77:
323:
85:
bookkeeper gathers and combines the reports from various places. The numbers in the reports are translated into data in
Schwarzschild coordinates, which provide a systematic means of evaluating and describing the events globally. Thus, the physicist can compare and interpret the data intelligently. He can find meaningful information from these data. The Schwarzschild form of the Schwarzschild metric using Schwarzschild coordinates is given by
34:. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial slices are flat. There is no coordinate singularity at the Schwarzschild radius (event horizon). The outgoing ones are simply the time reverse of ingoing coordinates (the time is the proper time along outgoing particles that reach infinity with zero velocity).
731:
2324:. This is an indication that the Schwarzschild black hole has two horizons, a past horizon, and a future horizon. The Original form of the GP coordinates is regular across the future horizon (where particles fall into when they fall into a black hole) while the alternative negative version is regular across the past horizon (from which particles come out of the black hole if they do so).
4220:
have no meaning. Finally, Einstein came to Paris in early April. On the 5th of April 1922, in a debate at the "Collège de France" with
Painlevé, Becquerel, Brillouin, Cartan, De Donder, Hadamard, Langevin and Nordmann on "the infinite potentials", Einstein, baffled by the non quadratic cross term in the line element, rejected the Painlevé solution.
3485:
3069:
419:
4212:
in the radial than the tangential directions). The "trick" of the
Painlevé proposal was that he no longer stuck to a full quadratic (static) form but instead, allowed a cross time-space product making the metric form no longer static but stationary and no longer direction symmetric but preferentially oriented.
278:
4219:
Painlevé wrote to
Einstein to introduce his solution and invited Einstein to Paris for a debate. In Einstein's reply letter (December 7), he apologized for not being in a position to visit soon and explained why he was not pleased with Painlevé's arguments, emphasising that the coordinates themselves
4211:
Both
Painlevé and Gullstrand used this solution to argue that Einstein's theory was incomplete in that it gave multiple solutions for the gravitational field of a spherical body, and moreover gave different physics (they argued that the lengths of rods could sometimes be longer and sometimes shorter
4092:
At r/M = 30, the black hole has become much bigger, spanning a diametrical angle of ~15 degrees in the sky. The band of secondary images has also grown to 10 degrees. It’s now possible to find faint tertiary images in the band, which are produced by the light rays that have looped around the black
2271:
to be the negative of what we choose above is also called the GP coordinate system. The only change in the metric is that cross term changes sign. This metric is regular for outgoing raindrops—i.e. particles which leave the black hole travelling outward with just escape velocity so that their speed
1822:
the speed increases as the raindrop gets closer to the singularity. Eventually, the speed becomes infinite at the singularity. As shown below the speed is always less than the speed of light. The results may not be correctly predicted by the equation at and very near the singularity since the true
84:
A Schwarzschild observer is a far observer or a bookkeeper. He does not directly make measurements of events that occur in different places. Instead, he is far away from the black hole and the events. Observers local to the events are enlisted to make measurements and send the results to him. The
3638:
1685:
The speed tends to 0 as r approaches the event horizon. The raindrop appears to have slowed as it gets nearer the event horizon and halted at the event horizon as measured by the bookkeeper. Indeed, an observer outside the event horizon would see the raindrop plunge slower and slower. Its image
1139:
4215:
In a second, longer paper (November 14, 1921), Painlevé explains how he derived his solution by directly solving
Einstein's equations for a generic spherically symmetric form of the metric. The result, equation (4) of his paper, depended on two arbitrary functions of the r coordinate yielding a
4207:
Although the publication of
Gullstrand's paper came after Painlevé's, Gullstrand's paper was dated 25 May 1921, whereas Painlevé's publication was a writeup of his presentation before the Academie des Sciences in Paris on 24 October 1921. In this way, Gullstrand's work appears to have priority.
4096:
At r/M = 2, the event horizon, the black hole now occupies a substantial portion of the sky. The rain observer would see an area up to 42 degrees from the radially inward direction that is pitch dark. The band of secondary and tertiary images, rather than increasing, has decreased in size to 5
4109:
At r/M=0.001, the curve of distant star angle versus view angle appears to form a right angle at the 90 degrees view angle. Almost all of the star images are congregated in a narrow ring 90 degrees from the radially inward direction. Between the ring and the radially inward direction is the
2532:
3853: is the impact parameter. Each incoming light ray can be backtraced to a corresponding ray at infinity. The Impact parameter for the incoming light ray is the distance between the corresponding ray at infinity and a ray parallel to it that plunges directly into the black hole.
1686:
infinitely redshifts and never makes it through the event horizon. However, the bookkeeper does not physically measure the speed directly. He translates data relayed by the shell observer into
Schwarzschild values and computes the speed. The result is only an accounting entry.
1462:
4101:
is now quite dominant. The speed of plunging has reached the light speed. The distribution pattern of primary images is changing drastically. The primary images are shifting toward the boundary of the band. The edge near the band is now crowded with stars. Due to
3317:
2908:
1788:. At points very far away from the black hole, the speed is extremely small. As the raindrop plunges toward the black hole, the speed increases. At the event horizon, the speed has the value 1. There is no discontinuity or singularity at the event horizon.
3741:
726:{\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt_{r}^{2}+2\left(1-{\frac {2M}{r}}\right)\,a'dt_{r}dr-\left({\frac {1}{1-{\frac {2M}{r}}}}-\left(1-{\frac {2M}{r}}\right)\,{a'}^{2}\right)dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}
4089:, these obstructed stars are now deflected 5 degrees away from the back. In between these stars and the black hole is a circular band of secondary images of the stars. The duplicate images are instrumental in the identification of the black hole.
3198:
2775:
It can not escape from the event horizon. Instead, it gets stuck at the event horizon. Since light moves faster than all others, matter can only move inward at the event horizon. Everything inside the event horizon is hidden from the outside
1949:
1679:
4198:. Most of the stars and their images caused by multiple orbits of the light around the black hole are squeezed to a narrow band at the 90° viewing angle. The observer sees a magnificent bright ring of stars bisecting the dark sky.
91:
4196:
4085:= 500, the black hole is still very far away. It subtends a diametrical angle of ~ 1 degree in the sky. The stars are not distorted much by the presence of the black hole, except for the stars directly behind it. Due to
3491:
942:
931:
2892:
2221:
3796:
2374:
1771:
2611:
2905:
No object should have speed greater than the speed of light as measured in the same reference frame. Thus, the principle of causality is preserved. Indeed, the speed of raindrop is less than that of light:
1310:
4106:, the primary image of the stars which were originally located behind the rain observer have their images appreciably red-shifted, while those that were in front are blue-shifted and appear very bright.
57:
The derivation of GP coordinates requires defining the following coordinate systems and understanding how data measured for events in one coordinate system is interpreted in another coordinate system.
3480:{\displaystyle \cos {\boldsymbol {\Phi }}_{r}={\frac {dr_{r}}{dt_{r}}}={\frac {{\sqrt {\dfrac {2M}{r}}}+\cos {\boldsymbol {\Phi }}_{s}}{1+{\sqrt {\dfrac {2M}{r}}}\ \cos {\boldsymbol {\Phi }}_{s}}},\,}
2682:
3064:{\displaystyle {\frac {\left({\dfrac {dr}{dt_{r}}}\right)_{\text{raindrop}}}{\left({\dfrac {dr}{dt_{r}}}\right)_{\text{light}}}}={\frac {\sqrt {\dfrac {2M}{r}}}{1+{\sqrt {\dfrac {2M}{r}}}}}<1.}
3302:, is one of the largest known supermassive black holes. With a mass of 3 billion solar masses, it takes about 3 hours for light to travel to the central singularity and 5 hours for a raindrop.
3992:
3267:
3890:
1220:, since, although the temporal term goes to zero, the off-diagonal term in the metric is still non-zero and ensures that the metric is still invertible (the determinant of the metric is
791:
3644:
1505:
2773:
1996:
1270:
2811:
the rain observer measures that the light moves toward the center with speed greater than 2. This is plausible. Even in special relativity, the proper speed of a moving object is
2318:
4139:
3926:
3851:
64:. Time and mass have units in meters. The speed of light in flat spacetime has a value of 1. The gravitational constant has a value of 1. The metric is expressed in the +−−−
2253:
is exactly the proper time along the trajectory. One could have defined the GP coordinates by this requirement, rather than by demanding that the spatial surfaces be flat.
3080:
1838:
1187:
379:
4017:
3823:
1820:
4071:
3955:
2366:
4583:
4544:
3857:
Because of spherical symmetry, the trajectory of light always lies in a plane passing through the center of sphere. It's possible to simplify the metric by assuming
2809:
1826:
Despite the problem with the singularity, it's still possible to compute the travel time for the raindrop from the horizon to the center of black hole mathematically.
4042:
850:
2721:
2251:
2029:
1214:
1570:
1534:
1302:
820:
411:
4093:
hole once already. The primary images are distributed more tightly in the rest of the sky. The pattern of distribution is similar to that previously exhibited.
3073:
The time of travel for light shining inward from event horizon to the center of black hole can be obtained by integrating the equation for the velocity of light,
273:{\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt^{2}-{\frac {dr^{2}}{\left(1-{\frac {2M}{r}}\right)}}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}
1589:
3633:{\displaystyle \cos {\boldsymbol {\Phi }}_{s}={\frac {dr_{s}}{dt_{s}}}=\pm {\sqrt {1-\left(1-{\frac {2M}{r}}\right){\frac {{\mathit {I}}^{2}}{r^{2}}}}},\,\!}
1134:{\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt_{r}^{2}-2{\sqrt {\frac {2M}{r}}}dt_{r}dr-dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}
4144:
2816:
45:
in 1922. It was not explicitly shown that these solutions were simply coordinate transformations of the usual
Schwarzschild solution until 1933 in
2038:
2527:{\displaystyle 0=\left(dr+\left(1+{\sqrt {\frac {2M}{r}}}\ \right)dt_{r}\right)\left(dr-\left(1-{\sqrt {\frac {2M}{r}}}\ \right)\,dt_{r}\right),}
3896:
3311:
How does the universe look like as seen by a rain observer plunging into the black hole? The view can be described by the following equations:
3752:
858:
4234:
1696:
4363:
Tony
Rothman; Richard Matzner; Bill Unruh (1985). "Grand Illusions: Further conversations on the edge of Spacetime". In Tony Rothman (ed.).
2540:
1777:
2003:
Using this result for the speed of the raindrop we can find the proper time along the trajectory of the raindrop in terms of the time
1457:{\displaystyle a(r)=-\int {\frac {\sqrt {\frac {2M}{r}}}{1-{\frac {2M}{r}}}}dr=2M\left(-2y+\ln \left({\frac {y+1}{y-1}}\right)\right)}
4528:
4216:
double infinity of solutions. We now know that these simply represent a variety of choices of both the time and radial coordinates.
4239:
2328:
1216:
is constant) is simply the flat metric expressed in spherical polar coordinates. This metric is regular along the horizon where
2620:
4588:
4276:
Gullstrand, Allvar (1922). "Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie".
4463:
Jean Eisenstaedt (1987). "The Early Interpretation of the Schwarzschild solution". In Don Howqard; John Stachel (eds.).
3960:
1572:
lead to other coordinate charts for the Schwarzschild vacuum; a general treatment is given in Francis & Kosowsky.
3206:
4449:
3860:
3736:{\displaystyle \phi ={\mathit {I}}\int _{r_{0}}^{\infty }{\frac {dr}{r^{2}\ \cos {\boldsymbol {\Phi }}_{s}}},\,\!}
2331:
are regular across both horizons at the expense of making the metric strongly dependent on the time coordinate.
1470:
27:
2726:
1954:
1784:
The speed of the raindrop is inversely proportional to the square root of the radius and equals the negative
1223:
4244:
3284:
2257:
739:
61:
3798:
are the rain observer's and shell observer's viewing angles with respect to the radially outward direction.
76:
2279:
322:
4229:
4116:
3905:
3830:
4351:
There is no published hard copy textbook of EBH2e. Instead, you may freely download the online version
4339:
4321:
Matthew R. Francis and Arthur Kosowsky (2004). "Geodesics in the Generalized Schwarzschild Solution",
2264:
changes as the raindrops fall, this metric is time dependent while the GP metric is time independent.
4593:
4487:
4397:
4304:
3193:{\displaystyle \int _{0}^{T_{r}}\,dt=-\int _{2M}^{0}\left({\sqrt {\frac {2M}{r}}}+1\right)^{-1}\,dr.}
1944:{\displaystyle \int _{0}^{T_{r}}\,dt=-\int _{2M}^{0}\left({\sqrt {\frac {2M}{r}}}\right)^{-1}\,dr.\,}
23:
2260:, in which the "radial" coordinate is chosen to be constant along the paths of the raindrops. Since
4098:
46:
1147:
336:
4538:
4503:
4478:
Jean Eisenstaedt (1982). "Histoire et Singularités de la Solution de Schwarzschild (1915–1923)".
4413:
4387:
4322:
4086:
3997:
3803:
3273:
1794:
1580:
Define a raindrop as an object which plunges radially toward a black hole from rest at infinity.
4047:
3931:
2342:
4524:
2782:
42:
4495:
4405:
4022:
825:
38:
2694:
2229:
2007:
1192:
1785:
1674:{\displaystyle {\frac {dr}{dt}}=-\left(1-{\frac {2M}{r}}\right){\sqrt {\frac {2M}{r}}}.\,}
1546:
1510:
1278:
796:
387:
65:
4431:
4263:
2272:
at infinity is zero. In the usual GP coordinates, such particles cannot be described for
4491:
4401:
4308:
3895:
4103:
3280:
2685:
16:
Coordinates suitable for following a free-falling observer of a Schwarzchild black hole
4563:
4378:
Hamilton, Andrew J.S.; Lisle, Jason P. (June 2008). "The river model of black holes".
4577:
4507:
4417:
4191:{\displaystyle \cos {\boldsymbol {\Phi }}_{r}\rightarrow {\sqrt {\frac {r}{2M}}}\,\!}
3299:
1776:
3291:, with mass of 3.7 million solar masses, the light travel time is about 14 seconds.
4568:
3295:
31:
4430:"La gravitation dans la mécanique de Newton et dans la mécanique d'Einstein"
2887:{\displaystyle {\frac {dr_{ff}}{d\tau }}={\frac {v}{\sqrt {1-v^{2}}}}\geq 1.}
3288:
2216:{\displaystyle d\tau ^{2}=g|_{\text{trajectory}}=dt_{r}^{2}\left=dt_{r}^{2}}
4110:
enormous black hole. On the opposite side, only a few stars shine faintly.
3825:
is the angle between the distant star and the radially outward direction.
3791:{\displaystyle {\boldsymbol {\Phi }}_{r},\ {\boldsymbol {\Phi }}_{s}\,\!}
2723:
the speed of light shining outward away from the center of black hole is
926:{\displaystyle a'=-{\frac {1}{1-{\frac {2M}{r}}}}{\sqrt {\frac {2M}{r}}}}
4262:
Paul Painlevé, "La mécanique classique et la théorie de la relativité",
1766:{\displaystyle {\frac {dr}{dt_{r}}}=-\beta =-{\sqrt {\frac {2M}{r}}}.\,}
4499:
1823:
solution may be quite different when quantum mechanics is incorporated.
4558:
4409:
4392:
4326:
1540:
as it must be to remove that singularity in the Schwarzschild metric.
1583:
In Schwarzschild coordinates, the velocity of a raindrop is given by
2606:{\displaystyle {\frac {dr}{dt_{r}}}=\pm 1-{\sqrt {\frac {2M}{r}}}.}
326:
Free falling worldlines in Gullstrand–Painlevé raindrop coordinates
80:
Free falling worldlines in classic Schwarzschild-Droste coordinates
4340:"Chapter 6: Diving; Exploring Black Holes, Second Edition (EBH2e)"
321:
75:
49:'s paper, although Einstein immediately believed that to be true.
4073:
to infinity. A chart of the sample results is shown at right.
3276:
with a typical size of 3 solar masses is about 11 microseconds.
4019:
of the distant star, is determined by numerically integrating
3911:
3836:
3656:
3600:
2267:
The metric obtained if, in the above, we take the function
3928:
can be computed knowing the rain observer's r-coordinate
2677:{\displaystyle r\to \infty ,{\tfrac {dr}{dt_{r}}}=\pm 1.}
2226:
I.e. along the rain drops trajectory, the elapse of time
1144:
The spatial metric (i.e. the restriction of the metric
2731:
2637:
4147:
4119:
4050:
4025:
4000:
3963:
3934:
3908:
3863:
3833:
3806:
3755:
3647:
3494:
3432:
3384:
3320:
3209:
3083:
3035:
3010:
2964:
2920:
2911:
2819:
2785:
2729:
2697:
2623:
2543:
2377:
2345:
2282:
2232:
2041:
2010:
1957:
1841:
1797:
1699:
1592:
1549:
1513:
1473:
1313:
1281:
1226:
1195:
1150:
945:
861:
828:
799:
742:
422:
390:
339:
94:
4564:
Dr. Andrew J S Hamilton's video "Inside Black Holes"
413:. Substituting in the Schwarzschild metric one gets
4190:
4133:
4065:
4036:
4011:
3986:
3949:
3920:
3884:
3845:
3817:
3790:
3735:
3632:
3479:
3261:
3192:
3063:
2886:
2803:
2767:
2715:
2676:
2605:
2526:
2360:
2312:
2245:
2215:
2023:
1990:
1943:
1814:
1765:
1673:
1564:
1528:
1499:
1456:
1296:
1264:
1208:
1181:
1133:
925:
844:
814:
785:
725:
405:
373:
272:
4187:
4130:
4113:As the rain observer approaches the singularity,
4062:
4033:
4008:
3983:
3946:
3917:
3881:
3842:
3814:
3787:
3732:
3629:
1811:
4367:. Dover Publications (New York). pp. 49–81.
3987:{\displaystyle \ {\boldsymbol {\Phi }}_{r0}\,\!}
60:Convention: The units for the variables are all
4338:Bertschinger, Edmund; Taylor, Edwin F. (2020).
4297:Annales de la Société Scientifique de Bruxelles
4569:Black hole orbit simulation in GP coordinates.
4465:Einstein and the History of General Relativity
4295:G. Lemaître (1933). "L'Univers en expansion".
3262:{\displaystyle T_{r}=4M\ln 2-2M\approx 0.77M.}
3885:{\displaystyle \theta ={\frac {\pi }{2}}\,\!}
3294:The supermassive black hole at the center of
2617:At places very far away from the black hole,
8:
4543:: CS1 maint: multiple names: authors list (
2256:A closely related set of coordinates is the
1690:In GP coordinates, the velocity is given by
22:are a particular set of coordinates for the
4264:C. R. Acad. Sci. (Paris) 173, 677–680(1921)
2900:There are two important points to consider:
37:The solution was proposed independently by
4444:Diana Buchwald; et al., eds. (2009).
4432:C.R Acad. Sci. (Paris) 173, 873-886(1921)
4391:
4186:
4169:
4160:
4155:
4146:
4129:
4118:
4061:
4055:
4049:
4032:
4024:
4007:
3999:
3982:
3973:
3968:
3962:
3945:
3939:
3933:
3916:
3910:
3909:
3907:
3880:
3870:
3862:
3841:
3835:
3834:
3832:
3813:
3805:
3786:
3780:
3775:
3762:
3757:
3754:
3731:
3719:
3714:
3698:
3683:
3677:
3670:
3665:
3655:
3654:
3646:
3628:
3615:
3605:
3599:
3598:
3595:
3575:
3556:
3541:
3526:
3516:
3507:
3502:
3493:
3476:
3464:
3459:
3430:
3416:
3411:
3382:
3379:
3367:
3352:
3342:
3333:
3328:
3319:
3214:
3208:
3180:
3171:
3143:
3131:
3123:
3106:
3098:
3093:
3088:
3082:
3033:
3007:
2996:
2982:
2963:
2952:
2938:
2919:
2912:
2910:
2869:
2853:
2830:
2820:
2818:
2784:
2749:
2730:
2728:
2696:
2655:
2636:
2622:
2583:
2562:
2544:
2542:
2510:
2502:
2477:
2441:
2409:
2376:
2344:
2301:
2283:
2281:
2237:
2231:
2207:
2202:
2181:
2164:
2143:
2126:
2105:
2088:
2083:
2067:
2062:
2049:
2040:
2015:
2009:
1987:
1971:
1962:
1956:
1940:
1930:
1921:
1900:
1889:
1881:
1864:
1856:
1851:
1846:
1840:
1810:
1796:
1762:
1742:
1718:
1700:
1698:
1670:
1650:
1630:
1593:
1591:
1548:
1512:
1500:{\displaystyle y={\sqrt {\frac {r}{2M}}}}
1480:
1472:
1419:
1359:
1335:
1312:
1280:
1256:
1234:
1225:
1200:
1194:
1171:
1160:
1155:
1149:
1130:
1124:
1116:
1104:
1094:
1081:
1073:
1067:
1054:
1032:
1008:
996:
991:
983:
963:
944:
906:
888:
876:
860:
836:
827:
798:
763:
741:
722:
716:
708:
696:
686:
673:
665:
659:
646:
628:
618:
616:
596:
564:
552:
532:
516:
496:
473:
468:
460:
440:
421:
389:
344:
338:
269:
263:
255:
243:
233:
220:
212:
206:
177:
159:
149:
140:
132:
112:
93:
4467:. Birkhauser (Berlin). pp. 222–223.
4278:Arkiv för Matematik, Astronomi och Fysik
3894:
2768:{\displaystyle {\tfrac {dr}{dt_{r}}}=0.}
2688:is 1, the same as in special relativity.
1991:{\displaystyle T_{r}={\frac {4}{3}}M.\,}
1775:
4584:Coordinate charts in general relativity
4446:The Collected papers of Albert Einstein
4255:
1265:{\displaystyle -r^{4}\sin(\theta )^{2}}
4536:
3307:A rain observer's view of the universe
786:{\displaystyle a'(r)={\frac {da}{dr}}}
4480:Archive for History of Exact Sciences
7:
3306:
2313:{\displaystyle {\frac {dr}{dt_{r}}}}
3298:, a giant elliptical galaxy in the
4134:{\displaystyle r\rightarrow 0\,\!}
3678:
3279:Ignoring effects of rotation, for
2630:
2339:Assume radial motion. For light,
1830:Integrate the equation of motion:
307:are the Schwarzschild coordinates,
14:
4235:Eddington–Finkelstein coordinates
3921:{\displaystyle {\mathit {I}}\,\!}
3846:{\displaystyle {\mathit {I}}\,\!}
4519:Misner, Thorne, Wheeler (1973).
4156:
3969:
3776:
3758:
3715:
3503:
3460:
3412:
3329:
330:Define a new time coordinate by
822:such that the term multiplying
20:Gullstrand–Painlevé coordinates
4559:The River Model of Black Holes
4166:
4123:
3287:residing at the center of the
2627:
2063:
1559:
1553:
1523:
1517:
1323:
1317:
1291:
1285:
1253:
1246:
1156:
809:
803:
757:
751:
400:
394:
368:
362:
313:is the mass of the black hole.
1:
2276:. They have a zero value for
4240:Kruskal–Szekeres coordinates
3272:The light travel time for a
2329:Kruskal–Szekeres coordinates
1182:{\displaystyle g|_{t=t_{r}}}
384:for some arbitrary function
374:{\displaystyle t_{r}=t+a(r)}
4523:. W H Freeman and Company.
4380:American Journal of Physics
4365:Frontiers of Modern Physics
4012:{\displaystyle \ \phi \,\!}
3818:{\displaystyle \ \phi \,\!}
1815:{\displaystyle r<2M\,\!}
4610:
4450:Princeton University Press
3994:. Then, the actual angle
2779:Inside the event horizon,
1791:Inside the event horizon,
4066:{\displaystyle r_{0}\,\!}
3950:{\displaystyle r_{0}\,\!}
2361:{\displaystyle d\tau =0.}
1786:newtonian escape velocity
72:Schwarzschild coordinates
2804:{\displaystyle r<2M,}
28:Einstein field equations
3285:supermassive black hole
1536:is clearly singular at
936:and the metric becomes
4192:
4135:
4067:
4038:
4037:{\displaystyle dr\,\!}
4013:
3988:
3951:
3922:
3899:
3886:
3847:
3819:
3792:
3737:
3634:
3481:
3263:
3194:
3065:
2888:
2805:
2769:
2717:
2691:At the event horizon,
2678:
2607:
2528:
2362:
2314:
2247:
2217:
2025:
1992:
1945:
1816:
1780:
1767:
1675:
1566:
1530:
1501:
1458:
1298:
1266:
1210:
1189:on the surface where
1183:
1135:
927:
846:
845:{\displaystyle dr^{2}}
816:
787:
727:
407:
375:
327:
274:
81:
4230:Isotropic coordinates
4193:
4136:
4087:gravitational lensing
4068:
4039:
4014:
3989:
3952:
3923:
3902:The impact parameter
3898:
3887:
3848:
3820:
3793:
3738:
3635:
3482:
3264:
3195:
3066:
2889:
2806:
2770:
2718:
2716:{\displaystyle r=2M,}
2679:
2608:
2529:
2363:
2315:
2248:
2246:{\displaystyle t_{r}}
2218:
2026:
2024:{\displaystyle t_{r}}
1993:
1946:
1817:
1779:
1768:
1676:
1567:
1531:
1502:
1459:
1299:
1267:
1211:
1209:{\displaystyle t_{r}}
1184:
1136:
928:
847:
817:
788:
728:
408:
376:
325:
275:
79:
4589:Lorentzian manifolds
4245:Lemaître coordinates
4145:
4117:
4048:
4023:
3998:
3961:
3932:
3906:
3861:
3831:
3804:
3753:
3645:
3492:
3318:
3207:
3081:
2909:
2817:
2783:
2727:
2695:
2621:
2541:
2375:
2343:
2280:
2258:Lemaître coordinates
2230:
2039:
2008:
1955:
1839:
1795:
1697:
1590:
1565:{\displaystyle a(r)}
1547:
1529:{\displaystyle a(r)}
1511:
1471:
1311:
1297:{\displaystyle a(r)}
1279:
1224:
1193:
1148:
943:
859:
826:
815:{\displaystyle a(r)}
797:
740:
420:
406:{\displaystyle a(r)}
388:
337:
92:
26:– a solution to the
24:Schwarzschild metric
4492:1982AHES...27..157E
4452:. pp. 368–370.
4402:2008AmJPh..76..519H
4309:1933ASSB...53...51L
3682:
3136:
3105:
2212:
2093:
1894:
1863:
1001:
793:. If we now choose
478:
4500:10.1007/BF00348347
4188:
4131:
4063:
4034:
4009:
3984:
3957:and viewing angle
3947:
3918:
3900:
3882:
3843:
3815:
3788:
3733:
3661:
3630:
3477:
3446:
3398:
3274:stellar black hole
3259:
3190:
3119:
3084:
3061:
3049:
3024:
2990:
2946:
2884:
2801:
2765:
2757:
2713:
2674:
2663:
2603:
2524:
2358:
2310:
2243:
2213:
2198:
2079:
2021:
1988:
1941:
1877:
1842:
1812:
1781:
1763:
1671:
1576:Motion of raindrop
1562:
1543:Other choices for
1526:
1497:
1454:
1294:
1262:
1206:
1179:
1131:
987:
923:
842:
812:
783:
723:
464:
403:
371:
328:
270:
82:
30:which describes a
4410:10.1119/1.2830526
4184:
4183:
4099:aberration effect
4003:
3966:
3878:
3809:
3773:
3726:
3706:
3623:
3621:
3588:
3548:
3471:
3451:
3447:
3445:
3399:
3397:
3374:
3158:
3157:
3053:
3050:
3048:
3025:
3023:
3002:
2999:
2989:
2955:
2945:
2876:
2875:
2848:
2756:
2662:
2598:
2597:
2569:
2496:
2492:
2491:
2428:
2424:
2423:
2308:
2179:
2178:
2158:
2157:
2141:
2140:
2118:
2070:
1979:
1915:
1914:
1757:
1756:
1725:
1665:
1664:
1643:
1611:
1495:
1494:
1443:
1375:
1372:
1351:
1350:
1023:
1022:
976:
921:
920:
904:
901:
852:is unity, we get
781:
609:
580:
577:
509:
453:
197:
190:
125:
43:Allvar Gullstrand
4601:
4548:
4542:
4534:
4512:
4511:
4475:
4469:
4468:
4460:
4454:
4453:
4441:
4435:
4428:
4422:
4421:
4395:
4375:
4369:
4368:
4360:
4354:
4353:
4344:
4335:
4329:
4319:
4313:
4312:
4292:
4286:
4285:
4273:
4267:
4260:
4197:
4195:
4194:
4189:
4185:
4182:
4171:
4170:
4165:
4164:
4159:
4140:
4138:
4137:
4132:
4072:
4070:
4069:
4064:
4060:
4059:
4043:
4041:
4040:
4035:
4018:
4016:
4015:
4010:
4001:
3993:
3991:
3990:
3985:
3981:
3980:
3972:
3964:
3956:
3954:
3953:
3948:
3944:
3943:
3927:
3925:
3924:
3919:
3915:
3914:
3891:
3889:
3888:
3883:
3879:
3871:
3852:
3850:
3849:
3844:
3840:
3839:
3824:
3822:
3821:
3816:
3807:
3797:
3795:
3794:
3789:
3785:
3784:
3779:
3771:
3767:
3766:
3761:
3742:
3740:
3739:
3734:
3727:
3725:
3724:
3723:
3718:
3704:
3703:
3702:
3692:
3684:
3681:
3676:
3675:
3674:
3660:
3659:
3639:
3637:
3636:
3631:
3624:
3622:
3620:
3619:
3610:
3609:
3604:
3603:
3596:
3594:
3590:
3589:
3584:
3576:
3557:
3549:
3547:
3546:
3545:
3532:
3531:
3530:
3517:
3512:
3511:
3506:
3486:
3484:
3483:
3478:
3472:
3470:
3469:
3468:
3463:
3449:
3448:
3441:
3433:
3431:
3422:
3421:
3420:
3415:
3400:
3393:
3385:
3383:
3380:
3375:
3373:
3372:
3371:
3358:
3357:
3356:
3343:
3338:
3337:
3332:
3268:
3266:
3265:
3260:
3219:
3218:
3199:
3197:
3196:
3191:
3179:
3178:
3170:
3166:
3159:
3153:
3145:
3144:
3135:
3130:
3104:
3103:
3102:
3092:
3070:
3068:
3067:
3062:
3054:
3052:
3051:
3044:
3036:
3034:
3019:
3011:
3009:
3008:
3003:
3001:
3000:
2997:
2995:
2991:
2988:
2987:
2986:
2973:
2965:
2957:
2956:
2953:
2951:
2947:
2944:
2943:
2942:
2929:
2921:
2913:
2893:
2891:
2890:
2885:
2877:
2874:
2873:
2858:
2854:
2849:
2847:
2839:
2838:
2837:
2821:
2810:
2808:
2807:
2802:
2774:
2772:
2771:
2766:
2758:
2755:
2754:
2753:
2740:
2732:
2722:
2720:
2719:
2714:
2683:
2681:
2680:
2675:
2664:
2661:
2660:
2659:
2646:
2638:
2612:
2610:
2609:
2604:
2599:
2593:
2585:
2584:
2570:
2568:
2567:
2566:
2553:
2545:
2533:
2531:
2530:
2525:
2520:
2516:
2515:
2514:
2501:
2497:
2494:
2493:
2487:
2479:
2478:
2451:
2447:
2446:
2445:
2433:
2429:
2426:
2425:
2419:
2411:
2410:
2367:
2365:
2364:
2359:
2319:
2317:
2316:
2311:
2309:
2307:
2306:
2305:
2292:
2284:
2252:
2250:
2249:
2244:
2242:
2241:
2222:
2220:
2219:
2214:
2211:
2206:
2191:
2187:
2186:
2185:
2180:
2174:
2166:
2165:
2159:
2153:
2145:
2144:
2142:
2136:
2128:
2127:
2119:
2114:
2106:
2092:
2087:
2072:
2071:
2068:
2066:
2054:
2053:
2030:
2028:
2027:
2022:
2020:
2019:
1997:
1995:
1994:
1989:
1980:
1972:
1967:
1966:
1950:
1948:
1947:
1942:
1929:
1928:
1920:
1916:
1910:
1902:
1901:
1893:
1888:
1862:
1861:
1860:
1850:
1821:
1819:
1818:
1813:
1772:
1770:
1769:
1764:
1758:
1752:
1744:
1743:
1726:
1724:
1723:
1722:
1709:
1701:
1680:
1678:
1677:
1672:
1666:
1660:
1652:
1651:
1649:
1645:
1644:
1639:
1631:
1612:
1610:
1602:
1594:
1571:
1569:
1568:
1563:
1535:
1533:
1532:
1527:
1506:
1504:
1503:
1498:
1496:
1493:
1482:
1481:
1463:
1461:
1460:
1455:
1453:
1449:
1448:
1444:
1442:
1431:
1420:
1376:
1374:
1373:
1368:
1360:
1346:
1338:
1337:
1336:
1303:
1301:
1300:
1295:
1271:
1269:
1268:
1263:
1261:
1260:
1239:
1238:
1215:
1213:
1212:
1207:
1205:
1204:
1188:
1186:
1185:
1180:
1178:
1177:
1176:
1175:
1159:
1140:
1138:
1137:
1132:
1129:
1128:
1109:
1108:
1099:
1098:
1086:
1085:
1072:
1071:
1059:
1058:
1037:
1036:
1024:
1018:
1010:
1009:
1000:
995:
982:
978:
977:
972:
964:
932:
930:
929:
924:
922:
916:
908:
907:
905:
903:
902:
897:
889:
877:
869:
851:
849:
848:
843:
841:
840:
821:
819:
818:
813:
792:
790:
789:
784:
782:
780:
772:
764:
750:
732:
730:
729:
724:
721:
720:
701:
700:
691:
690:
678:
677:
664:
663:
651:
650:
638:
634:
633:
632:
627:
626:
615:
611:
610:
605:
597:
581:
579:
578:
573:
565:
553:
537:
536:
524:
515:
511:
510:
505:
497:
477:
472:
459:
455:
454:
449:
441:
412:
410:
409:
404:
380:
378:
377:
372:
349:
348:
279:
277:
276:
271:
268:
267:
248:
247:
238:
237:
225:
224:
211:
210:
198:
196:
192:
191:
186:
178:
165:
164:
163:
150:
145:
144:
131:
127:
126:
121:
113:
4609:
4608:
4604:
4603:
4602:
4600:
4599:
4598:
4574:
4573:
4555:
4535:
4531:
4518:
4515:
4477:
4476:
4472:
4462:
4461:
4457:
4443:
4442:
4438:
4429:
4425:
4377:
4376:
4372:
4362:
4361:
4357:
4342:
4337:
4336:
4332:
4320:
4316:
4294:
4293:
4289:
4275:
4274:
4270:
4261:
4257:
4253:
4226:
4205:
4175:
4154:
4143:
4142:
4115:
4114:
4051:
4046:
4045:
4021:
4020:
3996:
3995:
3967:
3959:
3958:
3935:
3930:
3929:
3904:
3903:
3859:
3858:
3829:
3828:
3802:
3801:
3774:
3756:
3751:
3750:
3713:
3694:
3693:
3685:
3666:
3643:
3642:
3611:
3597:
3577:
3568:
3564:
3537:
3533:
3522:
3518:
3501:
3490:
3489:
3458:
3434:
3423:
3410:
3386:
3381:
3363:
3359:
3348:
3344:
3327:
3316:
3315:
3309:
3210:
3205:
3204:
3146:
3142:
3138:
3137:
3094:
3079:
3078:
3037:
3026:
3012:
2978:
2974:
2966:
2959:
2958:
2934:
2930:
2922:
2915:
2914:
2907:
2906:
2865:
2840:
2826:
2822:
2815:
2814:
2781:
2780:
2745:
2741:
2733:
2725:
2724:
2693:
2692:
2651:
2647:
2639:
2619:
2618:
2586:
2558:
2554:
2546:
2539:
2538:
2506:
2480:
2470:
2466:
2456:
2452:
2437:
2412:
2402:
2398:
2388:
2384:
2373:
2372:
2341:
2340:
2337:
2335:Speeds of light
2297:
2293:
2285:
2278:
2277:
2233:
2228:
2227:
2167:
2163:
2146:
2129:
2107:
2098:
2094:
2061:
2045:
2037:
2036:
2011:
2006:
2005:
1958:
1953:
1952:
1903:
1896:
1895:
1852:
1837:
1836:
1793:
1792:
1745:
1714:
1710:
1702:
1695:
1694:
1653:
1632:
1623:
1619:
1603:
1595:
1588:
1587:
1578:
1545:
1544:
1509:
1508:
1507:. The function
1486:
1469:
1468:
1432:
1421:
1415:
1396:
1392:
1361:
1352:
1339:
1309:
1308:
1277:
1276:
1252:
1230:
1222:
1221:
1196:
1191:
1190:
1167:
1154:
1146:
1145:
1120:
1100:
1090:
1077:
1063:
1050:
1028:
1011:
965:
956:
952:
941:
940:
909:
890:
881:
862:
857:
856:
832:
824:
823:
795:
794:
773:
765:
743:
738:
737:
712:
692:
682:
669:
655:
642:
619:
617:
598:
589:
585:
566:
557:
551:
547:
528:
517:
498:
489:
485:
442:
433:
429:
418:
417:
386:
385:
340:
335:
334:
320:
259:
239:
229:
216:
202:
179:
170:
166:
155:
151:
136:
114:
105:
101:
90:
89:
74:
66:sign convention
55:
17:
12:
11:
5:
4607:
4605:
4597:
4596:
4591:
4586:
4576:
4575:
4572:
4571:
4566:
4561:
4554:
4553:External links
4551:
4550:
4549:
4529:
4514:
4513:
4486:(2): 157–198.
4470:
4455:
4436:
4423:
4386:(6): 519–532.
4370:
4355:
4330:
4314:
4287:
4268:
4254:
4252:
4249:
4248:
4247:
4242:
4237:
4232:
4225:
4222:
4204:
4201:
4200:
4199:
4181:
4178:
4174:
4168:
4163:
4158:
4153:
4150:
4128:
4125:
4122:
4111:
4107:
4104:Doppler effect
4097:degrees. The
4094:
4090:
4058:
4054:
4031:
4028:
4006:
3979:
3976:
3971:
3942:
3938:
3913:
3877:
3874:
3869:
3866:
3855:
3854:
3838:
3826:
3812:
3799:
3783:
3778:
3770:
3765:
3760:
3744:
3743:
3730:
3722:
3717:
3712:
3709:
3701:
3697:
3691:
3688:
3680:
3673:
3669:
3664:
3658:
3653:
3650:
3640:
3627:
3618:
3614:
3608:
3602:
3593:
3587:
3583:
3580:
3574:
3571:
3567:
3563:
3560:
3555:
3552:
3544:
3540:
3536:
3529:
3525:
3521:
3515:
3510:
3505:
3500:
3497:
3487:
3475:
3467:
3462:
3457:
3454:
3444:
3440:
3437:
3429:
3426:
3419:
3414:
3409:
3406:
3403:
3396:
3392:
3389:
3378:
3370:
3366:
3362:
3355:
3351:
3347:
3341:
3336:
3331:
3326:
3323:
3308:
3305:
3304:
3303:
3292:
3281:Sagittarius A*
3277:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3217:
3213:
3203:The result is
3201:
3200:
3189:
3186:
3183:
3177:
3174:
3169:
3165:
3162:
3156:
3152:
3149:
3141:
3134:
3129:
3126:
3122:
3118:
3115:
3112:
3109:
3101:
3097:
3091:
3087:
3075:
3074:
3071:
3060:
3057:
3047:
3043:
3040:
3032:
3029:
3022:
3018:
3015:
3006:
2994:
2985:
2981:
2977:
2972:
2969:
2962:
2950:
2941:
2937:
2933:
2928:
2925:
2918:
2902:
2901:
2897:
2896:
2895:
2894:
2883:
2880:
2872:
2868:
2864:
2861:
2857:
2852:
2846:
2843:
2836:
2833:
2829:
2825:
2800:
2797:
2794:
2791:
2788:
2777:
2764:
2761:
2752:
2748:
2744:
2739:
2736:
2712:
2709:
2706:
2703:
2700:
2689:
2686:speed of light
2673:
2670:
2667:
2658:
2654:
2650:
2645:
2642:
2635:
2632:
2629:
2626:
2614:
2613:
2602:
2596:
2592:
2589:
2582:
2579:
2576:
2573:
2565:
2561:
2557:
2552:
2549:
2535:
2534:
2523:
2519:
2513:
2509:
2505:
2500:
2490:
2486:
2483:
2476:
2473:
2469:
2465:
2462:
2459:
2455:
2450:
2444:
2440:
2436:
2432:
2422:
2418:
2415:
2408:
2405:
2401:
2397:
2394:
2391:
2387:
2383:
2380:
2357:
2354:
2351:
2348:
2336:
2333:
2304:
2300:
2296:
2291:
2288:
2240:
2236:
2224:
2223:
2210:
2205:
2201:
2197:
2194:
2190:
2184:
2177:
2173:
2170:
2162:
2156:
2152:
2149:
2139:
2135:
2132:
2125:
2122:
2117:
2113:
2110:
2104:
2101:
2097:
2091:
2086:
2082:
2078:
2075:
2065:
2060:
2057:
2052:
2048:
2044:
2018:
2014:
2001:
2000:
1999:
1998:
1986:
1983:
1978:
1975:
1970:
1965:
1961:
1951:The result is
1939:
1936:
1933:
1927:
1924:
1919:
1913:
1909:
1906:
1899:
1892:
1887:
1884:
1880:
1876:
1873:
1870:
1867:
1859:
1855:
1849:
1845:
1828:
1827:
1824:
1809:
1806:
1803:
1800:
1789:
1774:
1773:
1761:
1755:
1751:
1748:
1741:
1738:
1735:
1732:
1729:
1721:
1717:
1713:
1708:
1705:
1688:
1687:
1682:
1681:
1669:
1663:
1659:
1656:
1648:
1642:
1638:
1635:
1629:
1626:
1622:
1618:
1615:
1609:
1606:
1601:
1598:
1577:
1574:
1561:
1558:
1555:
1552:
1525:
1522:
1519:
1516:
1492:
1489:
1485:
1479:
1476:
1465:
1464:
1452:
1447:
1441:
1438:
1435:
1430:
1427:
1424:
1418:
1414:
1411:
1408:
1405:
1402:
1399:
1395:
1391:
1388:
1385:
1382:
1379:
1371:
1367:
1364:
1358:
1355:
1349:
1345:
1342:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1293:
1290:
1287:
1284:
1259:
1255:
1251:
1248:
1245:
1242:
1237:
1233:
1229:
1203:
1199:
1174:
1170:
1166:
1163:
1158:
1153:
1142:
1141:
1127:
1123:
1119:
1115:
1112:
1107:
1103:
1097:
1093:
1089:
1084:
1080:
1076:
1070:
1066:
1062:
1057:
1053:
1049:
1046:
1043:
1040:
1035:
1031:
1027:
1021:
1017:
1014:
1007:
1004:
999:
994:
990:
986:
981:
975:
971:
968:
962:
959:
955:
951:
948:
934:
933:
919:
915:
912:
900:
896:
893:
887:
884:
880:
875:
872:
868:
865:
839:
835:
831:
811:
808:
805:
802:
779:
776:
771:
768:
762:
759:
756:
753:
749:
746:
734:
733:
719:
715:
711:
707:
704:
699:
695:
689:
685:
681:
676:
672:
668:
662:
658:
654:
649:
645:
641:
637:
631:
625:
622:
614:
608:
604:
601:
595:
592:
588:
584:
576:
572:
569:
563:
560:
556:
550:
546:
543:
540:
535:
531:
527:
523:
520:
514:
508:
504:
501:
495:
492:
488:
484:
481:
476:
471:
467:
463:
458:
452:
448:
445:
439:
436:
432:
428:
425:
402:
399:
396:
393:
382:
381:
370:
367:
364:
361:
358:
355:
352:
347:
343:
319:
318:GP coordinates
316:
315:
314:
308:
290:
281:
280:
266:
262:
258:
254:
251:
246:
242:
236:
232:
228:
223:
219:
215:
209:
205:
201:
195:
189:
185:
182:
176:
173:
169:
162:
158:
154:
148:
143:
139:
135:
130:
124:
120:
117:
111:
108:
104:
100:
97:
73:
70:
54:
51:
41:in 1921 and
15:
13:
10:
9:
6:
4:
3:
2:
4606:
4595:
4592:
4590:
4587:
4585:
4582:
4581:
4579:
4570:
4567:
4565:
4562:
4560:
4557:
4556:
4552:
4546:
4540:
4532:
4530:0-7167-0344-0
4526:
4522:
4517:
4516:
4509:
4505:
4501:
4497:
4493:
4489:
4485:
4481:
4474:
4471:
4466:
4459:
4456:
4451:
4447:
4440:
4437:
4433:
4427:
4424:
4419:
4415:
4411:
4407:
4403:
4399:
4394:
4393:gr-qc/0411060
4389:
4385:
4381:
4374:
4371:
4366:
4359:
4356:
4352:
4348:
4341:
4334:
4331:
4328:
4327:gr-qc/0311038
4324:
4318:
4315:
4310:
4306:
4302:
4298:
4291:
4288:
4283:
4279:
4272:
4269:
4265:
4259:
4256:
4250:
4246:
4243:
4241:
4238:
4236:
4233:
4231:
4228:
4227:
4223:
4221:
4217:
4213:
4209:
4202:
4179:
4176:
4172:
4161:
4151:
4148:
4126:
4120:
4112:
4108:
4105:
4100:
4095:
4091:
4088:
4084:
4080:
4076:
4075:
4074:
4056:
4052:
4029:
4026:
4004:
3977:
3974:
3940:
3936:
3897:
3893:
3875:
3872:
3867:
3864:
3827:
3810:
3800:
3781:
3768:
3763:
3749:
3748:
3747:
3728:
3720:
3710:
3707:
3699:
3695:
3689:
3686:
3671:
3667:
3662:
3651:
3648:
3641:
3625:
3616:
3612:
3606:
3591:
3585:
3581:
3578:
3572:
3569:
3565:
3561:
3558:
3553:
3550:
3542:
3538:
3534:
3527:
3523:
3519:
3513:
3508:
3498:
3495:
3488:
3473:
3465:
3455:
3452:
3442:
3438:
3435:
3427:
3424:
3417:
3407:
3404:
3401:
3394:
3390:
3387:
3376:
3368:
3364:
3360:
3353:
3349:
3345:
3339:
3334:
3324:
3321:
3314:
3313:
3312:
3301:
3300:Virgo Cluster
3297:
3293:
3290:
3286:
3282:
3278:
3275:
3271:
3270:
3269:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3215:
3211:
3187:
3184:
3181:
3175:
3172:
3167:
3163:
3160:
3154:
3150:
3147:
3139:
3132:
3127:
3124:
3120:
3116:
3113:
3110:
3107:
3099:
3095:
3089:
3085:
3077:
3076:
3072:
3058:
3055:
3045:
3041:
3038:
3030:
3027:
3020:
3016:
3013:
3004:
2992:
2983:
2979:
2975:
2970:
2967:
2960:
2948:
2939:
2935:
2931:
2926:
2923:
2916:
2904:
2903:
2899:
2898:
2881:
2878:
2870:
2866:
2862:
2859:
2855:
2850:
2844:
2841:
2834:
2831:
2827:
2823:
2813:
2812:
2798:
2795:
2792:
2789:
2786:
2778:
2762:
2759:
2750:
2746:
2742:
2737:
2734:
2710:
2707:
2704:
2701:
2698:
2690:
2687:
2671:
2668:
2665:
2656:
2652:
2648:
2643:
2640:
2633:
2624:
2616:
2615:
2600:
2594:
2590:
2587:
2580:
2577:
2574:
2571:
2563:
2559:
2555:
2550:
2547:
2537:
2536:
2521:
2517:
2511:
2507:
2503:
2498:
2488:
2484:
2481:
2474:
2471:
2467:
2463:
2460:
2457:
2453:
2448:
2442:
2438:
2434:
2430:
2420:
2416:
2413:
2406:
2403:
2399:
2395:
2392:
2389:
2385:
2381:
2378:
2371:
2370:
2369:
2355:
2352:
2349:
2346:
2334:
2332:
2330:
2325:
2323:
2302:
2298:
2294:
2289:
2286:
2275:
2270:
2265:
2263:
2259:
2254:
2238:
2234:
2208:
2203:
2199:
2195:
2192:
2188:
2182:
2175:
2171:
2168:
2160:
2154:
2150:
2147:
2137:
2133:
2130:
2123:
2120:
2115:
2111:
2108:
2102:
2099:
2095:
2089:
2084:
2080:
2076:
2073:
2058:
2055:
2050:
2046:
2042:
2035:
2034:
2033:
2031:
2016:
2012:
1984:
1981:
1976:
1973:
1968:
1963:
1959:
1937:
1934:
1931:
1925:
1922:
1917:
1911:
1907:
1904:
1897:
1890:
1885:
1882:
1878:
1874:
1871:
1868:
1865:
1857:
1853:
1847:
1843:
1835:
1834:
1833:
1832:
1831:
1825:
1807:
1804:
1801:
1798:
1790:
1787:
1783:
1782:
1778:
1759:
1753:
1749:
1746:
1739:
1736:
1733:
1730:
1727:
1719:
1715:
1711:
1706:
1703:
1693:
1692:
1691:
1684:
1683:
1667:
1661:
1657:
1654:
1646:
1640:
1636:
1633:
1627:
1624:
1620:
1616:
1613:
1607:
1604:
1599:
1596:
1586:
1585:
1584:
1581:
1575:
1573:
1556:
1550:
1541:
1539:
1520:
1514:
1490:
1487:
1483:
1477:
1474:
1450:
1445:
1439:
1436:
1433:
1428:
1425:
1422:
1416:
1412:
1409:
1406:
1403:
1400:
1397:
1393:
1389:
1386:
1383:
1380:
1377:
1369:
1365:
1362:
1356:
1353:
1347:
1343:
1340:
1332:
1329:
1326:
1320:
1314:
1307:
1306:
1305:
1288:
1282:
1275:The function
1273:
1257:
1249:
1243:
1240:
1235:
1231:
1227:
1219:
1201:
1197:
1172:
1168:
1164:
1161:
1151:
1125:
1121:
1117:
1113:
1110:
1105:
1101:
1095:
1091:
1087:
1082:
1078:
1074:
1068:
1064:
1060:
1055:
1051:
1047:
1044:
1041:
1038:
1033:
1029:
1025:
1019:
1015:
1012:
1005:
1002:
997:
992:
988:
984:
979:
973:
969:
966:
960:
957:
953:
949:
946:
939:
938:
937:
917:
913:
910:
898:
894:
891:
885:
882:
878:
873:
870:
866:
863:
855:
854:
853:
837:
833:
829:
806:
800:
777:
774:
769:
766:
760:
754:
747:
744:
717:
713:
709:
705:
702:
697:
693:
687:
683:
679:
674:
670:
666:
660:
656:
652:
647:
643:
639:
635:
629:
623:
620:
612:
606:
602:
599:
593:
590:
586:
582:
574:
570:
567:
561:
558:
554:
548:
544:
541:
538:
533:
529:
525:
521:
518:
512:
506:
502:
499:
493:
490:
486:
482:
479:
474:
469:
465:
461:
456:
450:
446:
443:
437:
434:
430:
426:
423:
416:
415:
414:
397:
391:
365:
359:
356:
353:
350:
345:
341:
333:
332:
331:
324:
317:
312:
309:
306:
302:
298:
294:
291:
289:
286:
285:
284:
264:
260:
256:
252:
249:
244:
240:
234:
230:
226:
221:
217:
213:
207:
203:
199:
193:
187:
183:
180:
174:
171:
167:
160:
156:
152:
146:
141:
137:
133:
128:
122:
118:
115:
109:
106:
102:
98:
95:
88:
87:
86:
78:
71:
69:
67:
63:
58:
52:
50:
48:
44:
40:
39:Paul Painlevé
35:
33:
29:
25:
21:
4520:
4483:
4479:
4473:
4464:
4458:
4445:
4439:
4426:
4383:
4379:
4373:
4364:
4358:
4350:
4347:eftaylor.com
4346:
4333:
4317:
4300:
4296:
4290:
4281:
4277:
4271:
4258:
4218:
4214:
4210:
4206:
4082:
4078:
3901:
3856:
3745:
3310:
3202:
2338:
2326:
2321:
2273:
2268:
2266:
2261:
2255:
2225:
2004:
2002:
1829:
1689:
1582:
1579:
1542:
1537:
1466:
1304:is given by
1274:
1217:
1143:
935:
735:
383:
329:
310:
304:
300:
296:
292:
287:
282:
83:
59:
56:
36:
19:
18:
4594:Black holes
4521:Gravitation
2368:Therefore,
62:geometrized
4578:Categories
4284:(8): 1–15.
4251:References
3296:Messier 87
2069:trajectory
2032:. We have
53:Derivation
32:black hole
4539:cite book
4508:116541975
4418:119467298
4303:: 51–85.
4167:→
4157:Φ
4152:
4124:→
4005:ϕ
3970:Φ
3873:π
3865:θ
3811:ϕ
3777:Φ
3759:Φ
3716:Φ
3711:
3679:∞
3663:∫
3649:ϕ
3573:−
3562:−
3554:±
3504:Φ
3499:
3461:Φ
3456:
3413:Φ
3408:
3330:Φ
3325:
3289:Milky Way
3248:≈
3239:−
3233:
3173:−
3121:∫
3117:−
3086:∫
2879:≥
2863:−
2845:τ
2669:±
2631:∞
2628:→
2581:−
2575:±
2475:−
2464:−
2350:τ
2161:−
2103:−
2047:τ
1923:−
1879:∫
1875:−
1844:∫
1740:−
1734:β
1731:−
1628:−
1617:−
1437:−
1413:
1398:−
1357:−
1333:∫
1330:−
1250:θ
1244:
1228:−
1122:ϕ
1114:θ
1111:
1088:−
1079:θ
1061:−
1045:−
1003:−
961:−
886:−
874:−
714:ϕ
706:θ
703:
680:−
671:θ
653:−
594:−
583:−
562:−
545:−
494:−
438:−
261:ϕ
253:θ
250:
227:−
218:θ
200:−
175:−
147:−
110:−
4224:See also
2954:raindrop
867:′
748:′
624:′
522:′
47:Lemaître
4488:Bibcode
4398:Bibcode
4305:Bibcode
4203:History
3746:where
2274:r<2M
4527:
4506:
4416:
4141:, and
4002:
3965:
3808:
3772:
3705:
3450:
3283:, the
2776:world.
2495:
2427:
1467:where
736:where
305:φ
301:θ
283:where
4504:S2CID
4414:S2CID
4388:arXiv
4343:(PDF)
4323:arXiv
4044:from
2998:light
288:G=1=c
4545:link
4525:ISBN
3892:.
3251:0.77
3056:<
2790:<
2684:The
2327:The
2322:r=2M
2269:f(r)
1802:<
1538:r=2M
1218:r=2M
4496:doi
4406:doi
4301:A53
4149:cos
4077:At
3708:cos
3496:cos
3453:cos
3405:cos
3322:cos
2320:at
1272:).
1241:sin
1102:sin
694:sin
241:sin
4580::
4541:}}
4537:{{
4502:.
4494:.
4484:27
4482:.
4448:.
4412:.
4404:.
4396:.
4384:76
4382:.
4349:.
4345:.
4299:.
4282:16
4280:.
3230:ln
3059:1.
2882:1.
2763:0.
2672:1.
2356:0.
1410:ln
303:,
299:,
295:,
68:.
4547:)
4533:.
4510:.
4498::
4490::
4434:.
4420:.
4408::
4400::
4390::
4325::
4311:.
4307::
4266:.
4180:M
4177:2
4173:r
4162:r
4127:0
4121:r
4083:M
4081:/
4079:r
4057:0
4053:r
4030:r
4027:d
3978:0
3975:r
3941:0
3937:r
3912:I
3876:2
3868:=
3837:I
3782:s
3769:,
3764:r
3729:,
3721:s
3700:2
3696:r
3690:r
3687:d
3672:0
3668:r
3657:I
3652:=
3626:,
3617:2
3613:r
3607:2
3601:I
3592:)
3586:r
3582:M
3579:2
3570:1
3566:(
3559:1
3551:=
3543:s
3539:t
3535:d
3528:s
3524:r
3520:d
3514:=
3509:s
3474:,
3466:s
3443:r
3439:M
3436:2
3428:+
3425:1
3418:s
3402:+
3395:r
3391:M
3388:2
3377:=
3369:r
3365:t
3361:d
3354:r
3350:r
3346:d
3340:=
3335:r
3257:.
3254:M
3245:M
3242:2
3236:2
3227:M
3224:4
3221:=
3216:r
3212:T
3188:.
3185:r
3182:d
3176:1
3168:)
3164:1
3161:+
3155:r
3151:M
3148:2
3140:(
3133:0
3128:M
3125:2
3114:=
3111:t
3108:d
3100:r
3096:T
3090:0
3046:r
3042:M
3039:2
3031:+
3028:1
3021:r
3017:M
3014:2
3005:=
2993:)
2984:r
2980:t
2976:d
2971:r
2968:d
2961:(
2949:)
2940:r
2936:t
2932:d
2927:r
2924:d
2917:(
2871:2
2867:v
2860:1
2856:v
2851:=
2842:d
2835:f
2832:f
2828:r
2824:d
2799:,
2796:M
2793:2
2787:r
2760:=
2751:r
2747:t
2743:d
2738:r
2735:d
2711:,
2708:M
2705:2
2702:=
2699:r
2666:=
2657:r
2653:t
2649:d
2644:r
2641:d
2634:,
2625:r
2601:.
2595:r
2591:M
2588:2
2578:1
2572:=
2564:r
2560:t
2556:d
2551:r
2548:d
2522:,
2518:)
2512:r
2508:t
2504:d
2499:)
2489:r
2485:M
2482:2
2472:1
2468:(
2461:r
2458:d
2454:(
2449:)
2443:r
2439:t
2435:d
2431:)
2421:r
2417:M
2414:2
2407:+
2404:1
2400:(
2396:+
2393:r
2390:d
2386:(
2382:=
2379:0
2353:=
2347:d
2303:r
2299:t
2295:d
2290:r
2287:d
2262:r
2239:r
2235:t
2209:2
2204:r
2200:t
2196:d
2193:=
2189:]
2183:2
2176:r
2172:M
2169:2
2155:r
2151:M
2148:2
2138:r
2134:M
2131:2
2124:2
2121:+
2116:r
2112:M
2109:2
2100:1
2096:[
2090:2
2085:r
2081:t
2077:d
2074:=
2064:|
2059:g
2056:=
2051:2
2043:d
2017:r
2013:t
1985:.
1982:M
1977:3
1974:4
1969:=
1964:r
1960:T
1938:.
1935:r
1932:d
1926:1
1918:)
1912:r
1908:M
1905:2
1898:(
1891:0
1886:M
1883:2
1872:=
1869:t
1866:d
1858:r
1854:T
1848:0
1808:M
1805:2
1799:r
1760:.
1754:r
1750:M
1747:2
1737:=
1728:=
1720:r
1716:t
1712:d
1707:r
1704:d
1668:.
1662:r
1658:M
1655:2
1647:)
1641:r
1637:M
1634:2
1625:1
1621:(
1614:=
1608:t
1605:d
1600:r
1597:d
1560:)
1557:r
1554:(
1551:a
1524:)
1521:r
1518:(
1515:a
1491:M
1488:2
1484:r
1478:=
1475:y
1451:)
1446:)
1440:1
1434:y
1429:1
1426:+
1423:y
1417:(
1407:+
1404:y
1401:2
1394:(
1390:M
1387:2
1384:=
1381:r
1378:d
1370:r
1366:M
1363:2
1354:1
1348:r
1344:M
1341:2
1327:=
1324:)
1321:r
1318:(
1315:a
1292:)
1289:r
1286:(
1283:a
1258:2
1254:)
1247:(
1236:4
1232:r
1202:r
1198:t
1173:r
1169:t
1165:=
1162:t
1157:|
1152:g
1126:2
1118:d
1106:2
1096:2
1092:r
1083:2
1075:d
1069:2
1065:r
1056:2
1052:r
1048:d
1042:r
1039:d
1034:r
1030:t
1026:d
1020:r
1016:M
1013:2
1006:2
998:2
993:r
989:t
985:d
980:)
974:r
970:M
967:2
958:1
954:(
950:=
947:g
918:r
914:M
911:2
899:r
895:M
892:2
883:1
879:1
871:=
864:a
838:2
834:r
830:d
810:)
807:r
804:(
801:a
778:r
775:d
770:a
767:d
761:=
758:)
755:r
752:(
745:a
718:2
710:d
698:2
688:2
684:r
675:2
667:d
661:2
657:r
648:2
644:r
640:d
636:)
630:2
621:a
613:)
607:r
603:M
600:2
591:1
587:(
575:r
571:M
568:2
559:1
555:1
549:(
542:r
539:d
534:r
530:t
526:d
519:a
513:)
507:r
503:M
500:2
491:1
487:(
483:2
480:+
475:2
470:r
466:t
462:d
457:)
451:r
447:M
444:2
435:1
431:(
427:=
424:g
401:)
398:r
395:(
392:a
369:)
366:r
363:(
360:a
357:+
354:t
351:=
346:r
342:t
311:M
297:r
293:t
265:2
257:d
245:2
235:2
231:r
222:2
214:d
208:2
204:r
194:)
188:r
184:M
181:2
172:1
168:(
161:2
157:r
153:d
142:2
138:t
134:d
129:)
123:r
119:M
116:2
107:1
103:(
99:=
96:g
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