2336:
712:
42:
830:. Shapiro proposed an observational test of his prediction: bounce radar beams off the surface of Venus and Mercury and measure the round-trip travel time. When the Earth, Sun, and Venus are most favorably aligned, Shapiro showed that the expected time delay, due to the presence of the Sun, of a radar signal traveling from the Earth to Venus and back, would be about 200 microseconds, well within the limitations of 1960s-era technology.
3789:
725:
2322:
801:
Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path, these time delays should thereby be increased by almost 2Ă10 sec when the radar pulses pass near the sun. Such a change, equivalent to 60 km in distance, could
1340:
1955:
The doubling of the
Shapiro factor can be explained by the fact that there is not only the gravitational time dilation, but also the radial stretching of space, both of which contribute equally in general relativity for the time delay as they also do for the deflection of light.
1921:
2230:
1419:
represents the distance along the path from this point to the planet. The right-hand side of this equation is primarily due to the variable speed of the light ray; the contribution from the change in path, being of second order in
2298:, the Shapiro delay for gravitational waves is much smaller than that for neutrinos or photons. The observed 1.7-second difference in arrival times seen between gravitational wave and gamma ray arrivals from neutron star merger
1560:
899:
861:. The measured elapsed time of a light signal in a gravitational field is longer than it would be without the field, and for moderate-strength nearly static fields the difference is directly proportional to the classical
113:
1749:
2136:
856:
In a nearly static gravitational field of moderate strength (say, of stars and planets, but not one of a black hole or close binary system of neutron stars) the effect may be considered as a special case of
2073:
2012:
3492:
483:
785:
signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The time delay is caused by
849:
1807:
2478:
2286:
and other metric theories of gravity, though, the
Shapiro delay for gravitational waves is expected to be the same as that for light and neutrinos. However, in theories such as
1673:
756:
1462:
1611:
1585:
1950:
135:
1826:
1417:
1390:
852:
Left: unperturbed lightrays in a flat spacetime, right: Shapiro-delayed and deflected lightrays in the vicinity of a gravitating mass (click to start the animation)
1631:
1438:
1363:
891:
340:
3539:
1477:
3774:
3563:
3309:
2142:
693:
503:
345:
458:
2435:
Irwin I. Shapiro; Gordon H. Pettengill; Michael E. Ash; Melvin L. Stone; et al. (1968). "Fourth Test of
General Relativity: Preliminary Results".
1335:{\displaystyle \Delta t\approx {\frac {4GM}{c^{3}}}\left(\ln \left-{\frac {1}{2}}\left\right)+{\mathcal {O}}\left({\frac {G^{2}M^{2}}{c^{5}d}}\right),}
840:, were successful, matching the predicted amount of time delay. The experiments have been repeated many times since then, with increasing accuracy.
749:
3345:
3299:
2242:
Shapiro delay must be considered along with ranging data when trying to accurately determine the distance to interplanetary probes such as the
1817:
371:
205:
2335:
711:
3483:
2636:
2287:
834:
789:, which increases the time it takes light to travel a given distance from the perspective of an outside observer. In a 1964 article entitled
410:
48:
2475:
3816:
2967:
1681:
2270:, the Shapiro delay for high-energy neutrinos must be the same as that for photons to within 10%, consistent with recent estimates of the
810:
as the speed of light and calculates the time delay of the passage of light waves or rays over finite coordinate distance according to a
742:
3591:
3220:
122:
3225:
148:
2079:
3545:
428:
41:
873:
Shapiro's original formulation was derived from the
Schwarzschild solution and included terms to the first order in solar mass (
3258:
335:
3361:
3439:
1962:
3502:
2809:
Lawrence M. Krauss; Scott
Tremaine (January 18, 1988). "Test of the Weak Equivalence Principle for Neutrinos and Photons".
468:
3792:
3534:
3369:
422:
130:
2018:
488:
3356:
2657:
3116:
1392:
is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and
3811:
3397:
3131:
2719:
2291:
858:
778:
729:
220:
140:
3684:
3126:
3079:
2848:
S. Desai; E. Kahya; R. P. Woodard (2008). "Reduced time delay for gravitational waves with dark matter emulators".
2373:
2303:
633:
181:
433:
3714:
3340:
3021:
245:
1757:
508:
3739:
3289:
3049:
2960:
815:
668:
658:
325:
3468:
3390:
893:) for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:
387:
3010:
3001:
2811:
2772:
2437:
2400:
862:
802:
now be measured over the required path length to within about 5 to 10% with presently obtainable equipment.
653:
3557:
3463:
2897:
E. Kahya; S. Desai (2016). "Constraints on frequency-dependent violations of
Shapiro delay from GW150914".
453:
3694:
3421:
3095:
3059:
2349:
848:
811:
623:
608:
215:
3699:
3304:
3243:
1643:
598:
166:
3604:
3530:
3374:
2976:
2918:
2869:
2820:
2781:
2738:
2676:
2575:
2514:
2446:
2409:
1810:
837:
418:
1443:
3679:
3674:
3669:
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3496:
3445:
3263:
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3074:
3005:
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2358:
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663:
548:
320:
225:
210:
171:
33:
1929:
1471:
For a signal going around a massive object, the time delay can be calculated as the following:
3754:
3624:
3586:
3578:
3179:
3136:
2899:
2850:
2836:
2797:
2704:
2632:
2599:
1916:{\displaystyle \Delta t=-(1+\gamma ){\frac {R_{s}}{2c}}\ln(1-\mathbf {R} \cdot \mathbf {x} ),}
543:
493:
268:
3719:
3659:
3649:
3596:
3573:
3151:
2926:
2877:
2828:
2789:
2746:
2694:
2684:
2591:
2583:
2522:
2471:
2454:
2417:
2327:
2302:
was far less than the estimated
Shapiro delay of about 1000 days. This rules out a class of
794:
638:
618:
573:
553:
498:
2282:
in 2016, the one-way
Shapiro delay was calculated by two groups and is about 1800 days. In
1395:
1368:
3734:
3709:
3634:
3629:
3512:
3473:
3435:
3379:
3253:
3189:
2628:
2482:
2247:
2243:
673:
648:
533:
528:
392:
273:
235:
3517:
443:
2922:
2873:
2824:
2785:
2742:
2680:
2579:
2518:
2450:
2413:
3759:
3431:
3415:
3411:
3314:
3284:
3141:
3044:
2699:
2652:
2621:
2553:
2368:
2341:
2275:
1616:
1465:
1423:
1348:
876:
716:
683:
678:
366:
230:
3805:
3749:
3729:
3724:
3639:
3507:
3335:
3279:
3111:
3064:
2889:
2758:
2534:
2271:
827:
786:
643:
603:
558:
538:
463:
361:
240:
2938:
1555:{\displaystyle \Delta t=-{\frac {2GM}{c^{3}}}\ln(1-\mathbf {R} \cdot \mathbf {x} ).}
3769:
3689:
3654:
3184:
3146:
2930:
2611:
2225:{\displaystyle T={\frac {s'}{c'}}={\frac {s}{c\left(1-{\tfrac {R_{s}}{r}}\right)}}}
588:
568:
2647:
for an excellent advanced undergraduate level introduction to the
Shapiro effect.
3553:
3522:
3069:
2458:
2363:
2307:
2295:
1634:
1588:
1365:
is the distance of closest approach of the radar wave to the center of the Sun,
478:
448:
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2832:
2793:
2526:
2421:
3764:
3330:
3174:
3169:
2317:
688:
250:
176:
17:
2495:
Sibel Boran; et al. (2018). "GW170817 Falsifies Dark Matter
Emulators".
3161:
2353:
2259:
259:
2840:
2801:
2708:
2603:
2717:
John C. Baez; Emory F. Bunn (2005). "The Meaning of Einstein's Equation".
3704:
3054:
2570:
2299:
2267:
628:
438:
278:
2768:"New Precision Tests of the Einstein Equivalence Principle from Sn1987a"
2689:
2595:
2476:
Tests of General Relativity from observations of planets and spacecraft
2263:
865:, precisely as given by standard gravitational time dilation formulas.
108:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
2750:
2733:
1613:
is the unit vector pointing from the observer to the gravitating mass
3384:
2587:
1744:{\displaystyle \Delta x=-R_{s}\ln(1-\mathbf {R} \cdot \mathbf {x} ),}
2767:
2274:, which imply that those neutrinos were moving at very close to the
2913:
2509:
2864:
2671:
847:
782:
3365:
2552:
van Straten W; Bailes M; Britton M; et al. (12 July 2001).
1754:
which is a fictive extra distance the light has to travel. Here
806:
Throughout this article discussing the time delay, Shapiro uses
2949:
27:
Time delay caused by space-time distortion near massive objects
2398:
Irwin I. Shapiro (1964). "Fourth Test of General Relativity".
2945:
2653:"The Confrontation between General Relativity and Experiment"
2131:{\displaystyle s'={\frac {s}{\sqrt {1-{\tfrac {R_{s}}{r}}}}}}
2713:
A graduate level survey of the solar system tests, and more.
1449:
1275:
2196:
2107:
2045:
2007:{\displaystyle \tau =t{\sqrt {1-{\tfrac {R_{s}}{r}}}}}
1984:
869:
Time delay due to light traveling around a single mass
833:
The first tests, performed in 1966 and 1967 using the
826:
The time delay effect was first predicted in 1964, by
2145:
2082:
2021:
1965:
1932:
1829:
1760:
1684:
1646:
1619:
1597:
1571:
1480:
1446:
1426:
1398:
1371:
1351:
902:
879:
51:
3617:
3482:
3454:
3323:
3272:
3234:
3213:
3202:
3160:
3104:
3088:
3030:
2994:
2983:
2068:{\displaystyle c'=c{\sqrt {1-{\tfrac {R_{s}}{r}}}}}
2620:
2254:Shapiro delay of neutrinos and gravitational waves
2224:
2130:
2067:
2006:
1944:
1915:
1801:
1743:
1667:
1625:
1605:
1579:
1554:
1456:
1432:
1411:
1384:
1357:
1334:
885:
107:
2290:and other modified GR theories, which reproduce
777:effect, is one of the four classic Solar System
1926:which is twice the Newtonian prediction (with
1591:pointing from the observer to the source, and
2961:
2393:
2391:
2389:
2258:From the nearly simultaneous observations of
750:
8:
3210:
2991:
2968:
2954:
2946:
1802:{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}
757:
743:
309:
199:
29:
2912:
2863:
2732:
2698:
2688:
2670:
2569:
2508:
2202:
2195:
2175:
2152:
2144:
2113:
2106:
2094:
2081:
2051:
2044:
2036:
2020:
1990:
1983:
1975:
1964:
1931:
1902:
1894:
1863:
1857:
1828:
1791:
1774:
1765:
1759:
1730:
1722:
1701:
1683:
1645:
1618:
1598:
1596:
1572:
1570:
1541:
1533:
1510:
1493:
1479:
1448:
1447:
1445:
1425:
1403:
1397:
1376:
1370:
1350:
1313:
1301:
1291:
1284:
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1273:
1247:
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1233:
1220:
1215:
1200:
1187:
1177:
1161:
1157:
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1134:
1129:
1115:
1109:
1094:
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1060:
1047:
1042:
1026:
1007:
1003:
993:
980:
975:
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929:
912:
901:
878:
96:
87:
75:
56:
50:
2385:
379:
353:
312:
258:
32:
1675:, this formula can also be written as
1633:. The dot denotes the usual Euclidean
2766:Michael J. Longo (January 18, 1988).
7:
3540:TolmanâOppenheimerâVolkoff equation
3493:FriedmannâLemaĂźtreâRobertsonâWalker
1830:
1685:
1668:{\displaystyle \Delta x=c\Delta t}
1659:
1647:
1481:
903:
68:
25:
3310:HamiltonâJacobiâEinstein equation
2623:Introducing Einstein's Relativity
791:Fourth Test of General Relativity
3788:
3787:
2334:
2320:
2306:that dispense with the need for
2278:. After the direct detection of
1903:
1895:
1731:
1723:
1599:
1573:
1542:
1534:
724:
723:
710:
40:
3117:Massâenergy equivalence (E=mc)
2931:10.1016/j.physletb.2016.03.033
2554:"Boost for General Relativity"
1907:
1885:
1854:
1842:
1735:
1713:
1546:
1524:
1457:{\displaystyle {\mathcal {O}}}
1240:
1208:
1154:
1122:
1067:
1035:
1000:
968:
1:
2658:Living Reviews in Relativity
2288:tensorâvectorâscalar gravity
1606:{\displaystyle \mathbf {x} }
1580:{\displaystyle \mathbf {R} }
3817:Tests of general relativity
3132:Relativistic Doppler effect
2720:American Journal of Physics
2459:10.1103/PhysRevLett.20.1265
859:gravitational time dilation
779:tests of general relativity
221:Gravitational time dilation
3833:
3603:In computational physics:
3127:Relativity of simultaneity
2882:10.1103/PhysRevD.77.124041
2833:10.1103/PhysRevLett.60.176
2794:10.1103/PhysRevLett.60.173
2651:Will, Clifford M. (2014).
2527:10.1103/PhysRevD.97.041501
2422:10.1103/PhysRevLett.13.789
2374:Gravitomagnetic time delay
2304:modified models of gravity
341:MathissonâPapapetrouâDixon
182:Pseudo-Riemannian manifold
3785:
3440:LenseâThirring precession
3022:Doubly special relativity
1945:{\displaystyle \gamma =0}
3300:Post-Newtonian formalism
3290:Einstein field equations
3226:Mathematical formulation
3050:Hyperbolic orthogonality
816:Einstein field equations
775:gravitational time delay
346:HamiltonâJacobiâEinstein
326:Einstein field equations
149:Mathematical formulation
3011:Galilean transformation
3002:Principle of relativity
2812:Physical Review Letters
2773:Physical Review Letters
2619:d'Inverno, Ray (1992).
2438:Physical Review Letters
2401:Physical Review Letters
2294:and avoid the need for
863:gravitational potential
3096:Lorentz transformation
2350:Gravitational redshift
2226:
2132:
2069:
2008:
1946:
1917:
1803:
1745:
1669:
1627:
1607:
1581:
1556:
1458:
1434:
1413:
1386:
1359:
1336:
887:
853:
844:Calculating time delay
838:Haystack radar antenna
812:Schwarzschild solution
804:
216:Gravitational redshift
109:
3564:WeylâLewisâPapapetrou
3305:Raychaudhuri equation
3244:Equivalence principle
2238:Interplanetary probes
2227:
2133:
2070:
2009:
1947:
1918:
1804:
1746:
1670:
1628:
1608:
1582:
1557:
1459:
1435:
1414:
1412:{\displaystyle x_{p}}
1387:
1385:{\displaystyle x_{e}}
1360:
1337:
888:
851:
799:
504:WeylâLewisâPapapetrou
459:KerrâNewmanâde Sitter
279:EinsteinâRosen bridge
211:Gravitational lensing
167:Equivalence principle
110:
3605:Numerical relativity
3446:pulsar timing arrays
2143:
2080:
2019:
1963:
1930:
1827:
1811:Schwarzschild radius
1758:
1682:
1644:
1617:
1595:
1569:
1478:
1444:
1424:
1396:
1369:
1349:
900:
877:
434:EinsteinâRosen waves
160:Fundamental concepts
49:
3497:Friedmann equations
3391:HulseâTaylor binary
3353:Gravitational waves
3249:Riemannian geometry
3075:Proper acceleration
3060:Maxwell's equations
3006:Galilean relativity
2923:2016PhLB..756..265K
2874:2008PhRvD..77l4041D
2825:1988PhRvL..60..176K
2786:1988PhRvL..60..173L
2743:2005AmJPh..73..644B
2690:10.12942/lrr-2014-4
2681:2014LRR....17....4W
2580:2001Natur.412..158V
2519:2018PhRvD..97d1501B
2451:1968PhRvL..20.1265S
2414:1964PhRvL..13..789S
2280:gravitational waves
1468:of order of error.
1225:
1139:
1052:
985:
388:KaluzaâKlein theory
274:Minkowski spacetime
226:Gravitational waves
3812:Effects of gravity
3546:ReissnerâNordström
3464:BransâDicke theory
3295:Linearized gravity
3122:Length contraction
3040:Frame of reference
3017:Special relativity
2481:2012-04-26 at the
2359:Gravitational lens
2284:general relativity
2222:
2212:
2128:
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1125:
1038:
971:
883:
854:
771:Shapiro time delay
717:Physics portal
489:OppenheimerâSnyder
429:ReissnerâNordström
321:Linearized gravity
269:Spacetime diagrams
172:Special relativity
105:
34:General relativity
3799:
3798:
3613:
3612:
3592:OzsvĂĄthâSchĂŒcking
3198:
3197:
3180:Minkowski diagram
3137:Thomas precession
3080:Relativistic mass
2900:Physics Letters B
2851:Physical Review D
2751:10.1119/1.1852541
2638:978-0-19-859686-8
2485:(slides undated).
2445:(22): 1265â1269.
2220:
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2002:
1999:
1877:
1797:
1626:{\displaystyle M}
1516:
1440:, is negligible.
1433:{\displaystyle M}
1358:{\displaystyle d}
1323:
1258:
1172:
1102:
1085:
935:
886:{\displaystyle M}
767:
766:
400:
399:
286:
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16:(Redirected from
3824:
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3574:van Stockum dust
3346:Two-body problem
3264:Mach's principle
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3152:Terrell rotation
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2588:10.1038/35084015
2573:
2571:astro-ph/0108254
2564:(6843): 158â60.
2539:
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2512:
2492:
2486:
2472:Elena V. Pitjeva
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2328:Astronomy portal
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1251:
1238:
1237:
1224:
1219:
1206:
1205:
1204:
1192:
1191:
1178:
1173:
1171:
1170:
1169:
1165:
1152:
1151:
1138:
1133:
1120:
1119:
1110:
1103:
1095:
1090:
1086:
1084:
1083:
1082:
1078:
1065:
1064:
1051:
1046:
1031:
1030:
1017:
1016:
1015:
1011:
998:
997:
984:
979:
964:
963:
953:
936:
934:
933:
924:
913:
892:
890:
889:
884:
759:
752:
745:
732:
727:
726:
719:
715:
714:
499:van Stockum dust
484:RobertsonâWalker
310:
200:
114:
112:
111:
106:
104:
103:
91:
83:
82:
64:
63:
44:
30:
21:
3832:
3831:
3827:
3826:
3825:
3823:
3822:
3821:
3802:
3801:
3800:
3795:
3781:
3609:
3513:BKL singularity
3503:LemaĂźtreâTolman
3478:
3474:Quantum gravity
3456:
3450:
3436:geodetic effect
3410:(together with
3380:LISA Pathfinder
3319:
3268:
3254:Penrose diagram
3236:
3230:
3205:
3194:
3190:Minkowski space
3156:
3100:
3084:
3032:
3026:
2986:
2979:
2974:
2896:
2847:
2808:
2765:
2716:
2650:
2639:
2629:Clarendon Press
2618:
2551:
2548:
2546:Further reading
2543:
2542:
2494:
2493:
2489:
2483:Wayback Machine
2470:
2466:
2434:
2433:
2429:
2408:(26): 789â791.
2397:
2396:
2387:
2382:
2340:
2333:
2326:
2321:
2319:
2316:
2256:
2240:
2198:
2188:
2184:
2180:
2162:
2154:
2141:
2140:
2109:
2083:
2078:
2077:
2047:
2022:
2017:
2016:
1986:
1961:
1960:
1928:
1927:
1869:
1859:
1825:
1824:
1787:
1776:
1761:
1756:
1755:
1697:
1680:
1679:
1642:
1641:
1615:
1614:
1593:
1592:
1567:
1566:
1506:
1495:
1476:
1475:
1442:
1441:
1422:
1421:
1399:
1394:
1393:
1372:
1367:
1366:
1347:
1346:
1309:
1308:
1297:
1287:
1286:
1280:
1239:
1229:
1207:
1196:
1183:
1179:
1153:
1143:
1121:
1111:
1108:
1104:
1066:
1056:
1022:
1018:
999:
989:
955:
954:
948:
941:
937:
925:
914:
898:
897:
875:
874:
871:
846:
824:
763:
722:
709:
708:
701:
700:
524:
523:
514:
513:
469:LemaĂźtreâTolman
414:
413:
402:
401:
393:Quantum gravity
380:Advanced theory
307:
306:
305:
288:
287:
236:Geodetic effect
197:
196:
187:
186:
162:
161:
145:
115:
92:
71:
52:
47:
46:
28:
23:
22:
15:
12:
11:
5:
3830:
3828:
3820:
3819:
3814:
3804:
3803:
3797:
3796:
3786:
3783:
3782:
3780:
3779:
3772:
3767:
3762:
3757:
3752:
3747:
3742:
3737:
3732:
3727:
3722:
3717:
3712:
3707:
3702:
3700:Choquet-Bruhat
3697:
3692:
3687:
3682:
3677:
3672:
3667:
3662:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3621:
3619:
3615:
3614:
3611:
3610:
3608:
3607:
3600:
3599:
3594:
3589:
3582:
3581:
3576:
3571:
3566:
3561:
3552:Axisymmetric:
3549:
3548:
3543:
3537:
3526:
3525:
3520:
3515:
3510:
3505:
3500:
3491:Cosmological:
3488:
3486:
3480:
3479:
3477:
3476:
3471:
3466:
3460:
3458:
3452:
3451:
3449:
3448:
3443:
3432:frame-dragging
3429:
3424:
3419:
3416:Einstein rings
3412:Einstein cross
3405:
3394:
3393:
3388:
3382:
3377:
3372:
3359:
3349:
3348:
3343:
3338:
3333:
3327:
3325:
3321:
3320:
3318:
3317:
3315:Ernst equation
3312:
3307:
3302:
3297:
3292:
3287:
3285:BSSN formalism
3282:
3276:
3274:
3270:
3269:
3267:
3266:
3261:
3256:
3251:
3246:
3240:
3238:
3232:
3231:
3229:
3228:
3223:
3217:
3215:
3208:
3200:
3199:
3196:
3195:
3193:
3192:
3187:
3182:
3177:
3172:
3166:
3164:
3158:
3157:
3155:
3154:
3149:
3144:
3142:Ladder paradox
3139:
3134:
3129:
3124:
3119:
3114:
3108:
3106:
3102:
3101:
3099:
3098:
3092:
3090:
3086:
3085:
3083:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3045:Speed of light
3042:
3036:
3034:
3028:
3027:
3025:
3024:
3019:
3014:
3008:
2998:
2996:
2989:
2981:
2980:
2975:
2973:
2972:
2965:
2958:
2950:
2944:
2943:
2894:
2858:(12): 124041.
2845:
2819:(3): 176â177.
2806:
2780:(3): 173â175.
2763:
2727:(7): 644â652.
2714:
2648:
2637:
2616:
2547:
2544:
2541:
2540:
2487:
2464:
2427:
2384:
2383:
2381:
2378:
2377:
2376:
2371:
2369:VSOP (planets)
2366:
2361:
2356:
2346:
2345:
2342:Physics portal
2331:
2315:
2312:
2276:speed of light
2255:
2252:
2239:
2236:
2233:
2232:
2217:
2210:
2205:
2201:
2194:
2191:
2187:
2183:
2179:
2174:
2168:
2165:
2160:
2157:
2151:
2148:
2138:
2121:
2116:
2112:
2105:
2102:
2098:
2093:
2089:
2086:
2075:
2059:
2054:
2050:
2043:
2040:
2035:
2032:
2028:
2025:
2014:
1998:
1993:
1989:
1982:
1979:
1974:
1971:
1968:
1941:
1938:
1935:
1924:
1923:
1912:
1909:
1905:
1901:
1897:
1893:
1890:
1887:
1884:
1881:
1875:
1872:
1866:
1862:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1818:PPN parameters
1794:
1790:
1785:
1782:
1779:
1773:
1768:
1764:
1752:
1751:
1740:
1737:
1733:
1729:
1725:
1721:
1718:
1715:
1712:
1709:
1704:
1700:
1696:
1693:
1690:
1687:
1664:
1661:
1658:
1655:
1652:
1649:
1622:
1601:
1575:
1563:
1562:
1551:
1548:
1544:
1540:
1536:
1532:
1529:
1526:
1523:
1520:
1513:
1509:
1504:
1501:
1498:
1492:
1489:
1486:
1483:
1451:
1429:
1406:
1402:
1379:
1375:
1354:
1343:
1342:
1331:
1327:
1321:
1316:
1312:
1304:
1300:
1294:
1290:
1283:
1277:
1272:
1268:
1263:
1254:
1250:
1246:
1242:
1236:
1232:
1228:
1223:
1218:
1214:
1210:
1203:
1199:
1195:
1190:
1186:
1182:
1176:
1168:
1164:
1160:
1156:
1150:
1146:
1142:
1137:
1132:
1128:
1124:
1118:
1114:
1107:
1101:
1098:
1093:
1089:
1081:
1077:
1073:
1069:
1063:
1059:
1055:
1050:
1045:
1041:
1037:
1034:
1029:
1025:
1021:
1014:
1010:
1006:
1002:
996:
992:
988:
983:
978:
974:
970:
967:
962:
958:
951:
947:
944:
940:
932:
928:
923:
920:
917:
911:
908:
905:
882:
870:
867:
845:
842:
823:
820:
765:
764:
762:
761:
754:
747:
739:
736:
735:
734:
733:
720:
703:
702:
699:
698:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
525:
521:
520:
519:
516:
515:
512:
511:
506:
501:
496:
491:
486:
481:
476:
471:
466:
461:
456:
451:
446:
441:
436:
431:
426:
415:
409:
408:
407:
404:
403:
398:
397:
396:
395:
390:
382:
381:
377:
376:
375:
374:
372:Post-Newtonian
369:
364:
356:
355:
351:
350:
349:
348:
343:
338:
333:
328:
323:
315:
314:
308:
304:
303:
300:
296:
295:
294:
293:
290:
289:
284:
283:
282:
281:
276:
271:
263:
262:
256:
255:
254:
253:
248:
243:
238:
233:
231:Frame-dragging
228:
223:
218:
213:
208:
206:Kepler problem
198:
194:
193:
192:
189:
188:
185:
184:
179:
174:
169:
163:
159:
158:
157:
154:
153:
152:
151:
146:
144:
143:
138:
133:
127:
125:
117:
116:
102:
99:
95:
90:
86:
81:
78:
74:
70:
67:
62:
59:
55:
45:
37:
36:
26:
24:
18:Shapiro effect
14:
13:
10:
9:
6:
4:
3:
2:
3829:
3818:
3815:
3813:
3810:
3809:
3807:
3794:
3784:
3778:
3777:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3685:Chandrasekhar
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3645:Schwarzschild
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3622:
3620:
3616:
3606:
3602:
3601:
3598:
3595:
3593:
3590:
3588:
3584:
3583:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3559:
3555:
3551:
3550:
3547:
3544:
3541:
3538:
3536:
3532:
3531:Schwarzschild
3528:
3527:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3498:
3494:
3490:
3489:
3487:
3485:
3481:
3475:
3472:
3470:
3467:
3465:
3462:
3461:
3459:
3453:
3447:
3444:
3441:
3437:
3433:
3430:
3428:
3427:Shapiro delay
3425:
3423:
3420:
3417:
3413:
3409:
3406:
3403:
3399:
3396:
3395:
3392:
3389:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3370:collaboration
3367:
3363:
3360:
3358:
3354:
3351:
3350:
3347:
3344:
3342:
3339:
3337:
3336:Event horizon
3334:
3332:
3329:
3328:
3326:
3322:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3280:ADM formalism
3278:
3277:
3275:
3271:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3241:
3239:
3233:
3227:
3224:
3222:
3219:
3218:
3216:
3212:
3209:
3207:
3201:
3191:
3188:
3186:
3185:Biquaternions
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3163:
3159:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3112:Time dilation
3110:
3109:
3107:
3103:
3097:
3094:
3093:
3091:
3087:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3065:Proper length
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3037:
3035:
3029:
3023:
3020:
3018:
3015:
3012:
3009:
3007:
3003:
3000:
2999:
2997:
2993:
2990:
2988:
2982:
2978:
2971:
2966:
2964:
2959:
2957:
2952:
2951:
2948:
2940:
2936:
2932:
2928:
2924:
2920:
2915:
2910:
2906:
2902:
2901:
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2866:
2861:
2857:
2853:
2852:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2813:
2807:
2803:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2774:
2769:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2735:
2734:gr-qc/0103044
2730:
2726:
2722:
2721:
2715:
2710:
2706:
2701:
2696:
2691:
2686:
2682:
2678:
2673:
2668:
2664:
2660:
2659:
2654:
2649:
2646:
2640:
2634:
2630:
2625:
2624:
2617:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2572:
2567:
2563:
2559:
2555:
2550:
2549:
2545:
2536:
2532:
2528:
2524:
2520:
2516:
2511:
2506:
2503:(4): 041501.
2502:
2498:
2491:
2488:
2484:
2480:
2477:
2473:
2468:
2465:
2460:
2456:
2452:
2448:
2444:
2440:
2439:
2431:
2428:
2423:
2419:
2415:
2411:
2407:
2403:
2402:
2394:
2392:
2390:
2386:
2379:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2351:
2348:
2347:
2343:
2337:
2332:
2329:
2318:
2313:
2311:
2309:
2305:
2301:
2297:
2293:
2292:Milgrom's law
2289:
2285:
2281:
2277:
2273:
2272:neutrino mass
2269:
2265:
2261:
2253:
2251:
2249:
2245:
2237:
2235:
2215:
2208:
2203:
2199:
2192:
2189:
2185:
2181:
2177:
2172:
2166:
2163:
2158:
2155:
2149:
2146:
2139:
2119:
2114:
2110:
2103:
2100:
2096:
2091:
2087:
2084:
2076:
2057:
2052:
2048:
2041:
2038:
2033:
2030:
2026:
2023:
2015:
1996:
1991:
1987:
1980:
1977:
1972:
1969:
1966:
1959:
1958:
1957:
1953:
1939:
1936:
1933:
1910:
1899:
1891:
1888:
1882:
1879:
1873:
1870:
1864:
1860:
1851:
1848:
1845:
1839:
1836:
1833:
1823:
1822:
1821:
1819:
1814:
1812:
1792:
1788:
1783:
1780:
1777:
1771:
1766:
1762:
1738:
1727:
1719:
1716:
1710:
1707:
1702:
1698:
1694:
1691:
1688:
1678:
1677:
1676:
1662:
1656:
1653:
1650:
1638:
1636:
1620:
1590:
1549:
1538:
1530:
1527:
1521:
1518:
1511:
1507:
1502:
1499:
1496:
1490:
1487:
1484:
1474:
1473:
1472:
1469:
1467:
1466:Landau symbol
1427:
1404:
1400:
1377:
1373:
1352:
1329:
1325:
1319:
1314:
1310:
1302:
1298:
1292:
1288:
1281:
1270:
1266:
1261:
1252:
1248:
1244:
1234:
1230:
1226:
1221:
1216:
1212:
1201:
1197:
1193:
1188:
1184:
1180:
1174:
1166:
1162:
1158:
1148:
1144:
1140:
1135:
1130:
1126:
1116:
1112:
1105:
1099:
1096:
1091:
1087:
1079:
1075:
1071:
1061:
1057:
1053:
1048:
1043:
1039:
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1027:
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1004:
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986:
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909:
906:
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895:
894:
880:
868:
866:
864:
860:
850:
843:
841:
839:
836:
831:
829:
828:Irwin Shapiro
821:
819:
817:
813:
809:
803:
798:
796:
795:Irwin Shapiro
792:
788:
787:time dilation
784:
780:
776:
772:
760:
755:
753:
748:
746:
741:
740:
738:
737:
731:
721:
718:
713:
707:
706:
705:
704:
697:
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692:
690:
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685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
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642:
640:
637:
635:
634:Chandrasekhar
632:
630:
627:
625:
622:
620:
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615:
612:
610:
607:
605:
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585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
549:Schwarzschild
547:
545:
542:
540:
537:
535:
532:
530:
527:
526:
518:
517:
510:
509:HartleâThorne
507:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
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460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
430:
427:
424:
420:
419:Schwarzschild
417:
416:
412:
406:
405:
394:
391:
389:
386:
385:
384:
383:
378:
373:
370:
368:
365:
363:
360:
359:
358:
357:
352:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
318:
317:
316:
311:
301:
298:
297:
292:
291:
280:
277:
275:
272:
270:
267:
266:
265:
264:
261:
257:
252:
249:
247:
244:
242:
241:Event horizon
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
203:
202:
201:
191:
190:
183:
180:
178:
175:
173:
170:
168:
165:
164:
156:
155:
150:
147:
142:
139:
137:
134:
132:
129:
128:
126:
124:
121:
120:
119:
118:
100:
97:
93:
88:
84:
79:
76:
72:
65:
60:
57:
53:
43:
39:
38:
35:
31:
19:
3775:
3469:KaluzaâKlein
3426:
3221:Introduction
3147:Twin paradox
2904:
2898:
2855:
2849:
2816:
2810:
2777:
2771:
2724:
2718:
2665:(1): 4â107.
2662:
2656:
2645:Section 15.6
2644:
2622:
2561:
2557:
2500:
2497:Phys. Rev. D
2496:
2490:
2467:
2442:
2436:
2430:
2405:
2399:
2257:
2250:spacecraft.
2241:
2234:
1954:
1925:
1815:
1753:
1639:
1564:
1470:
1344:
872:
855:
832:
825:
807:
805:
800:
790:
774:
770:
768:
694:
654:Raychaudhuri
123:Introduction
3558:KerrâNewman
3529:Spherical:
3398:Other tests
3341:Singularity
3273:Formulation
3235:Fundamental
3089:Formulation
3070:Proper time
3031:Fundamental
2907:: 265â267.
2596:1959.3/1820
2364:Proper time
2308:dark matter
2296:dark matter
1635:dot product
1589:unit vector
773:effect, or
669:van Stockum
599:Oppenheimer
454:KerrâNewman
246:Singularity
3806:Categories
3710:Zel'dovich
3618:Scientists
3597:Alcubierre
3404:of Mercury
3402:precession
3331:Black hole
3214:Background
3206:relativity
3175:World line
3170:Light cone
2995:Background
2987:relativity
2977:Relativity
2914:1602.04779
2510:1710.06168
2380:References
522:Scientists
354:Formalisms
302:Formalisms
251:Black hole
177:World line
3680:Robertson
3665:Friedmann
3660:Eddington
3650:de Sitter
3484:Solutions
3362:detectors
3357:astronomy
3324:Phenomena
3259:Geodesics
3162:Spacetime
3105:Phenomena
2890:118785933
2865:0804.3804
2759:119456465
2672:1403.7377
2535:119468128
2354:blueshift
2260:neutrinos
2193:−
2104:−
2042:−
1981:−
1967:τ
1934:γ
1900:⋅
1892:−
1883:
1852:γ
1840:−
1831:Δ
1728:⋅
1720:−
1711:
1695:−
1686:Δ
1660:Δ
1648:Δ
1539:⋅
1531:−
1522:
1491:−
1482:Δ
1092:−
1020:−
946:
910:≈
904:Δ
614:Robertson
579:Friedmann
574:Eddington
564:Nordström
554:de Sitter
411:Solutions
336:Geodesics
331:Friedmann
313:Equations
299:Equations
260:Spacetime
195:Phenomena
101:ν
98:μ
89:κ
80:ν
77:μ
69:Λ
61:ν
58:μ
3793:Category
3670:LemaĂźtre
3635:Einstein
3625:Poincaré
3585:Others:
3569:TaubâNUT
3535:interior
3457:theories
3455:Advanced
3422:redshift
3237:concepts
3055:Rapidity
3033:concepts
2939:54657234
2841:10038467
2802:10038466
2709:28179848
2604:11449265
2479:Archived
2314:See also
2300:GW170817
2268:SN 1987A
2167:′
2159:′
2088:′
2027:′
730:Category
594:LemaĂźtre
559:Reissner
544:Poincaré
529:Einstein
474:TaubâNUT
439:Wormhole
423:interior
136:Timeline
3735:Hawking
3730:Penrose
3715:Novikov
3695:Wheeler
3640:Hilbert
3630:Lorentz
3587:pp-wave
3408:lensing
3204:General
2985:Special
2919:Bibcode
2870:Bibcode
2821:Bibcode
2782:Bibcode
2739:Bibcode
2700:5255900
2677:Bibcode
2612:4363384
2576:Bibcode
2515:Bibcode
2447:Bibcode
2410:Bibcode
2264:photons
2248:Pioneer
2244:Voyager
1809:is the
1587:is the
1464:is the
822:History
814:to the
797:wrote:
649:Hawking
644:Penrose
619:Bardeen
609:Wheeler
539:Hilbert
534:Lorentz
494:pp-wave
131:History
3776:others
3765:Thorne
3755:Misner
3740:Taylor
3725:Geroch
3720:Ehlers
3690:Zwicky
3508:Kasner
2937:
2888:
2839:
2800:
2757:
2707:
2697:
2635:
2610:
2602:
2558:Nature
2533:
1640:Using
1345:where
728:
695:others
689:Thorne
679:Newman
659:Taylor
639:Ehlers
624:Walker
589:Zwicky
464:Kasner
3770:Weiss
3750:Bondi
3745:Hulse
3675:Milne
3579:discs
3523:Milne
3518:Gödel
3375:Virgo
2935:S2CID
2909:arXiv
2886:S2CID
2860:arXiv
2755:S2CID
2729:arXiv
2667:arXiv
2608:S2CID
2566:arXiv
2531:S2CID
2505:arXiv
2266:from
1565:Here
783:Radar
664:Hulse
604:Gödel
584:Milne
479:Milne
444:Gödel
141:Tests
3705:Kerr
3655:Weyl
3554:Kerr
3414:and
3368:and
3366:LIGO
2837:PMID
2798:PMID
2705:PMID
2643:See
2633:ISBN
2600:PMID
2352:and
2262:and
2246:and
769:The
674:Taub
629:Kerr
569:Weyl
449:Kerr
367:BSSN
3760:Yau
3385:GEO
2927:doi
2905:756
2878:doi
2829:doi
2790:doi
2747:doi
2695:PMC
2685:doi
2592:hdl
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2523:doi
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2418:doi
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835:MIT
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2404:.
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