22:
1737:
189:). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere). Conventionally, the metric on the 2-sphere is written in
1918:
758:
1547:
3240:
As these equations are effectively two-dimensional, they can be solved without overwhelming difficulty for a variety of assumptions about the nature of the infalling material (that is, for the assumption of a spherically symmetric black hole that is accreting charged or neutral dust, gas, plasma or
2668:
1748:
2302:
which describes the gradient (in the free-falling tetrad frame) of the circumferential radius along the radial direction. This is not in general unity; compare, for example, to the standard
Swarschild solution, or the Reissner–Nordström solution. The sign of
1197:
is the inverse vierbein. The convention here and in what follows is that the roman indexes refer to the flat orthonormal tetrad frame, while the greek indexes refer to the coordinate frame. The inverse vierbein can be directly read off of the above metric as
1374:
551:
2037:
1732:{\displaystyle e_{\;\mu }^{i}={\begin{bmatrix}{\frac {1}{\alpha }}&0&0&0\\-{\frac {\beta _{t}}{\alpha \beta _{r}}}&{\frac {1}{\beta _{r}}}&0&0\\0&0&r&0\\0&0&0&r\sin \theta \\\end{bmatrix}}}
2814:
2741:
2118:
2539:
270:
1159:
2869:
2970:
1913:{\displaystyle e_{i}^{\;\mu }={\begin{bmatrix}\alpha &\beta _{t}&0&0\\0&\beta _{r}&0&0\\0&0&{\frac {1}{r}}&0\\0&0&0&{\frac {1}{r\sin \theta }}\\\end{bmatrix}}}
3235:
1501:
1263:
2533:
1433:
3023:
493:
2297:
2248:
2426:
These two relations on the circumferential radius provide another reason why this particular parameterization of the metric is convenient: it has a simple intuitive characterization.
828:
753:{\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }=-{\frac {dt^{2}}{\alpha ^{2}}}+{\frac {1}{\beta _{r}^{2}}}\left(dr-\beta _{t}{\frac {dt}{\alpha }}\right)^{2}+r^{2}\,g(\Omega ).}
1269:
968:
2421:
2388:
1926:
1195:
2472:
2175:
2148:
3053:
398:
2355:
2328:
2202:
1076:
1019:
921:
790:
39:
3073:
1539:
894:
988:
3150:
357:
3294:
Andrew J. S. Hamilton and Pedro P. Avelino, "The physics of the relativistic counter-streaming instability that drives mass inflation inside black holes" (2008),
3121:
3097:
868:
848:
325:
130:
in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition irrotational, they are not realistic models of
2747:
2674:
2051:
2663:{\displaystyle \Gamma _{rtr}=-\beta _{t}{\frac {\partial \ln \alpha }{\partial r}}+{\frac {\partial \beta _{t}}{\partial r}}-\partial _{t}\ln \beta _{r}}
199:
86:
1084:
58:
2820:
2900:
292:
65:
1038:
408:
3166:
1439:
3326:
105:
1204:
1021:
can be interpreted as the radial derivative of the circumferential radius in a freely-falling frame; this becomes explicit in the
134:
in nature. However, their metrics are considerably simpler than those of rotating spacetimes, making them much easier to analyze.
300:
72:
2480:
1380:
43:
54:
3349:
280:
127:
1033:
Note that the above metric is written as a sum of squares, and therefore it can be understood as explicitly encoding a
3318:
2976:
526:
178:
441:
32:
507:
2259:
2210:
328:
1369:{\displaystyle e_{\;\mu }^{r}dx^{\mu }={\frac {1}{\beta _{r}}}\left(dr-\beta _{t}{\frac {dt}{\alpha }}\right)}
795:
79:
3076:
2042:
The particularly simple form of the above is a prime motivating factor for working with the given metric.
926:
416:
412:
288:
850:
is defined so that it is the circumferential radius, that is, so that the proper circumference at radius
3153:
512:
436:
296:
295:. A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of
2032:{\displaystyle (e_{\;\mu }^{i})^{T}e_{i}^{\;\nu }=e_{\mu }^{\;\;i}e_{i}^{\;\nu }=\delta _{\mu }^{\nu }}
327:, there are precisely 3 rotational Killing vector fields. Stated in another way, the dimension of the
307:
of
Killing vector fields and thus generate these vector fields. For a spherically symmetric spacetime
3268:
3263:
2393:
2360:
304:
174:
2045:
The vierbein relates vector fields in the coordinate frame to vector fields in the tetrad frame, as
1167:
542:
146:
3157:
2444:
2439:
2204:
was the proper rate of change of the circumferential radius; this can now be explicitly written as
2153:
2126:
424:
3295:
3246:
3031:
520:
362:
284:
2333:
2306:
2180:
1051:
997:
899:
766:
419:. This means that the exterior region around a spherically symmetric gravitating object must be
141:
similar to those of rotating spacetimes, and these typically have qualitative features (such as
3322:
190:
3058:
3258:
1509:
1046:
1022:
873:
500:
420:
401:
973:
3273:
3126:
2887:
2435:
333:
186:
138:
3311:
3106:
3082:
2891:
2883:
2809:{\displaystyle \Gamma _{\theta r\theta }=\Gamma _{\phi r\phi }={\frac {\beta _{r}}{r}}}
2736:{\displaystyle \Gamma _{\theta t\theta }=\Gamma _{\phi t\phi }={\frac {\beta _{t}}{r}}}
853:
833:
310:
166:
142:
2113:{\displaystyle \partial _{i}=e_{i}^{\;\mu }{\frac {\partial \;\;}{\partial x^{\mu }}}}
3343:
2177:
which is the radial derivative in the rest frame. By construction, as noted earlier,
496:
149:
due to counter-moving streams of infalling matter in the interior of a black hole.
991:
516:
170:
137:
Spherically symmetric models are not entirely inappropriate: many of them have
21:
2894:
scalar can be found in
Hamilton & Avelino. The Einstein equations become
131:
532:
Circumferential radius, given below, convenient for studying mass inflation.
162:
970:
is the proper rate of change of the circumferential radius (that is, where
529:, sometimes used for studying static spherically symmetric perfect fluids.
1042:
1034:
145:) that are unaffected by rotation. One such application is the study of
265:{\displaystyle g_{\Omega }=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}}
1742:
The vierbein itself is the inverse(-transpose) of the inverse vierbein
1154:{\displaystyle g_{\mu \nu }=\eta _{ij}\,e_{\;\mu }^{i}\,e_{\;\nu }^{j}}
119:
2864:{\displaystyle \Gamma _{\phi \theta \phi }={\frac {\cot \theta }{r}}}
182:
279:
Spherical symmetry is a characteristic feature of many solutions of
3299:
2965:{\displaystyle \nabla _{t}\beta _{t}=-{\frac {M}{r^{2}}}-4\pi rp}
400:. In general, none of these are time-like, as that would imply a
126:
are commonly used to obtain analytic and numerical solutions to
3230:{\displaystyle {\frac {2M}{r}}-1=\beta _{t}^{2}-\beta _{r}^{2}}
1496:{\displaystyle e_{\;\mu }^{\phi }dx^{\mu }=r\sin \theta d\phi }
415:
is necessarily isometric to a subset of the maximally extended
15:
1258:{\displaystyle e_{\;\mu }^{t}dx^{\mu }={\frac {dt}{\alpha }}}
276:
and so the full metric includes a term proportional to this.
2330:
effectively determines "which way is down"; the sign of
2528:{\displaystyle \Gamma _{rtt}=-\partial _{r}\ln \alpha }
1428:{\displaystyle e_{\;\mu }^{\theta }dx^{\mu }=rd\theta }
3334:
See
Section 6.1 for a discussion of spherical symmetry
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202:
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distinguishes incoming and outgoing frames, so that
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of this group are 2-spheres (ordinary 2-dimensional
2438:in the tetrad frame can be written in terms of the
792:is the standard metric on the unit radius 2-sphere
46:. Unsourced material may be challenged and removed.
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1041:. That is, the metric tensor can be written as a
411:) that any spherically symmetric solution of the
2150:which is the proper time in the rest frame, and
303:. The isometries referred to above are actually
1541:. Written as a matrix, the inverse vierbein is
3018:{\displaystyle \nabla _{t}\beta _{r}=4\pi rf}
8:
896:. In this coordinate choice, the parameter
488:{\displaystyle x^{\mu }=(t,r,\theta ,\phi )}
3290:
3288:
2091:
2090:
2078:
2005:
1989:
1988:
1969:
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1213:
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1123:
3241:dark matter, of high or low temperature,
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2292:{\displaystyle \beta _{r}=\partial _{r}r}
2280:
2267:
2261:
2243:{\displaystyle \beta _{t}=\partial _{t}r}
2231:
2218:
2212:
2188:
2182:
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2134:
2128:
2101:
2084:
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1441:
1407:
1394:
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1340:
1314:
1305:
1296:
1283:
1277:
1271:
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1206:
1181:
1175:
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1145:
1139:
1134:
1128:
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1117:
1108:
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1059:
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1005:
999:
975:
949:
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855:
835:
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728:
715:
695:
689:
662:
657:
648:
637:
626:
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604:
591:
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553:
541:One popular metric, used in the study of
449:
443:
364:
335:
312:
256:
248:
236:
223:
207:
201:
106:Learn how and when to remove this message
2474:in the tetrad frame, which are given by
519:are round, and thus useful for studying
3284:
823:{\displaystyle \Omega =(\theta ,\phi )}
3055:is the covariant time derivative (and
2882:A complete set of expressions for the
2123:The most interesting of these two are
7:
963:{\displaystyle \beta _{t}=dr/d\tau }
44:adding citations to reliable sources
1506:where the signature was take to be
3123:the radial energy flux. The mass
3062:
3036:
2981:
2905:
2825:
2771:
2752:
2698:
2679:
2635:
2622:
2607:
2592:
2578:
2544:
2507:
2485:
2449:
2277:
2228:
2158:
2131:
2094:
2087:
2056:
799:
776:
741:
299:, which, in a very precise sense,
208:
14:
55:"Spherically symmetric spacetime"
124:spherically symmetric spacetimes
20:
2416:{\displaystyle \beta _{r}<0}
2383:{\displaystyle \beta _{r}>0}
159:spherically symmetric spacetime
31:needs additional citations for
3139:
3133:
3103:the isotropic pressure!), and
1950:
1930:
1528:
1513:
1190:{\displaystyle e_{\;\mu }^{i}}
817:
805:
779:
773:
744:
738:
482:
458:
381:
375:
346:
340:
1:
2467:{\displaystyle \Gamma _{ijk}}
2170:{\displaystyle \partial _{r}}
2143:{\displaystyle \partial _{t}}
537:Circumferential radius metric
503:are possible; these include:
431:Spherically symmetric metrics
169:contains a subgroup which is
1029:Orthonormal tetrad formalism
3319:University of Chicago Press
3048:{\displaystyle \nabla _{t}}
495:, to write the metric (the
393:{\displaystyle \dim K(M)=3}
293:Reissner–Nordström solution
3366:
2350:{\displaystyle \beta _{r}}
2323:{\displaystyle \beta _{r}}
2197:{\displaystyle \beta _{t}}
1071:{\displaystyle \eta _{ij}}
1014:{\displaystyle \beta _{r}}
916:{\displaystyle \beta _{t}}
785:{\displaystyle g(\Omega )}
527:Gaussian polar coordinates
305:local flow diffeomorphisms
281:Einstein's field equations
128:Einstein's field equations
2390:is an ingoing frame, and
1037:, and, in particular, an
508:Schwarzschild coordinates
435:Conventionally, one uses
3309:Wald, Robert M. (1984).
2039:is the identity matrix.
830:. The radial coordinate
3068:{\displaystyle \nabla }
3245:material with various
3231:
3146:
3117:
3093:
3077:Levi-Civita connection
3069:
3049:
3019:
2966:
2865:
2810:
2737:
2664:
2529:
2468:
2423:is an outgoing frame.
2417:
2384:
2351:
2324:
2293:
2244:
2198:
2171:
2144:
2114:
2033:
1914:
1733:
1535:
1534:{\displaystyle (-+++)}
1497:
1429:
1370:
1259:
1191:
1155:
1072:
1015:
984:
964:
917:
890:
889:{\displaystyle 2\pi r}
864:
844:
824:
786:
754:
489:
417:Schwarzschild solution
413:vacuum field equations
394:
353:
321:
289:Schwarzschild solution
266:
3232:
3147:
3118:
3099:the radial pressure (
3094:
3070:
3050:
3020:
2967:
2874:and all others zero.
2866:
2811:
2738:
2665:
2530:
2469:
2418:
2385:
2352:
2325:
2294:
2245:
2199:
2172:
2145:
2115:
2034:
1915:
1734:
1536:
1498:
1430:
1371:
1260:
1192:
1156:
1073:
1016:
985:
983:{\displaystyle \tau }
965:
918:
891:
865:
845:
825:
787:
755:
513:Isotropic coordinates
490:
437:spherical coordinates
395:
354:
322:
297:Killing vector fields
267:
3350:Lorentzian manifolds
3269:Spacetime symmetries
3264:Stationary spacetime
3167:
3145:{\displaystyle M(r)}
3127:
3107:
3083:
3059:
3032:
2977:
2901:
2821:
2748:
2675:
2540:
2481:
2445:
2394:
2361:
2334:
2307:
2260:
2211:
2181:
2154:
2127:
2052:
1927:
1749:
1548:
1510:
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1270:
1205:
1168:
1085:
1052:
998:
974:
927:
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874:
854:
834:
796:
767:
552:
442:
363:
352:{\displaystyle K(M)}
334:
311:
200:
175:rotation group SO(3)
40:improve this article
3226:
3208:
2440:Christoffel symbols
2253:Similarly, one has
2083:
2028:
2010:
1994:
1974:
1948:
1767:
1566:
1458:
1399:
1288:
1223:
1186:
1150:
1133:
923:is defined so that
667:
425:asymptotically flat
301:preserve the metric
3313:General Relativity
3247:equations of state
3227:
3212:
3194:
3154:Misner-Thorne mass
3142:
3113:
3089:
3065:
3045:
3015:
2962:
2878:Einstein equations
2861:
2806:
2733:
2660:
2525:
2464:
2413:
2380:
2347:
2320:
2289:
2240:
2194:
2167:
2140:
2110:
2068:
2029:
2014:
1995:
1978:
1959:
1933:
1910:
1904:
1752:
1729:
1723:
1551:
1531:
1493:
1443:
1425:
1384:
1366:
1273:
1255:
1208:
1187:
1171:
1151:
1135:
1118:
1068:
1039:orthonormal tetrad
1011:
980:
960:
913:
886:
860:
840:
820:
782:
750:
653:
485:
409:Birkhoff's theorem
390:
349:
317:
285:general relativity
262:
3183:
3116:{\displaystyle f}
3092:{\displaystyle p}
2945:
2859:
2804:
2731:
2629:
2599:
2108:
1900:
1855:
1656:
1637:
1586:
1359:
1320:
1253:
994:). The parameter
863:{\displaystyle r}
843:{\displaystyle r}
708:
668:
643:
501:coordinate charts
407:It is known (see
320:{\displaystyle M}
287:, especially the
191:polar coordinates
185:in 3-dimensional
153:Formal definition
116:
115:
108:
90:
3357:
3332:
3316:
3302:
3292:
3259:Static spacetime
3236:
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3225:
3220:
3207:
3202:
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3179:
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2166:
2165:
2149:
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1217:
1196:
1194:
1193:
1188:
1185:
1180:
1160:
1158:
1157:
1152:
1149:
1144:
1132:
1127:
1116:
1115:
1100:
1099:
1077:
1075:
1074:
1069:
1067:
1066:
1047:Minkowski metric
1023:tetrad formalism
1020:
1018:
1017:
1012:
1010:
1009:
989:
987:
986:
981:
969:
967:
966:
961:
953:
939:
938:
922:
920:
919:
914:
912:
911:
895:
893:
892:
887:
869:
867:
866:
861:
849:
847:
846:
841:
829:
827:
826:
821:
791:
789:
788:
783:
759:
757:
756:
751:
733:
732:
720:
719:
714:
710:
709:
704:
696:
694:
693:
669:
666:
661:
649:
644:
642:
641:
632:
631:
630:
617:
609:
608:
596:
595:
583:
582:
567:
566:
494:
492:
491:
486:
454:
453:
402:static spacetime
399:
397:
396:
391:
358:
356:
355:
350:
326:
324:
323:
318:
271:
269:
268:
263:
261:
260:
241:
240:
228:
227:
212:
211:
139:Penrose diagrams
111:
104:
100:
97:
91:
89:
48:
24:
16:
3365:
3364:
3360:
3359:
3358:
3356:
3355:
3354:
3340:
3339:
3329:
3308:
3305:
3293:
3286:
3282:
3274:De Sitter space
3255:
3172:
3165:
3164:
3125:
3124:
3105:
3104:
3081:
3080:
3057:
3056:
3035:
3030:
3029:
2990:
2980:
2975:
2974:
2935:
2914:
2904:
2899:
2898:
2888:Einstein tensor
2880:
2845:
2824:
2819:
2818:
2791:
2770:
2751:
2746:
2745:
2718:
2697:
2678:
2673:
2672:
2650:
2634:
2621:
2610:
2606:
2591:
2577:
2565:
2543:
2538:
2537:
2506:
2484:
2479:
2478:
2448:
2443:
2442:
2436:connection form
2432:
2430:Connection form
2397:
2392:
2391:
2364:
2359:
2358:
2337:
2332:
2331:
2310:
2305:
2304:
2276:
2263:
2258:
2257:
2227:
2214:
2209:
2208:
2184:
2179:
2178:
2157:
2152:
2151:
2130:
2125:
2124:
2097:
2093:
2086:
2055:
2050:
2049:
1949:
1925:
1924:
1903:
1902:
1886:
1879:
1874:
1869:
1863:
1862:
1857:
1845:
1840:
1834:
1833:
1828:
1823:
1813:
1811:
1805:
1804:
1799:
1794:
1784:
1782:
1772:
1747:
1746:
1722:
1721:
1707:
1702:
1697:
1691:
1690:
1685:
1680:
1675:
1669:
1668:
1663:
1658:
1646:
1639:
1626:
1622:
1612:
1604:
1603:
1598:
1593:
1588:
1571:
1546:
1545:
1508:
1507:
1462:
1438:
1437:
1403:
1379:
1378:
1348:
1336:
1326:
1322:
1310:
1292:
1268:
1267:
1242:
1227:
1203:
1202:
1166:
1165:
1104:
1088:
1083:
1082:
1055:
1050:
1049:
1031:
1001:
996:
995:
972:
971:
930:
925:
924:
903:
898:
897:
872:
871:
852:
851:
832:
831:
794:
793:
765:
764:
724:
697:
685:
675:
671:
670:
633:
622:
618:
600:
587:
571:
558:
550:
549:
539:
445:
440:
439:
433:
361:
360:
359:is 3; that is,
332:
331:
329:Killing algebra
309:
308:
252:
232:
219:
203:
198:
197:
187:Euclidean space
155:
143:Cauchy horizons
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
3363:
3361:
3353:
3352:
3342:
3341:
3338:
3337:
3327:
3304:
3303:
3283:
3281:
3278:
3277:
3276:
3271:
3266:
3261:
3254:
3251:
3238:
3237:
3224:
3219:
3215:
3211:
3206:
3201:
3197:
3193:
3190:
3187:
3182:
3178:
3175:
3141:
3138:
3135:
3132:
3112:
3088:
3064:
3042:
3038:
3026:
3025:
3014:
3011:
3008:
3005:
3002:
2997:
2993:
2987:
2983:
2972:
2961:
2958:
2955:
2952:
2949:
2942:
2938:
2934:
2929:
2926:
2921:
2917:
2911:
2907:
2892:Weyl curvature
2884:Riemann tensor
2879:
2876:
2872:
2871:
2858:
2854:
2851:
2848:
2842:
2837:
2834:
2831:
2827:
2816:
2803:
2798:
2794:
2788:
2783:
2780:
2777:
2773:
2769:
2764:
2761:
2758:
2754:
2743:
2730:
2725:
2721:
2715:
2710:
2707:
2704:
2700:
2696:
2691:
2688:
2685:
2681:
2670:
2657:
2653:
2649:
2646:
2641:
2637:
2633:
2627:
2624:
2617:
2613:
2609:
2603:
2597:
2594:
2589:
2586:
2583:
2580:
2572:
2568:
2564:
2561:
2556:
2553:
2550:
2546:
2535:
2524:
2521:
2518:
2513:
2509:
2505:
2502:
2497:
2494:
2491:
2487:
2461:
2458:
2455:
2451:
2431:
2428:
2412:
2409:
2404:
2400:
2379:
2376:
2371:
2367:
2344:
2340:
2317:
2313:
2300:
2299:
2288:
2283:
2279:
2275:
2270:
2266:
2251:
2250:
2239:
2234:
2230:
2226:
2221:
2217:
2191:
2187:
2164:
2160:
2137:
2133:
2121:
2120:
2104:
2100:
2096:
2089:
2081:
2075:
2071:
2067:
2062:
2058:
2026:
2021:
2017:
2013:
2008:
2002:
1998:
1992:
1985:
1981:
1977:
1972:
1966:
1962:
1956:
1952:
1946:
1941:
1936:
1932:
1921:
1920:
1907:
1898:
1895:
1892:
1889:
1885:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1864:
1861:
1858:
1854:
1851:
1846:
1844:
1841:
1839:
1836:
1835:
1832:
1829:
1827:
1824:
1820:
1816:
1812:
1810:
1807:
1806:
1803:
1800:
1798:
1795:
1791:
1787:
1783:
1781:
1778:
1777:
1775:
1770:
1765:
1759:
1755:
1740:
1739:
1726:
1720:
1717:
1714:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1692:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1670:
1667:
1664:
1662:
1659:
1653:
1649:
1645:
1640:
1633:
1629:
1625:
1619:
1615:
1609:
1606:
1605:
1602:
1599:
1597:
1594:
1592:
1589:
1585:
1582:
1577:
1576:
1574:
1569:
1564:
1559:
1554:
1530:
1527:
1524:
1521:
1518:
1515:
1504:
1503:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1469:
1465:
1461:
1456:
1451:
1446:
1435:
1424:
1421:
1418:
1415:
1410:
1406:
1402:
1397:
1392:
1387:
1376:
1364:
1358:
1354:
1351:
1343:
1339:
1335:
1332:
1329:
1325:
1317:
1313:
1309:
1304:
1299:
1295:
1291:
1286:
1281:
1276:
1265:
1252:
1248:
1245:
1239:
1234:
1230:
1226:
1221:
1216:
1211:
1184:
1179:
1174:
1162:
1161:
1148:
1143:
1138:
1131:
1126:
1121:
1114:
1111:
1107:
1103:
1098:
1095:
1091:
1065:
1062:
1058:
1030:
1027:
1008:
1004:
979:
959:
956:
952:
948:
945:
942:
937:
933:
910:
906:
885:
882:
879:
859:
839:
819:
816:
813:
810:
807:
804:
801:
781:
778:
775:
772:
761:
760:
749:
746:
743:
740:
737:
731:
727:
723:
718:
713:
707:
703:
700:
692:
688:
684:
681:
678:
674:
665:
660:
656:
652:
647:
640:
636:
629:
625:
621:
615:
612:
607:
603:
599:
594:
590:
586:
581:
578:
574:
570:
565:
561:
557:
543:mass inflation
538:
535:
534:
533:
530:
524:
510:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
452:
448:
432:
429:
389:
386:
383:
380:
377:
374:
371:
368:
348:
345:
342:
339:
316:
274:
273:
259:
255:
251:
247:
244:
239:
235:
231:
226:
222:
218:
215:
210:
206:
167:isometry group
154:
151:
147:mass inflation
114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3362:
3351:
3348:
3347:
3345:
3335:
3330:
3328:0-226-87033-2
3324:
3320:
3315:
3314:
3307:
3306:
3301:
3297:
3291:
3289:
3285:
3279:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3256:
3252:
3250:
3248:
3244:
3222:
3217:
3213:
3209:
3204:
3199:
3195:
3191:
3188:
3185:
3180:
3176:
3173:
3163:
3162:
3161:
3159:
3158:interior mass
3155:
3136:
3130:
3110:
3102:
3086:
3078:
3040:
3012:
3009:
3006:
3003:
3000:
2995:
2991:
2985:
2973:
2959:
2956:
2953:
2950:
2947:
2940:
2936:
2932:
2927:
2924:
2919:
2915:
2909:
2897:
2896:
2895:
2893:
2889:
2885:
2877:
2875:
2856:
2852:
2849:
2846:
2840:
2835:
2832:
2829:
2817:
2801:
2796:
2792:
2786:
2781:
2778:
2775:
2767:
2762:
2759:
2756:
2744:
2728:
2723:
2719:
2713:
2708:
2705:
2702:
2694:
2689:
2686:
2683:
2671:
2655:
2651:
2647:
2644:
2639:
2631:
2625:
2615:
2611:
2601:
2595:
2587:
2584:
2581:
2570:
2566:
2562:
2559:
2554:
2551:
2548:
2536:
2522:
2519:
2516:
2511:
2503:
2500:
2495:
2492:
2489:
2477:
2476:
2475:
2459:
2456:
2453:
2441:
2437:
2429:
2427:
2424:
2410:
2407:
2402:
2398:
2377:
2374:
2369:
2365:
2342:
2338:
2315:
2311:
2286:
2281:
2273:
2268:
2264:
2256:
2255:
2254:
2237:
2232:
2224:
2219:
2215:
2207:
2206:
2205:
2189:
2185:
2162:
2135:
2102:
2098:
2079:
2073:
2069:
2065:
2060:
2048:
2047:
2046:
2043:
2040:
2024:
2019:
2015:
2011:
2006:
2000:
1996:
1990:
1983:
1979:
1975:
1970:
1964:
1960:
1954:
1944:
1939:
1934:
1905:
1896:
1893:
1890:
1887:
1883:
1876:
1871:
1866:
1859:
1852:
1849:
1842:
1837:
1830:
1825:
1818:
1814:
1808:
1801:
1796:
1789:
1785:
1779:
1773:
1768:
1763:
1757:
1753:
1745:
1744:
1743:
1724:
1718:
1715:
1712:
1709:
1704:
1699:
1694:
1687:
1682:
1677:
1672:
1665:
1660:
1651:
1647:
1643:
1631:
1627:
1623:
1617:
1613:
1607:
1600:
1595:
1590:
1583:
1580:
1572:
1567:
1562:
1557:
1552:
1544:
1543:
1542:
1525:
1522:
1519:
1516:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1467:
1463:
1459:
1454:
1449:
1444:
1436:
1422:
1419:
1416:
1413:
1408:
1404:
1400:
1395:
1390:
1385:
1377:
1362:
1356:
1352:
1349:
1341:
1337:
1333:
1330:
1327:
1323:
1315:
1311:
1307:
1302:
1297:
1293:
1289:
1284:
1279:
1274:
1266:
1250:
1246:
1243:
1237:
1232:
1228:
1224:
1219:
1214:
1209:
1201:
1200:
1199:
1182:
1177:
1172:
1146:
1141:
1136:
1129:
1124:
1119:
1112:
1109:
1105:
1101:
1096:
1093:
1089:
1081:
1080:
1079:
1063:
1060:
1056:
1048:
1044:
1040:
1036:
1028:
1026:
1024:
1006:
1002:
993:
977:
957:
954:
950:
946:
943:
940:
935:
931:
908:
904:
883:
880:
877:
857:
837:
814:
811:
808:
802:
770:
747:
735:
729:
725:
721:
716:
711:
705:
701:
698:
690:
686:
682:
679:
676:
672:
663:
658:
654:
650:
645:
638:
634:
627:
623:
619:
613:
610:
605:
601:
597:
592:
588:
584:
579:
576:
572:
568:
563:
559:
555:
548:
547:
546:
544:
536:
531:
528:
525:
522:
518:
514:
511:
509:
506:
505:
504:
502:
498:
479:
476:
473:
470:
467:
464:
461:
455:
450:
446:
438:
430:
428:
426:
422:
418:
414:
410:
405:
403:
387:
384:
378:
372:
369:
366:
343:
337:
330:
314:
306:
302:
298:
294:
290:
286:
282:
277:
257:
253:
249:
245:
242:
237:
233:
229:
224:
220:
216:
213:
204:
196:
195:
194:
192:
188:
184:
180:
176:
172:
168:
164:
160:
152:
150:
148:
144:
140:
135:
133:
129:
125:
121:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
3333:
3312:
3242:
3239:
3100:
3027:
2881:
2873:
2433:
2425:
2301:
2252:
2122:
2044:
2041:
1922:
1741:
1505:
1163:
1032:
762:
540:
497:line element
434:
406:
278:
275:
158:
156:
136:
123:
117:
102:
96:October 2020
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
3317:. Chicago:
3160:, given by
992:proper time
517:light cones
515:, in which
499:). Several
132:black holes
3280:References
1164:where the
521:null dusts
171:isomorphic
66:newspapers
3300:0811.1926
3214:β
3210:−
3196:β
3186:−
3063:∇
3037:∇
3007:π
2992:β
2982:∇
2954:π
2948:−
2928:−
2916:β
2906:∇
2853:θ
2850:
2836:ϕ
2833:θ
2830:ϕ
2826:Γ
2793:β
2782:ϕ
2776:ϕ
2772:Γ
2763:θ
2757:θ
2753:Γ
2720:β
2709:ϕ
2703:ϕ
2699:Γ
2690:θ
2684:θ
2680:Γ
2652:β
2648:
2636:∂
2632:−
2623:∂
2612:β
2608:∂
2593:∂
2588:α
2585:
2579:∂
2567:β
2563:−
2545:Γ
2523:α
2520:
2508:∂
2504:−
2486:Γ
2450:Γ
2399:β
2366:β
2339:β
2312:β
2278:∂
2265:β
2229:∂
2216:β
2186:β
2159:∂
2132:∂
2103:μ
2095:∂
2088:∂
2080:μ
2057:∂
2025:ν
2020:μ
2016:δ
2007:ν
1984:μ
1971:ν
1940:μ
1923:That is,
1897:θ
1894:
1815:β
1786:β
1780:α
1764:μ
1719:θ
1716:
1648:β
1628:β
1624:α
1614:β
1608:−
1584:α
1558:μ
1517:−
1491:ϕ
1485:θ
1482:
1468:μ
1455:ϕ
1450:μ
1423:θ
1409:μ
1396:θ
1391:μ
1357:α
1338:β
1334:−
1312:β
1298:μ
1280:μ
1251:α
1233:μ
1215:μ
1178:μ
1142:ν
1125:μ
1106:η
1097:ν
1094:μ
1057:η
1003:β
978:τ
958:τ
932:β
905:β
881:π
815:ϕ
809:θ
800:Ω
777:Ω
742:Ω
706:α
687:β
683:−
655:β
635:α
614:−
606:ν
593:μ
580:ν
577:μ
480:ϕ
474:θ
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370:
254:φ
246:θ
243:
221:θ
209:Ω
163:spacetime
3344:Category
3253:See also
2890:and the
1043:pullback
1035:vierbein
291:and the
177:and the
3152:is the
1045:of the
990:is the
183:spheres
173:to the
120:physics
80:scholar
3325:
3028:where
2886:, the
763:Here,
421:static
179:orbits
165:whose
82:
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68:
61:
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3296:arXiv
545:, is
161:is a
87:JSTOR
73:books
3323:ISBN
3243:i.e.
3075:the
2434:The
2408:<
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423:and
59:news
3249:.)
3156:or
3101:not
3079:),
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1891:sin
1713:sin
1479:sin
870:is
367:dim
283:of
234:sin
193:as
118:In
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2645:ln
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1078::
1025:.
427:.
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2066:=
2061:i
2012:=
2001:i
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1980:e
1976:=
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1906:]
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774:(
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