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2604:, 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'.
2582:. 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.
140:
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2596: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
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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.
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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
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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
2537:
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
1310:
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
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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
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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
1982:
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):
1008:(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
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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
1322:
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).
1124:
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
1911:
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
1080:
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.
2443:
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
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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
2011:
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
1306:
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.
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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.
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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
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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
211:
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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
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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.
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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.
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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
1920:
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.
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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.
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1846:{\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)}
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2003:, 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.
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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
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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
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2642:"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..."
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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
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of the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates.
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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:
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has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields.
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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
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1239:. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.
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of the given event is formed by all events that can be reached through time-like curves lying within the future light cone.
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A world line of an object (generally approximated as a point in space, e.g., a particle or observer) is the sequence of
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World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are called
1597:. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed
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may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve
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involves a long discussion of worldlines over dinner in the midst of a philosophical debate between
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Schubert, Christian (2001). "Perturbative quantum field theory in the string-inspired formalism".
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lines. This preceded much of his work on the formulation which later became more standard. The
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Once the object is not approximated as a mere point but has extended volume, it traces not a
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is thick; it is not a 3-dimensional volume but is instead a 4-dimensional spacetime region.
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of the particle. If the worldline M is a line segment, then the particle is said to be in
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in spacetime (a curve in a four-dimensional space) and does not return to the same point.
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and are dependent on the mass-energy distribution in spacetime. Again the metric defines
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puts some constraints on possible world lines. In special relativity the description of
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Three different world lines representing travel at different constant four-velocities.
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Almost all science-fiction stories which use this concept actively, such as to enable
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if its tangent is future timelike at each point. The arclength parameter is called
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in the early days of relativity. As described by
Toronto lawyer Norman Robertson:
2007:
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333:
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2892:; Kosower, David A. (1991). "Efficient calculation of one-loop QCD amplitudes".
2706:
Absolute Choice depicts different world lines as a sub-plot and setting device.
1960:, but since distinct events sometimes result in a zero value, unlike metrics in
1940:
coordinate systems that do not accelerate (and so do not rotate either), termed
541:
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2811:"Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction"
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1945:
948:. The term is now used most often in the context of relativity theories (i.e.,
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2775:"Special Relativity" section of chapter "Euclidean / Lorentzian Vector Spaces"
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wrote an essay "What is the fourth dimension ?", which he published as a
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in which he related a simplified explanation of the hypothesis for laymen.
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curve in spacetime. This terminology causes confusions. More properly, a
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The concept of a "world line" is distinguished from concepts such as an "
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19:"Worldline" redirects here. For the French payment services company, see
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Two world lines that start out separately and then intersect, signify a
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206:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
3062:(1982). "Pair production at strong coupling in weak external fields".
3552:
2847:"An operator calculus having applications in quantum electrodynamics"
2682:
2624:
2578:
I remember lecturing at one of the
Saturday evening lectures at the
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as it was about 8 minutes ago, not as it is "right now". Unlike the
2013:
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883:
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357:
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defining a world line, are real number functions of a real variable
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is the region between the two light cones. Points in an observer's
1972:
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World line, worldsheet, and world volume, as they are derived from
1076:
A world line traces out the path of a single point in spacetime. A
1005:
726:
536:
376:
3192:
The
Mathematics Department in the University of Toronto, 1827â1978
2738:, curves that represent a variety of different types of world line
3101:"Worldline instantons and pair production in inhomogenous fields"
2692:
1956:
for each pair of events. The bilinear form is sometimes termed a
1012:, which is a plane usually plotted with the time coordinate, say
941:
887:
1326:
1219:
usually denotes the time coordinate) depending on one parameter
3775:
1096:
1091:
1995:
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of a particle, observer or small object. One usually uses the
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2105:
1948:
is a constant. The structure of spacetime is determined by a
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898:
1311:
the line is tilted from the vertical, the larger the speed.
3756:
3321:
2681:" series. World lines and other physical concepts like the
2674:
1971:
World lines of freely falling particles/objects are called
2496:
curves. Also, in general relativity, world lines include
2436:{\displaystyle {\frac {du}{d\tau }}={\frac {dw}{d\tau }},}
1064:
and usually denoted τ. The length of M is called the
2086:
2566:
A popular description of human world lines was given by
2120:
often means the single spacetime event being considered.
1577:. It is a four-dimensional vector, defined in the point
16:
Unique path of an object as it travels through spacetime
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1975:. In special relativity these are straight lines in
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in depth text on world lines and special relativity
1262:of an object or an observer as the curve parameter
921:
are rendered as curves in spacetime to show their (
56:. Unsourced material may be challenged and removed.
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913:of a car on a road) by inclusion of the dimension
205:
2592:, describes the worldlines of stars and people:
1855:such that the derivatives are taken at the point
4194:
2936:
3218:
3099:Dunne, Gerald V.; Schubert, Christian (2005).
2540:world line formulation of quantum field theory
2480:exists and its dynamics are determined by the
2372:with respect to η. When two world lines
3352:
3098:
2954:Annual Review of Nuclear and Particle Science
2104:, which is defined for a given observer by a
2038:At a given event on a world line, spacetime (
2026:world lines are time-like curves in spacetime
1465:and a point on the curve a little (parameter
1327:Tangent vector to a world line: four-velocity
969:spatial position of the object at that time.
848:
3286:Various English translations on Wikisource:
2545:
1254:is a curve in spacetime that traces out the
1092:World lines as a method of describing events
1052:, horizontally. As expressed by F.R. Harvey
1032:, vertically, and the space coordinate, say
2888:
2500:curves and null curves in spacetime, where
3359:
3345:
2126:
1807:
1776:
1745:
1360:
1154:
940:The idea of world lines was originated by
855:
841:
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3015:
2984:
2966:
116:Learn how and when to remove this message
3212:
3186:
2999:
2006:
1994:
1391:{\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
1289:
1185:{\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
1095:
2946:"Progress in one-loop QCD computations"
2841:
2805:
886:. It is an important concept of modern
4195:
3160:
2772:
2632:describes the world line of a person:
2216:that is time-like, the Minkowski form
2209:{\displaystyle v={\frac {dw}{d\tau }}}
1494:{\displaystyle \tau _{0}+\Delta \tau }
3340:
2685:are also used throughout the series.
1964:of mathematics, the bilinear form is
2321:{\displaystyle x\mapsto \eta (v,x).}
1968:a mathematical metric on spacetime.
1438:on the curve at the parameter value
1286:Trivial examples of spacetime curves
54:adding citations to reliable sources
25:
3931:TolmanâOppenheimerâVolkoff equation
3884:FriedmannâLemaĂźtreâRobertsonâWalker
2533:World lines in quantum field theory
959:
13:
2277:{\displaystyle R^{4}\rightarrow R}
2093:in Galilean/Newtonian theory, the
1944:. In such coordinate systems, the
1537:
1508:
1485:
166:
14:
4224:
3701:HamiltonâJacobiâEinstein equation
3311:
3170:Scientific Romances: First Series
2520:Eddington-Finkelstein coordinates
2456:World lines in general relativity
2166:{\displaystyle w(\tau )\in R^{4}}
1924:World lines in special relativity
1523:{\displaystyle \Delta \tau \to 0}
4179:
4178:
2336:of this linear functional. Then
822:
821:
808:
138:
30:
3237:
3188:Robinson, Gilbert de Beauregard
3180:
3166:"What is the fourth dimension?"
2173:determines a velocity 4-vector
2100:Included in "elsewhere" is the
2042:) is divided into three parts.
1901:{\displaystyle \tau =\tau _{0}}
1557:of the world line at the point
988:Spacetime is the collection of
41:needs additional citations for
3508:Massâenergy equivalence (E=mc)
3366:
3245:"Technovelgy: Chronovitameter"
3154:
3092:
3048:
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2722:Specific types of world lines
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2300:
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2226:
2147:
2141:
1642:
1613:
1514:
1354:
1348:
1331:The four coordinate functions
1148:
1142:
1:
3034:10.1016/S0370-1573(01)00013-8
2986:10.1146/annurev.nucl.46.1.109
2759:
2509:) of general relativity. Any
2251:determines a linear function
1530:, this difference divided by
1501:) farther away. In the limit
3086:10.1016/0550-3213(82)90455-2
2944:; Kosower, David A. (1996).
2586:Kurt Vonnegut, in his novel
2464:is basically the same as in
2072:within the entire spacetime.
1546:{\displaystyle \Delta \tau }
7:
3523:Relativistic Doppler effect
3264:Minkowski, Hermann (1909),
3196:University of Toronto Press
2916:10.1103/PhysRevLett.66.1669
2716:
2468:, with the difference that
1942:inertial coordinate systems
879:that an object traces in 4-
319:Gravitational time dilation
10:
4229:
3994:In computational physics:
3518:Relativity of simultaneity
3140:10.1103/PhysRevD.72.105004
2460:The use of world lines in
2350:relativity of simultaneity
2244:{\displaystyle \eta (v,x)}
1622:{\displaystyle {\vec {v}}}
1056:A curve M in is called a
981:of the Earth is therefore
439:MathissonâPapapetrouâDixon
280:Pseudo-Riemannian manifold
18:
4176:
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3873:
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3831:LenseâThirring precession
3714:
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3479:
3421:
3413:Doubly special relativity
3385:
3374:
3272:Physikalische Zeitschrift
2773:Harvey, F. Reese (1990).
2546:World lines in literature
2507:diffeomorphism invariance
1458:{\displaystyle \tau _{0}}
1088:but rather a world tube.
929:âto reveal the nature of
3691:Post-Newtonian formalism
3681:Einstein field equations
3617:Mathematical formulation
3441:Hyperbolic orthogonality
3219:Oliver Franklin (2008).
2779:Spinors and Calibrations
2662:(expanded from "Beep").
2580:Royal Canadian Institute
2482:Einstein field equations
444:HamiltonâJacobiâEinstein
424:Einstein field equations
247:Mathematical formulation
3402:Galilean transformation
3393:Principle of relativity
2895:Physical Review Letters
2652:uses the term, as does
2342:simultaneous hyperplane
2127:Simultaneous hyperplane
2102:simultaneous hyperplane
1246:is used informally for
1058:worldline of a particle
3487:Lorentz transformation
2876:10.1103/PhysRev.84.108
2829:10.1103/PhysRev.80.440
2731:Closed timelike curves
2615:work in 2008 entitled
2437:
2322:
2278:
2245:
2210:
2167:
2017:
2004:
1952:η, which gives a
1902:
1869:
1847:
1623:
1591:
1571:
1553:defines a vector, the
1547:
1524:
1495:
1459:
1432:
1412:
1392:
1303:
1282:along the world line.
1276:
1233:
1213:
1186:
1113:
1046:
1026:
875:) of an object is the
314:Gravitational redshift
207:
3955:WeylâLewisâPapapetrou
3696:Raychaudhuri equation
3635:Equivalence principle
3305:Macmillan and Company
3267:"Raum und Zeit"
2649:Methuselah's Children
2572:University of Toronto
2438:
2366:orthogonal complement
2323:
2279:
2246:
2211:
2168:
2110:hyperbolic-orthogonal
2061:to the given event).
2010:
1998:
1903:
1870:
1848:
1624:
1592:
1572:
1548:
1525:
1496:
1460:
1433:
1413:
1411:{\displaystyle \tau }
1393:
1293:
1277:
1275:{\displaystyle \tau }
1234:
1232:{\displaystyle \tau }
1214:
1212:{\displaystyle x^{0}}
1187:
1099:
1047:
1027:
944:and was pioneered by
602:WeylâLewisâPapapetrou
557:KerrâNewmanâde Sitter
377:EinsteinâRosen bridge
309:Gravitational lensing
265:Equivalence principle
208:
4203:Theory of relativity
3996:Numerical relativity
3837:pulsar timing arrays
3301:Theory of Relativity
3058:; Alvarez, Orlando;
2659:The Quincunx of Time
2384:
2352:is a statement that
2288:
2255:
2220:
2177:
2135:
1879:
1859:
1633:
1629:, or in components:
1604:
1581:
1561:
1534:
1505:
1469:
1442:
1422:
1402:
1335:
1266:
1242:Sometimes, the term
1223:
1196:
1129:
1036:
1016:
905:" (e.g., a planet's
532:EinsteinâRosen waves
258:Fundamental concepts
147:
50:improve this article
4208:Minkowski spacetime
3888:Friedmann equations
3782:HulseâTaylor binary
3744:Gravitational waves
3640:Riemannian geometry
3466:Proper acceleration
3451:Maxwell's equations
3397:Galilean relativity
3132:2005PhRvD..72j5004D
3078:1982NuPhB.197..509A
3060:Manton, Nicholas S.
3026:2001PhR...355...73S
2977:1996ARNPS..46..109B
2908:1991PhRvL..66.1669B
2868:1951PhRv...84..108F
2843:Feynman, Richard P.
2807:Feynman, Richard P.
2622:In the short story
2589:Slaughterhouse-Five
2450:retarded potentials
2131:Since a world line
892:theoretical physics
890:, and particularly
486:KaluzaâKlein theory
372:Minkowski spacetime
324:Gravitational waves
3937:ReissnerâNordström
3855:BransâDicke theory
3686:Linearized gravity
3513:Length contraction
3431:Frame of reference
3408:Special relativity
3297:Ludwik Silberstein
2785:. pp. 62â67.
2630:Robert A. Heinlein
2556:scientific romance
2466:special relativity
2462:general relativity
2433:
2318:
2274:
2241:
2206:
2163:
2018:
2005:
1930:special relativity
1898:
1865:
1843:
1619:
1587:
1567:
1543:
1520:
1491:
1455:
1428:
1408:
1388:
1304:
1272:
1229:
1209:
1182:
1116:A one-dimensional
1114:
1042:
1022:
992:, together with a
954:general relativity
950:special relativity
931:special relativity
815:Physics portal
587:OppenheimerâSnyder
527:ReissnerâNordström
419:Linearized gravity
367:Spacetime diagrams
270:Special relativity
203:
132:General relativity
4190:
4189:
4004:
4003:
3983:OzsvĂĄthâSchĂŒcking
3589:
3588:
3571:Minkowski diagram
3528:Thomas precession
3471:Relativistic mass
3230:978-1-906557-00-3
3109:Physical Review D
3065:Nuclear Physics B
2902:(13): 1669â1672.
2679:Science Adventure
2515:comoving observer
2428:
2405:
2204:
1868:{\displaystyle p}
1836:
1805:
1774:
1743:
1645:
1616:
1590:{\displaystyle p}
1570:{\displaystyle p}
1431:{\displaystyle p}
1045:{\displaystyle x}
1025:{\displaystyle t}
1010:Minkowski diagram
1001:coordinate system
972:For example, the
946:Hermann Minkowski
865:
864:
498:
497:
384:
383:
126:
125:
118:
100:
4220:
4182:
4181:
3965:van Stockum dust
3737:Two-body problem
3655:Mach's principle
3602:
3601:
3543:Terrell rotation
3383:
3382:
3361:
3354:
3347:
3338:
3337:
3279:
3269:
3256:
3255:
3253:
3251:
3241:
3235:
3234:
3216:
3210:
3209:
3184:
3178:
3177:
3176:. pp. 1â32.
3158:
3152:
3151:
3125:
3105:
3096:
3090:
3089:
3052:
3046:
3045:
3019:
2997:
2991:
2990:
2988:
2970:
2950:
2934:
2928:
2927:
2886:
2880:
2879:
2851:
2839:
2833:
2832:
2803:
2797:
2796:
2770:
2736:Causal structure
2697:Platonic realism
2442:
2440:
2439:
2434:
2429:
2427:
2419:
2411:
2406:
2404:
2396:
2388:
2344:with respect to
2327:
2325:
2324:
2319:
2283:
2281:
2280:
2275:
2267:
2266:
2250:
2248:
2247:
2242:
2215:
2213:
2212:
2207:
2205:
2203:
2195:
2187:
2172:
2170:
2169:
2164:
2162:
2161:
1999:An example of a
1958:spacetime metric
1907:
1905:
1904:
1899:
1897:
1896:
1874:
1872:
1871:
1866:
1852:
1850:
1849:
1844:
1842:
1838:
1837:
1835:
1827:
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1806:
1804:
1796:
1795:
1794:
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1453:
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1434:
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1409:
1397:
1395:
1394:
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1281:
1279:
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1238:
1236:
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1218:
1216:
1215:
1210:
1208:
1207:
1191:
1189:
1188:
1183:
1141:
1140:
1051:
1049:
1048:
1043:
1031:
1029:
1028:
1023:
960:Usage in physics
925:) more absolute
857:
850:
843:
830:
825:
824:
817:
813:
812:
597:van Stockum dust
582:RobertsonâWalker
408:
407:
298:
297:
212:
210:
209:
204:
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189:
181:
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162:
161:
142:
128:
127:
121:
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107:
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99:
58:
34:
26:
4228:
4227:
4223:
4222:
4221:
4219:
4218:
4217:
4213:Time in science
4193:
4192:
4191:
4186:
4172:
4000:
3904:BKL singularity
3894:LemaĂźtreâTolman
3869:
3865:Quantum gravity
3847:
3841:
3827:geodetic effect
3801:(together with
3771:LISA Pathfinder
3710:
3659:
3645:Penrose diagram
3627:
3621:
3596:
3585:
3581:Minkowski space
3547:
3491:
3475:
3423:
3417:
3377:
3370:
3365:
3314:
3260:
3259:
3249:
3247:
3243:
3242:
3238:
3231:
3217:
3213:
3206:
3185:
3181:
3174:S. Sonnenschein
3159:
3155:
3103:
3097:
3093:
3056:Affleck, Ian K.
3053:
3049:
3010:(2â3): 73â234.
3003:Physics Reports
2998:
2994:
2948:
2935:
2931:
2887:
2883:
2855:Physical Review
2849:
2840:
2836:
2816:Physical Review
2804:
2800:
2793:
2771:
2767:
2762:
2749:Feynman diagram
2719:
2689:Neal Stephenson
2613:science fiction
2609:Oliver Franklin
2548:
2535:
2513:curve admits a
2458:
2420:
2412:
2410:
2397:
2389:
2387:
2385:
2382:
2381:
2380:are related by
2289:
2286:
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2262:
2258:
2256:
2253:
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2221:
2218:
2217:
2196:
2188:
2186:
2178:
2175:
2174:
2157:
2153:
2136:
2133:
2132:
2129:
2040:Minkowski space
1977:Minkowski space
1926:
1892:
1888:
1880:
1877:
1876:
1860:
1857:
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1828:
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1582:
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1506:
1503:
1502:
1476:
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1467:
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1423:
1420:
1419:
1403:
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1399:
1342:
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1332:
1329:
1288:
1267:
1264:
1263:
1224:
1221:
1220:
1203:
1199:
1197:
1194:
1193:
1136:
1132:
1130:
1127:
1126:
1094:
1037:
1034:
1033:
1017:
1014:
1013:
962:
927:position states
861:
820:
807:
806:
799:
798:
622:
621:
612:
611:
567:LemaĂźtreâTolman
512:
511:
500:
499:
491:Quantum gravity
478:Advanced theory
405:
404:
403:
386:
385:
334:Geodetic effect
295:
294:
285:
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260:
259:
243:
213:
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185:
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111:
105:
102:
59:
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35:
24:
17:
12:
11:
5:
4226:
4216:
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4205:
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4177:
4174:
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4170:
4163:
4158:
4153:
4148:
4143:
4138:
4133:
4128:
4123:
4118:
4113:
4108:
4103:
4098:
4093:
4091:Choquet-Bruhat
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4028:
4023:
4018:
4012:
4010:
4006:
4005:
4002:
4001:
3999:
3998:
3991:
3990:
3985:
3980:
3973:
3972:
3967:
3962:
3957:
3952:
3943:Axisymmetric:
3940:
3939:
3934:
3928:
3917:
3916:
3911:
3906:
3901:
3896:
3891:
3882:Cosmological:
3879:
3877:
3871:
3870:
3868:
3867:
3862:
3857:
3851:
3849:
3843:
3842:
3840:
3839:
3834:
3823:frame-dragging
3820:
3815:
3810:
3807:Einstein rings
3803:Einstein cross
3796:
3785:
3784:
3779:
3773:
3768:
3763:
3750:
3740:
3739:
3734:
3729:
3724:
3718:
3716:
3712:
3711:
3709:
3708:
3706:Ernst equation
3703:
3698:
3693:
3688:
3683:
3678:
3676:BSSN formalism
3673:
3667:
3665:
3661:
3660:
3658:
3657:
3652:
3647:
3642:
3637:
3631:
3629:
3623:
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3614:
3608:
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3599:
3591:
3590:
3587:
3586:
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3583:
3578:
3573:
3568:
3563:
3557:
3555:
3549:
3548:
3546:
3545:
3540:
3535:
3533:Ladder paradox
3530:
3525:
3520:
3515:
3510:
3505:
3499:
3497:
3493:
3492:
3490:
3489:
3483:
3481:
3477:
3476:
3474:
3473:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3436:Speed of light
3433:
3427:
3425:
3419:
3418:
3416:
3415:
3410:
3405:
3399:
3389:
3387:
3380:
3372:
3371:
3364:
3363:
3356:
3349:
3341:
3335:
3334:
3326:
3325:
3313:
3312:External links
3310:
3309:
3308:
3293:
3292:
3291:
3290:
3288:Space and Time
3281:
3280:
3258:
3257:
3236:
3229:
3223:. Epic Press.
3211:
3204:
3198:. p. 19.
3179:
3153:
3123:hep-th/0507174
3116:(10): 105004.
3091:
3072:(3): 509â519.
3047:
3017:hep-th/0101036
2992:
2968:hep-ph/9602280
2929:
2881:
2862:(3): 440â457.
2834:
2823:(1): 108â128.
2798:
2791:
2783:Academic Press
2764:
2763:
2761:
2758:
2757:
2756:
2754:Time geography
2751:
2746:
2745:
2744:
2742:Isotropic line
2739:
2733:
2728:
2718:
2715:
2711:Charles Stross
2673:, produced by
2644:
2643:
2639:
2638:
2598:
2597:
2584:
2583:
2564:
2563:
2547:
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2457:
2454:
2432:
2426:
2423:
2418:
2415:
2409:
2403:
2400:
2395:
2392:
2340:is called the
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2036:
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2029:
1993:
1992:
1946:speed of light
1936:is limited to
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1555:tangent vector
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1256:(time) history
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1021:
961:
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937:interactions.
907:orbit in space
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470:Post-Newtonian
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329:Frame-dragging
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304:Kepler problem
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4185:
4175:
4169:
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4159:
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4149:
4147:
4144:
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4119:
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4109:
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4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
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4076:Chandrasekhar
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4036:Schwarzschild
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4017:
4014:
4013:
4011:
4007:
3997:
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3922:Schwarzschild
3919:
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3900:
3897:
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3821:
3819:
3818:Shapiro delay
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3808:
3804:
3800:
3797:
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3790:
3787:
3786:
3783:
3780:
3777:
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3764:
3762:
3761:collaboration
3758:
3754:
3751:
3749:
3745:
3742:
3741:
3738:
3735:
3733:
3730:
3728:
3727:Event horizon
3725:
3723:
3720:
3719:
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3713:
3707:
3704:
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3699:
3697:
3694:
3692:
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3674:
3672:
3671:ADM formalism
3669:
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3576:Biquaternions
3574:
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3506:
3504:
3503:Time dilation
3501:
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3498:
3494:
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3485:
3484:
3482:
3478:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3456:Proper length
3454:
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3306:
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3273:
3268:
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3222:
3215:
3207:
3205:0-7727-1600-5
3201:
3197:
3193:
3189:
3183:
3175:
3171:
3167:
3163:
3162:Hinton, C. H.
3157:
3149:
3145:
3141:
3137:
3133:
3129:
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3119:
3115:
3111:
3110:
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3018:
3013:
3009:
3005:
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2794:
2792:9780080918631
2788:
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2446:Coulomb's law
2430:
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2033:
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2027:
2023:
2020:
2019:
2015:
2009:
2002:
1997:
1989:
1986:
1985:
1984:
1980:
1978:
1974:
1969:
1967:
1963:
1962:metric spaces
1959:
1955:
1951:
1950:bilinear form
1947:
1943:
1939:
1935:
1931:
1921:
1919:
1918:tangent space
1916:, termed the
1915:
1909:
1893:
1889:
1885:
1882:
1862:
1853:
1839:
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1648:
1639:
1610:
1600:
1599:four-velocity
1584:
1564:
1556:
1540:
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1511:
1488:
1482:
1477:
1473:
1450:
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1123:
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1103:
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1071:
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1063:
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1039:
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991:
986:
984:
980:
975:
970:
967:
957:
955:
951:
947:
943:
938:
936:
935:gravitational
932:
928:
924:
920:
916:
912:
908:
904:
900:
895:
893:
889:
885:
882:
878:
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858:
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846:
844:
839:
838:
836:
835:
829:
819:
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811:
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803:
802:
795:
794:
790:
788:
785:
783:
780:
778:
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773:
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768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
732:Chandrasekhar
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
647:Schwarzschild
645:
643:
640:
638:
635:
633:
630:
628:
625:
624:
616:
615:
608:
607:HartleâThorne
605:
603:
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598:
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588:
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583:
580:
578:
575:
573:
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553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
522:
518:
517:Schwarzschild
515:
514:
510:
504:
503:
492:
489:
487:
484:
483:
482:
481:
477:
476:
471:
468:
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378:
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373:
370:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
345:
342:
340:
339:Event horizon
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
301:
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218:
217:
216:
198:
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191:
186:
182:
177:
174:
170:
163:
158:
155:
151:
141:
137:
136:
133:
130:
129:
120:
117:
109:
106:November 2023
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: â
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
4166:
3860:KaluzaâKlein
3612:Introduction
3565:
3538:Twin paradox
3327:
3300:
3275:
3271:
3248:. Retrieved
3239:
3220:
3214:
3191:
3182:
3169:
3156:
3113:
3107:
3094:
3069:
3063:
3050:
3007:
3001:
2995:
2958:
2952:
2942:Dixon, Lance
2932:
2899:
2893:
2884:
2859:
2853:
2837:
2820:
2814:
2801:
2778:
2768:
2708:
2705:
2687:
2667:visual novel
2664:
2657:
2647:
2645:
2623:
2621:
2616:
2611:published a
2606:
2599:
2587:
2585:
2568:J. C. Fields
2565:
2552:C. H. Hinton
2549:
2536:
2524:
2459:
2377:
2373:
2369:
2361:
2357:
2353:
2345:
2341:
2337:
2329:
2130:
2117:
2101:
2094:
2090:
2081:
2077:
2065:
2054:
2047:
2037:
2031:
2025:
2021:
1987:
1981:
1970:
1965:
1957:
1937:
1927:
1914:linear space
1910:
1854:
1554:
1330:
1321:
1315:
1313:
1309:
1305:
1299:
1298:is time and
1295:
1255:
1251:
1247:
1243:
1241:
1121:
1117:
1115:
1085:
1083:
1075:
1065:
1057:
987:
978:
973:
971:
963:
939:
919:perceptually
914:
910:
906:
896:
872:
868:
866:
792:
752:Raychaudhuri
274:
221:Introduction
112:
103:
93:
86:
79:
72:
65:"World line"
60:
48:Please help
43:verification
40:
21:Worldline SA
3949:KerrâNewman
3920:Spherical:
3789:Other tests
3732:Singularity
3664:Formulation
3626:Fundamental
3480:Formulation
3461:Proper time
3422:Fundamental
3320:article on
3318:World lines
3250:8 September
3221:World Lines
2961:: 109â148.
2671:Steins;Gate
2654:James Blish
2646:Heinlein's
2617:World Lines
2602:time travel
2558:. He wrote
2356:depends on
1991:suppressed.
1954:real number
1260:proper time
1078:world sheet
1066:proper time
1062:proper time
881:dimensional
767:van Stockum
697:Oppenheimer
552:KerrâNewman
344:Singularity
4197:Categories
4101:Zel'dovich
4009:Scientists
3988:Alcubierre
3795:of Mercury
3793:precession
3722:Black hole
3605:Background
3597:relativity
3566:World line
3561:Light cone
3386:Background
3378:relativity
3368:Relativity
2760:References
2701:nominalism
2360:. Indeed,
2334:null space
2070:light cone
2059:light cone
2032:space-like
2001:light cone
1988:light-like
1252:world line
1244:world line
1086:world line
994:continuous
979:world line
942:physicists
923:relatively
911:trajectory
903:trajectory
869:world line
620:Scientists
452:Formalisms
400:Formalisms
349:Black hole
275:World line
76:newspapers
4071:Robertson
4056:Friedmann
4051:Eddington
4041:de Sitter
3875:Solutions
3753:detectors
3748:astronomy
3715:Phenomena
3650:Geodesics
3553:Spacetime
3496:Phenomena
3303:, p 130,
3148:119357180
3042:118891361
2938:Bern, Zvi
2890:Bern, Zvi
2726:Geodesics
2691:'s novel
2683:Dirac Sea
2628:, author
2625:Life-Line
2527:geodesics
2490:spacelike
2486:lightlike
2470:spacetime
2425:τ
2402:τ
2298:η
2295:↦
2269:→
2224:η
2201:τ
2151:∈
2145:τ
2095:elsewhere
2082:elsewhere
2078:Elsewhere
2066:lightcone
2022:time-like
2014:spacetime
1973:geodesics
1934:spacetime
1890:τ
1883:τ
1833:τ
1802:τ
1771:τ
1740:τ
1643:→
1614:→
1541:τ
1538:Δ
1515:→
1512:τ
1509:Δ
1489:τ
1486:Δ
1474:τ
1447:τ
1406:τ
1352:τ
1316:collision
1302:distance.
1270:τ
1227:τ
1146:τ
1102:particles
1070:free fall
966:spacetime
884:spacetime
873:worldline
712:Robertson
677:Friedmann
672:Eddington
662:Nordström
652:de Sitter
509:Solutions
434:Geodesics
429:Friedmann
411:Equations
397:Equations
358:Spacetime
293:Phenomena
199:ν
196:μ
187:κ
178:ν
175:μ
167:Λ
159:ν
156:μ
4184:Category
4061:LemaĂźtre
4026:Einstein
4016:Poincaré
3976:Others:
3960:TaubâNUT
3926:interior
3848:theories
3846:Advanced
3813:redshift
3628:concepts
3446:Rapidity
3424:concepts
3190:(1979).
3164:(1884).
2924:10043277
2845:(1951).
2809:(1950).
2717:See also
2550:In 1884
2511:timelike
2502:timelike
2498:timelike
2494:timelike
2488:(null),
2108:that is
1875:, so at
1006:manifold
901:" or a "
828:Category
692:LemaĂźtre
657:Reissner
642:Poincaré
627:Einstein
572:TaubâNUT
537:Wormhole
521:interior
234:Timeline
4126:Hawking
4121:Penrose
4106:Novikov
4086:Wheeler
4031:Hilbert
4021:Lorentz
3978:pp-wave
3799:lensing
3595:General
3376:Special
3299:(1914)
3278:: 75â88
3128:Bibcode
3074:Bibcode
3022:Bibcode
2973:Bibcode
2904:Bibcode
2864:Bibcode
2693:Anathem
2607:Author
2570:at the
2472:can be
2364:is the
2332:be the
2118:present
2091:present
1938:special
1192:(where
1106:strings
983:helical
909:or the
888:physics
747:Hawking
742:Penrose
717:Bardeen
707:Wheeler
637:Hilbert
632:Lorentz
592:pp-wave
229:History
90:scholar
4167:others
4156:Thorne
4146:Misner
4131:Taylor
4116:Geroch
4111:Ehlers
4081:Zwicky
3899:Kasner
3227:
3202:
3146:
3040:
2922:
2789:
2669:named
2492:, and
2478:metric
2474:curved
2348:. The
2048:future
1110:branes
1108:, and
998:smooth
990:events
826:
793:others
787:Thorne
777:Newman
757:Taylor
737:Ehlers
722:Walker
687:Zwicky
562:Kasner
92:
85:
78:
71:
63:
4161:Weiss
4141:Bondi
4136:Hulse
4066:Milne
3970:discs
3914:Milne
3909:Gödel
3766:Virgo
3144:S2CID
3118:arXiv
3104:(PDF)
3038:S2CID
3012:arXiv
2963:arXiv
2949:(PDF)
2850:(PDF)
2328:Let
2106:space
1122:curve
974:orbit
899:orbit
762:Hulse
702:Gödel
682:Milne
577:Milne
542:Gödel
239:Tests
97:JSTOR
83:books
4096:Kerr
4046:Weyl
3945:Kerr
3805:and
3759:and
3757:LIGO
3322:h2g2
3252:2010
3225:ISBN
3200:ISBN
2920:PMID
2787:ISBN
2699:and
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