821:
2615:, oversimplify this concept to a one-dimensional timeline to fit a linear structure, which does not fit models of reality. Such time machines are often portrayed as being instantaneous, with its contents departing one time and arriving in anotherâbut at the same literal geographic point in space. This is often carried out without note of a reference frame, or with the implicit assumption that the reference frame is local; as such, this would require either accurate teleportation, as a rotating planet, being under acceleration, is not an inertial frame, or for the time machine to remain in the same place, its contents 'frozen'.
2593:. It was advertised to be a "Mathematical Fantasy"âand it was! The substance of the exercise was as follows: He postulated that, commencing with his birth, every human being had some kind of spiritual aura with a long filament or thread attached, that traveled behind him throughout his life. He then proceeded in imagination to describe the complicated entanglement every individual became involved in his relationship to other individuals, comparing the simple entanglements of youth to those complicated knots that develop in later life.
151:
4191:
834:
2607:âBilly Pilgrim says that the Universe does not look like a lot of bright little dots to the creatures from Tralfamadore. The creatures can see where each star has been and where it is going, so that the heavens are filled with rarefied, luminous spaghetti. And Tralfamadorians don't see human beings as two-legged creatures, either. They see them as great millepedes - "with babies' legs at one end and old people's legs at the other," says Billy Pilgrim.â
43:
2007:
1108:
2648:
behind you more of this space-time event, reaching to perhaps nineteen-sixteen, of which we see a cross-section here at right angles to the time axis, and as thick as the present. At the far end is a baby, smelling of sour milk and drooling its breakfast on its bib. At the other end lies, perhaps, an old man someplace in the nineteen-eighties.
1862:
2647:
He stepped up to one of the reporters. "Suppose we take you as an example. Your name is Rogers, is it not? Very well, Rogers, you are a space-time event having duration four ways. You are not quite six feet tall, you are about twenty inches wide and perhaps ten inches thick. In time, there stretches
979:
events corresponding to the history of the object. A world line is a special type of curve in spacetime. Below an equivalent definition will be explained: A world line is either a time-like or a null curve in spacetime. Each point of a world line is an event that can be labeled with the time and the
2548:
Quantum field theory, the framework in which all of modern particle physics is described, is usually described as a theory of quantized fields. However, although not widely appreciated, it has been known since
Feynman that many quantum field theories may equivalently be described in terms of world
1321:
A line at constant space coordinate (a vertical line using the convention adopted above) may represent a particle at rest (or a stationary observer). A tilted line represents a particle with a constant coordinate speed (constant change in space coordinate with increasing time coordinate). The more
2095:
are inaccessible to them; only points in the past can send signals to the observer. In ordinary laboratory experience, using common units and methods of measurement, it may seem that we look at the present, but in fact there is always a delay time for light to propagate. For example, we see the
2123:
to their world line. It is really three-dimensional, though it would be a 2-plane in the diagram because we had to throw away one dimension to make an intelligible picture. Although the light cones are the same for all observers at a given spacetime event, different observers, with differing
1993:
Often the time units are chosen such that the speed of light is represented by lines at a fixed angle, usually at 45 degrees, forming a cone with the vertical (time) axis. In general, useful curves in spacetime can be of three types (the other types would be partly one, partly another type):
1019:(a topological space that locally resembles Euclidean space near each point). The concept may be applied as well to a higher-dimensional space. For easy visualizations of four dimensions, two space coordinates are often suppressed. An event is then represented by a point in a
1643:
2001:
curves, having at each point the speed of light. They form a cone in spacetime, dividing it into two parts. The cone is three-dimensional in spacetime, appears as a line in drawings with two dimensions suppressed, and as a cone in drawings with one spatial dimension
1333:
World lines of a particle and an observer may be interconnected with the world line of a photon (the path of light) and form a diagram depicting the emission of a photon by a particle that is subsequently observed by the observer (or absorbed by another particle).
1135:
can be represented by the coordinates as a function of one parameter. Each value of the parameter corresponds to a point in spacetime and varying the parameter traces out a line. So in mathematical terms a curve is defined by four coordinate functions
1922:
All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. Therefore, all tangent vectors for a point p span a
1091:
is the analogous two-dimensional surface traced out by a one-dimensional line (like a string) traveling through spacetime. The world sheet of an open string (with loose ends) is a strip; that of a closed string (a loop) resembles a tube.
2454:
then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve the movement of information by light. For instance, the traditional electro-static force described by
987:
of the Earth in space is approximately a circle, a three-dimensional (closed) curve in space: the Earth returns every year to the same point in space relative to the sun. However, it arrives there at a different (later) time. The
2022:
The momentarily co-moving inertial frames along the trajectory ("world line") of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the
1317:
A curve that consists of a horizontal line segment (a line at constant coordinate time), may represent a rod in spacetime and would not be a world line in the proper sense. The parameter simply traces the length of the rod.
1329:
or "encounter". Two world lines starting at the same event in spacetime, each following its own path afterwards, may represent e.g. the decay of a particle into two others or the emission of one particle by another.
1014:
identifying the events. Each event can be labeled by four numbers: a time coordinate and three space coordinates; thus spacetime is a four-dimensional space. The mathematical term for spacetime is a four-dimensional
2528:
whose "time axis" corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis. See also for example
222:
2515:
curves fall within the lightcone. However, a lightcone is not necessarily inclined at 45 degrees to the time axis. However, this is an artifact of the chosen coordinate system, and reflects the coordinate freedom
2124:
velocities but coincident at the event (point) in the spacetime, have world lines that cross each other at an angle determined by their relative velocities, and thus they have different simultaneous hyperplanes.
2452:
2019:
2045:
curves falling outside the light cone. Such curves may describe, for example, the length of a physical object. The circumference of a cylinder and the length of a rod are space-like curves.
1939:
So far a world line (and the concept of tangent vectors) has been described without a means of quantifying the interval between events. The basic mathematics is as follows: The theory of
1931:
at point p. For example, taking a 2-dimensional space, like the (curved) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.
2573:
Why, then, should not the four-dimensional beings be ourselves, and our successive states the passing of them through the three-dimensional space to which our consciousness is confined.
1407:
1201:
1857:{\displaystyle {\vec {v}}=\left(v^{0},v^{1},v^{2},v^{3}\right)=\left({\frac {dx^{0}}{d\tau }}\;,{\frac {dx^{1}}{d\tau }}\;,{\frac {dx^{2}}{d\tau }}\;,{\frac {dx^{3}}{d\tau }}\right)}
2225:
1510:
2337:
2293:
1302:
2182:
1539:
1917:
2014:, the three-dimensional surface of all possible light rays arriving at and departing from a point in spacetime. Here, it is depicted with one spatial dimension suppressed.
1562:
2260:
1638:
1474:
1429:
and can simply be differentiated by the usual calculus. Without the existence of a metric (this is important to realize) one can imagine the difference between a point
2079:
at the given event is formed by all events that can be connected through light rays with the event. When we observe the sky at night, we basically see only the past
3894:
1427:
1291:
1248:
1228:
592:
2653:"Imagine this space-time event that we call Rogers as a long pink worm, continuous through the years, one end in his mother's womb, and the other at the grave..."
1884:
1606:
1586:
1447:
1061:
1041:
2720:
A space armada trying to complete a (nearly) closed time-like path as a strategic maneuver forms the backdrop and a main plot device of "Singularity Sky" by
865:
2027:
of the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates.
2619:
937:
244:
2035:
curves, with a speed less than the speed of light. These curves must fall within a cone defined by light-like curves. In our definition above:
2964:
157:
2553:
has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields.
449:
3941:
4176:
2068:
of the given event is formed by all events that can influence the event (that is, that can be connected by world lines within the past
802:
3965:
3711:
612:
454:
1250:. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.
858:
567:
2394:
3747:
3701:
480:
314:
3885:
3239:
519:
3369:
2061:
of the given event is formed by all events that can be reached through time-like curves lying within the future light cone.
2550:
851:
107:
3342:
975:
A world line of an object (generally approximated as a point in space, e.g., a particle or observer) is the sequence of
3622:
231:
79:
2536:
World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are called
1608:. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed
3993:
3627:
3214:
2801:
2530:
257:
126:
3947:
820:
537:
150:
86:
3660:
444:
3763:
3198:
64:
1345:
1139:
4213:
4194:
3936:
3841:
3771:
531:
239:
3904:
2187:
1479:
577:
93:
4218:
3758:
597:
2298:
3799:
3533:
3518:
3206:
3176:
1952:
838:
329:
249:
60:
2265:
4086:
3528:
3481:
2360:
2145:
1515:
742:
290:
75:
4116:
3742:
3423:
2517:
2459:
may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve
1889:
542:
354:
17:
2706:
involves a long discussion of worldlines over dinner in the midst of a philosophical debate between
4223:
4141:
3691:
3451:
3362:
2659:
2590:
2492:
777:
767:
617:
434:
3011:
Schubert, Christian (2001). "Perturbative quantum field theory in the string-inspired formalism".
1544:
3870:
3792:
3412:
3403:
2956:
2905:
2344:
933:
762:
496:
53:
3111:
2230:
1614:
4096:
3959:
3865:
3823:
3497:
3461:
2549:
lines. This preceded much of his work on the formulation which later became more standard. The
1452:
732:
717:
562:
324:
2821:
4101:
3706:
3645:
3315:
2785:
2741:
2582:
2376:
2120:
1095:
Once the object is not approximated as a mere point but has extended volume, it traces not a
707:
275:
2108:
is thick; it is not a 3-dimensional volume but is instead a 4-dimensional spacetime region.
4006:
3932:
3776:
3378:
3138:
3084:
3032:
2983:
2914:
2874:
2669:
1412:
1276:
1233:
1206:
527:
2746:
1079:
of the particle. If the worldline M is a line segment, then the particle is said to be in
996:
in spacetime (a curve in a four-dimensional space) and does not return to the same point.
8:
4081:
4076:
4066:
3898:
3847:
3665:
3650:
3476:
3407:
3355:
2857:
2599:
2495:
and are dependent on the mass-energy distribution in spacetime. Again the metric defines
1120:
1112:
1004:
902:
722:
692:
687:
439:
4071:
3142:
3088:
3036:
2987:
2918:
2878:
1943:
puts some constraints on possible world lines. In special relativity the description of
702:
672:
100:
4146:
4046:
3970:
3828:
3809:
3803:
3754:
3696:
3605:
3523:
3441:
3418:
3386:
3307:
3154:
3128:
3070:
3048:
3022:
2973:
2640:
2566:
2476:
2472:
2460:
1940:
1869:
1591:
1571:
1432:
1305:
Three different world lines representing travel at different constant four-velocities.
1046:
1026:
1000:
964:
960:
941:
928:, and typically encompasses a large area of spacetime wherein paths which are straight
772:
657:
582:
429:
334:
319:
280:
142:
3255:
3044:
2611:
Almost all science-fiction stories which use this concept actively, such as to enable
4156:
3988:
3980:
3581:
3538:
3298:
3235:
3210:
3184:
3158:
3119:
3096:
3075:
3052:
2930:
2797:
2689:
2688:, tells a story based on the shifting of world lines. Steins;Gate is a part of the "
1020:
1011:
956:
602:
377:
4026:
652:
4061:
4051:
3998:
3975:
3553:
3146:
3092:
3040:
2991:
2922:
2882:
2835:
2707:
2456:
1071:
if its tangent is future timelike at each point. The arclength parameter is called
727:
682:
662:
607:
4121:
2996:
2585:
in the early days of relativity. As described by
Toronto lawyer Norman Robertson:
2018:
747:
4136:
4111:
4036:
4031:
3914:
3875:
3837:
3781:
3655:
3591:
3013:
2865:
2853:
2826:
2817:
2759:
2699:
2623:
2050:
1987:
1008:
887:
782:
757:
642:
637:
501:
382:
344:
3919:
2926:
2903:; Kosower, David A. (1991). "Efficient calculation of one-loop QCD amplitudes".
2717:
Absolute Choice depicts different world lines as a sub-plot and setting device.
1971:, but since distinct events sometimes result in a zero value, unlike metrics in
1951:
coordinate systems that do not accelerate (and so do not rotate either), termed
552:
4161:
3833:
3817:
3813:
3716:
3686:
3543:
3446:
3150:
2952:
2822:"Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction"
2793:
2764:
2752:
2721:
1956:
959:. The term is now used most often in the context of relativity theories (i.e.,
825:
792:
787:
475:
339:
2786:"Special Relativity" section of chapter "Euclidean / Lorentzian Vector Spaces"
2565:
wrote an essay "What is the fourth dimension ?", which he published as a
4207:
4151:
4131:
4126:
4041:
3909:
3737:
3681:
3513:
3466:
2488:
1960:
1928:
1609:
1116:
752:
667:
647:
572:
470:
349:
712:
4171:
4091:
4056:
3586:
3548:
3172:
2934:
2886:
2839:
2677:
2630:
in which he related a simplified explanation of the hypothesis for laymen.
2578:
2562:
2525:
1972:
1924:
697:
677:
31:
3328:
3277:
3955:
3924:
3471:
3066:
2681:
2664:
2612:
1964:
1270:
1261:
curve in spacetime. This terminology causes confusions. More properly, a
1088:
1072:
945:
908:
The concept of a "world line" is distinguished from concepts such as an "
587:
557:
30:"Worldline" redirects here. For the French payment services company, see
1325:
Two world lines that start out separately and then intersect, signify a
4166:
3732:
3571:
3133:
3027:
2978:
2711:
2080:
2069:
2011:
929:
913:
797:
359:
217:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
3073:(1982). "Pair production at strong coupling in weak external fields".
3563:
2858:"An operator calculus having applications in quantum electrodynamics"
2693:
2635:
2589:
I remember lecturing at one of the
Saturday evening lectures at the
2500:
2496:
2484:
2480:
2100:
as it was about 8 minutes ago, not as it is "right now". Unlike the
2024:
1944:
1080:
976:
894:
891:
368:
1409:
defining a world line, are real number functions of a real variable
42:
4106:
3456:
2948:
2900:
2736:
2537:
2521:
2512:
2508:
2504:
2091:
is the region between the two light cones. Points in an observer's
1983:
1111:
World line, worldsheet, and world volume, as they are derived from
1087:
A world line traces out the path of a single point in spacetime. A
1016:
737:
547:
387:
3203:
The
Mathematics Department in the University of Toronto, 1827â1978
2749:, curves that represent a variety of different types of world line
3112:"Worldline instantons and pair production in inhomogenous fields"
2703:
1967:
for each pair of events. The bilinear form is sometimes termed a
1023:, which is a plane usually plotted with the time coordinate, say
952:
898:
1337:
1230:
usually denotes the time coordinate) depending on one parameter
3786:
1107:
1102:
2006:
1301:
1269:
of a particle, observer or small object. One usually uses the
3347:
2116:
1959:
is a constant. The structure of spacetime is determined by a
993:
909:
1322:
the line is tilted from the vertical, the larger the speed.
3767:
3332:
2692:" series. World lines and other physical concepts like the
2685:
1982:
World lines of freely falling particles/objects are called
2507:
curves. Also, in general relativity, world lines include
2447:{\displaystyle {\frac {du}{d\tau }}={\frac {dw}{d\tau }},}
1075:
and usually denoted τ. The length of M is called the
2097:
2577:
A popular description of human world lines was given by
2131:
often means the single spacetime event being considered.
1588:. It is a four-dimensional vector, defined in the point
27:
Unique path of an object as it travels through spacetime
1296:
3276:
2543:
2397:
2301:
2268:
2233:
2190:
2148:
1892:
1872:
1646:
1617:
1594:
1574:
1547:
1518:
1482:
1455:
1435:
1415:
1348:
1279:
1236:
1209:
1142:
1049:
1029:
160:
3065:
2466:
1986:. In special relativity these are straight lines in
1934:
3343:
in depth text on world lines and special relativity
1273:of an object or an observer as the curve parameter
932:
are rendered as curves in spacetime to show their (
67:. Unsourced material may be challenged and removed.
3274:
2446:
2331:
2287:
2254:
2219:
2176:
1911:
1878:
1856:
1632:
1600:
1580:
1556:
1533:
1504:
1468:
1441:
1421:
1401:
1285:
1242:
1222:
1195:
1055:
1035:
924:of a car on a road) by inclusion of the dimension
216:
2603:, describes the worldlines of stars and people:
1866:such that the derivatives are taken at the point
4205:
2947:
3229:
3110:Dunne, Gerald V.; Schubert, Christian (2005).
2551:world line formulation of quantum field theory
2491:exists and its dynamics are determined by the
2383:with respect to η. When two world lines
3363:
3109:
2965:Annual Review of Nuclear and Particle Science
2115:, which is defined for a given observer by a
2049:At a given event on a world line, spacetime (
2037:world lines are time-like curves in spacetime
1476:and a point on the curve a little (parameter
1338:Tangent vector to a world line: four-velocity
980:spatial position of the object at that time.
859:
3297:Various English translations on Wikisource:
2556:
1265:is a curve in spacetime that traces out the
1103:World lines as a method of describing events
1063:, horizontally. As expressed by F.R. Harvey
1043:, vertically, and the space coordinate, say
2899:
2511:curves and null curves in spacetime, where
3370:
3356:
2137:
1818:
1787:
1756:
1371:
1165:
951:The idea of world lines was originated by
866:
852:
3132:
3026:
2995:
2977:
127:Learn how and when to remove this message
3223:
3197:
3010:
2017:
2005:
1402:{\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
1300:
1196:{\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
1106:
2957:"Progress in one-loop QCD computations"
2852:
2816:
897:. It is an important concept of modern
14:
4206:
3171:
2783:
2643:describes the world line of a person:
2227:that is time-like, the Minkowski form
2220:{\displaystyle v={\frac {dw}{d\tau }}}
1505:{\displaystyle \tau _{0}+\Delta \tau }
3351:
2696:are also used throughout the series.
1975:of mathematics, the bilinear form is
2332:{\displaystyle x\mapsto \eta (v,x).}
1979:a mathematical metric on spacetime.
1449:on the curve at the parameter value
1297:Trivial examples of spacetime curves
65:adding citations to reliable sources
36:
3942:TolmanâOppenheimerâVolkoff equation
3895:FriedmannâLemaĂźtreâRobertsonâWalker
2544:World lines in quantum field theory
970:
24:
2288:{\displaystyle R^{4}\rightarrow R}
2104:in Galilean/Newtonian theory, the
1955:. In such coordinate systems, the
1548:
1519:
1496:
177:
25:
4235:
3712:HamiltonâJacobiâEinstein equation
3322:
3181:Scientific Romances: First Series
2531:Eddington-Finkelstein coordinates
2467:World lines in general relativity
2177:{\displaystyle w(\tau )\in R^{4}}
1935:World lines in special relativity
1534:{\displaystyle \Delta \tau \to 0}
4190:
4189:
2347:of this linear functional. Then
833:
832:
819:
149:
41:
3248:
3199:Robinson, Gilbert de Beauregard
3191:
3177:"What is the fourth dimension?"
2184:determines a velocity 4-vector
2111:Included in "elsewhere" is the
2053:) is divided into three parts.
1912:{\displaystyle \tau =\tau _{0}}
1568:of the world line at the point
999:Spacetime is the collection of
52:needs additional citations for
3519:Massâenergy equivalence (E=mc)
3377:
3256:"Technovelgy: Chronovitameter"
3165:
3103:
3059:
3004:
2941:
2893:
2846:
2810:
2777:
2733:Specific types of world lines
2323:
2311:
2305:
2279:
2249:
2237:
2158:
2152:
1653:
1624:
1525:
1365:
1359:
1342:The four coordinate functions
1159:
1153:
13:
1:
3045:10.1016/S0370-1573(01)00013-8
2997:10.1146/annurev.nucl.46.1.109
2770:
2520:) of general relativity. Any
2262:determines a linear function
1541:, this difference divided by
1512:) farther away. In the limit
3097:10.1016/0550-3213(82)90455-2
2955:; Kosower, David A. (1996).
2597:Kurt Vonnegut, in his novel
2475:is basically the same as in
2083:within the entire spacetime.
1557:{\displaystyle \Delta \tau }
7:
3534:Relativistic Doppler effect
3275:Minkowski, Hermann (1909),
3207:University of Toronto Press
2927:10.1103/PhysRevLett.66.1669
2727:
2479:, with the difference that
1953:inertial coordinate systems
890:that an object traces in 4-
330:Gravitational time dilation
10:
4240:
4005:In computational physics:
3529:Relativity of simultaneity
3151:10.1103/PhysRevD.72.105004
2471:The use of world lines in
2361:relativity of simultaneity
2255:{\displaystyle \eta (v,x)}
1633:{\displaystyle {\vec {v}}}
1067:A curve M in is called a
992:of the Earth is therefore
450:MathissonâPapapetrouâDixon
291:Pseudo-Riemannian manifold
29:
4187:
4019:
3884:
3856:
3842:LenseâThirring precession
3725:
3674:
3636:
3615:
3604:
3562:
3506:
3490:
3432:
3424:Doubly special relativity
3396:
3385:
3283:Physikalische Zeitschrift
2784:Harvey, F. Reese (1990).
2557:World lines in literature
2518:diffeomorphism invariance
1469:{\displaystyle \tau _{0}}
1099:but rather a world tube.
940:âto reveal the nature of
3702:Post-Newtonian formalism
3692:Einstein field equations
3628:Mathematical formulation
3452:Hyperbolic orthogonality
3230:Oliver Franklin (2008).
2790:Spinors and Calibrations
2673:(expanded from "Beep").
2591:Royal Canadian Institute
2493:Einstein field equations
455:HamiltonâJacobiâEinstein
435:Einstein field equations
258:Mathematical formulation
3413:Galilean transformation
3404:Principle of relativity
2906:Physical Review Letters
2663:uses the term, as does
2353:simultaneous hyperplane
2138:Simultaneous hyperplane
2113:simultaneous hyperplane
1257:is used informally for
1069:worldline of a particle
3498:Lorentz transformation
2887:10.1103/PhysRev.84.108
2840:10.1103/PhysRev.80.440
2742:Closed timelike curves
2626:work in 2008 entitled
2448:
2333:
2289:
2256:
2221:
2178:
2028:
2015:
1963:η, which gives a
1913:
1880:
1858:
1634:
1602:
1582:
1564:defines a vector, the
1558:
1535:
1506:
1470:
1443:
1423:
1403:
1314:
1293:along the world line.
1287:
1244:
1224:
1197:
1124:
1057:
1037:
886:) of an object is the
325:Gravitational redshift
218:
3966:WeylâLewisâPapapetrou
3707:Raychaudhuri equation
3646:Equivalence principle
3316:Macmillan and Company
3278:"Raum und Zeit"
2660:Methuselah's Children
2583:University of Toronto
2449:
2377:orthogonal complement
2334:
2290:
2257:
2222:
2179:
2121:hyperbolic-orthogonal
2072:to the given event).
2021:
2009:
1914:
1881:
1859:
1635:
1603:
1583:
1559:
1536:
1507:
1471:
1444:
1424:
1422:{\displaystyle \tau }
1404:
1304:
1288:
1286:{\displaystyle \tau }
1245:
1243:{\displaystyle \tau }
1225:
1223:{\displaystyle x^{0}}
1198:
1110:
1058:
1038:
955:and was pioneered by
613:WeylâLewisâPapapetrou
568:KerrâNewmanâde Sitter
388:EinsteinâRosen bridge
320:Gravitational lensing
276:Equivalence principle
219:
4214:Theory of relativity
4007:Numerical relativity
3848:pulsar timing arrays
3312:Theory of Relativity
3069:; Alvarez, Orlando;
2670:The Quincunx of Time
2395:
2363:is a statement that
2299:
2266:
2231:
2188:
2146:
1890:
1870:
1644:
1640:, or in components:
1615:
1592:
1572:
1545:
1516:
1480:
1453:
1433:
1413:
1346:
1277:
1253:Sometimes, the term
1234:
1207:
1140:
1047:
1027:
916:" (e.g., a planet's
543:EinsteinâRosen waves
269:Fundamental concepts
158:
61:improve this article
4219:Minkowski spacetime
3899:Friedmann equations
3793:HulseâTaylor binary
3755:Gravitational waves
3651:Riemannian geometry
3477:Proper acceleration
3462:Maxwell's equations
3408:Galilean relativity
3143:2005PhRvD..72j5004D
3089:1982NuPhB.197..509A
3071:Manton, Nicholas S.
3037:2001PhR...355...73S
2988:1996ARNPS..46..109B
2919:1991PhRvL..66.1669B
2879:1951PhRv...84..108F
2854:Feynman, Richard P.
2818:Feynman, Richard P.
2633:In the short story
2600:Slaughterhouse-Five
2461:retarded potentials
2142:Since a world line
903:theoretical physics
901:, and particularly
497:KaluzaâKlein theory
383:Minkowski spacetime
335:Gravitational waves
3948:ReissnerâNordström
3866:BransâDicke theory
3697:Linearized gravity
3524:Length contraction
3442:Frame of reference
3419:Special relativity
3308:Ludwik Silberstein
2796:. pp. 62â67.
2641:Robert A. Heinlein
2567:scientific romance
2477:special relativity
2473:general relativity
2444:
2329:
2285:
2252:
2217:
2174:
2029:
2016:
1941:special relativity
1909:
1876:
1854:
1630:
1598:
1578:
1554:
1531:
1502:
1466:
1439:
1419:
1399:
1315:
1283:
1240:
1220:
1193:
1127:A one-dimensional
1125:
1053:
1033:
1003:, together with a
965:general relativity
961:special relativity
942:special relativity
826:Physics portal
598:OppenheimerâSnyder
538:ReissnerâNordström
430:Linearized gravity
378:Spacetime diagrams
281:Special relativity
214:
143:General relativity
4201:
4200:
4015:
4014:
3994:OzsvĂĄthâSchĂŒcking
3600:
3599:
3582:Minkowski diagram
3539:Thomas precession
3482:Relativistic mass
3241:978-1-906557-00-3
3120:Physical Review D
3076:Nuclear Physics B
2913:(13): 1669â1672.
2690:Science Adventure
2526:comoving observer
2439:
2416:
2215:
1879:{\displaystyle p}
1847:
1816:
1785:
1754:
1656:
1627:
1601:{\displaystyle p}
1581:{\displaystyle p}
1442:{\displaystyle p}
1056:{\displaystyle x}
1036:{\displaystyle t}
1021:Minkowski diagram
1012:coordinate system
983:For example, the
957:Hermann Minkowski
876:
875:
509:
508:
395:
394:
137:
136:
129:
111:
16:(Redirected from
4231:
4193:
4192:
3976:van Stockum dust
3748:Two-body problem
3666:Mach's principle
3613:
3612:
3554:Terrell rotation
3394:
3393:
3372:
3365:
3358:
3349:
3348:
3290:
3280:
3267:
3266:
3264:
3262:
3252:
3246:
3245:
3227:
3221:
3220:
3195:
3189:
3188:
3187:. pp. 1â32.
3169:
3163:
3162:
3136:
3116:
3107:
3101:
3100:
3063:
3057:
3056:
3030:
3008:
3002:
3001:
2999:
2981:
2961:
2945:
2939:
2938:
2897:
2891:
2890:
2862:
2850:
2844:
2843:
2814:
2808:
2807:
2781:
2747:Causal structure
2708:Platonic realism
2453:
2451:
2450:
2445:
2440:
2438:
2430:
2422:
2417:
2415:
2407:
2399:
2355:with respect to
2338:
2336:
2335:
2330:
2294:
2292:
2291:
2286:
2278:
2277:
2261:
2259:
2258:
2253:
2226:
2224:
2223:
2218:
2216:
2214:
2206:
2198:
2183:
2181:
2180:
2175:
2173:
2172:
2010:An example of a
1969:spacetime metric
1918:
1916:
1915:
1910:
1908:
1907:
1885:
1883:
1882:
1877:
1863:
1861:
1860:
1855:
1853:
1849:
1848:
1846:
1838:
1837:
1836:
1823:
1817:
1815:
1807:
1806:
1805:
1792:
1786:
1784:
1776:
1775:
1774:
1761:
1755:
1753:
1745:
1744:
1743:
1730:
1720:
1716:
1715:
1714:
1702:
1701:
1689:
1688:
1676:
1675:
1658:
1657:
1649:
1639:
1637:
1636:
1631:
1629:
1628:
1620:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1563:
1561:
1560:
1555:
1540:
1538:
1537:
1532:
1511:
1509:
1508:
1503:
1492:
1491:
1475:
1473:
1472:
1467:
1465:
1464:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1358:
1357:
1292:
1290:
1289:
1284:
1249:
1247:
1246:
1241:
1229:
1227:
1226:
1221:
1219:
1218:
1202:
1200:
1199:
1194:
1152:
1151:
1062:
1060:
1059:
1054:
1042:
1040:
1039:
1034:
971:Usage in physics
936:) more absolute
868:
861:
854:
841:
836:
835:
828:
824:
823:
608:van Stockum dust
593:RobertsonâWalker
419:
418:
309:
308:
223:
221:
220:
215:
213:
212:
200:
192:
191:
173:
172:
153:
139:
138:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
4239:
4238:
4234:
4233:
4232:
4230:
4229:
4228:
4224:Time in science
4204:
4203:
4202:
4197:
4183:
4011:
3915:BKL singularity
3905:LemaĂźtreâTolman
3880:
3876:Quantum gravity
3858:
3852:
3838:geodetic effect
3812:(together with
3782:LISA Pathfinder
3721:
3670:
3656:Penrose diagram
3638:
3632:
3607:
3596:
3592:Minkowski space
3558:
3502:
3486:
3434:
3428:
3388:
3381:
3376:
3325:
3271:
3270:
3260:
3258:
3254:
3253:
3249:
3242:
3228:
3224:
3217:
3196:
3192:
3185:S. Sonnenschein
3170:
3166:
3114:
3108:
3104:
3067:Affleck, Ian K.
3064:
3060:
3021:(2â3): 73â234.
3014:Physics Reports
3009:
3005:
2959:
2946:
2942:
2898:
2894:
2866:Physical Review
2860:
2851:
2847:
2827:Physical Review
2815:
2811:
2804:
2782:
2778:
2773:
2760:Feynman diagram
2730:
2700:Neal Stephenson
2624:science fiction
2620:Oliver Franklin
2559:
2546:
2524:curve admits a
2469:
2431:
2423:
2421:
2408:
2400:
2398:
2396:
2393:
2392:
2391:are related by
2300:
2297:
2296:
2273:
2269:
2267:
2264:
2263:
2232:
2229:
2228:
2207:
2199:
2197:
2189:
2186:
2185:
2168:
2164:
2147:
2144:
2143:
2140:
2051:Minkowski space
1988:Minkowski space
1937:
1903:
1899:
1891:
1888:
1887:
1871:
1868:
1867:
1839:
1832:
1828:
1824:
1822:
1808:
1801:
1797:
1793:
1791:
1777:
1770:
1766:
1762:
1760:
1746:
1739:
1735:
1731:
1729:
1728:
1724:
1710:
1706:
1697:
1693:
1684:
1680:
1671:
1667:
1666:
1662:
1648:
1647:
1645:
1642:
1641:
1619:
1618:
1616:
1613:
1612:
1593:
1590:
1589:
1573:
1570:
1569:
1546:
1543:
1542:
1517:
1514:
1513:
1487:
1483:
1481:
1478:
1477:
1460:
1456:
1454:
1451:
1450:
1434:
1431:
1430:
1414:
1411:
1410:
1353:
1349:
1347:
1344:
1343:
1340:
1299:
1278:
1275:
1274:
1235:
1232:
1231:
1214:
1210:
1208:
1205:
1204:
1147:
1143:
1141:
1138:
1137:
1105:
1048:
1045:
1044:
1028:
1025:
1024:
973:
938:position states
872:
831:
818:
817:
810:
809:
633:
632:
623:
622:
578:LemaĂźtreâTolman
523:
522:
511:
510:
502:Quantum gravity
489:Advanced theory
416:
415:
414:
397:
396:
345:Geodetic effect
306:
305:
296:
295:
271:
270:
254:
224:
205:
201:
196:
184:
180:
165:
161:
159:
156:
155:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4237:
4227:
4226:
4221:
4216:
4199:
4198:
4188:
4185:
4184:
4182:
4181:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4124:
4119:
4114:
4109:
4104:
4102:Choquet-Bruhat
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4029:
4023:
4021:
4017:
4016:
4013:
4012:
4010:
4009:
4002:
4001:
3996:
3991:
3984:
3983:
3978:
3973:
3968:
3963:
3954:Axisymmetric:
3951:
3950:
3945:
3939:
3928:
3927:
3922:
3917:
3912:
3907:
3902:
3893:Cosmological:
3890:
3888:
3882:
3881:
3879:
3878:
3873:
3868:
3862:
3860:
3854:
3853:
3851:
3850:
3845:
3834:frame-dragging
3831:
3826:
3821:
3818:Einstein rings
3814:Einstein cross
3807:
3796:
3795:
3790:
3784:
3779:
3774:
3761:
3751:
3750:
3745:
3740:
3735:
3729:
3727:
3723:
3722:
3720:
3719:
3717:Ernst equation
3714:
3709:
3704:
3699:
3694:
3689:
3687:BSSN formalism
3684:
3678:
3676:
3672:
3671:
3669:
3668:
3663:
3658:
3653:
3648:
3642:
3640:
3634:
3633:
3631:
3630:
3625:
3619:
3617:
3610:
3602:
3601:
3598:
3597:
3595:
3594:
3589:
3584:
3579:
3574:
3568:
3566:
3560:
3559:
3557:
3556:
3551:
3546:
3544:Ladder paradox
3541:
3536:
3531:
3526:
3521:
3516:
3510:
3508:
3504:
3503:
3501:
3500:
3494:
3492:
3488:
3487:
3485:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3447:Speed of light
3444:
3438:
3436:
3430:
3429:
3427:
3426:
3421:
3416:
3410:
3400:
3398:
3391:
3383:
3382:
3375:
3374:
3367:
3360:
3352:
3346:
3345:
3337:
3336:
3324:
3323:External links
3321:
3320:
3319:
3304:
3303:
3302:
3301:
3299:Space and Time
3292:
3291:
3269:
3268:
3247:
3240:
3234:. Epic Press.
3222:
3215:
3209:. p. 19.
3190:
3164:
3134:hep-th/0507174
3127:(10): 105004.
3102:
3083:(3): 509â519.
3058:
3028:hep-th/0101036
3003:
2979:hep-ph/9602280
2940:
2892:
2873:(3): 440â457.
2845:
2834:(1): 108â128.
2809:
2802:
2794:Academic Press
2775:
2774:
2772:
2769:
2768:
2767:
2765:Time geography
2762:
2757:
2756:
2755:
2753:Isotropic line
2750:
2744:
2739:
2729:
2726:
2722:Charles Stross
2684:, produced by
2655:
2654:
2650:
2649:
2609:
2608:
2595:
2594:
2575:
2574:
2558:
2555:
2545:
2542:
2468:
2465:
2443:
2437:
2434:
2429:
2426:
2420:
2414:
2411:
2406:
2403:
2351:is called the
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2284:
2281:
2276:
2272:
2251:
2248:
2245:
2242:
2239:
2236:
2213:
2210:
2205:
2202:
2196:
2193:
2171:
2167:
2163:
2160:
2157:
2154:
2151:
2139:
2136:
2135:
2134:
2133:
2132:
2125:
2086:
2085:
2084:
2062:
2047:
2046:
2040:
2004:
2003:
1957:speed of light
1947:is limited to
1936:
1933:
1906:
1902:
1898:
1895:
1875:
1852:
1845:
1842:
1835:
1831:
1827:
1821:
1814:
1811:
1804:
1800:
1796:
1790:
1783:
1780:
1773:
1769:
1765:
1759:
1752:
1749:
1742:
1738:
1734:
1727:
1723:
1719:
1713:
1709:
1705:
1700:
1696:
1692:
1687:
1683:
1679:
1674:
1670:
1665:
1661:
1655:
1652:
1626:
1623:
1597:
1577:
1566:tangent vector
1553:
1550:
1530:
1527:
1524:
1521:
1501:
1498:
1495:
1490:
1486:
1463:
1459:
1438:
1418:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1370:
1367:
1364:
1361:
1356:
1352:
1339:
1336:
1298:
1295:
1282:
1267:(time) history
1239:
1217:
1213:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1164:
1161:
1158:
1155:
1150:
1146:
1104:
1101:
1085:
1084:
1052:
1032:
972:
969:
948:interactions.
918:orbit in space
874:
873:
871:
870:
863:
856:
848:
845:
844:
843:
842:
829:
812:
811:
808:
807:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
634:
630:
629:
628:
625:
624:
621:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
524:
518:
517:
516:
513:
512:
507:
506:
505:
504:
499:
491:
490:
486:
485:
484:
483:
481:Post-Newtonian
478:
473:
465:
464:
460:
459:
458:
457:
452:
447:
442:
437:
432:
424:
423:
417:
413:
412:
409:
405:
404:
403:
402:
399:
398:
393:
392:
391:
390:
385:
380:
372:
371:
365:
364:
363:
362:
357:
352:
347:
342:
340:Frame-dragging
337:
332:
327:
322:
317:
315:Kepler problem
307:
303:
302:
301:
298:
297:
294:
293:
288:
283:
278:
272:
268:
267:
266:
263:
262:
261:
260:
255:
253:
252:
247:
242:
236:
234:
226:
225:
211:
208:
204:
199:
195:
190:
187:
183:
179:
176:
171:
168:
164:
154:
146:
145:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
4236:
4225:
4222:
4220:
4217:
4215:
4212:
4211:
4209:
4196:
4186:
4180:
4179:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4087:Chandrasekhar
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4047:Schwarzschild
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4024:
4022:
4018:
4008:
4004:
4003:
4000:
3997:
3995:
3992:
3990:
3986:
3985:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3961:
3957:
3953:
3952:
3949:
3946:
3943:
3940:
3938:
3934:
3933:Schwarzschild
3930:
3929:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3900:
3896:
3892:
3891:
3889:
3887:
3883:
3877:
3874:
3872:
3869:
3867:
3864:
3863:
3861:
3855:
3849:
3846:
3843:
3839:
3835:
3832:
3830:
3829:Shapiro delay
3827:
3825:
3822:
3819:
3815:
3811:
3808:
3805:
3801:
3798:
3797:
3794:
3791:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3772:collaboration
3769:
3765:
3762:
3760:
3756:
3753:
3752:
3749:
3746:
3744:
3741:
3739:
3738:Event horizon
3736:
3734:
3731:
3730:
3728:
3724:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3682:ADM formalism
3680:
3679:
3677:
3673:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3643:
3641:
3635:
3629:
3626:
3624:
3621:
3620:
3618:
3614:
3611:
3609:
3603:
3593:
3590:
3588:
3587:Biquaternions
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3569:
3567:
3565:
3561:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3515:
3514:Time dilation
3512:
3511:
3509:
3505:
3499:
3496:
3495:
3493:
3489:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3467:Proper length
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3439:
3437:
3431:
3425:
3422:
3420:
3417:
3414:
3411:
3409:
3405:
3402:
3401:
3399:
3395:
3392:
3390:
3384:
3380:
3373:
3368:
3366:
3361:
3359:
3354:
3353:
3350:
3344:
3341:
3340:
3339:
3334:
3330:
3327:
3326:
3317:
3313:
3309:
3306:
3305:
3300:
3296:
3295:
3294:
3293:
3288:
3284:
3279:
3273:
3272:
3257:
3251:
3243:
3237:
3233:
3226:
3218:
3216:0-7727-1600-5
3212:
3208:
3204:
3200:
3194:
3186:
3182:
3178:
3174:
3173:Hinton, C. H.
3168:
3160:
3156:
3152:
3148:
3144:
3140:
3135:
3130:
3126:
3122:
3121:
3113:
3106:
3098:
3094:
3090:
3086:
3082:
3078:
3077:
3072:
3068:
3062:
3054:
3050:
3046:
3042:
3038:
3034:
3029:
3024:
3020:
3016:
3015:
3007:
2998:
2993:
2989:
2985:
2980:
2975:
2971:
2967:
2966:
2958:
2954:
2950:
2944:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2907:
2902:
2896:
2888:
2884:
2880:
2876:
2872:
2868:
2867:
2859:
2855:
2849:
2841:
2837:
2833:
2829:
2828:
2823:
2819:
2813:
2805:
2803:9780080918631
2799:
2795:
2791:
2787:
2780:
2776:
2766:
2763:
2761:
2758:
2754:
2751:
2748:
2745:
2743:
2740:
2738:
2735:
2734:
2732:
2731:
2725:
2723:
2718:
2715:
2713:
2709:
2705:
2701:
2697:
2695:
2691:
2687:
2683:
2679:
2674:
2672:
2671:
2666:
2662:
2661:
2652:
2651:
2646:
2645:
2644:
2642:
2638:
2637:
2631:
2629:
2625:
2621:
2616:
2614:
2606:
2605:
2604:
2602:
2601:
2592:
2588:
2587:
2586:
2584:
2580:
2572:
2571:
2570:
2568:
2564:
2554:
2552:
2541:
2539:
2534:
2532:
2527:
2523:
2519:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2464:
2462:
2458:
2457:Coulomb's law
2441:
2435:
2432:
2427:
2424:
2418:
2412:
2409:
2404:
2401:
2390:
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2326:
2320:
2317:
2314:
2308:
2302:
2282:
2274:
2270:
2246:
2243:
2240:
2234:
2211:
2208:
2203:
2200:
2194:
2191:
2169:
2165:
2161:
2155:
2149:
2130:
2126:
2122:
2118:
2114:
2110:
2109:
2107:
2103:
2099:
2094:
2090:
2087:
2082:
2078:
2074:
2073:
2071:
2067:
2063:
2060:
2056:
2055:
2054:
2052:
2044:
2041:
2038:
2034:
2031:
2030:
2026:
2020:
2013:
2008:
2000:
1997:
1996:
1995:
1991:
1989:
1985:
1980:
1978:
1974:
1973:metric spaces
1970:
1966:
1962:
1961:bilinear form
1958:
1954:
1950:
1946:
1942:
1932:
1930:
1929:tangent space
1927:, termed the
1926:
1920:
1904:
1900:
1896:
1893:
1873:
1864:
1850:
1843:
1840:
1833:
1829:
1825:
1819:
1812:
1809:
1802:
1798:
1794:
1788:
1781:
1778:
1771:
1767:
1763:
1757:
1750:
1747:
1740:
1736:
1732:
1725:
1721:
1717:
1711:
1707:
1703:
1698:
1694:
1690:
1685:
1681:
1677:
1672:
1668:
1663:
1659:
1650:
1621:
1611:
1610:four-velocity
1595:
1575:
1567:
1551:
1528:
1522:
1499:
1493:
1488:
1484:
1461:
1457:
1436:
1416:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1368:
1362:
1354:
1350:
1335:
1331:
1328:
1323:
1319:
1312:
1308:
1303:
1294:
1280:
1272:
1268:
1264:
1260:
1256:
1251:
1237:
1215:
1211:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1162:
1156:
1148:
1144:
1134:
1130:
1122:
1118:
1114:
1109:
1100:
1098:
1093:
1090:
1082:
1078:
1074:
1070:
1066:
1065:
1064:
1050:
1030:
1022:
1018:
1013:
1010:
1006:
1002:
997:
995:
991:
986:
981:
978:
968:
966:
962:
958:
954:
949:
947:
946:gravitational
943:
939:
935:
931:
927:
923:
919:
915:
911:
906:
904:
900:
896:
893:
889:
885:
881:
869:
864:
862:
857:
855:
850:
849:
847:
846:
840:
830:
827:
822:
816:
815:
814:
813:
806:
805:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
743:Chandrasekhar
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
658:Schwarzschild
656:
654:
651:
649:
646:
644:
641:
639:
636:
635:
627:
626:
619:
618:HartleâThorne
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
533:
529:
528:Schwarzschild
526:
525:
521:
515:
514:
503:
500:
498:
495:
494:
493:
492:
488:
487:
482:
479:
477:
474:
472:
469:
468:
467:
466:
462:
461:
456:
453:
451:
448:
446:
443:
441:
438:
436:
433:
431:
428:
427:
426:
425:
421:
420:
410:
407:
406:
401:
400:
389:
386:
384:
381:
379:
376:
375:
374:
373:
370:
367:
366:
361:
358:
356:
353:
351:
350:Event horizon
348:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
312:
311:
310:
300:
299:
292:
289:
287:
284:
282:
279:
277:
274:
273:
265:
264:
259:
256:
251:
248:
246:
243:
241:
238:
237:
235:
233:
230:
229:
228:
227:
209:
206:
202:
197:
193:
188:
185:
181:
174:
169:
166:
162:
152:
148:
147:
144:
141:
140:
131:
128:
120:
117:November 2023
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: â
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
4177:
3871:KaluzaâKlein
3623:Introduction
3576:
3549:Twin paradox
3338:
3311:
3286:
3282:
3259:. Retrieved
3250:
3231:
3225:
3202:
3193:
3180:
3167:
3124:
3118:
3105:
3080:
3074:
3061:
3018:
3012:
3006:
2969:
2963:
2953:Dixon, Lance
2943:
2910:
2904:
2895:
2870:
2864:
2848:
2831:
2825:
2812:
2789:
2779:
2719:
2716:
2698:
2678:visual novel
2675:
2668:
2658:
2656:
2634:
2632:
2627:
2622:published a
2617:
2610:
2598:
2596:
2579:J. C. Fields
2576:
2563:C. H. Hinton
2560:
2547:
2535:
2470:
2388:
2384:
2380:
2372:
2368:
2364:
2356:
2352:
2348:
2340:
2141:
2128:
2112:
2105:
2101:
2092:
2088:
2076:
2065:
2058:
2048:
2042:
2036:
2032:
1998:
1992:
1981:
1976:
1968:
1948:
1938:
1925:linear space
1921:
1865:
1565:
1341:
1332:
1326:
1324:
1320:
1316:
1310:
1309:is time and
1306:
1266:
1262:
1258:
1254:
1252:
1132:
1128:
1126:
1096:
1094:
1086:
1076:
1068:
998:
989:
984:
982:
974:
950:
930:perceptually
925:
921:
917:
907:
883:
879:
877:
803:
763:Raychaudhuri
285:
232:Introduction
123:
114:
104:
97:
90:
83:
76:"World line"
71:
59:Please help
54:verification
51:
32:Worldline SA
3960:KerrâNewman
3931:Spherical:
3800:Other tests
3743:Singularity
3675:Formulation
3637:Fundamental
3491:Formulation
3472:Proper time
3433:Fundamental
3331:article on
3329:World lines
3261:8 September
3232:World Lines
2972:: 109â148.
2682:Steins;Gate
2665:James Blish
2657:Heinlein's
2628:World Lines
2613:time travel
2569:. He wrote
2367:depends on
2002:suppressed.
1965:real number
1271:proper time
1089:world sheet
1077:proper time
1073:proper time
892:dimensional
778:van Stockum
708:Oppenheimer
563:KerrâNewman
355:Singularity
4208:Categories
4112:Zel'dovich
4020:Scientists
3999:Alcubierre
3806:of Mercury
3804:precession
3733:Black hole
3616:Background
3608:relativity
3577:World line
3572:Light cone
3397:Background
3389:relativity
3379:Relativity
2771:References
2712:nominalism
2371:. Indeed,
2345:null space
2081:light cone
2070:light cone
2043:space-like
2012:light cone
1999:light-like
1263:world line
1255:world line
1097:world line
1005:continuous
990:world line
953:physicists
934:relatively
922:trajectory
914:trajectory
880:world line
631:Scientists
463:Formalisms
411:Formalisms
360:Black hole
286:World line
87:newspapers
4082:Robertson
4067:Friedmann
4062:Eddington
4052:de Sitter
3886:Solutions
3764:detectors
3759:astronomy
3726:Phenomena
3661:Geodesics
3564:Spacetime
3507:Phenomena
3314:, p 130,
3159:119357180
3053:118891361
2949:Bern, Zvi
2901:Bern, Zvi
2737:Geodesics
2702:'s novel
2694:Dirac Sea
2639:, author
2636:Life-Line
2538:geodesics
2501:spacelike
2497:lightlike
2481:spacetime
2436:τ
2413:τ
2309:η
2306:↦
2280:→
2235:η
2212:τ
2162:∈
2156:τ
2106:elsewhere
2093:elsewhere
2089:Elsewhere
2077:lightcone
2033:time-like
2025:spacetime
1984:geodesics
1945:spacetime
1901:τ
1894:τ
1844:τ
1813:τ
1782:τ
1751:τ
1654:→
1625:→
1552:τ
1549:Δ
1526:→
1523:τ
1520:Δ
1500:τ
1497:Δ
1485:τ
1458:τ
1417:τ
1363:τ
1327:collision
1313:distance.
1281:τ
1238:τ
1157:τ
1113:particles
1081:free fall
977:spacetime
895:spacetime
884:worldline
723:Robertson
688:Friedmann
683:Eddington
673:Nordström
663:de Sitter
520:Solutions
445:Geodesics
440:Friedmann
422:Equations
408:Equations
369:Spacetime
304:Phenomena
210:ν
207:μ
198:κ
189:ν
186:μ
178:Λ
170:ν
167:μ
18:Worldline
4195:Category
4072:LemaĂźtre
4037:Einstein
4027:Poincaré
3987:Others:
3971:TaubâNUT
3937:interior
3859:theories
3857:Advanced
3824:redshift
3639:concepts
3457:Rapidity
3435:concepts
3201:(1979).
3175:(1884).
2935:10043277
2856:(1951).
2820:(1950).
2728:See also
2561:In 1884
2522:timelike
2513:timelike
2509:timelike
2505:timelike
2499:(null),
2119:that is
1886:, so at
1017:manifold
912:" or a "
839:Category
703:LemaĂźtre
668:Reissner
653:Poincaré
638:Einstein
583:TaubâNUT
548:Wormhole
532:interior
245:Timeline
4137:Hawking
4132:Penrose
4117:Novikov
4097:Wheeler
4042:Hilbert
4032:Lorentz
3989:pp-wave
3810:lensing
3606:General
3387:Special
3310:(1914)
3289:: 75â88
3139:Bibcode
3085:Bibcode
3033:Bibcode
2984:Bibcode
2915:Bibcode
2875:Bibcode
2704:Anathem
2618:Author
2581:at the
2483:can be
2375:is the
2343:be the
2129:present
2102:present
1949:special
1203:(where
1117:strings
994:helical
920:or the
899:physics
758:Hawking
753:Penrose
728:Bardeen
718:Wheeler
648:Hilbert
643:Lorentz
603:pp-wave
240:History
101:scholar
4178:others
4167:Thorne
4157:Misner
4142:Taylor
4127:Geroch
4122:Ehlers
4092:Zwicky
3910:Kasner
3238:
3213:
3157:
3051:
2933:
2800:
2680:named
2503:, and
2489:metric
2485:curved
2359:. The
2059:future
1121:branes
1119:, and
1009:smooth
1001:events
837:
804:others
798:Thorne
788:Newman
768:Taylor
748:Ehlers
733:Walker
698:Zwicky
573:Kasner
103:
96:
89:
82:
74:
4172:Weiss
4152:Bondi
4147:Hulse
4077:Milne
3981:discs
3925:Milne
3920:Gödel
3777:Virgo
3155:S2CID
3129:arXiv
3115:(PDF)
3049:S2CID
3023:arXiv
2974:arXiv
2960:(PDF)
2861:(PDF)
2339:Let
2117:space
1133:curve
985:orbit
910:orbit
773:Hulse
713:Gödel
693:Milne
588:Milne
553:Gödel
250:Tests
108:JSTOR
94:books
4107:Kerr
4057:Weyl
3956:Kerr
3816:and
3770:and
3768:LIGO
3333:h2g2
3263:2010
3236:ISBN
3211:ISBN
2931:PMID
2798:ISBN
2710:and
2686:5pb.
2487:. A
2387:and
2127:The
2075:The
2066:past
2064:The
2057:The
1129:line
1007:and
963:and
926:time
888:path
882:(or
878:The
783:Taub
738:Kerr
678:Weyl
558:Kerr
476:BSSN
80:news
4162:Yau
3787:GEO
3147:doi
3093:doi
3081:197
3041:doi
3019:355
2992:doi
2923:doi
2883:doi
2836:doi
2667:'s
2379:of
2295:by
2098:Sun
1977:not
1259:any
1131:or
967:).
944:or
793:Yau
471:ADM
63:by
4210::
3836:/
3802::
3757::
3287:10
3285:,
3281:,
3205:.
3183:.
3179:.
3153:.
3145:.
3137:.
3125:72
3123:.
3117:.
3091:.
3079:.
3047:.
3039:.
3031:.
3017:.
2990:.
2982:.
2970:46
2968:.
2962:.
2951:;
2929:.
2921:.
2911:66
2909:.
2881:.
2871:84
2869:.
2863:.
2832:80
2830:.
2824:.
2792:.
2788:.
2724:.
2714:.
2676:A
2540:.
2533:.
2463:.
1990:.
1919:.
1115:,
905:.
3962:)
3958:(
3944:)
3935:(
3901:)
3897:(
3844:)
3840:(
3820:)
3789:)
3766:(
3415:)
3406:(
3371:e
3364:t
3357:v
3335:.
3318:.
3265:.
3244:.
3219:.
3161:.
3149::
3141::
3131::
3099:.
3095::
3087::
3055:.
3043::
3035::
3025::
3000:.
2994::
2986::
2976::
2937:.
2925::
2917::
2889:.
2885::
2877::
2842:.
2838::
2806:.
2516:(
2442:,
2433:d
2428:w
2425:d
2419:=
2410:d
2405:u
2402:d
2389:w
2385:u
2381:v
2373:N
2369:v
2365:N
2357:v
2349:N
2341:N
2327:.
2324:)
2321:x
2318:,
2315:v
2312:(
2303:x
2283:R
2275:4
2271:R
2250:)
2247:x
2244:,
2241:v
2238:(
2209:d
2204:w
2201:d
2195:=
2192:v
2170:4
2166:R
2159:)
2153:(
2150:w
2039:.
1905:0
1897:=
1874:p
1851:)
1841:d
1834:3
1830:x
1826:d
1820:,
1810:d
1803:2
1799:x
1795:d
1789:,
1779:d
1772:1
1768:x
1764:d
1758:,
1748:d
1741:0
1737:x
1733:d
1726:(
1722:=
1718:)
1712:3
1708:v
1704:,
1699:2
1695:v
1691:,
1686:1
1682:v
1678:,
1673:0
1669:v
1664:(
1660:=
1651:v
1622:v
1596:p
1576:p
1529:0
1494:+
1489:0
1462:0
1437:p
1397:3
1394:,
1391:2
1388:,
1385:1
1382:,
1379:0
1376:=
1373:a
1369:,
1366:)
1360:(
1355:a
1351:x
1311:x
1307:t
1216:0
1212:x
1191:3
1188:,
1185:2
1182:,
1179:1
1176:,
1173:0
1170:=
1167:a
1163:,
1160:)
1154:(
1149:a
1145:x
1123:.
1083:.
1051:x
1031:t
867:e
860:t
853:v
534:)
530:(
203:T
194:=
182:g
175:+
163:G
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
