Spacetime diagram - Knowledge (XXG)

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located on a straight line parallel to its space axis. This line passes through A and B, so A and B are simultaneous from the reference frame of the observer with black axes. However, the clock that is moving relative to the black observer marks off time along the blue time axis. This is represented by the distance from O to B. Therefore, the observer at A with the black axes notices their clock as reading the distance from O to A while they observe the clock moving relative him or her to read the distance from O to B. Due to the distance from O to B being smaller than the distance from O to A, they conclude that the time passed on the clock moving relative to them is smaller than that passed on their own clock.

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any inertial reference frame. Therefore, any point above the origin and between the world lines of both photons can be reached with a speed smaller than that of the light and can have a cause-and-effect relationship with the origin. This area is the absolute future, because any event there happens later compared to the event represented by the origin regardless of the observer, which is obvious graphically from the Minkowski diagram in Fig 4-6.

358:, position vs. time graphs (called x-t graphs for short) provide a useful means to describe motion. Kinematic features besides the object's position are visible by the slope and shape of the lines. In Fig 1-1, the plotted object moves away from the origin at a positive constant velocity (1.66 m/s) for 6 seconds, halts for 5 seconds, then returns to the origin over a period of 7 seconds at a non-constant speed (but negative velocity). 5275: 241: 56: 1536: 1522: 4422: 828: 2792:-axis and a parallel line passing through C and D he concludes the same way this object to be contracted from OD to OC. Each observer estimates objects moving with the other observer to be contracted. This apparently paradoxical situation is again a consequence of the relativity of simultaneity as demonstrated by the analysis via Minkowski diagram. 1353: 2728: 2691: 2346: 1855: 3602:

Minkowski's insight is central to the understanding of the physical world. It focuses attention on those quantities, such as interval, which are the same in all frames of reference. It brings out the relative character of quantities, such as velocity, energy, time, distance, which depend on the frame

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system, who sends it back, again faster than light, arriving at B. But B is in the past relative to O. The absurdity of this process becomes obvious when both observers subsequently confirm that they received no message at all, but all messages were directed towards the other observer as can be seen

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Straight lines passing the origin which are steeper than both photon world lines correspond with objects moving more slowly than the speed of light. If this applies to an object, then it applies from the viewpoint of all observers, because the world lines of these photons are the angle bisectors for

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Several authors showed that there is a frame of reference between the resting and moving ones where their symmetry would be apparent ("median frame"). In this frame, the two other frames are moving in opposite directions with equal speed. Using such coordinates makes the units of length and time the

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axis (not drawn) is horizontal. The dashed line is the spacetime trajectory ("world line") of the particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and they intersect at that event.

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Consider the animation in Fig 5-1. The curved line represents the world line of a particle that undergoes continuous acceleration, including complete changes of direction in the positive and negative x-directions. The red axes are the axes of the MCRF for each point along the particle's trajectory.

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Another postulate of special relativity is the constancy of the speed of light. It says that any observer in an inertial reference frame measuring the vacuum speed of light relative to themself obtains the same value regardless of his own motion and that of the light source. This statement seems to

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A second observer, having moved together with the clock from O to B, will argue that the black axis clock has only reached C and therefore runs slower. The reason for these apparently paradoxical statements is the different determination of the events happening synchronously at different locations.

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Further coordinate systems corresponding to observers with arbitrary velocities can be added to this Minkowski diagram. For all these systems both photon world lines represent the angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the

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In Fig 4-2, the observer whose reference frame is given by the black axes is assumed to move from the origin O towards A. The moving clock has the reference frame given by the blue axes and moves from O to B. For the black observer, all events happening simultaneously with the event at A are

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frame axes are warped by the same factor relative to the median frame and hence have identical unit lengths. This implies that, for a Loedel spacetime diagram, we can directly compare spacetime lengths between different frames as they appear on the page; no unit length scaling/conversion between

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because they have a finite spatial distance different from zero for all observers. On the other hand, a straight line connecting such events is always the space coordinate axis of a possible observer for whom they happen at the same time. By a slight variation of the velocity of this coordinate

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The second observer will argue that the first observer has evaluated the endpoints of the object at O and A respectively and therefore at different times, leading to a wrong result due to his motion in the meantime. If the second observer investigates the length of another object with endpoints

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Given an object moving faster than light, say from O to A in Fig 4-7, then for any observer watching the object moving from O to A, another observer can be found (moving at less than the speed of light with respect to the first) for whom the object moves from A to O. The question of which

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Let's take the example of the fall of an object dropped without initial velocity from a rocket. The rocket has a uniformly accelerated motion with respect to an inertial reference frame. As can be seen from Fig 6-2 of a Minkowski diagram in a non-inertial reference frame, the object once

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The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. Bends in the world line represent particle acceleration. As the particle accelerates, its view of spacetime changes. These changes in view are governed by the Lorentz

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in collaboration with Josef Sauter in two papers in 1921. Relativistic effects such as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them. Gruner extended this method in subsequent papers (1922–1924), and gave credit to

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These considerations show that the speed of light as a limit is a consequence of the properties of spacetime, and not of the properties of objects such as technologically imperfect space ships. The prohibition of faster-than-light motion, therefore, has nothing in particular to do with

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in its rest frame) that moves relative to an observer is observed to contract/shorten. The situation is depicted in symmetric Loedel diagrams in Fig 4-3. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram.

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in its rest frame) that moves relative to an observer is observed to run slower. The situation is depicted in the symmetric Loedel diagrams of Fig 4-1. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram.

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Gruner, Paul (1924). "Geometrische Darstellungen der speziellen Relativitätstheorie, insbesondere des elektromagnetischen Feldes bewegter Körper" [Geometrich representations of the special theory of relativity, in particular the electromagnetic field of moving bodies].

2955:, because they have a time distance greater than zero for all observers. A straight line connecting these two events is always the time axis of a possible observer for whom they happen at the same place. Two events which can be connected just with the speed of light are called 3153:

If one imagines each event to be the flashing of a light, then the events that are within the past light cone of the observer are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the velocity relative to the observer.

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At its most basic level, a spacetime diagram is merely a time vs position graph, with the directions of the axes in a usual p-t graph exchanged; that is, the vertical axis refers to temporal and the horizontal axis to spatial coordinate values. Especially when used in

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graphically in the Minkowski diagram. Furthermore, if it were possible to accelerate an observer to the speed of light, their space and time axes would coincide with their angle bisector. The coordinate system would collapse, in concordance with the fact that due to

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Determining position and time of the event A as an example in the diagram leads to the same time for both observers, as expected. Only for the position different values result, because the moving observer has approached the position of the event A since

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to each other and the scale on their time axis is stretched. To determine the coordinates of a certain event, two lines, each parallel to one of the two axes, must be constructed passing through the event, and their intersections with the axes read off.

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It is often, incorrectly, asserted that special relativity cannot handle accelerating particles or accelerating reference frames. In reality, accelerating particles present no difficulty at all in special relativity. On the other hand, accelerating

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While a frame at rest in a Minkowski diagram has orthogonal spacetime axes, a frame moving relative to the rest frame in a Minkowski diagram has spacetime axes which form an acute angle. This asymmetry of Minkowski diagrams can be misleading, since

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Gruner, Paul (1921). "Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie" [An elementary geometric representation of the transformation formulae of the special theory of relativity].

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regardless of the speed of the observer. Therefore, no event outside the light cones can be reached from the origin, even by a light-signal, nor by any object or signal moving with less than the speed of light. Such pairs of events are called

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Gruner, Paul (1922). "Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt II" [Graphical representation of the special theory of relativity in the four-dimensional spacetime world II].

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For all these considerations it was assumed, that both observers take into account the speed of light and their distance to all events they see in order to determine the actual times at which these events happen from their point of view.

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Gruner, Paul (1922). "Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt I" [Graphical representation of the special theory of relativity in the four-dimensional spacetime world I].

3635: 4327: 634: 2649:(1921) showed that there is always a median frame with respect to two relatively moving frames, and derived the relations between them from the Lorentz transformation. However, he didn't give a graphical representation in a diagram. 2947:

Following the same argument the range below the origin and between the photon world lines is the absolute past relative to the origin. Any event there belongs definitely to the past and can be the cause of an effect at the origin.

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In Minkowski's 1908 paper there were three diagrams, first to illustrate the Lorentz transformation, then the partition of the plane by the light-cone, and finally illustration of worldlines. The first diagram used a branch of the

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do require some special treatment, However, as long as one is dealing with flat, Minkowskian spacetime, special relativity can handle the situation. It is only in the presence of gravitation that general relativity is required.

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An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity. This is not a difficult situation to handle. Accelerating frames require that one understand the concept of a

3949: 2552: 1267: 582:, modelling the existence of one common position axis. On the other hand, due to two different time axes the observers usually measure different coordinates for the same event. This graphical translation from 328:

consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space

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axis is always more flat and the time axis more steep than the photon world lines. The scales on both axes are always identical, but usually different from those of the other coordinate systems.

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The term Minkowski diagram refers to a specific form of spacetime diagram frequently used in special relativity. A Minkowski diagram is a two-dimensional graphical depiction of a portion of

867:. This follows from the second postulate of special relativity, which says that the speed of light is the same for all observers, regardless of their relative motion (see below). The angle 2341:{\displaystyle {\begin{aligned}\sin \varphi =\cos \theta &=\beta ,\\\cos \varphi =\sin \theta &={\frac {1}{\gamma }},\\\tan \varphi =\cot \theta &=\beta \gamma .\end{aligned}}} 2986:

Following the same argument, all straight lines passing through the origin and which are more nearly horizontal than the photon world lines, would correspond to objects or signals moving

409:(i.e., conventional 3-space frames), S and S′ (pronounced "S prime"), each with observers O and O′ at rest in their respective frames, but measuring the other as moving with speeds ± 545:

axis, which is identical for both observers, it represents their coordinate system. Since the reference frames are in standard configuration, both observers agree on the location of the

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Fig 4-7 Sending a message at superluminal speed from O via A to B into the past. Both observers consider the temporal order of the pairs of events O and A as well as A and B different.

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are the unit lengths of the rest frame axes and moving frame axes, respectively, in a Minkowski diagram, then the two unit lengths are warped relative to each other via the formula:

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In a Minkowski diagram, lengths on the page cannot be directly compared to each other, due to warping factor between the axes' unit lengths in a Minkowski diagram. In particular, if

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Also, any general technical means of sending signals faster than light would permit information to be sent into the originator's own past. In the diagram, an observer at O in the

1850:{\displaystyle {\begin{aligned}&(1)&\beta &={\frac {2\beta _{0}}{1+{\beta _{0}}^{2}}},\\&(2)&\beta _{0}&={\frac {\gamma -1}{\beta \gamma }}.\end{aligned}}} 1428:

depending on velocity, thus illustrating time dilation. The second diagram showed the conjugate hyperbola to calibrate space, where a similar stretching leaves the impression of

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In principle a further dimension of space can be added to the Minkowski diagram leading to a three-dimensional representation. In this case the ranges of future and past become

963: 2872:-axes of equal absolute value. From the rule for reading off coordinates in coordinate system with tilted axes follows that the two world lines are the angle bisectors of the 3075:

The coordinates of events in the unprimed (stationary) frame can be related to their coordinates in any momentarily co-moving primed frame using the Lorentz transformations.

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Fig 6-1 Minkowski diagram in an inertial reference frame. On the left, the vertical world line of the falling object. On the right, the hyperbolic world line of the rocket.

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time axis. Each parallel line to this axis would correspond also to an object at rest but at another position. The blue line describes an object moving with constant speed

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Fig 4-4 Relativistic length contraction, as depicted in a single Loedel spacetime diagram. Both observers consider objects moving with the other observer as being shorter.

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system in both directions it is always possible to find two inertial reference frames whose observers estimate the chronological order of these events to be different.

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Fig 4-3 Relativistic length contraction, as depicted in two Loedel spacetime diagrams. Both observers consider objects moving with the other observer as being shorter.

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In a further step of simplification it is often sufficient to consider just the direction of the observed motion and ignore the other two spatial components, allowing

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Fig 6-2 Minkowski diagram in a non-inertial reference frame. On the left, the world line of the falling object. On the right, the vertical world line of the rocket.

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However, it turns out that when drawing such a symmetric diagram, it is possible to derive the diagram's relations even without mentioning the median frame and

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must be physically equivalent. The Loedel diagram is an alternative spacetime diagram that makes the symmetry of inertial references frames much more manifest.

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be paradoxical, but it follows immediately from the differential equation yielding this, and the Minkowski diagram agrees. It explains also the result of the

3938:[a) Graphical representation of the four-dimensional spacetime universe. b) Graphical representation of universal time in the theory of relativity]. 3936:"a) Représentation graphique de l'univers espace-temps à quatre dimensions. b) Représentation graphique du temps universel dans la théorie de la relativité" 1491:

for their spacetime geometry. Instead they included an acknowledgement of Minkowski's contribution to philosophy by the totality of his innovation of 1908.

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events which were simultaneous before an acceleration (horizontally spaced events) are at different times afterwards due to the relativity of simultaneity,

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which was considered to be a mystery before the theory of relativity was discovered, when photons were thought to be waves through an undetectable medium.

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observer is right has no unique answer, and therefore makes no physical sense. Any such moving object or signal would violate the principle of causality.

1444:. Since the 1960s a version of this more complete configuration has been referred to as The Minkowski Diagram, and used as a standard illustration of the 5260: 5049: 4795: 184: 3050: 2702:

Fig 4-2. Relativistic time dilation, as depicted in a single Loedel spacetime diagram. Both observers consider the clock of the other as running slower.

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Fig 4-1. Relativistic time dilation, as depicted in two Loedel spacetime diagrams. Both observers consider the clock of the other as running slower.

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events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and

2643:-axis, in order to demonstrate length contraction and time dilation in the symmetric case of two rods and two clocks moving in opposite direction. 637:

Fig 2-2 Minkowski diagram for various speeds of the primed frame, which is moving relative to the unprimed frame. The dashed lines represent the

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Special relativity, a geometric approach: course with exercises and answers followed by the conference "Interstellar travel and antimatter"

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are O and A. For a second observer moving together with the object, so that for him the object is at rest, it has the proper length OB at

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The construction of symmetric Minkowski diagrams was later independently rediscovered by several authors. For instance, starting in 1948,

1195: 273: 5077: 4706: 4351: 3657: 3412: 124: 1578: 1079: 4711: 4406: 3786: 3691: 3336: 204: 649:, usually where space has been curtailed to a single dimension. The units of measurement in these diagrams are taken such that the 5031: 333:

are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.

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This spatial setting is displayed in the Fig 1-2, in which the temporal coordinates are separately annotated as quantities

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The history of an object's location through time traces out a line or curve on a spacetime diagram, referred to as the object's

4744: 730:-axis. Both the original set of axes and the primed set of axes have the property that they are orthogonal with respect to the 3966:[The importance of "reduced" orthogonal coordinate-systems for tensor analysis and the special theory of relativity]. 4847: 4925: 4245: 3576: 1436:

included a diagram of "Minkowski's representation of the Lorentz transformation". This diagram included the unit hyperbola,

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Fig 4-6 Past and future relative to the origin. For the grey areas a corresponding temporal classification is not possible.

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also published a paper presenting such relations, and gave credit to Loedel in a subsequent paper in 1957. Some authors of

5278: 5020: 4855: 3964:"Die Bedeutung "reduzierter" orthogonaler Koordinatensysteme für die Tensoranalysis und die spezielle Relativitätstheorie" 3013:

system sends a message moving faster than light to A. At A, it is received by another observer, moving so as to be in the

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are deformed according to the position. In an inertial reference frame a free particle has a straight world line. In a

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As illustrated in Fig 2-3, the boosted and unboosted spacetime axes will in general have unequal unit lengths. If

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Fig 5-2 The momentarily co-moving inertial frames along the world line of a rapidly accelerating observer (origin).

4084: 4080: 3812:]. Naturwissenschaftliche monographien und lehrbücher ... 3. Bd. (First ed.). Springer. pp. 177–180. 3045:

Fig 5-1 The momentarily co-moving reference frames of an accelerating particle as observed from a stationary frame

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Fig 5-2 illustrates the changing views of spacetime along the world line of a rapidly accelerating particle. The

1505: 5225: 4775: 4535: 4446: 3542:(1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics". 3162: 406: 134: 4954: 4876: 4496: 4487: 3140:

the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.

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An elementary geometrical representation of the transformation formulas of the special theory of relativity

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dropped, gains speed, reaches a maximum, and then sees its speed decrease and asymptotically cancel on the

486: 5180: 4907: 4581: 4545: 3041: 2757:-axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the 2661: 1429: 1066:{\displaystyle {\begin{aligned}ct'&=\gamma (ct-\beta x),\\x'&=\gamma (x-\beta ct)\\\end{aligned}}} 954: 665: 629:

Fig 2-1 In the theory of relativity each observer assigns the event at A to a different time and location.

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A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

2884:-axes. As shown in Fig 4-5, the Minkowski diagram illustrates them as being angle bisectors of the 2719:

Due to the principle of relativity, the question of who is right has no answer and does not make sense.

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published a series of papers in Spanish language, presenting the details of such an approach. In 1955,

2764:-axis and the parallel line passing through A and B. For this observer the endpoints of the object at 2698: 490:

Fig 1-3. In Newtonian physics for both observers the event at A is assigned to the same point in time.

5090: 5016: 4860: 4462: 4140: 4105: 4019: 3975: 3900: 3856: 3445: 766: 546: 382:>, in accordance with the dimension associated with the spatial axis, which is frequently labeled 330: 5165: 5160: 5155: 5150: 4982: 4931: 4749: 4734: 4560: 4491: 4439: 3361: 3049: 2978: 2808:

Fig 4-5 Minkowski diagram for 3 coordinate systems. For the speeds relative to the system in black

1441: 1437: 1381: 1147: 5054: 3521: 3226: 625: 5230: 5130: 4912: 4893: 4887: 4838: 4780: 4689: 4607: 4525: 4502: 4470: 4400: 4035: 3991: 3916: 3872: 3559: 2580: 2452: 2121: 2022: 1968: 1890: 1664: 1501: 1469: 1433: 1364: 363: 310: 299: 259: 245: 209: 71: 66: 2935: 2804: 2665: 4131:

Amar, Henri; Loedel, Enrique (1957). "Geometric Representation of the Lorentz Transformation".

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plus or minus one through that event. The horizontal lines correspond to the usual notion of

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in 1908. Minkowski diagrams are two-dimensional graphs that depict events as happening in a

19: 4382: 3935: 3714:[Elementary geometric representation of the formulae of the theory of relativity]. 3711: 3621: 819:

With that, light paths are represented by lines parallel to the bisector between the axes.

718:. In the new frame of reference the simultaneous events lie parallel to a line inclined by 5220: 5195: 5120: 5115: 4998: 4959: 4921: 4865: 4739: 4675: 3535: 3280: 3275: 3149:

the world line always remains within the future and past light cones of the current event.

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electromagnetic waves or light, but comes as a consequence of the structure of spacetime.

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representing the duration between two events happening on this worldline, also called the

646: 104: 5003: 4368: 3515: 3081: 4358: 4144: 4109: 4023: 3979: 3904: 3860: 3449: 3109: 366:(SR), the temporal axes of a spacetime diagram are often scaled with the speed of light 5245: 4917: 4901: 4897: 4800: 4770: 4627: 4530: 4278: 4222: 3771: 3726:

Elementary geometric representation of the formulas of the special theory of relativity

3594: 3267: 2910: 2560: 2432: 2101: 2002: 1948: 1870: 1644: 1376: 1143: 232: 290:

is a graphical illustration of locations in space at various times, especially in the

5291: 5235: 5215: 5210: 5125: 4993: 4821: 4765: 4597: 4550: 4039: 3995: 3920: 3876: 3579:, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by 3500: 3220: 3026: 2744: 1473: 1307: 550: 295: 3712:"Représentation géométrique élémentaire des formules de la théorie de la relativité" 405:, compare with each other, it is useful to standardize and simplify the setup. Two 5255: 5175: 5140: 4670: 4632: 3465: 812: 401:

To ease insight into how spacetime coordinates, measured by observers in different

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Amar, Henri (1955). "New Geometric Representation of the Lorentz Transformation".

690:-axis. As shown in Fig 2-1, the new time axis of the observer forms an angle 55: 3803: 3622:"La transformation de Lorentz-Einstein et le temps universel de M. Ed. Guillaume" 3374: 5039: 5008: 4555: 2967: 2743:

Relativistic length contraction refers to the fact that a ruler (indicating its

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can directly be used in the following construction, providing the same result:

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axis happen simultaneously for both observers. There is only one universal time

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axes of frame S are oriented parallel to the respective primed axes of frame S′.

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may be interpreted as the time axis for the second observer. Together with the

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Fig 1-2. Galilean diagram of two frames of reference in standard configuration.

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Mermin, N. David (1968). "17: Minkowski diagrams: The Geometry of Spacetime".

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holds. That means any position on such a world line corresponds with steps on

803: 792: 650: 638: 355: 306: 4269: 3458: 3250:. The velocity is measured by an observer at rest in the accelerated rocket. 1688:, then these expressions are connected with the values in their median frame 4647: 1456:

is tantamount to the arbitrariness of what hyperbola radius is selected for

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The Lorentz–Einstein transformation and the universal time of Ed. Guillaume

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Relativistic time dilation refers to the fact that a clock (indicating its

4421: 3483: 5190: 4540: 4229:. The MIT introductory physics series. New York: Norton pp. 82–83. 4194: 3285: 2669: 2614: 2547:{\displaystyle U^{\prime }=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}} 681: 506:

on Fig 1-3 are the coordinate system of an observer, referred to as

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to be plotted in 2-dimensional spacetime diagrams, as introduced above.

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frames is necessary due to the symmetric nature of the Loedel diagram.

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This changes the dimension of the addressed physical quantity from <

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Understanding Relativity: A simplified approach to Einstein's theories

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For world lines of photons passing the origin in different directions

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of their coordinate systems. The axes for the moving observer are not

4870: 4311:(3. impr., rev ed.). London: Butterworth. page 256, Figure 6.4. 3652:(Reprint of 1975 ed.). Courier Dover Publications. p. 460. 3555: 294:. Spacetime diagrams can show the geometry underlying phenomena like 24: 3963: 3950:

Graphical representation of the four-dimensional space-time universe

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with respect to the orthogonal axes of the median frame (Fig. 3-1).

4261: 1262:{\displaystyle U'=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}\,.} 3169: 3161: 3048: 3040: 2977: 2934: 2803: 2734: 2726: 2697: 2689: 1351: 826: 654: 632: 624: 485: 392: 345: 18: 3777:(Reprint of 1968 ed.). Courier Dover Publications. pp. 4851: 2897:-axes as well. That means both observers measure the same speed 1634:{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}} 1457: 1135:{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}} 807: 4435: 3405:

Special Relativity for Beginners: A Textbook for Undergraduates

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with apexes touching each other at the origin. They are called

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Reprint (2013) of third edition (1922) at Google books, p. 187

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The small dots are other arbitrary events in the spacetime.

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and intervals of about 8 minutes and 19 seconds (499 seconds)

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The most well-known class of spacetime diagrams are known as

4335:(2. ed., 9. printing ed.). New York, NY: W.H. Freeman. 3178:

The photon world lines are determined using the metric with

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In Fig 4-4, the observer is assumed again to move along the

2589: 2491: 2461: 2130: 2031: 1977: 1899: 1673: 765:

One frequently encounters Minkowski diagrams where the time

608:

and vice versa is described mathematically by the so-called

2951:

The relationship between any such pairs of event is called

2351:

Two methods of construction are obvious from Fig. 3-2: the

664:

A particular Minkowski diagram illustrates the result of a

3626:

Archives des sciences physiques et naturelles (Supplement)

1316:. The same interpretation can also be applied to distance 565:. Generally stated, all events on a line parallel to the 3544:

Proceedings of the American Academy of Arts and Sciences

854:

axes will be identical with that between the time axes

724:

to the previous lines of simultaneity. This is the new

3482: 1581: 1082: 4329:

Spacetime physics: introduction to special relativity

4085:, Kapelusz Editorial, Buenos Aires, Argentina (1955). 3229: 3184: 3112: 3084: 2913: 2583: 2563: 2557:

By contrast, in a symmetric Loedel diagram, both the

2485: 2455: 2435: 2209: 2124: 2104: 2046:

is done in accordance with the ordinary method using

2025: 2005: 1971: 1951: 1893: 1873: 1704: 1667: 1647: 1384: 1278:-axis represents the worldline of a clock resting in 1198: 966: 882: 2652:

Symmetric diagrams were systematically developed by

1914:, then by (2) they are moving in their median frame 5103: 4968: 4940: 4809: 4758: 4720: 4699: 4688: 4646: 4590: 4574: 4516: 4480: 4469: 4352:"The non-Euclidean style of Minkowskian relativity" 2782:. the object is contracted for the first observer. 1928:each in opposite directions. On the other hand, if 917:{\displaystyle \tan \alpha ={\frac {v}{c}}=\beta .} 517:. This observer's world line is identical with the 4363:. Oxford University PressOxford. pp. 91–127. 4326:Taylor, Edwin F.; Wheeler, John Archibald (2003). 3770: 3686:. University of California Press. pp. 151ff. 3600:. San Francisco : W. H. Freeman. p. 37. 3593: 3432:[On the electrodynamics of moving bodies] 3379:(illustrated ed.). Springer. pp. 92–93. 3242: 3199: 3123: 3098: 2919: 2596: 2569: 2546: 2468: 2441: 2340: 2137: 2110: 2038: 2011: 1984: 1957: 1906: 1879: 1849: 1680: 1653: 1633: 1416: 1261: 1134: 1065: 953:and vice versa is described mathematically by the 916: 3029:, time would effectively stop passing for them. 1371:providing his graphical representation in 1908. 436:The origins of frames S and S′ coincide at time 4283:. New York : McGraw-Hill. p. 155-199. 4290:Relativity: special, general, and cosmological 2672:use symmetric Minkowski diagrams, denoting as 2617:(1920) drew Minkowski diagrams by placing the 4447: 4168:"Can Special Relativity Handle Acceleration?" 3940:Archives des sciences physiques et naturelles 3716:Archives des sciences physiques et naturelles 3615: 3613: 3611: 3215:the world line of a free particle is curved. 1356:Light cone and hyperbolas in Minkowski (1908) 1175:respectively, the unit length on the axes of 267: 16:Graph of space and time in special relativity 8: 3499:Various English translations on Wikisource: 1945:, then by (1) the relative velocity between 4309:An introduction to the theory of relativity 4696: 4477: 4454: 4440: 4432: 4059:Anales de la Sociedad Cientifica Argentina 3736: 3734: 3705: 3703: 3677: 3675: 3673: 389:Standard configuration of reference frames 274: 260: 38: 4292:(2nd ed.). Oxford University Press. 4202:. Querrien: Mathieu Rouaud. p. 534. 3457: 3234: 3228: 3183: 3111: 3083: 2912: 2588: 2582: 2562: 2534: 2516: 2502: 2490: 2484: 2460: 2454: 2434: 2280: 2210: 2208: 2129: 2123: 2103: 2030: 2024: 2004: 1976: 1970: 1950: 1898: 1892: 1872: 1817: 1804: 1770: 1763: 1758: 1743: 1733: 1705: 1703: 1672: 1666: 1646: 1619: 1615: 1604: 1580: 1402: 1389: 1383: 1255: 1245: 1227: 1213: 1197: 1120: 1116: 1105: 1081: 967: 965: 895: 881: 668:. The Lorentz transformation relates two 661:for a stationary observer at the origin. 529:to the right, such as a moving observer. 3764: 3762: 3331:(3rd ed.). Incomprehensible Books. 3106:axis (not drawn) is vertical, while the 791:Approximately 30 centimetres length and 4246:"Lorentz boosts and Minkowski diagrams" 3476: 3474: 3297: 2072:at all. Instead, the relative velocity 46: 4398: 4174:. University of California, Riverside 4057:[Aberration and Relativity]. 3158:Case of non-inertial reference frames 1337:-axes for clocks and rods resting in 831:Fig 2-3 Different scales on the axes. 653:at an event consists of the lines of 482:Non-relativistic "spacetime diagrams" 160:Newton's law of universal gravitation 7: 4280:Space and time in special relativity 3833:Elements of the theory of relativity 3710:Gruner, Paul; Sauter, Josef (1921). 3430:"Zur Elektrodynamik bewegter Körper" 3306:"What are position vs. time graphs?" 3069:momentarily comoving reference frame 2406:-axis is added perpendicular to the 2357:-axis is drawn perpendicular to the 1306:-axis represents the rest length or 787:. Such a diagram may have units of 451:-direction of frame S with velocity 5026:Tolman–Oppenheimer–Volkoff equation 4979:Friedmann–Lemaître–Robertson–Walker 1476:plane that has Minkowski diagrams. 4369:10.1093/oso/9780198500889.003.0007 3592:Taylor, Edwin F.; Wheeler (1966). 3136:transformations. Also note that: 2907:corresponding angle bisector. The 2681:Relativistic phenomena in diagrams 2624:-axis almost perpendicular to the 1999:. The construction of the axes of 1527:Fig. 3-1: View in the median frame 1460:in the Minkowski diagram. In 1912 1163:is the unit length on the axes of 696:with the previous time axis, with 641:of a flash of light at the origin. 125:Introduction to general relativity 23:The world line (yellow path) of a 14: 4796:Hamilton–Jacobi–Einstein equation 3805:Die Relativitätstheorie Einsteins 3223:where its proper time freezes at 2154:is the angle between the axes of 1479:When Taylor and Wheeler composed 1472:to develop the properties of the 205:Mathematics of general relativity 130:Mathematics of general relativity 5274: 5273: 4420: 3829:Elemente der Relativitätstheorie 3407:. World Scientific. p. 49. 3260: 1534: 1520: 370:, and thus are often labeled by 350:Fig 1-1. Position vs. time graph 337:Introduction to kinetic diagrams 302:without mathematical equations. 240: 239: 226: 54: 4172:The Original Usenet Physics FAQ 3810:Einstein's Theory of Relativity 2800:Constancy of the speed of light 2657:Mirimanoff's treatment as well. 1424:to show the locus of a unit of 4603:Mass–energy equivalence (E=mc) 4307:Rosser, William G. V. (1971). 1795: 1789: 1718: 1712: 1056: 1038: 1010: 992: 354:In the study of 1-dimensional 1: 4405:: CS1 maint: date and year ( 2974:The speed of light as a limit 1417:{\displaystyle t^{2}-x^{2}=1} 1294:between these events. Length 927:The corresponding boost from 4350:Walter, Scott (1999-07-22). 3514:Silberstein, Ludwik (1914). 3376:Mechanics and Thermodynamics 3373:Demtröder, Wolfgang (2016). 3243:{\displaystyle t_{\text{H}}} 3213:non-inertial reference frame 2931:Speed of light and causality 1992:in their own rest frames is 1512:Formulation via median frame 762:, as shown in Fig 2-2. 670:inertial frames of reference 292:special theory of relativity 4618:Relativistic Doppler effect 4250:American Journal of Physics 4244:Glass, E. N. (1975-11-01). 4133:American Journal of Physics 4098:American Journal of Physics 3620:Mirimanoff, Dmitry (1921). 3481:Minkowski, Hermann (1909). 2835:Michelson–Morley experiment 2597:{\displaystyle S^{\prime }} 2469:{\displaystyle U^{\prime }} 2419:-axis perpendicular to the 2138:{\displaystyle S^{\prime }} 2039:{\displaystyle S^{\prime }} 1985:{\displaystyle S^{\prime }} 1907:{\displaystyle S^{\prime }} 1681:{\displaystyle S^{\prime }} 1541:Fig. 3-2: Symmetric diagram 510:, and who is positioned at 440:= 0 in frame S and also at 342:Position versus time graphs 155:Introduction to gravitation 5324: 5089:In computational physics: 4613:Relativity of simultaneity 4288:Rindler, Wolfgang (2001). 4055:"Aberración y Relatividad" 3769:Shadowitz, Albert (1988). 3648:Shadowitz, Albert (2012). 3487:[Space and time]. 769:are scaled by a factor of 744:Whatever the magnitude of 5271: 4926:Lense–Thirring precession 4508:Doubly special relativity 4357:. In Gray, Jeremy (ed.). 3744:Physikalische Zeitschrift 3650:The Electromagnetic Field 3489:Physikalische Zeitschrift 3428:Einstein, Albert (1905). 2387:′-axis is drawn at angle 2377:-axes are added at angle 1506:inertial reference frames 1452:has pointed out that the 407:Galilean reference frames 185:Derivations of relativity 4786:Post-Newtonian formalism 4776:Einstein field equations 4712:Mathematical formulation 4536:Hyperbolic orthogonality 4193:Rouaud, Mathieu (2021). 4053:Loedel, Enrique (1948). 3517:The Theory of Relativity 3459:10.1002/andp.19053221004 3351:Mermin (1968) Chapter 17 3200:{\displaystyle d\tau =0} 1504:postulates that any two 1363:announced his theory of 676:stationary at the event 532:This blue line labelled 494:The black axes labelled 135:Einstein field equations 4497:Galilean transformation 4488:Principle of relativity 4411:(see page 10 of e-link) 3403:Freund, Jürgen (2008). 3327:Collier, Peter (2017). 1549:same for both axes. If 1468:applied the methods of 1454:principle of relativity 1448:of special relativity. 1446:transformation geometry 957:, which can be written 732:Minkowski inner product 610:Galilean transformation 455:as measured in frame S. 100:Lorentz transformations 27:, which is at location 4582:Lorentz transformation 4012:Zeitschrift für Physik 3968:Zeitschrift für Physik 3893:Zeitschrift für Physik 3849:Zeitschrift für Physik 3835:]. Bern: P. Haupt. 3244: 3201: 3175: 3167: 3125: 3100: 3054: 3046: 3037:Accelerating observers 2983: 2940: 2921: 2829: 2740: 2732: 2703: 2695: 2662:Enrique Loedel Palumbo 2630:-axis, as well as the 2598: 2571: 2548: 2470: 2443: 2342: 2139: 2112: 2040: 2013: 1986: 1959: 1908: 1881: 1851: 1682: 1655: 1635: 1430:FitzGerald contraction 1418: 1357: 1263: 1136: 1067: 955:Lorentz transformation 918: 832: 775:such that one unit of 666:Lorentz transformation 642: 630: 491: 447:Frame S′ moves in the 415:standard configuration 398: 351: 36: 5050:Weyl−Lewis−Papapetrou 4791:Raychaudhuri equation 4730:Equivalence principle 4429:at Wikimedia Commons 4360:The Symbolic Universe 3962:Gruner, Paul (1922). 3934:Gruner, Paul (1921). 3827:Gruner, Paul (1922). 3682:Sartori, Leo (1996). 3484:"Raum und Zeit" 3245: 3202: 3173: 3165: 3126: 3101: 3052: 3044: 2981: 2938: 2922: 2807: 2738: 2730: 2701: 2693: 2599: 2572: 2549: 2471: 2444: 2343: 2140: 2113: 2041: 2014: 1987: 1960: 1909: 1882: 1852: 1683: 1656: 1636: 1419: 1355: 1264: 1137: 1068: 919: 830: 636: 628: 489: 396: 349: 195:Differential geometry 95:Equivalence principle 22: 5091:Numerical relativity 4932:pulsar timing arrays 3666:Google books, p. 460 3227: 3182: 3110: 3082: 2911: 2581: 2561: 2483: 2453: 2433: 2393:with respect to the 2207: 2186:between the axes of 2122: 2102: 2023: 2003: 1969: 1949: 1891: 1871: 1702: 1665: 1645: 1579: 1382: 1310:of a rod resting in 1196: 1080: 964: 880: 823:Mathematical details 767:units of measurement 758:forms the universal 331:units of measurement 173:Relevant mathematics 4983:Friedmann equations 4877:Hulse–Taylor binary 4839:Gravitational waves 4735:Riemannian geometry 4561:Proper acceleration 4546:Maxwell's equations 4492:Galilean relativity 4145:1957AmJPh..25..326A 4110:1955AmJPh..23..487A 4024:1924ZPhy...21..366G 3980:1922ZPhy...10..236G 3905:1922ZPhy...10..227G 3861:1922ZPhy...10...22G 3577:Synthetic Spacetime 3466:English translation 3450:1905AnP...322..891E 3362:Vladimir Karapetoff 3099:{\displaystyle ct'} 1921:with approximately 1442:conjugate diameters 1148:conjugate diameters 781:equals one unit of 659:simultaneous events 42:Part of a series on 5298:Special relativity 5032:Reissner–Nordström 4950:Brans–Dicke theory 4781:Linearized gravity 4608:Length contraction 4526:Frame of reference 4503:Special relativity 4427:Minkowski diagrams 4227:Special relativity 4081:Fisica relativista 4032:10.1007/BF01328285 3988:10.1007/BF01332564 3913:10.1007/BF01332563 3869:10.1007/BF01332542 3802:Born, Max (1920). 3773:Special Relativity 3438:Annalen der Physik 3392:Extract of page 93 3240: 3197: 3176: 3168: 3124:{\displaystyle x'} 3121: 3096: 3055: 3047: 2984: 2941: 2917: 2903:for both photons. 2830: 2741: 2733: 2723:Length contraction 2704: 2696: 2594: 2567: 2544: 2466: 2439: 2338: 2336: 2135: 2108: 2036: 2009: 1982: 1955: 1904: 1877: 1847: 1845: 1678: 1651: 1641:are given between 1631: 1502:special relativity 1470:synthetic geometry 1434:Ludwik Silberstein 1414: 1365:special relativity 1358: 1259: 1132: 1063: 1061: 914: 833: 798:Astronomical units 680:makes a change of 643: 631: 616:Minkowski diagrams 492: 444:′ = 0 in frame S′. 413:are said to be in 399: 364:special relativity 352: 318:Minkowski diagrams 300:length contraction 233:Physics portal 210:Spacetime topology 190:Spacetime diagrams 118:General relativity 90:Spacetime manifold 83:Spacetime concepts 72:General relativity 67:Special relativity 37: 5285: 5284: 5099: 5098: 5078:Ozsváth–Schücking 4684: 4683: 4666:Minkowski diagram 4623:Thomas precession 4566:Relativistic mass 4425:Media related to 4378:978-0-19-850088-9 4342:978-0-7167-2327-1 4318:978-0-408-55700-9 4299:978-0-19-856732-5 4256:(11): 1013–1014. 4236:978-0-17-177075-9 4209:978-2-9549309-3-0 4153:10.1119/1.1934453 4118:10.1119/1.1934074 3596:Spacetime physics 3540:Lewis, Gilbert N. 3386:978-3-319-27877-3 3237: 2988:faster than light 2920:{\displaystyle x} 2786:moving along the 2647:Dmitry Mirimanoff 2570:{\displaystyle S} 2542: 2541: 2442:{\displaystyle U} 2288: 2111:{\displaystyle S} 2012:{\displaystyle S} 1958:{\displaystyle S} 1880:{\displaystyle S} 1860:For instance, if 1838: 1777: 1654:{\displaystyle S} 1627: 1489:Minkowski diagram 1483:(1966), they did 1481:Spacetime Physics 1369:Hermann Minkowski 1253: 1252: 1128: 903: 322:Hermann Minkowski 288:spacetime diagram 284: 283: 148:Classical gravity 5315: 5277: 5276: 5060:van Stockum dust 4832:Two-body problem 4750:Mach's principle 4697: 4638:Terrell rotation 4478: 4456: 4449: 4442: 4433: 4424: 4410: 4404: 4396: 4394: 4393: 4387: 4381:. Archived from 4356: 4346: 4334: 4322: 4303: 4284: 4273: 4240: 4214: 4213: 4201: 4190: 4184: 4183: 4181: 4179: 4163: 4157: 4156: 4128: 4122: 4121: 4093: 4087: 4077: 4071: 4070: 4050: 4044: 4043: 4006: 4000: 3999: 3959: 3953: 3947: 3931: 3925: 3924: 3887: 3881: 3880: 3843: 3837: 3836: 3824: 3818: 3813: 3799: 3793: 3792: 3776: 3766: 3757: 3751: 3738: 3729: 3723: 3707: 3698: 3697: 3679: 3668: 3663: 3645: 3639: 3633: 3617: 3606: 3605: 3599: 3589: 3583: 3574: 3568: 3567: 3556:10.2307/20022840 3536:Wilson, Edwin B. 3532: 3526: 3525: 3511: 3505: 3496: 3486: 3478: 3469: 3463: 3461: 3435: 3425: 3419: 3418: 3400: 3394: 3390: 3370: 3364: 3358: 3352: 3349: 3343: 3342: 3324: 3318: 3317: 3315: 3313: 3302: 3270: 3265: 3264: 3249: 3247: 3246: 3241: 3239: 3238: 3235: 3206: 3204: 3203: 3198: 3130: 3128: 3127: 3122: 3120: 3105: 3103: 3102: 3097: 3095: 3023: 3012: 2926: 2924: 2923: 2918: 2902: 2896: 2889: 2883: 2877: 2871: 2865: 2859: 2849: 2827: 2817: 2791: 2781: 2777: 2770: 2763: 2756: 2642: 2635: 2629: 2623: 2603: 2601: 2600: 2595: 2593: 2592: 2576: 2574: 2573: 2568: 2553: 2551: 2550: 2545: 2543: 2540: 2539: 2538: 2522: 2521: 2520: 2504: 2503: 2495: 2494: 2475: 2473: 2472: 2467: 2465: 2464: 2448: 2446: 2445: 2440: 2425: 2418: 2412: 2405: 2399: 2392: 2382: 2376: 2370: 2363: 2356: 2347: 2345: 2344: 2339: 2337: 2289: 2281: 2199: 2192: 2185: 2179: 2172: 2166: 2160: 2153: 2144: 2142: 2141: 2136: 2134: 2133: 2117: 2115: 2114: 2109: 2097: 2096: 2094: 2093: 2088: 2085: 2071: 2059: 2045: 2043: 2042: 2037: 2035: 2034: 2018: 2016: 2015: 2010: 1998: 1991: 1989: 1988: 1983: 1981: 1980: 1964: 1962: 1961: 1956: 1937: 1927: 1913: 1911: 1910: 1905: 1903: 1902: 1886: 1884: 1883: 1878: 1866: 1856: 1854: 1853: 1848: 1846: 1839: 1837: 1829: 1818: 1809: 1808: 1785: 1778: 1776: 1775: 1774: 1769: 1768: 1767: 1749: 1748: 1747: 1734: 1708: 1687: 1685: 1684: 1679: 1677: 1676: 1660: 1658: 1657: 1652: 1640: 1638: 1637: 1632: 1630: 1629: 1628: 1620: 1614: 1610: 1609: 1608: 1574: 1573: 1571: 1570: 1565: 1562: 1538: 1524: 1462:Gilbert N. Lewis 1440:, and a pair of 1423: 1421: 1420: 1415: 1407: 1406: 1394: 1393: 1343: 1336: 1329: 1322: 1315: 1305: 1299: 1289: 1283: 1277: 1268: 1266: 1265: 1260: 1254: 1251: 1250: 1249: 1233: 1232: 1231: 1215: 1214: 1206: 1188: 1181: 1174: 1168: 1162: 1141: 1139: 1138: 1133: 1131: 1130: 1129: 1121: 1115: 1111: 1110: 1109: 1072: 1070: 1069: 1064: 1062: 1027: 981: 952: 945: 938: 932: 923: 921: 920: 915: 904: 896: 872: 866: 859: 853: 846: 840: 786: 780: 774: 757: 747: 729: 723: 717: 716: 714: 713: 710: 707: 695: 689: 679: 607: 600: 593: 587: 581: 570: 564: 544: 538: 528: 522: 516: 505: 499: 403:reference frames 385: 373: 369: 276: 269: 262: 248: 243: 242: 235: 231: 230: 200:Curved spacetime 58: 39: 5323: 5322: 5318: 5317: 5316: 5314: 5313: 5312: 5288: 5287: 5286: 5281: 5267: 5095: 4999:BKL singularity 4989:Lemaître–Tolman 4964: 4960:Quantum gravity 4942: 4936: 4922:geodetic effect 4896:(together with 4866:LISA Pathfinder 4805: 4754: 4740:Penrose diagram 4722: 4716: 4691: 4680: 4676:Minkowski space 4642: 4586: 4570: 4518: 4512: 4472: 4465: 4460: 4418: 4397: 4391: 4389: 4385: 4379: 4354: 4349: 4343: 4332: 4325: 4319: 4306: 4300: 4287: 4276: 4243: 4237: 4221: 4218: 4217: 4210: 4199: 4192: 4191: 4187: 4177: 4175: 4166:Gibbs, Philip. 4165: 4164: 4160: 4130: 4129: 4125: 4095: 4094: 4090: 4078: 4074: 4052: 4051: 4047: 4008: 4007: 4003: 3961: 3960: 3956: 3933: 3932: 3928: 3889: 3888: 3884: 3845: 3844: 3840: 3826: 3825: 3821: 3801: 3800: 3796: 3789: 3768: 3767: 3760: 3740: 3739: 3732: 3709: 3708: 3701: 3694: 3681: 3680: 3671: 3660: 3647: 3646: 3642: 3619: 3618: 3609: 3591: 3590: 3586: 3575: 3571: 3550:(11): 387–507. 3534: 3533: 3529: 3513: 3512: 3508: 3480: 3479: 3472: 3444:(10): 891–921. 3433: 3427: 3426: 3422: 3415: 3402: 3401: 3397: 3387: 3372: 3371: 3367: 3359: 3355: 3350: 3346: 3339: 3326: 3325: 3321: 3311: 3309: 3304: 3303: 3299: 3294: 3281:Penrose diagram 3276:Minkowski space 3266: 3259: 3256: 3230: 3225: 3224: 3180: 3179: 3160: 3113: 3108: 3107: 3088: 3080: 3079: 3039: 3014: 3004: 2976: 2933: 2909: 2908: 2898: 2891: 2885: 2879: 2873: 2867: 2861: 2851: 2841: 2819: 2809: 2802: 2787: 2779: 2772: 2765: 2758: 2752: 2725: 2688: 2683: 2674:Loedel diagrams 2637: 2631: 2625: 2618: 2611: 2584: 2579: 2578: 2559: 2558: 2530: 2523: 2512: 2505: 2486: 2481: 2480: 2456: 2451: 2450: 2431: 2430: 2420: 2414: 2407: 2401: 2394: 2388: 2378: 2372: 2365: 2358: 2352: 2335: 2334: 2318: 2294: 2293: 2273: 2249: 2248: 2235: 2205: 2204: 2200:, it is given: 2194: 2187: 2181: 2174: 2168: 2162: 2155: 2149: 2125: 2120: 2119: 2100: 2099: 2089: 2086: 2081: 2080: 2078: 2073: 2070: 2064: 2058: 2047: 2026: 2021: 2020: 2001: 2000: 1993: 1972: 1967: 1966: 1947: 1946: 1944: 1935: 1929: 1922: 1920: 1894: 1889: 1888: 1869: 1868: 1861: 1844: 1843: 1830: 1819: 1810: 1800: 1798: 1783: 1782: 1759: 1757: 1750: 1739: 1735: 1726: 1721: 1700: 1699: 1694: 1668: 1663: 1662: 1643: 1642: 1600: 1593: 1589: 1588: 1577: 1576: 1566: 1563: 1558: 1557: 1555: 1550: 1546: 1545: 1544: 1543: 1542: 1539: 1530: 1529: 1528: 1525: 1514: 1497: 1495:Loedel diagrams 1466:Edwin B. Wilson 1450:E. T. Whittaker 1398: 1385: 1380: 1379: 1361:Albert Einstein 1350: 1338: 1331: 1324: 1317: 1311: 1301: 1295: 1285: 1279: 1273: 1241: 1234: 1223: 1216: 1199: 1194: 1193: 1183: 1176: 1170: 1164: 1158: 1101: 1094: 1090: 1089: 1078: 1077: 1060: 1059: 1028: 1020: 1017: 1016: 982: 974: 962: 961: 947: 940: 934: 928: 878: 877: 868: 861: 855: 848: 842: 836: 825: 782: 776: 770: 749: 745: 725: 719: 711: 708: 705: 704: 702: 697: 691: 685: 677: 647:Minkowski space 623: 618: 602: 595: 589: 583: 572: 566: 559: 540: 533: 524: 518: 511: 501: 495: 484: 391: 383: 371: 367: 344: 339: 320:, developed by 280: 251: 238: 225: 224: 216: 215: 214: 174: 166: 165: 164: 149: 141: 140: 139: 119: 111: 110: 109: 105:Minkowski space 84: 76: 17: 12: 11: 5: 5321: 5319: 5311: 5310: 5305: 5300: 5290: 5289: 5283: 5282: 5272: 5269: 5268: 5266: 5265: 5258: 5253: 5248: 5243: 5238: 5233: 5228: 5223: 5218: 5213: 5208: 5203: 5198: 5193: 5188: 5186:Choquet-Bruhat 5183: 5178: 5173: 5168: 5163: 5158: 5153: 5148: 5143: 5138: 5133: 5128: 5123: 5118: 5113: 5107: 5105: 5101: 5100: 5097: 5096: 5094: 5093: 5086: 5085: 5080: 5075: 5068: 5067: 5062: 5057: 5052: 5047: 5038:Axisymmetric: 5035: 5034: 5029: 5023: 5012: 5011: 5006: 5001: 4996: 4991: 4986: 4977:Cosmological: 4974: 4972: 4966: 4965: 4963: 4962: 4957: 4952: 4946: 4944: 4938: 4937: 4935: 4934: 4929: 4918:frame-dragging 4915: 4910: 4905: 4902:Einstein rings 4898:Einstein cross 4891: 4880: 4879: 4874: 4868: 4863: 4858: 4845: 4835: 4834: 4829: 4824: 4819: 4813: 4811: 4807: 4806: 4804: 4803: 4801:Ernst equation 4798: 4793: 4788: 4783: 4778: 4773: 4771:BSSN formalism 4768: 4762: 4760: 4756: 4755: 4753: 4752: 4747: 4742: 4737: 4732: 4726: 4724: 4718: 4717: 4715: 4714: 4709: 4703: 4701: 4694: 4686: 4685: 4682: 4681: 4679: 4678: 4673: 4668: 4663: 4658: 4652: 4650: 4644: 4643: 4641: 4640: 4635: 4630: 4628:Ladder paradox 4625: 4620: 4615: 4610: 4605: 4600: 4594: 4592: 4588: 4587: 4585: 4584: 4578: 4576: 4572: 4571: 4569: 4568: 4563: 4558: 4553: 4548: 4543: 4538: 4533: 4531:Speed of light 4528: 4522: 4520: 4514: 4513: 4511: 4510: 4505: 4500: 4494: 4484: 4482: 4475: 4467: 4466: 4461: 4459: 4458: 4451: 4444: 4436: 4417: 4416:External links 4414: 4413: 4412: 4377: 4347: 4341: 4323: 4317: 4304: 4298: 4285: 4274: 4262:10.1119/1.9953 4241: 4235: 4216: 4215: 4208: 4185: 4158: 4139:(5): 326–327. 4123: 4104:(8): 487–489. 4088: 4072: 4045: 4018:(1): 366–371. 4001: 3974:(1): 236–242. 3954: 3948:(translation: 3926: 3899:(1): 227–235. 3882: 3838: 3819: 3794: 3787: 3758: 3752:(translation: 3730: 3724:(Translation: 3699: 3692: 3669: 3659:978-0486132013 3658: 3640: 3634:(Translation: 3607: 3584: 3569: 3527: 3506: 3504: 3503: 3501:Space and Time 3470: 3420: 3414:978-9812771599 3413: 3395: 3385: 3365: 3353: 3344: 3337: 3319: 3308:. Khan Academy 3296: 3295: 3293: 3290: 3289: 3288: 3283: 3278: 3272: 3271: 3268:Physics portal 3255: 3252: 3233: 3196: 3193: 3190: 3187: 3159: 3156: 3151: 3150: 3147: 3144: 3141: 3119: 3116: 3094: 3091: 3087: 3038: 3035: 2975: 2972: 2932: 2929: 2916: 2801: 2798: 2724: 2721: 2687: 2684: 2682: 2679: 2678: 2677: 2658: 2650: 2644: 2610: 2607: 2591: 2587: 2566: 2555: 2554: 2537: 2533: 2529: 2526: 2519: 2515: 2511: 2508: 2501: 2498: 2493: 2489: 2463: 2459: 2438: 2413:-axis and the 2349: 2348: 2333: 2330: 2327: 2324: 2321: 2319: 2317: 2314: 2311: 2308: 2305: 2302: 2299: 2296: 2295: 2292: 2287: 2284: 2279: 2276: 2274: 2272: 2269: 2266: 2263: 2260: 2257: 2254: 2251: 2250: 2247: 2244: 2241: 2238: 2236: 2234: 2231: 2228: 2225: 2222: 2219: 2216: 2213: 2212: 2132: 2128: 2107: 2068: 2056: 2033: 2029: 2008: 1979: 1975: 1954: 1942: 1933: 1918: 1901: 1897: 1876: 1858: 1857: 1842: 1836: 1833: 1828: 1825: 1822: 1816: 1813: 1811: 1807: 1803: 1799: 1797: 1794: 1791: 1788: 1786: 1784: 1781: 1773: 1766: 1762: 1756: 1753: 1746: 1742: 1738: 1732: 1729: 1727: 1725: 1722: 1720: 1717: 1714: 1711: 1709: 1707: 1692: 1675: 1671: 1650: 1626: 1623: 1618: 1613: 1607: 1603: 1599: 1596: 1592: 1587: 1584: 1540: 1533: 1532: 1531: 1526: 1519: 1518: 1517: 1516: 1515: 1513: 1510: 1496: 1493: 1413: 1410: 1405: 1401: 1397: 1392: 1388: 1377:unit hyperbola 1367:in 1905, with 1349: 1346: 1270: 1269: 1258: 1248: 1244: 1240: 1237: 1230: 1226: 1222: 1219: 1212: 1209: 1205: 1202: 1144:Lorentz factor 1127: 1124: 1119: 1114: 1108: 1104: 1100: 1097: 1093: 1088: 1085: 1074: 1073: 1058: 1055: 1052: 1049: 1046: 1043: 1040: 1037: 1034: 1031: 1029: 1026: 1023: 1019: 1018: 1015: 1012: 1009: 1006: 1003: 1000: 997: 994: 991: 988: 985: 983: 980: 977: 973: 970: 969: 925: 924: 913: 910: 907: 902: 899: 894: 891: 888: 885: 824: 821: 817: 816: 810: 801: 795: 622: 619: 617: 614: 483: 480: 457: 456: 445: 434: 390: 387: 343: 340: 338: 335: 282: 281: 279: 278: 271: 264: 256: 253: 252: 250: 249: 236: 221: 218: 217: 213: 212: 207: 202: 197: 192: 187: 182: 176: 175: 172: 171: 168: 167: 163: 162: 157: 151: 150: 147: 146: 143: 142: 138: 137: 132: 127: 121: 120: 117: 116: 113: 112: 108: 107: 102: 97: 92: 86: 85: 82: 81: 78: 77: 75: 74: 69: 63: 60: 59: 51: 50: 44: 43: 15: 13: 10: 9: 6: 4: 3: 2: 5320: 5309: 5306: 5304: 5301: 5299: 5296: 5295: 5293: 5280: 5270: 5264: 5263: 5259: 5257: 5254: 5252: 5249: 5247: 5244: 5242: 5239: 5237: 5234: 5232: 5229: 5227: 5224: 5222: 5219: 5217: 5214: 5212: 5209: 5207: 5204: 5202: 5199: 5197: 5194: 5192: 5189: 5187: 5184: 5182: 5179: 5177: 5174: 5172: 5171:Chandrasekhar 5169: 5167: 5164: 5162: 5159: 5157: 5154: 5152: 5149: 5147: 5144: 5142: 5139: 5137: 5134: 5132: 5131:Schwarzschild 5129: 5127: 5124: 5122: 5119: 5117: 5114: 5112: 5109: 5108: 5106: 5102: 5092: 5088: 5087: 5084: 5081: 5079: 5076: 5074: 5070: 5069: 5066: 5063: 5061: 5058: 5056: 5053: 5051: 5048: 5045: 5041: 5037: 5036: 5033: 5030: 5027: 5024: 5022: 5018: 5017:Schwarzschild 5014: 5013: 5010: 5007: 5005: 5002: 5000: 4997: 4995: 4992: 4990: 4987: 4984: 4980: 4976: 4975: 4973: 4971: 4967: 4961: 4958: 4956: 4953: 4951: 4948: 4947: 4945: 4939: 4933: 4930: 4927: 4923: 4919: 4916: 4914: 4913:Shapiro delay 4911: 4909: 4906: 4903: 4899: 4895: 4892: 4889: 4885: 4882: 4881: 4878: 4875: 4872: 4869: 4867: 4864: 4862: 4859: 4857: 4856:collaboration 4853: 4849: 4846: 4844: 4840: 4837: 4836: 4833: 4830: 4828: 4825: 4823: 4822:Event horizon 4820: 4818: 4815: 4814: 4812: 4808: 4802: 4799: 4797: 4794: 4792: 4789: 4787: 4784: 4782: 4779: 4777: 4774: 4772: 4769: 4767: 4766:ADM formalism 4764: 4763: 4761: 4757: 4751: 4748: 4746: 4743: 4741: 4738: 4736: 4733: 4731: 4728: 4727: 4725: 4719: 4713: 4710: 4708: 4705: 4704: 4702: 4698: 4695: 4693: 4687: 4677: 4674: 4672: 4671:Biquaternions 4669: 4667: 4664: 4662: 4659: 4657: 4654: 4653: 4651: 4649: 4645: 4639: 4636: 4634: 4631: 4629: 4626: 4624: 4621: 4619: 4616: 4614: 4611: 4609: 4606: 4604: 4601: 4599: 4598:Time dilation 4596: 4595: 4593: 4589: 4583: 4580: 4579: 4577: 4573: 4567: 4564: 4562: 4559: 4557: 4554: 4552: 4551:Proper length 4549: 4547: 4544: 4542: 4539: 4537: 4534: 4532: 4529: 4527: 4524: 4523: 4521: 4515: 4509: 4506: 4504: 4501: 4498: 4495: 4493: 4489: 4486: 4485: 4483: 4479: 4476: 4474: 4468: 4464: 4457: 4452: 4450: 4445: 4443: 4438: 4437: 4434: 4430: 4428: 4423: 4415: 4408: 4402: 4388:on 2013-10-16 4384: 4380: 4374: 4370: 4366: 4362: 4361: 4353: 4348: 4344: 4338: 4331: 4330: 4324: 4320: 4314: 4310: 4305: 4301: 4295: 4291: 4286: 4282: 4281: 4275: 4271: 4267: 4263: 4259: 4255: 4251: 4247: 4242: 4238: 4232: 4228: 4224: 4223:French, A. P. 4220: 4219: 4211: 4205: 4198: 4197: 4189: 4186: 4173: 4169: 4162: 4159: 4154: 4150: 4146: 4142: 4138: 4134: 4127: 4124: 4119: 4115: 4111: 4107: 4103: 4099: 4092: 4089: 4086: 4083: 4082: 4076: 4073: 4068: 4064: 4060: 4056: 4049: 4046: 4041: 4037: 4033: 4029: 4025: 4021: 4017: 4013: 4005: 4002: 3997: 3993: 3989: 3985: 3981: 3977: 3973: 3969: 3965: 3958: 3955: 3951: 3945: 3941: 3937: 3930: 3927: 3922: 3918: 3914: 3910: 3906: 3902: 3898: 3894: 3886: 3883: 3878: 3874: 3870: 3866: 3862: 3858: 3854: 3850: 3842: 3839: 3834: 3830: 3823: 3820: 3817: 3811: 3807: 3806: 3798: 3795: 3790: 3788:0-486-65743-4 3784: 3780: 3775: 3774: 3765: 3763: 3759: 3755: 3749: 3745: 3737: 3735: 3731: 3727: 3721: 3717: 3713: 3706: 3704: 3700: 3695: 3693:0-520-20029-2 3689: 3685: 3678: 3676: 3674: 3670: 3667: 3661: 3655: 3651: 3644: 3641: 3637: 3631: 3627: 3623: 3616: 3614: 3612: 3608: 3604: 3603:of reference. 3598: 3597: 3588: 3585: 3582: 3578: 3573: 3570: 3565: 3561: 3557: 3553: 3549: 3545: 3541: 3537: 3531: 3528: 3523: 3519: 3518: 3510: 3507: 3502: 3498: 3497: 3494: 3490: 3485: 3477: 3475: 3471: 3467: 3460: 3455: 3451: 3447: 3443: 3439: 3431: 3424: 3421: 3416: 3410: 3406: 3399: 3396: 3393: 3388: 3382: 3378: 3377: 3369: 3366: 3363: 3357: 3354: 3348: 3345: 3340: 3338:9780957389465 3334: 3330: 3323: 3320: 3307: 3301: 3298: 3291: 3287: 3284: 3282: 3279: 3277: 3274: 3273: 3269: 3263: 3258: 3253: 3251: 3231: 3222: 3216: 3214: 3210: 3194: 3191: 3188: 3185: 3172: 3164: 3157: 3155: 3148: 3145: 3142: 3139: 3138: 3137: 3133: 3117: 3114: 3092: 3089: 3085: 3076: 3072: 3070: 3064: 3061: 3051: 3043: 3036: 3034: 3030: 3028: 3027:time dilation 3021: 3017: 3011: 3007: 3001: 2997: 2994: 2989: 2980: 2973: 2971: 2969: 2965: 2960: 2958: 2954: 2949: 2945: 2937: 2930: 2928: 2914: 2904: 2901: 2894: 2888: 2882: 2876: 2870: 2864: 2858: 2854: 2848: 2844: 2838: 2836: 2826: 2822: 2816: 2812: 2806: 2799: 2797: 2793: 2790: 2783: 2775: 2768: 2761: 2755: 2749: 2746: 2745:proper length 2737: 2729: 2722: 2720: 2716: 2712: 2709: 2700: 2692: 2686:Time dilation 2685: 2680: 2675: 2671: 2667: 2663: 2659: 2655: 2651: 2648: 2645: 2640: 2636:-axis to the 2634: 2628: 2621: 2616: 2613: 2612: 2608: 2606: 2585: 2564: 2535: 2531: 2527: 2524: 2517: 2513: 2509: 2506: 2499: 2496: 2487: 2479: 2478: 2477: 2457: 2436: 2427: 2423: 2417: 2410: 2404: 2397: 2391: 2386: 2381: 2375: 2368: 2361: 2355: 2331: 2328: 2325: 2322: 2320: 2315: 2312: 2309: 2306: 2303: 2300: 2297: 2290: 2285: 2282: 2277: 2275: 2270: 2267: 2264: 2261: 2258: 2255: 2252: 2245: 2242: 2239: 2237: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2203: 2202: 2201: 2197: 2190: 2184: 2177: 2171: 2165: 2158: 2152: 2146: 2126: 2105: 2092: 2084: 2076: 2067: 2061: 2055: 2051: 2027: 2006: 1997: 1973: 1952: 1941: 1932: 1926: 1917: 1895: 1874: 1864: 1840: 1834: 1831: 1826: 1823: 1820: 1814: 1812: 1805: 1801: 1792: 1787: 1779: 1771: 1764: 1760: 1754: 1751: 1744: 1740: 1736: 1730: 1728: 1723: 1715: 1710: 1698: 1697: 1696: 1691: 1669: 1648: 1624: 1621: 1616: 1611: 1605: 1601: 1597: 1594: 1590: 1585: 1582: 1569: 1561: 1553: 1537: 1523: 1511: 1509: 1507: 1503: 1494: 1492: 1490: 1487:use the term 1486: 1482: 1477: 1475: 1474:non-Euclidean 1471: 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1439: 1438:its conjugate 1435: 1431: 1427: 1411: 1408: 1403: 1399: 1395: 1390: 1386: 1378: 1372: 1370: 1366: 1362: 1354: 1347: 1345: 1341: 1334: 1327: 1320: 1314: 1309: 1308:proper length 1304: 1298: 1293: 1288: 1282: 1276: 1256: 1246: 1242: 1238: 1235: 1228: 1224: 1220: 1217: 1210: 1207: 1203: 1200: 1192: 1191: 1190: 1186: 1179: 1173: 1167: 1161: 1155: 1153: 1150:of a pair of 1149: 1145: 1125: 1122: 1117: 1112: 1106: 1102: 1098: 1095: 1091: 1086: 1083: 1053: 1050: 1047: 1044: 1041: 1035: 1032: 1030: 1024: 1021: 1013: 1007: 1004: 1001: 998: 995: 989: 986: 984: 978: 975: 971: 960: 959: 958: 956: 950: 943: 937: 931: 911: 908: 905: 900: 897: 892: 889: 886: 883: 876: 875: 874: 871: 864: 858: 851: 845: 839: 829: 822: 820: 814: 811: 809: 805: 802: 799: 796: 794: 790: 789: 788: 785: 779: 773: 768: 763: 761: 756: 752: 742: 740: 739: 736:relativistic 733: 728: 722: 700: 694: 688: 683: 675: 671: 667: 662: 660: 656: 652: 648: 640: 635: 627: 620: 615: 613: 611: 605: 598: 592: 586: 579: 575: 569: 562: 555: 552: 551:perpendicular 548: 543: 536: 530: 527: 521: 514: 509: 504: 498: 488: 481: 479: 477: 473: 468: 466: 462: 454: 450: 446: 443: 439: 435: 432: 428: 424: 420: 419: 418: 416: 412: 408: 404: 395: 388: 386: 381: 377: 365: 359: 357: 348: 341: 336: 334: 332: 327: 323: 319: 314: 312: 308: 303: 301: 297: 296:time dilation 293: 289: 277: 272: 270: 265: 263: 258: 257: 255: 254: 247: 237: 234: 229: 223: 222: 220: 219: 211: 208: 206: 203: 201: 198: 196: 193: 191: 188: 186: 183: 181: 178: 177: 170: 169: 161: 158: 156: 153: 152: 145: 144: 136: 133: 131: 128: 126: 123: 122: 115: 114: 106: 103: 101: 98: 96: 93: 91: 88: 87: 80: 79: 73: 70: 68: 65: 64: 62: 61: 57: 53: 52: 49: 45: 41: 40: 34: 30: 26: 21: 5261: 4955:Kaluza–Klein 4707:Introduction 4633:Twin paradox 4419: 4390:. Retrieved 4383:the original 4359: 4328: 4308: 4289: 4279: 4253: 4249: 4226: 4195: 4188: 4176:. Retrieved 4171: 4161: 4136: 4132: 4126: 4101: 4097: 4091: 4079: 4075: 4062: 4058: 4048: 4015: 4011: 4004: 3971: 3967: 3957: 3943: 3939: 3929: 3896: 3892: 3885: 3855:(1): 22–37. 3852: 3848: 3841: 3832: 3828: 3822: 3809: 3804: 3797: 3772: 3747: 3743: 3719: 3715: 3683: 3649: 3643: 3629: 3625: 3601: 3595: 3587: 3572: 3547: 3543: 3530: 3516: 3509: 3492: 3488: 3464:. See also: 3441: 3437: 3423: 3404: 3398: 3375: 3368: 3356: 3347: 3328: 3322: 3310:. 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