073:)
7067:2
7059:d
7047:2
7039:+
7034:2
7026:d
7022:(
7016:2
7012:r
7008:+
7003:2
6999:r
6995:d
6989:2
6985:)
6981:r
6978:,
6975:t
6972:(
6969:b
6966:+
6961:2
6957:t
6953:d
6947:2
6943:)
6939:r
6936:,
6933:t
6930:(
6927:a
6921:=
6918:g
6866:,
6827:)
6824:r
6821:(
6818:b
6814:1
6766:r
6762:1
6757:=
6752:3
6742:e
6704:r
6701:1
6696:=
6691:2
6681:e
6654:r
6642:)
6639:r
6636:(
6633:b
6629:1
6624:=
6619:1
6609:e
6582:t
6570:)
6567:r
6564:(
6561:a
6557:1
6552:=
6547:0
6537:e
6499:)
6496:r
6493:(
6486:3
6482:b
6473:b
6461:r
6458:1
6453:=
6444:]
6434:X
6428:[
6425:L
6422:=
6413:]
6403:X
6397:[
6394:L
6390:,
6387:)
6384:r
6381:(
6374:2
6370:b
6363:2
6359:b
6352:1
6342:2
6338:r
6334:1
6329:=
6320:]
6310:X
6304:[
6301:L
6272:X
6238:)
6235:r
6232:(
6224:2
6220:b
6215:a
6207:a
6199:r
6196:1
6191:=
6182:]
6172:X
6166:[
6163:E
6160:=
6151:]
6141:X
6135:[
6132:E
6128:,
6125:)
6122:r
6119:(
6111:3
6107:b
6102:a
6093:b
6084:a
6077:b
6069:a
6062:=
6053:]
6043:X
6037:[
6034:E
6005:t
5993:)
5990:r
5987:(
5984:a
5980:1
5975:=
5970:0
5960:e
5953:=
5944:X
5909:]
5902:L
5895:T
5891:B
5883:B
5878:E
5871:[
5867:=
5863:]
5850:R
5838:R
5826:R
5814:R
5802:R
5790:R
5776:R
5764:R
5752:R
5740:R
5728:R
5716:R
5702:R
5690:R
5678:R
5666:R
5654:R
5642:R
5628:R
5616:R
5604:R
5592:R
5580:R
5568:R
5554:R
5542:R
5530:R
5518:R
5506:R
5494:R
5480:R
5468:R
5456:R
5444:R
5432:R
5420:R
5412:[
5380:j
5368:i
5356:n
5344:m
5337:R
5313:)
5310:r
5307:(
5300:2
5296:b
5291:1
5283:2
5279:b
5268:2
5264:r
5260:1
5255:=
5243:2
5239:R
5206:1
5202:R
5196:=
5193:)
5190:r
5187:(
5180:3
5176:b
5168:b
5160:r
5157:1
5152:=
5140:1
5136:R
5103:0
5099:R
5093:=
5090:)
5087:r
5084:(
5076:2
5072:b
5067:a
5058:a
5046:r
5043:1
5038:=
5026:0
5022:R
4998:)
4995:r
4992:(
4984:3
4980:b
4975:a
4966:b
4957:a
4953:+
4950:b
4942:a
4932:=
4920:0
4916:R
4892:j
4890:,
4888:i
4822:|
4793:|
4783:n
4768:m
4762:R
4756:=
4746:n
4732:m
4690:n
4640:m
4618:n
4604:m
4592:d
4589:=
4579:n
4565:m
4513:d
4497:=
4492:3
4485:2
4454:)
4451:r
4448:(
4445:b
4437:d
4418:=
4413:3
4406:1
4375:)
4372:r
4369:(
4366:b
4358:d
4349:=
4344:2
4337:1
4309:0
4306:=
4301:3
4294:0
4266:0
4263:=
4258:2
4251:0
4223:t
4220:d
4216:)
4213:r
4210:(
4205:b
4198:a
4192:=
4187:1
4180:0
4143:n
4129:m
4094:d
4091:,
4085:d
4081:,
4078:r
4075:d
4071:,
4068:t
4065:d
4031:n
4010:n
3996:m
3981:=
3971:m
3961:d
3930:)
3924:2
3910:d
3897:+
3892:1
3878:)
3875:r
3872:(
3869:b
3861:d
3844:(
3837:=
3831:d
3822:d
3808:r
3805:+
3799:d
3793:r
3790:d
3777:=
3772:3
3764:d
3740:d
3734:r
3731:d
3728:=
3723:2
3715:d
3693:0
3690:=
3685:1
3677:d
3654:1
3643:t
3640:d
3633:)
3630:r
3627:(
3624:b
3619:)
3616:r
3613:(
3606:a
3599:=
3596:t
3593:d
3587:r
3584:d
3580:)
3577:r
3574:(
3567:a
3560:=
3555:0
3547:d
3521:r
3501:b
3497:a
3471:d
3458:r
3455:=
3450:3
3420:d
3417:r
3414:=
3409:2
3383:r
3380:d
3376:)
3373:r
3370:(
3367:b
3364:=
3359:1
3333:t
3330:d
3326:)
3323:r
3320:(
3317:a
3311:=
3306:0
3269:r
3265:g
3263:,
3261:f
3227:)
3224:r
3221:(
3212:=
3209:)
3206:r
3203:(
3200:f
3197:=
3194:)
3191:r
3188:(
3185:b
3150:)
3147:r
3144:(
3135:=
3132:)
3129:r
3126:(
3123:f
3120:=
3117:)
3114:r
3111:(
3108:b
3081:2
3077:)
3073:r
3070:(
3067:b
3061:1
3056:=
3053:)
3050:r
3047:(
3038:f
3014:)
3011:r
3008:(
3005:f
2963:,
2958:2
2954:r
2947:r
2939:1
2935:r
2930:,
2925:2
2917:d
2911:2
2907:r
2903:+
2898:2
2894:r
2890:d
2885:)
2879:2
2875:)
2871:r
2868:(
2859:f
2855:+
2852:1
2848:(
2844:=
2839:2
2831:d
2805:)
2793:r
2790:,
2778:r
2772:,
2769:0
2766:(
2763:=
2748:,
2745:)
2732:,
2719:,
2716:)
2713:r
2710:(
2701:f
2697:(
2694:=
2689:r
2658:)
2646:r
2642:,
2630:r
2626:,
2623:)
2620:r
2617:(
2614:f
2611:(
2605:)
2599:,
2596:r
2593:,
2590:z
2587:(
2572:.
2560:)
2557:r
2554:(
2551:f
2531:E
2491:,
2486:2
2482:r
2475:r
2467:1
2463:r
2457:,
2452:2
2444:d
2439:2
2435:r
2431:+
2426:2
2422:r
2418:d
2413:2
2409:)
2405:r
2402:(
2399:b
2396:=
2391:2
2387:/
2380:=
2374:,
2371:0
2368:=
2365:t
2360:|
2355:g
2332:2
2328:/
2321:=
2315:,
2312:0
2309:=
2306:t
2280:0
2276:t
2272:=
2269:t
2229:2
2225:r
2204:0
2196:1
2192:r
2114:0
2111:=
2108:r
2086:0
2082:t
2078:=
2075:t
2055:0
2052:=
2019:=
2012:,
2007:0
2003:r
1999:=
1996:r
1992:,
1987:0
1983:t
1979:=
1976:t
1952:0
1949:=
1942:,
1937:0
1933:r
1929:=
1926:r
1922:,
1917:0
1913:t
1909:=
1906:t
1878:t
1875:d
1872:)
1869:r
1866:(
1863:a
1856:2
1852:t
1844:1
1840:t
1831:=
1802:t
1799:d
1796:)
1793:r
1790:(
1787:a
1750:0
1742:=
1736:,
1731:0
1723:=
1717:,
1712:0
1708:r
1704:=
1701:r
1673:r
1670:d
1667:)
1664:r
1661:(
1658:b
1651:2
1647:r
1639:1
1635:r
1626:=
1597:r
1594:d
1591:)
1588:r
1585:(
1582:b
1553:r
1458:)
1452:,
1446:(
1443:=
1411:2
1406:0
1402:r
1397:/
1393:1
1390:=
1387:K
1361:2
1356:0
1352:r
1345:4
1342:=
1339:A
1303:g
1260:,
1245:0
1241:,
1237:)
1231:2
1223:d
1211:2
1203:+
1198:2
1190:d
1186:(
1180:2
1175:0
1171:r
1167:=
1158:g
1152:2
1147:0
1143:r
1139:=
1132:0
1128:r
1124:=
1121:r
1118:,
1113:0
1109:t
1105:=
1102:t
1097:|
1092:g
1063:0
1059:r
1055:=
1052:r
1048:,
1043:0
1039:t
1035:=
1032:t
1000:2
996:/
989:+
943:0
939:t
935:=
932:t
891:t
883:=
874:X
747:+
671:t
609:b
605:a
578:g
557:)
551:,
545:(
542:=
497:,
482:0
478:,
473:1
469:r
462:r
454:0
450:r
445:,
436:t
400:g
394:2
390:r
386:+
381:2
377:r
373:d
367:2
363:)
359:r
356:(
353:b
350:+
345:2
341:t
337:d
331:2
327:)
323:r
320:(
317:a
311:=
307:)
301:2
293:d
281:2
273:+
268:2
260:d
256:(
250:2
246:r
242:+
237:2
233:r
229:d
223:2
219:)
215:r
212:(
209:b
206:+
201:2
197:t
193:d
187:2
183:)
179:r
176:(
173:a
167:=
164:g
126:g
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