Spacetime - Knowledge

11771:, we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other. (a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source. (b) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of 3069:. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2-4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded. 11756:

S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.

16424: 17860: 239: 8180: 614: 879: 5292: 7446: 4541: 2720: 10222: 10208: 2712: 3328: 10030:. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero. 7966: 7202:, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere. All inertial observers will agree on what constitutes accelerating and non-accelerating motion. Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well. 4786: 3538: 10635:

the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as

7705: 6744: 528:) relates distances measured in space to distances measured in time. The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe that is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the 17872: 9235: 9877:" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig. 3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same. 10577:

paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm is convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques tend be used when solving gravitational wave problems, while curved spacetime techniques tend be used in the analysis of black holes.

542: 10391: 1148:. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because the moving point of view sees itself as stationary, and the position of the event as receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events. 9863: 9839: 830:(the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them by recognizing that the entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed. His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples. 755: 769: 3082: 11173: 16418: 725: 3584:. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation. The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower. 948:(superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital. In 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity. Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as 739: 10423:

moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.

17836: 16199: 252: 4160: 3766:. Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower. 62: 10428: 7936:, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.) Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d). 5229: 3273: 2659: 2119: 5403: 2976: 15038: 16209: 14908: 17884: 17908: 11145:(as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces 3004: 3590: 4230:

gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena. A few of these phenomena are described in the later sections of this article.

3792: 3781: 17896: 4221:, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together. The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory. 14187:...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages. 819:. He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world". Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform. 10411:

accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the

11563:(Gedanken–Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great 3600:, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference. 625: 4208:(i.e. between the bold blue lines in the figure), a shift takes place in the angle of her lines of simultaneity, corresponding to a rapid skip-over of the events in Terence's world line that Stella considers to be simultaneous with her own. Therefore, at the end of her trip, Stella finds that Terence has aged more than she has. 4168:

paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the

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that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."

6273: 10603:. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, 6088: 3638:′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted. 11797:

Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so

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By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break

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assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant

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This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of

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contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of

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is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is the difference between the

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Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light. Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c

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Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is

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Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame

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What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of

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The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer

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Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis,

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The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or

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The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements

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I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other

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In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks

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In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage

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electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion

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In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from

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since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different

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The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary

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A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks

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of general relativity does provide some additional insight into the subject. Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when

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The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore,

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The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime

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of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual

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Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig. 3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver

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is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at

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In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, will see a delay between the emission of a signal and its detection due to the speed of light. To

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Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime. Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an

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In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small

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that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space. In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout

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The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a

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These distinctions should result in a difference in the twins' ages. The spacetime diagram of Fig. 2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin

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axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary

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in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. No length

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involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each

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The (1+2)-dimensional boundary between space- and time-like hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in

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bodies or two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal

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Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse

3740:. The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation 630:

Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar

3866:) as the event, where the respective "moving clock" is collocated with the "first clock at rest" in both comparisons. Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right 8195:. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a lightlike world line and, in appropriate units, have equal space and time components for every observer. 347:

Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However,

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views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it

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so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.

9558: 3553:). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3 866:

was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.

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Fig. 3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is

8782:, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes, old-fashioned color television sets, etc.), has nevertheless not proven to be a 8789:

For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy. "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or

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While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a

6856:, otherwise whether a fast moving 1 m ball could fit through a 1 m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law. 10198:

so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.

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was the first to combine space and time into spacetime. He argued in 1898 that the simultaneity of two events is a matter of convention. In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly

4204:, are simultaneous with events on her world line. (Terence, in turn, would observe a set of horizontal lines of simultaneity.) Throughout both the outbound and the inbound legs of Stella's journey, she measures Terence's clocks as running slower than her own. 5299:

It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig. 3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.

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To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write

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in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing

11550:, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive. 8160:, since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches 6763:

The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates

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axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at

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Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively.

11073:(representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a 11423:

In 2019, James Scargill argued that complex life may be possible with two spatial dimensions. According to Scargill, a purely scalar theory of gravity may enable a local gravitational force, and 2D networks may be sufficient for complex neural

12940:(Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.) 11575:

might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as

1172:, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by 490:

The path of a particle through spacetime can be considered to be a sequence of events. The series of events can be linked together to form a curve that represents the particle's progress through spacetime. That path is called the particle's

11597:) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor. 1156:. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two 4136:

to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.

2622:(i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by 11077:

coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.

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she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of

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Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed

10446:. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be 3129:

Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig. 2-3. The three events

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of 1887 (Fig. 1-2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of

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A pure SR analysis would be as follows: Analyzed in Stella's rest frame, she is motionless for the entire trip. When she fires her rockets for the turnaround, she experiences a pseudo force which resembles a gravitational force.

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Fig. 2-10 illustrates the previous discussion of mutual time dilation with Minkowski diagrams. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with

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axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1 m. The endpoints O and B measured when

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Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the

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In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from

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Algebraic analyses of the energetics of this decay reaction are available online, so Fig. 3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is

1123: 6099: 3619: = 0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the 2983:

In Fig. 2–4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the

10568:

In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called

9256:

For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.

5924: 874:

led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré.

6715: 10626:. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at lightlike infinity to characterize what it means to say a metric is asymptotically flat, making no 4793:

The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,

8008:

In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector

7969:

Figure 3–8. Relativistic spacetime momentum vector. The coordinate axes of the rest frame are: momentum, p, and mass * c. For comparison, we have overlaid a spacetime coordinate system with axes: position, and time *

404:

as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers. Classical mechanics assumes that time has a constant rate of passage, independent of the

10235:

simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.

2924:

The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have slopes = 1 and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the

8700: 9872:

The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the

3177:

A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of

11533:

from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher

6497: 13741:

Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles

9054: 5889: 8045:. In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. 7911: 7830: 1140:

position of the two points using different coordinate systems, the distance between the points will be the same for both, assuming that they are measuring using the same units. The distance is "invariant".

708:(the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter. Lorentz's equations predicted a quantity that he called 7694: 3339:

Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski metric) form curves given by the two equations

3117:

From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are

1434: 9154: 2239:

itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard

2723:

Figure 2–3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse

549:: the position in space, and the time, which can be visualized as the reading of a clock located at each position in space. The 'observer' synchronizes the clocks according to their own reference frame. 6554: 12656: 9433: 8873: 3335:

In Euclidean space (having spatial dimensions only), the set of points equidistant (using the Euclidean metric) from some point form a circle (in two dimensions) or a sphere (in three dimensions). In

8367:

in a moving frame. This result can be derived by inspection of Fig. 3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.

1812: 5138: 8767: 910:

include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence."

10356:

which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.

7622: 7117: 7065: 4553:

in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.

14218: 890:

On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title,

6104: 5929: 666:, demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction. 10194: 3694:

This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the

573:

In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.

4148:

should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock. Nevertheless, the

2459: 1879: 9242:

To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.

7357: 7280:

both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of

7234:

to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with

2362: 7713:

that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:

3456:

In Fig. 2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.

3011:

A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more

8786:

concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.

8363: 7718:

Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S′ of the source.

850:. While it would appear that he did not at first think geometrically about spacetime, in the further development of general relativity, Einstein fully incorporated the spacetime formalism. 9608: 8559: 3811: > 0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with − 8156: 6373: 11261:

said that it "gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa" (Barrow 2002:204).

5548: 9885:

In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since

3062:) from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime. 8438: 3409: 8107: 3593:

Figure 2–9. In this spacetime diagram, the 1 m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.

3174:

being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.

8496: 3519:

Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate

2739:(i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime") belongs to a second observer O′. 9759: 9726: 8606: 7440: 6421: 3122:

separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the

1322: 796:

of clock synchronization assuming constant light speed. In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the

5552:

The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma

2541:. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If 11329:

can be expressed in terms of an action only for a four-dimensional manifold. Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron

8059:

We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that

2030: 2879: 2066:

Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general,

13782:. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along 11421:. However, a self-gravitating gas cannot be stably bound if the mass sphere is larger than ~10 solar masses, due to the small positivity of the cosmological constant observed. 7555: 3217: 11366:

would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) Lastly, if

9814: 7744:

In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where

3256: 1257: 463:, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event is represented by a set of coordinates 5339: 4156:

the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.

1144:

In special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of

12706: 10026: 9422: 1950: 9966: 5705:

greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.

4762: 4436: 2063:. The invariance of the spacetime interval between the same events for all inertial frames of reference is one of the fundamental results of special theory of relativity. 1646: 367:

presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as

10103: 9922: 4722: 570:

following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.

10599:, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in 9970:. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame. 9198:

In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915,

4829: 4535: 2109: 1699: 1282: 1006: 11143: 11081:

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event

10945: 10352: 4500: 4468: 2605: 2061: 1219: 16322: 11589:

In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the

8917: 3492:

these hyperbolae are spacelike intervals: No causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than

2160: 1979: 1907: 1670: 1193: 13646: 9689: 9650: 8929:

Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase

7934: 7518: 7475: 5695: 5570: 504:, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, the surface of a globe appears to be flat. A scale factor, 11435: 2217: 17539: 10885: 10827: 8274: 8236: 8041: 3439: 2566: 2535: 2508: 2265: 2187: 16245: 13260: 11311: 10062: 9180: 5397: 5368: 9323: 9265: 7748:

is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency

6719:

as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about

5578: 1445: 12398: 7916:

Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of

1596: 1345: 11099: 11071: 11051: 11028: 11008: 10988: 10965: 10850: 7495: 3510: 3482: 2640: 2237: 2084: 1616: 522: 12767: 4200:-plane) represent sets of events that are simultaneous in the observer frame. In Fig. 2-11, the blue lines connect events on Terence's world line which, 4181:

Stella is not in an inertial frame. Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:

2115:

of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.

870:

Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and

11822:, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of 135: 12428: 1649:

square of the spatial distance separating event B from event A and the square of the spatial distance traveled by a light signal in that same time interval

803:

While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various

3541:

Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds

2957:

of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.

8375:

Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several important conclusions.

2618:

A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2-1 presents a spacetime diagram illustrating the

2607:

In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed

14239: 6268:{\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'+vt'\right)\\y&=y'\\z&=z'\end{aligned}}} 5171:

becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities:

1018: 700:

By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later, i.e. the

11714: 13157: 11234:

differ from 1. The argument is often of an anthropic character and possibly the first of its kind, albeit before the complete concept came into vogue.

6083:{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}} 3772:

be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast.

14710: 1909:

can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval,

11676:, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group 9825:

spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.

424:, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's 14094:

Bondi, H.; Van der Burg, M. G. J.; Metzner, A. (1962). "Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems".

12329: 10399:

scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a

6636: 643: 3545:

Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately

3757:

Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.

800:. In 1905/1906 he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity. 13681: 13036: 12950:

Kastrup, H. A. (2008). "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics".

11767: 12610: 11571:

on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the

7124:

The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting

914:

included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the

10542:

to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime

13974: 8626: 3114:

From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O.

536:, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space. 16238: 9252:(2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision. 9226:, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective. 6598:

Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time

14447:. Also see Ehrenfest, P. (1917) "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?" 9821:

Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable

17586: 13191: 11997: 7362:

On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of

3655:(analogous to ordinary angles in Euclidean geometry). Because of this rotation, the projection of a primed meter-stick onto the unprimed 2615:. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage. 4278:). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks 2735:

With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-2, two

17821: 14969: 11785: 6752: 6738: 2122:

Figure 2–1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C

673:

implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether

190: 10733:. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of 8945: 5819: 584:, after one has factored out signal propagation delays, versus what one visually sees without such corrections. Failing to understand 17610: 17356: 12927: 11373:, gravitation of any kind becomes problematic, and the universe would probably be too simple to contain observers. For example, when 10573:", the "field-theoretical approach to general relativity", and so forth. Kip Thorne has provided a popular review of these theories. 4119:

The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion

2476:

A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.

7634: 7205:

A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.

3683:

in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −

1353: 922:), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity. 16776: 14329: 12352: 12256: 12218: 9202:

discovered that underlying each conservation law is a fundamental symmetry of nature. The fact that physical processes do not care

165: 12889: 9553:{\displaystyle {\boldsymbol {v_{cm}}}=\left(m_{1}{\boldsymbol {v_{1}}}+m_{2}{\boldsymbol {v_{2}}}\right)/\left(m_{1}+m_{2}\right)} 9061: 6426: 3523:

of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.

2785:

spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times

16231: 15808: 15644: 3101:

of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events.

8772:

The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.

944:

Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as

17392: 17346: 8800: 658:. The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the 11101:). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples 4310:. With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates 1710: 17530: 15273: 14761: 14233: 14078: 14031:

Weinberg, S. (1964). "Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of the S-Matrix".

13601: 13517: 13487: 13462: 13413: 13386: 13327: 13293: 13095: 12802: 12518: 12292: 12075: 12041: 11910: 10970:

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event

13768:, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 12298: 11974: 11788:

measures transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.

8704: 585: 17014: 16745: 14748: 9245:

In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:

7846: 7765: 7755:, but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by 6501: 5915:

Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.

3906:> 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the 3659:-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened. 3097:

the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the

12821:

Weinstein, Galina (2012). "Max Born, Albert Einstein and Hermann Minkowski's Space–Time Formalism of Special Relativity".

10440:

states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.

14937: 9615:

Fig. 3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components

3699:

clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses

3648: 284: 12100: 11033:

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing

10241:

Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where

10112: 7223:

to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250

4178:

Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.

2979:

Figure 2–4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere"

17267: 11765:

The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis.

5291: 2373: 938: 603: 130: 12390: 7560: 7380:

changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency

7304: 2279: 17638: 17272: 16496: 15072: 15022: 14868: 14842: 14790: 13841: 13809: 13719: 13222: 13132: 13068: 13020: 12687: 12593: 12557: 12466: 12367: 12174: 12142: 12007: 11968: 11879: 11445: 10726: 10619: 10586: 5268: 3312: 2698: 215: 13252: 8317: 3720:

Two events can be defined by the condition "two clocks are simultaneously at one place", i.e., when W′ passes each W

12524: 11572: 9565: 8508: 5057: 4362:. Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel 17859: 17592: 8116: 2118: 238: 17305: 15758: 15413: 14928: 13862: 11165: 10452:

An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation,

10400: 7071: 7019: 3728:. For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W 3054:

region of the event O, denoted "Elsewhere" in Fig. 2-4. Events on the light cone itself are said to be

678: 533: 31: 11347:

differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant

10947:

are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light

10442:

In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of

5478: 17408: 16569: 15637: 14962: 10794:

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected

8402: 8289:

Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in

8179: 5753:, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/ 5250: 5212:. The velocity of light is the same for all observers regardless their motions relative to the emitting source. 3346: 3294: 2680: 838: 17: 13624:

Byers, Nina (1998). "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws".

11280:. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a 10450:), there are no experiments that A and B can perform which will enable them to tell which setting they are in. 10406:

In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of

8064: 7158:. From the inverse transformation, the equations of the forwards transformation can be derived by solving for 6760:, which states that the influence that one particle exerts on another can not be transmitted instantaneously. 3089:

All observers will agree that for any given event, an event within the given event's future light cone occurs

2988:

of the event O, since adding a second spatial dimension (Fig. 2-5) makes the appearance that of two

17945: 17839: 17581: 17486: 17416: 16750: 13800:

The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity

11710: 11348: 8465: 7736:

Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?

7733:

Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?

6311: 5303:

To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for

3748:

temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the

599: 406: 17549: 13149: 17940: 17935: 17403: 15720: 11475: 11254: 10746: 9735: 9702: 8568: 7390: 1287: 10725:" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the 7725:

the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.

3042:

The region exterior to the light cone consists of events that are separated from the event O by more

878: 17444: 17178: 17163: 16705: 16685: 16631: 16516: 16352: 16011: 15057: 12209: 11351:. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if 7974:

In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity.

7955: 7625: 7257: 5145:

The relativistic formula for addition of velocities presented above exhibits several important features:

4218: 3093:

the given event. Likewise, for any given event, an event within the given event's past light cone occurs

2728: 647: 613: 444: 14918: 12321: 12068:

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

8614:

Another way of looking at the relationship between mass and energy is to consider a series expansion of

2835: 17960: 17850: 17731: 17173: 17126: 16372: 16357: 16295: 16274: 15363: 15343: 14912: 12420: 11960: 11837:

dimensions, the strength of the gravitational attraction between two bodies separated by a distance of

11658:, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since 11460: 9207: 5246: 4899:

of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3-2b:

3691:

than the other, is an important question that "goes to the heart of understanding special relativity."

3290: 3185: 3123: 2676: 39: 35: 14922: 10387:

space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.

9773: 6583:. So inverting the equations simply entails switching the primed and unprimed variables and replacing 6377: 3227: 1224: 17761: 17387: 17068: 16526: 16337: 16074: 15923: 15803: 15353: 15266: 14955: 14597: 11564: 11238: 11074: 10403:. No evidence of gravitation can be discovered following alongside the motions of a single particle. 9219: 7268: 5313: 4780: 689: 14349: 13673: 12737: 8794:, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula, 4192:

and 2-11 illustrate the concept of lines (planes) of simultaneity: Lines parallel to the observer's

2719: 1984: 17955: 17786: 17336: 17096: 17007: 15773: 14992: 14817: 12842:

Galison, Peter Louis (1979). "Minkowski's space–time: From visual thinking to the absolute world".

11547: 11242: 10758: 10066:, fail to behave properly under Lorentzian transformation. The linear transformation of velocities 9985: 9874: 9381: 9211: 8202:

of electromagnetism is that light carries energy and momentum, and that their ratio is a constant:

7445: 4560:

with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity

4540: 2736: 1820: 558:

is used to determine the time and position of events taking place within the whole frame. The term

376: 140: 11246: 7523: 6306:

terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.

5186:. The Galilean formula is a special case of the relativistic formula applicable to low velocities. 3589: 1621: 17515: 17437: 17057: 17048: 16786: 16564: 16466: 16382: 16377: 16029: 15680: 15200: 15195: 15175: 14753: 13938: 11384:

cannot cross without intersecting. Hence anthropic and other arguments rule out all cases except

11268:

showed that if there is only a single time dimension and more than three spatial dimensions, the

10789: 10738: 10730: 5239: 3283: 2669: 797: 325: 314: 277: 11276:

about its Sun cannot remain stable. The same is true of a star's orbit around the center of its

11226:

and undetectable to date, can be explained by appealing to the physical consequences of letting

10375: 8610:. Simplified to the case of zero velocity, this is Einstein's equation relating energy and mass. 4933:

The two yellow triangles are similar because they are right triangles that share a common angle

4800: 4152:

is not a true paradox because it is easily understood within the context of special relativity.

2089: 1675: 1262: 986: 861:

recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:

17741: 17604: 17510: 17468: 17142: 17106: 15383: 15185: 15180: 15160: 14344: 13317: 13012: 13004: 11742:, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the 11104: 10906: 10318: 9980:

both before and after collision. In the Newtonian analysis, conservation of mass dictates that

9223: 5902: 3791: 3780: 2711: 2575: 2126:

The equation above is similar to the Pythagorean theorem, except with a minus sign between the

1198: 1153: 701: 480: 353: 105: 13183: 11237:

The implicit notion that the dimensionality of the universe is special is first attributed to

10631:

the form of the boundary conditions appropriate for asymptotically flat gravitational fields.

8890: 3331:

Figure 2–7. (a) Families of invariant hyperbolae, (b) Hyperboloids of two sheets and one sheet

3327: 2781:

Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a

2129: 1955: 1912: 1652: 1175: 853:

When Einstein published in 1905, another of his competitors, his former mathematics professor

17746: 17351: 17290: 16939: 16212: 15844: 15789: 15743: 15728: 15373: 15190: 15170: 15165: 14421:[How do the fundamental laws of physics make manifest that space has 3 dimensions?]. 14175:

Strominger, Andrew (2017). "Lectures on the Infrared Structure of Gravity and Gauge Theory".

13591: 13283: 13102: 12714: 12585: 12579: 11418: 10766: 10754: 10666: 10437: 10221: 10207: 9931: 9659: 9620: 8503:

Likewise, comparing the time component of the relativistic momentum with that of the photon,

7919: 7500: 7457: 6757: 5680: 5555: 4727: 4401: 4213: 4175:

Complications arise if the twin paradox is analyzed from the traveling twin's point of view.

842: 200: 100: 14654: 14198: 11546:

The operational procedure adopted by Poincaré was essentially identical to what is known as

10558:

Are there any situations where a flat spacetime interpretation of general relativity may be

10382:

Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean

10071: 9890: 9367:{\displaystyle {\boldsymbol {p}}_{\boldsymbol {2}}=m_{2}{\boldsymbol {v}}_{\boldsymbol {2}}} 9309:{\displaystyle {\boldsymbol {p}}_{\boldsymbol {1}}=m_{1}{\boldsymbol {v}}_{\boldsymbol {1}}} 5659:{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}} 4690: 4212:

Although general relativity is not required to analyze the twin paradox, application of the

2192: 1565:{\displaystyle (\Delta s)^{2}=(\Delta ct)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}.} 17651: 17577: 17421: 17023: 16766: 16680: 16506: 16367: 16095: 16064: 15513: 15408: 15259: 14676: 14619: 14555: 14487: 14430: 14283: 14146: 14103: 14040: 14005: 13950: 13903: 13546: 12969: 12749: 12622: 12498: 12232: 11577: 11537:, who argued that time, simultaneity, and duration were matters of intuitive understanding. 10858: 10800: 8245: 8207: 8014: 6556:

etc. Although for brevity the Lorentz transformation equations are written without deltas,

4506: 3914:, where "two clocks are simultaneously at one place". In the lower picture the place for W′ 3537: 3417: 2544: 2513: 2486: 2243: 2165: 834: 705: 13939:"Exact Theory of the (Einstein) Gravitational Field in an Arbitrary Background Space–Time" 12915: 12353:"Fresnel's (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies." 11287: 10038: 9697:

of the resultant vector, meaning that energy is conserved. Likewise, the space components

9159: 5376: 5344: 4474: 4442: 3826:′) represents a line of simultaneity. All events on such a line have the same time value ( 2035: 8: 17726: 17721: 17711: 17543: 17492: 17310: 17295: 17121: 17052: 17000: 16976: 16771: 16626: 16491: 16443: 15969: 15428: 15388: 15067: 15062: 14832: 14419:"Welche Rolle spielt die Dreidimensionalität des Raumes in den Grundgesetzen der Physik?" 13778:

really represent tidal effects resulting from their differential attraction by mass

12284:

Catchers of the Light: Catching Space: Origins, Lunar, Solar, Solar System and Deep Space

12130: 11450: 11322: 11247:

the one which is at the same time the simplest in hypothesis and the richest in phenomena

10795: 10785: 10670: 10662: 10655: 9822: 8199: 7965: 6834:

There can be no such thing as length expansion/contraction in the transverse directions.

4785: 2954: 1884: 1145: 1009: 960: 930: 926: 713: 655: 433: 397: 17716: 16312: 14882: 14680: 14623: 14559: 14491: 14434: 14362: 14287: 14150: 14107: 14044: 14009: 13954: 13907: 13550: 12973: 12753: 12626: 12502: 12373: 12236: 12202:"Student understanding of time in special relativity: Simultaneity and reference frames" 12201: 9234: 7704: 5455:, and the red line representing the world line of a particle in motion has the equation 4856:. From the primed frame of the train, a passenger shoots a bullet with a speed given by 1578: 1327: 562:

refers to the whole ensemble of clocks associated with one inertial frame of reference.

17930: 17912: 17876: 17791: 17691: 17615: 17473: 17454: 17448: 17399: 17341: 17250: 17168: 17086: 17063: 17031: 16951: 16725: 16649: 16579: 15918: 15896: 15763: 15587: 15558: 15493: 15486: 15241: 15082: 15037: 14941: 14692: 14666: 14635: 14609: 14571: 14545: 14503: 14307: 14273: 14176: 14119: 13966: 13919: 13893: 13831: 13625: 13567: 13558: 13534: 13375: 13030: 12985: 12959: 12859: 12822: 12791: 12648: 12248: 12222: 11808: 11560: 11465: 11197: 11084: 11056: 11036: 11013: 10993: 10973: 10950: 10835: 10762: 10623: 10600: 10592: 10492: 7480: 6743: 5286: 4144: 3597: 3495: 3467: 3441:

some positive real constant. These equations describe two families of hyperbolae in an

3098: 3081: 2731:

compare with each other, it is useful to work with a simplified setup with frames in a

2625: 2222: 2069: 1601: 1165: 1157: 980: 934: 847: 683: 546: 507: 429: 421: 372: 357: 333: 310: 270: 256: 220: 77: 72: 10519:

general relativity, but it is not possible to demonstrate any non-trivial derivations.

8878:

This formula applies to all particles, massless as well as massive. For photons where

7729:

Two other scenarios are commonly examined in discussions of transverse Doppler shift:

1672:. If the event separation is due to a light signal, then this difference vanishes and 925:

The spacetime concept and the Lorentz group are closely connected to certain types of

17871: 17801: 17633: 17625: 17226: 17183: 16799: 16664: 16481: 15999: 15878: 15778: 15631: 15534: 15438: 15077: 14878: 14864: 14838: 14803: 14786: 14767: 14757: 14696: 14688: 14639: 14567: 14507: 14358: 14311: 14299: 14295: 14229: 14123: 14074: 14052: 13970: 13837: 13805: 13798: 13715: 13597: 13572: 13513: 13507: 13483: 13458: 13409: 13382: 13323: 13289: 13218: 13128: 13091: 13064: 13016: 12798: 12683: 12678:

Zahar, Elie (1989) , "Poincaré's Independent Discovery of the relativity principle",

12652: 12589: 12553: 12514: 12462: 12456: 12363: 12288: 12170: 12138: 12071: 12037: 12003: 11964: 11906: 11784:

Not all experiments characterize the effect in terms of a redshift. For example, the

11743: 11726: 11700:

is definitely finite, and only exceedingly large using the ordinary measuring units."

11440: 11193: 10900: 10288: 9768:

particles that collided. Indeed, it is larger than the sum of the individual masses:

8456: 8294: 854: 774: 659: 529: 436:

can slow the passage of time for an object as seen by an observer outside the field.

364: 329: 195: 17671: 14575: 14530: 13923: 13756:, concludes that a force of repulsion exists between the particles in scenario 12989: 12707:"Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910" 12643: 12031: 10903:

are used to represent observers in reference frames. Usually, Cartesian coordinates

10596: 8286:

must therefore be equated with the time component of the spacetime momentum vector.

5716:

represent equal spacetime intervals. Length contraction is represented by the ratio

4962:

The ratios of corresponding sides of the two yellow triangles are constant, so that

1704:

When the event considered is infinitesimally close to each other, then we may write

788: 744: 17888: 17706: 17696: 17643: 17620: 17198: 16944: 16690: 16669: 16621: 16584: 16397: 16117: 16105: 15933: 15901: 15733: 15602: 15518: 15508: 15423: 15007: 14684: 14627: 14563: 14495: 14438: 14354: 14291: 14154: 14111: 14048: 14013: 13958: 13911: 13562: 13554: 13113:

Landau, L. D., and Lifshitz, E. M. (2013). The classical theory of fields (Vol. 2).

12977: 12851: 12757: 12638: 12630: 12506: 12252: 12240: 11956:

Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity

11722: 11326: 10853: 10716: 10570: 9193: 7979: 3652: 2466: 1118:{\displaystyle (\Delta {d})^{2}=(\Delta {x})^{2}+(\Delta {y})^{2}+(\Delta {z})^{2}} 973: 663: 410: 322: 210: 17766: 11413:-dimensional spacetime is the marginal dimensionality. Moreover, it is the unique 10622:

problem in order to investigate the flow of energy at infinity due to propagating

4172:) to calculate the total amount of proper time experienced by the traveling twin. 554:

of the term, it does not make sense to speak of an observer as having a location.

17781: 17756: 17681: 17676: 17559: 17520: 17482: 17426: 17300: 17236: 16675: 16654: 16486: 16458: 16223: 16179: 16044: 16021: 15989: 15941: 15928: 15891: 15873: 15503: 15498: 15052: 14997: 14858: 14854: 14799: 14795: 14778: 14743: 14631: 13709: 13403: 13085: 12282: 12164: 11954: 11871: 11739: 10742: 10690: 10615: 10534: 10383: 7975: 7180:

The Lorentz transformations have a mathematical property called linearity, since

3603:

Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5

2950: 2715:

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

823: 760: 730: 693: 670: 541: 414: 368: 110: 17564: 14226:

Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations

11718: 5295:

Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction

2755:

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

2727:

To gain insight in how spacetime coordinates measured by observers in different

937:

and their transformation groups already developed in the 19th century, in which

440: 17864: 17806: 17478: 17462: 17458: 17361: 17331: 17188: 17091: 16971: 16730: 16720: 16659: 16616: 16448: 16438: 16407: 16307: 16279: 16122: 16112: 15911: 15783: 15753: 15738: 15523: 15481: 15476: 15348: 15134: 15119: 14739: 14414: 12738:"On the Electrodynamics of Moving Bodies ( Zur Elektrodynamik bewegter Körper)" 11485: 11414: 11334: 11330: 11265: 11258: 10990:. Another reference frame may be identified by a second coordinate chart about 10830: 10390: 9427: 8791: 7263: 7253: 5418:

represent equal spacetime intervals. Time dilation is represented by the ratio

4246: 4159: 3736:

is the temporal distance of the two events in S, and their spatial distance is

567: 243: 3634:(not illustrated) Drawing a vertical line through A so that it intersects the 545:

Figure 1-1. Each location in spacetime is marked by four numbers defined by a

17950: 17924: 17796: 17776: 17771: 17686: 17554: 17382: 17326: 17158: 17111: 16858: 16809: 16794: 16740: 16511: 16347: 16342: 16258: 16202: 16152: 16100: 16034: 16006: 15994: 15984: 15974: 15856: 15700: 15626: 15575: 15528: 15443: 15403: 15398: 15393: 15326: 15124: 14828: 14442: 14325: 14303: 14158: 12762: 12197: 12092: 11534: 11406: 11250: 11223: 10686: 10682: 10611: 9249:(1) The two bodies rebound from each other in a completely elastic collision. 8168: 8005:

is not affected by external forces, its total linear momentum cannot change.

8002: 7947: 6710:{\displaystyle t^{\prime }=\gamma \left(t-vx/c^{2}\right)=519.62\ {\text{s}}} 5906: 5282: 4834:

Fig. 3-2a illustrates a red train that is moving forward at a speed given by

4169: 4013:

To show the mutual time dilation immediately in the upper picture, the event

3581: 1161: 871: 812: 808: 768: 654:

medium; in the case of light waves, this was considered to be a hypothetical

624: 14884:

The Doom of Spacetime: Why It Must Dissolve Into More Fundamental Structures

12797:. Simon & Schuster. pp. 26–27, 122–127, 145–146, 345–349, 448–460. 12510: 9862: 9838: 3988:, connecting all events with the same spacetime interval from the origin as 906:), to the German Society of Scientists and Physicians. The opening words of 17900: 17816: 17736: 17701: 17231: 17193: 16956: 16843: 16838: 16833: 16820: 16559: 16174: 16168: 15834: 15768: 15690: 15621: 15570: 15471: 15418: 15358: 15144: 15109: 15002: 14931: 14807: 14115: 14017: 13576: 13184:"Being and Becoming in Modern Physics. 3. The Special Theory of Relativity" 12981: 11470: 11318: 11172: 10750: 10702: 10643:

reduce to special relativity in the case of weak fields at long distances.

9199: 5052:

in step 1 to yield Einstein's formula for the addition of velocities:

4140: 3984:-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through 3668: 205: 14137:

Sachs, Rainer K. (1962). "Asymptotic symmetries in gravitational theory".

13760:

while a force of attraction exists between the particles in scenario

11317:

is a positive whole number, then wave impulses become distorted. In 1922,

11010:. Two observers (one in each reference frame) may describe the same event 10555:

Is it possible to represent general relativity in terms of flat spacetime?

9973:

Conservation of momentum in the COM frame amounts to the requirement that

4021:′ = 0 (the location of clock W′ in S′), that is simultaneous to 3928: < 0, and so in this measurement the moving clock W passes W′ 2975: 1618:

units (like seconds) into space units (like meters). The squared interval

754: 17600: 17569: 17116: 16966: 16961: 16606: 16589: 16521: 16501: 16387: 16269: 16184: 16162: 16069: 16054: 15849: 15829: 15819: 15748: 15705: 15685: 15614: 15433: 15335: 15229: 15012: 14526: 11815:

and the proportional relationship of flux density and field strength. If

11340: 10899:

of particles and light beams. About each point (event) on this manifold,

10502: 10447: 7756: 3744:

in S, the temporal distance in S′ must be smaller than the one in S: the

3520: 2619: 882:

Figure 1–4. Hand-colored transparency presented by Minkowski in his 1908

804: 185: 30:"space and time" and "time and space" redirect here. For other uses, see 13630: 12881: 12863: 12483: 12227: 11343:

expands on the preceding argument in the following anthropic manner. If

11253:

argued that 3-dimensional space was a consequence of the inverse square

7271:). We are ignoring scenarios where they move along intermediate angles. 6612:, Stella's spaceship accelerates instantaneously to a speed of 0.5 857:, had also arrived at most of the basic elements of special relativity. 738: 646:

were considered to have proven the wave nature of light as opposed to a

17811: 17377: 17221: 17216: 16981: 16934: 16918: 16804: 16710: 16531: 16392: 16362: 16300: 16079: 15979: 15886: 15861: 15664: 15448: 15224: 15104: 14499: 13962: 13915: 13480:

The Classical Theory of Fields, Course of Theoretical Physics, Volume 2

12634: 11568: 11402: 10698: 10511:. The equivalence principle states that these two masses are identical. 7837: 5253: in this section. Unsourced material may be challenged and removed. 5160:

are both very small compared with the speed of light, then the product

3711:

at rest in two arbitrarily fixed points in S with the spatial distance

3297: in this section. Unsourced material may be challenged and removed. 2968: 2683: in this section. Unsourced material may be challenged and removed. 827: 816: 639: 14550: 14388: 14278: 13898: 13858: 13789: 12244: 8695:{\displaystyle E=\gamma mc^{2}={\frac {mc^{2}}{\sqrt {1-\beta ^{2}}}}} 3964:, thus giving the elapsed time between the two events, measured with W 3815: < 0", with unprimed, oblique axes, slanted to the left. 2953:, and should be considered no stranger than the manner in which, on a 724: 650:. Propagation of waves was then assumed to require the existence of a 439:

In ordinary space, a position is specified by three numbers, known as

16878: 16715: 16599: 16574: 16554: 16423: 16417: 16402: 16142: 15695: 15607: 15582: 15563: 15453: 15205: 15114: 14978: 14598:"Gravothermal phase transition, black holes and space dimensionality" 13783: 12855: 12134: 11455: 11333:

around nuclei cannot be stable; electrons would either fall into the

8183:

Figure 3–9. Energy and momentum of light in different inertial frames

8167:

We will use this information shortly to obtain an expression for the

3066: 2267:

as a distinct symbol in itself, rather than the square of something.

638:

By the mid-1800s, various experiments such as the observation of the

61: 14418: 11559:

A hallmark of Einstein's career, in fact, was his use of visualized

8939:

to denote the spatial momentum, the four-momentum may be written as

8397:

approaches 1, so the spatial component of the relativistic momentum

6564:, etc. We are, in general, always concerned with the space and time 5228: 3272: 2909:

from the light source (event Q), and returns to the light source at

2658: 2274:

There are two sign conventions in use in the relativity literature:

17751: 17101: 16913: 16893: 16735: 16641: 16541: 16332: 16327: 15710: 15553: 15463: 15321: 15311: 15129: 15092: 15017: 14671: 14614: 14264:

Tegmark, Max (1 April 1997). "On the dimensionality of spacetime".

14181: 13653:. California State University Affordable Learning Solutions Program 11363: 11149:

into relativity, by which all physical quantities are represented.

10896: 10781: 10734: 10710: 10674: 10427: 10363:. Most of the energy is carried off by the near-zero-mass neutrino. 8775:

The concept of relativistic mass that Einstein introduced in 1905,

7951: 3134:

are simultaneous from the reference frame of an observer moving at

858: 697:

changes occur in directions transverse to the direction of motion.

644:

differential measurements of the speed of light in air versus water

500: 425: 14711:"Life could exist in a 2D universe (according to physics, anyway)" 14521: 14519: 14517: 12964: 12827: 12611:"On the Dynamics of the Electron (Sur la dynamique de l'électron)" 12458:'Subtle is the Lord–': The Science and the Life of Albert Einstein 12036:(3rd ed.). Mineola, New York: Dover Publications. p. 7. 10693:

from point to point). This gives, in particular, local notions of

10213:

Figure 3-12a. Energy–momentum diagram for decay of a charged pion.

8278:, and since for photons, the space and time components are equal, 4884:, where the distance is measured along a line parallel to the red 16908: 16898: 16828: 16700: 16611: 16476: 16049: 16039: 15961: 15952: 15937: 15824: 15669: 15592: 15301: 15139: 13680:. Department of Physics and Astronomy, Georgia State University. 13090:(illustrated ed.). Oxford University Press. pp. 26–28. 11480: 11146: 10227:

Figure 3-12b. Graphing calculator analysis of charged pion decay.

5402: 5218: 4903:

From the platform, the composite speed of the bullet is given by

3569:, the elapsed time measured by the observer is 7.00 meters ( 3557:, the elapsed time measured by the observer is 5.24 meters ( 298: 160: 15839: 14771: 13509:

Introduction to Classical Mechanics: With Problems and Solutions

10639:. This implies the conclusion that General Relativity (GR) does 9049:{\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})} 5884:{\displaystyle OB/OK=\gamma (1-\beta ^{2})={\frac {1}{\gamma }}} 2933:

axes. Since OP = OQ = OR, the angle between

17431: 16903: 16853: 16848: 16147: 15306: 14907: 14514: 14380: 11823: 11359: 11277: 11273: 10706: 8446:, the classical term for momentum. Following this perspective, 7721:

Fig. 3-7b. What is the frequency measurement when the receiver

6572:

transformations are trivially related to each other, since the

2993: 2992:

meeting with their apices at O. One cone extends into the

841:, which was instrumental for his subsequent formulation of the 13796:

Lorentz, H. A.; Einstein, A.; Minkowski, H.; Weyl, H. (1952).

13083: 11721:

of the plane, and is isomorphic to the group of isometries in

8460:

of an object increases with velocity according to the formula

7689:{\displaystyle f={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{0}.} 6826:(or the other way around) it is easier at first to derive the 6616:. The distance from Earth to Mars is 300 light-seconds (about 3870:′-axis (upper pictures, clock W′) and the slanted to the left 1429:{\displaystyle (\Delta s)^{2}=(\Delta ct)^{2}-(\Delta x)^{2},} 588:

is the source of much confusion among students of relativity.

417:: it assumes that space follows the geometry of common sense. 16992: 16883: 16695: 16594: 16549: 16157: 15906: 15597: 15545: 15096: 14887:(Speech). The 2,384th Meeting Of The Society. Washington, D.C 14812:

The Principle of Relativity: A Collection of Original Memoirs

12713:. Berlin, Germany: Springer. pp. 193–252. Archived from 11381: 11269: 10737:

on them, with techniques that can be applied to the study of

10709:. From those, some other global quantities can be derived by 10694: 7212:

Terence observes Stella speeding away from him at 0.500

6631:. But Stella observes the time on her ship chronometer to be 6624:). Terence observes Stella crossing the finish-line clock at 6576:

frame can only be moving forwards or reverse with respect to

4556:

More generally, assuming that frame S′ is moving at velocity

3647:

illustrates that the primed and unprimed frames are mutually

3085:

Figure 2–6. Animation illustrating relativity of simultaneity

3003: 677:

flows at different speeds for different colors of light. The

349: 14947: 14318: 13795: 10415:

of a gravitational force acting at a long range from Earth.

9149:{\displaystyle P\equiv (E,{\vec {p}})=(E,p_{x},p_{y},p_{z})} 7449:

Figure 3–6. Spacetime diagram of relativistic Doppler effect

6887:

From previous results using similar triangles, we know that

6492:{\displaystyle \Delta t'=\gamma (\Delta t-v\Delta x/c^{2}),} 3980:, measured with W′, must be transformed to the scale of the 3662: 3107:

was drawn from the reference frame of an observer moving at

3027:

comprises the interior events of the past light cone. So in

17412: 16888: 16873: 16863: 16317: 16254: 15296: 15282: 14393:

Die philosophischen schriften von Gottfried Wilhelm Leibniz

13884:

Deser, S. (1970). "Self-Interaction and Gauge Invariance".

12920:

Encyclopédie des Sciences Mathématiques Pures et Appliquées

11812: 11281: 10722:

Ueber die Hypothesen, welche der Geometrie zu Grunde liegen

10268: 7477:

so that the separation between source and receiver at time

7175: 5918:

The Lorentz factor appears in the Lorentz transformations:

5781:). Since the tangent line through A and B has the equation 3834:′). Likewise, each line drawn parallel to a temporal axis ( 3762:

In this comparison the clock W is moving by with velocity −

3687:

in S', as running slower than hers. How two clocks can run

3623:-axis along their world lines. The projection of the rod's 3484:. On the other hand, the green hyperbolae, which cross the 3065:

The light cone has an essential role within the concept of

2997: 2989: 2905:. The pulse is reflected from a mirror situated a distance 1168:). Special relativity provides a new invariant, called the 401: 318: 17895: 16323:

International Earth Rotation and Reference Systems Service

14071:

Black Holes & Time Warps: Einstein's Outrageous Legacy

13703: 13701: 13699: 12875: 12873: 12287:. Paphos, Cyprus: ArtDeCiel Publishing. pp. 202–233. 12002:(2nd ed.). New York: Pergamon Press Ltd. p. 33. 11903:

Mathematics & Common Sense: A Case of Creative Tension

11200:(unidirectional). Let the number of spatial dimensions be 7294:, the wavelength is increased, and the observed frequency 4044:

In the lower picture the frame S is moving with velocity −

586:

the difference between what one measures and what one sees

16868: 13996:

Rosen, N. (1940). "General Relativity and Flat Space I".

13647:"Mechanics and Relativity. Chapter 4.3: Reference Frames" 13217:. Cambridge, UK: Cambridge University Press. p. 26. 13087:

Introducing Einstein's Relativity: A Deeper Understanding

12033:

Introduction to Tensor Calculus, Relativity and Cosmology

11717:

transforming planes into planes, it is isomorphic to the

11436:

Basic introduction to the mathematics of curved spacetime

8868:{\displaystyle E^{2}-p^{2}c^{2}=m_{\text{rest}}^{2}c^{4}} 7982:

quantity, possessing the same direction as the velocity:

4774: 4048:

in the frame S′ at rest. The worldline of clock W is the

2479:

In the following discussion, we use the first convention.

14093: 13861:. Internet Encyclopedia of Philosophy (ISSN 2161-0002). 13825: 13823: 13821: 13322:(1st ed.). Jones & Bartlett Pub. p. 1047. 13127:. New York: Penguin Random House LLC. pp. 155–156. 11417:

that can afford a "stable" gas sphere with a "positive"

11158: 7960: 6790:

in the red frame that is moving with velocity parameter

5373:

One also sees very frequently the use of the convention

3842:) represents a line of equal spatial coordinate values ( 3007:

Figure 2–5. Light cone in 2D space plus a time dimension

2797:′ axis passes through the events in frame S′ which have 2769:

The origins of frames S and S′ are coincident when time

1807:{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}.} 1259:

between the two events that are separated by a distance

15251: 13937:

Grishchuk, L. P.; Petrov, A. N.; Popova, A. D. (1984).

13764:. (ii) A second reporter, aware of the large mass 13696: 12870: 11222:, setting aside the compactified dimensions invoked by 9376:

collide to produce a single particle of conserved mass

8450:

can be interpreted as a relativistic generalization of

6732: 3532: 939:

invariant intervals analogous to the spacetime interval

898:). On 21 September 1908, Minkowski presented his talk, 14852: 13381:. Cambridge, Massachusetts: Harvard University Press. 12584:. New York: W. W. Norton & Company, Inc. pp. 11284:

impulse will travel at different speeds. If there are

10418:

Two central propositions underlie general relativity.

8762:{\displaystyle \approx mc^{2}+{\frac {1}{2}}mv^{2}...} 8297:. Infinity times zero is an ill-defined quantity, but 7978:, the product of a particle's mass and velocity, is a 5481: 4163:

Figure 2–11. Spacetime explanation of the twin paradox

2537:

is positive, the spacetime interval is referred to as

17848: 14478:

dimensions and the dimensionality of space problem".

13936: 13836:. Princeton, New Jersey: Princeton University Press. 13818: 13711:

Spacetime Physics: Introduction to Special Relativity

13596:. Cambridge: Cambridge University Press. p. 60. 13177: 13175: 12166:

Spacetime Physics: Introduction to Special Relativity

11905:. Wellesley, Massachusetts: A.K. Peters. p. 86. 11807:

This is because the law of gravitation (or any other

11709:

For instance, the Lorentz group is a subgroup of the

11290: 11107: 11087: 11059: 11039: 11016: 10996: 10976: 10953: 10909: 10861: 10838: 10803: 10321: 10115: 10074: 10041: 9988: 9934: 9893: 9776: 9738: 9705: 9662: 9623: 9568: 9436: 9384: 9326: 9268: 9162: 9064: 8948: 8893: 8803: 8707: 8629: 8571: 8511: 8468: 8405: 8320: 8248: 8210: 8119: 8067: 8017: 7922: 7906:{\displaystyle f=f'/\gamma =f'{\sqrt {1-\beta ^{2}}}} 7849: 7825:{\displaystyle f=f'\gamma =f'/{\sqrt {1-\beta ^{2}}}} 7768: 7637: 7563: 7526: 7503: 7483: 7460: 7393: 7307: 7074: 7022: 6639: 6549:{\displaystyle \Delta x'=\gamma (\Delta x-v\Delta t)} 6504: 6429: 6380: 6314: 6102: 5927: 5822: 5683: 5581: 5558: 5379: 5347: 5316: 5060: 4803: 4730: 4693: 4509: 4477: 4445: 4404: 3498: 3470: 3420: 3349: 3230: 3188: 2838: 2628: 2578: 2547: 2516: 2489: 2376: 2282: 2246: 2225: 2195: 2168: 2132: 2092: 2072: 2038: 1987: 1958: 1915: 1887: 1823: 1713: 1678: 1655: 1624: 1604: 1581: 1448: 1356: 1330: 1290: 1265: 1227: 1201: 1178: 1164:) or the distance between the two events (because of 1021: 989: 510: 13714:(2nd ed.). San Francisco, California: Freeman. 13482:(4th ed.). Amsterdam: Elsevier. pp. 1–24. 12886:

Jahresbericht der Deutschen Mathematiker-Vereinigung

12682:, Chicago, Illinois: Open Court Publishing Company, 12362:. Boston, Massachusetts: Birkhäuser. pp. 1–13. 12169:(2nd ed.). San Francisco, California: Freeman. 10741:

of higher dimensions. It enabled the formulation of

8187:

Light particles, or photons, travel at the speed of

7014:

into the equations in Step 2 immediately yield

5406:

Figure 3–4. Lorentz factor as a function of velocity

3976:. For a comparison, the length of the time interval 3874:-axes (lower pictures, clock W). The worldlines of W 1817:

In a different inertial frame, say with coordinates

919: 833:

Einstein in 1905 superseded previous attempts of an

14474:Tangherlini, F. R. (1963). "Schwarzschild field in 12581:

Einstein's Clocks, Poincaré's Maps: Empires of Time

11409:under its self-gravity, Wei-Xiang Feng showed that 11257:. While Kant's argument is historically important, 10366: 9880: 8308:By this analysis, if the energy of a photon equals 8191:, the constant that is conventionally known as the 7227:, and she can use the Lorentz transformations with 16253: 13797: 13708:Taylor, Edwin F.; Wheeler, John Archibald (1992). 13374: 13172: 13077: 12913: 12790: 12461:(11th ed.). Oxford: Oxford University Press. 12200:; Shaffer, Peter S.; Vokos, Stamatis (July 2001). 12196: 12163:Taylor, Edwin F.; Wheeler, John Archibald (1992). 11305: 11137: 11093: 11065: 11053:(representing an observer) and another containing 11045: 11022: 11002: 10982: 10959: 10939: 10879: 10844: 10821: 10715:Riemannian geometry originated with the vision of 10346: 10189:{\displaystyle v^{\prime }=(v-u)/(1-{vu}/{c^{2}})} 10188: 10097: 10056: 10020: 9960: 9916: 9808: 9753: 9720: 9683: 9644: 9602: 9552: 9416: 9366: 9308: 9238:Figure 3–10. Relativistic conservation of momentum 9174: 9148: 9048: 8911: 8867: 8761: 8694: 8600: 8553: 8490: 8432: 8357: 8268: 8230: 8150: 8101: 8035: 7928: 7905: 7824: 7688: 7616: 7549: 7512: 7489: 7469: 7434: 7351: 7216:, and he can use the Lorentz transformations with 7111: 7059: 6709: 6548: 6491: 6415: 6367: 6267: 6082: 5883: 5689: 5658: 5564: 5542: 5391: 5362: 5333: 5132: 4823: 4789:Figure 3–2. Relativistic composition of velocities 4756: 4716: 4529: 4494: 4462: 4430: 3504: 3476: 3433: 3403: 3250: 3211: 3156:From the reference frame of an observer moving at 3141:From the reference frame of an observer moving at 3015:(temporal distance) than necessary to cross their 2873: 2634: 2599: 2568:is negative, the spacetime interval is said to be 2560: 2529: 2502: 2453: 2356: 2259: 2231: 2211: 2181: 2154: 2103: 2078: 2055: 2024: 1973: 1944: 1901: 1873: 1806: 1693: 1664: 1640: 1610: 1590: 1564: 1428: 1339: 1316: 1276: 1251: 1213: 1187: 1117: 1000: 954: 669:Among other issues, the dependence of the partial 516: 336:effects, such as how different observers perceive 14170: 14168: 13535:"Optics of high-performance electron microscopes" 13190:. Metaphysics Research Lab, Stanford University. 13156:. Metaphysics Research Lab, Stanford University. 12916:"La théorie des groupes continus et la géométrie" 11878:. Metaphysics Research Lab, Stanford University. 11631:"With respect to this it is clear that the group 6966:Length contraction and similar triangles give us 5701:is the maximum possible speed in nature. For any 4233: 3488:axis, are spacelike curves because all intervals 2817:axis at any time other than zero. Therefore, the 2454:{\displaystyle s^{2}=-(ct)^{2}+x^{2}+y^{2}+z^{2}} 413:, or anything external. It assumes that space is 17922: 13455:Special Relativity for the Enthusiastic Beginner 13002: 7836:the source, emitted from this closest point, is 7617:{\displaystyle T=w+\beta w=\gamma w'(1+\beta ).} 7352:{\displaystyle f={\frac {1}{1+\beta _{s}}}f_{0}} 6747:Figure 3–5. Derivation of Lorentz Transformation 4052:-axis (slanted to the left), the worldline of W′ 2357:{\displaystyle s^{2}=(ct)^{2}-x^{2}-y^{2}-z^{2}} 14087: 13457:. CreateSpace Independent Publishing Platform. 10562:than the usual curved spacetime interpretation? 8933:to represent the four-momentum and a lowercase 8563:, so that Einstein arrived at the relationship 7708:Figure 3–7. Transverse Doppler effect scenarios 4395:, the coordinate transformation is as follows: 4041:, showing that the "moving" clock runs slower. 3046:than can be crossed at lightspeed in the given 2762:-direction of frame S with a constant velocity 2465:These sign conventions are associated with the 487:observer to be in motion relative to an event. 14216: 14165: 13477: 13319:Physics for Scientists and Engineers, Volume 2 12949: 12709:. In Renn, Jürgen; Schemmel, Matthias (eds.). 12573: 12571: 12569: 12550:Albert Einstein's Special Theory of Relativity 10528: 8358:{\displaystyle E^{\prime }=(1-\beta )\gamma E} 7557:Since the slope of the green light ray is −1, 7274: 5697:becomes physically meaningless, implying that 5219:Time dilation and length contraction revisited 3855:One may designate in both pictures the origin 2189:terms. The spacetime interval is the quantity 17008: 16239: 15267: 14963: 14064: 14062: 13930: 13707: 13377:Very Special Relativity: An Illustrated Guide 13054: 13052: 13050: 13048: 13046: 12162: 11166:Anthropic principle § Dimensions of spacetime 9828: 9603:{\displaystyle {\boldsymbol {p=p_{1}+p_{2}}}} 8554:{\displaystyle \gamma mc=m_{\text{rel}}c=E/c} 7241:to relate Ursula's measurements with his own. 6772:) in the black "rest system" and coordinates 5133:{\displaystyle u={v+u' \over 1+(vu'/c^{2})}.} 3074: 2572:. Spacetime intervals are equal to zero when 716:, the Fizeau experiment and other phenomena. 278: 14877: 14738: 14096:Proceedings of the Royal Society of London A 14024: 13539:Science and Technology of Advanced Materials 13501: 13499: 13401: 13116: 12816: 12814: 12782: 12729: 12615:Rendiconti del Circolo Matematico di Palermo 12481: 10720: 8151:{\displaystyle p^{\prime }=-\beta \gamma mc} 7699: 3818:Each line drawn parallel to a spatial axis ( 1008:between two points can be defined using the 837:–energy relation by introducing the general 332:are useful in visualizing and understanding 14473: 14217:Bär, Christian; Fredenhagen, Klaus (2009). 13829: 12844:Historical Studies in the Physical Sciences 12711:The Genesis of General Relativity, Volume 3 12680:Einstein's Revolution: A Study in Heuristic 12566: 12358:. In Kox, A. J.; Eisenstaedt, Jean (eds.). 11872:"Newton's Views on Space, Time, and Motion" 11869: 9214:, the fact that such processes do not care 8370: 7112:{\displaystyle x=\gamma x'+\beta \gamma w'} 7060:{\displaystyle w=\gamma w'+\beta \gamma x'} 5724:. The invariant hyperbola has the equation 5543:{\textstyle OB=OK/{\sqrt {1-v^{2}/c^{2}}}.} 5426:. The invariant hyperbola has the equation 4769:Galilean law for the addition of velocities 4240: 4033:) in S′. This shows that the time interval 3565:), while for a clock traveling at 0.7 2813:, so that they are not coincident with the 27:Mathematical model combining space and time 17015: 17001: 16246: 16232: 15274: 15260: 14970: 14956: 14929:Encyclopedia of Space–time and gravitation 14925:Historical: Albert Einstein's 1926 article 14212: 14210: 14208: 14206: 14174: 14059: 13638: 13206: 13147: 13063:. Massachusetts: Harcourt/Academic Press. 13043: 13035:: CS1 maint: location missing publisher ( 12322:"What is Einstein's Theory of Relativity?" 12023: 10246:mass and the muon mass is 33.91 MeV. 6753:derivations of the Lorentz transformations 6739:Derivations of the Lorentz transformations 5896: 4937:. In the large yellow triangle, the ratio 4100:in the upper pictures) as simultaneous to 3882:are the corresponding vertical time axes ( 3166:, the events appear to occur in the order 2888:′ axis is also tilted with respect to the 1952:. For any other infinitesimal event where 371:. This interpretation proved vital to the 285: 271: 14670: 14613: 14549: 14413: 14395:. Vol. 4. Weidmann. pp. 427–463 14348: 14277: 14180: 14130: 13989: 13897: 13877: 13629: 13589: 13566: 13496: 12963: 12909: 12907: 12879: 12826: 12820: 12811: 12761: 12700: 12698: 12642: 12484:"The Genesis of the theory of relativity" 12226: 12061: 12059: 12057: 12055: 12053: 11999:The Theory of Space, Time and Gravitation 11948: 11946: 11944: 11942: 11725:which is often expressed in terms of the 11713:. The Lorentz group is isomorphic to the 11667:is mathematically more intelligible than 11204:and the number of temporal dimensions be 8433:{\displaystyle \beta \gamma mc=\gamma mv} 7672: 6093:The inverse Lorentz transformations are: 5475:. A bit of algebraic manipulation yields 5269:Learn how and when to remove this message 3663:Mutual time dilation and the twin paradox 3641:In regards to mutual length contraction, 3631:axis yields the foreshortened length OC. 3580:This illustrates the phenomenon known as 3515:Fig. 2-7b reflects the situation in 3404:{\displaystyle (ct)^{2}-x^{2}=\pm s^{2},} 3313:Learn how and when to remove this message 3019:at lightspeed; these events comprise the 2699:Learn how and when to remove this message 16777:International Commission on Stratigraphy 14652: 14030: 13830:Mook, Delo E.; Vargish, Thoma s (1987). 13593:Revolutions in Twentieth-Century Physics 13448: 13446: 13444: 13442: 13440: 13438: 13436: 13434: 13432: 13368: 13366: 13246: 13244: 13242: 13240: 13238: 13236: 13234: 12788: 12735: 12608: 12319: 12219:American Association of Physics Teachers 12158: 12156: 12154: 12120: 12118: 12070:(3rd ed.). Incomprehensible Books. 11989: 11940: 11938: 11936: 11934: 11932: 11930: 11928: 11926: 11924: 11922: 11171: 10614:, M. G. van der Burg, A. W. Metzner and 10580: 10426: 10389: 9233: 8178: 8102:{\displaystyle (mc)^{\prime }=\gamma mc} 7964: 7703: 7444: 7176:Linearity of the Lorentz transformations 6742: 5401: 5290: 4784: 4539: 4158: 4076:. The invariant hyperbola through event 3588: 3536: 3459:The magenta hyperbolae, which cross the 3326: 3152:the events appear to occur in the order 3080: 3002: 2974: 2718: 2710: 2117: 2032:which in turn upon integration leads to 877: 826:analyzed special relativity in terms of 576:Physicists distinguish between what one 540: 14827: 14655:"Existence of life in 2 + 1 dimensions" 14653:Scargill, J. H. C. (26 February 2020). 14525: 14386: 14330:"What makes time different from space?" 14263: 14228:. Dordrecht: Springer. pp. 39–58. 14203: 13478:Landau, L. D.; Lifshitz, E. M. (2006). 13402:Forshaw, Jeffrey; Smith, Gavin (2014). 13395: 13364: 13362: 13360: 13358: 13356: 13354: 13352: 13350: 13348: 13346: 13188:The Stanford Encyclopedia of Philosophy 13122: 12841: 12835: 12577: 12450: 12448: 12446: 12350: 12344: 12065: 11401:On the other hand, in view of creating 11398:, which describes the world around us. 9741: 9708: 9590: 9577: 9570: 9497: 9472: 9446: 9443: 9439: 9354: 9329: 9296: 9271: 9260:In case (2), two masses with momentums 8491:{\displaystyle m_{\text{rel}}=\gamma m} 8293:must be zero, meaning that photons are 7194:are obtained as linear combinations of 6368:{\displaystyle t'=\gamma (t-vx/c^{2}),} 4891:axis rather than parallel to the black 4548:Spacetime and composition of velocities 3673: 662:of 1851, conducted by French physicist 352:and time took on new meanings with the 14: 17923: 14068: 13943:Communications in Mathematical Physics 13408:. John Wiley & Sons. p. 118. 13315: 13288:(1st ed.). Springer. p. 42. 13212: 12904: 12704: 12695: 12547: 12541: 12280: 12262:from the original on 28 September 2018 12124: 12050: 12029: 11952: 11870:Rynasiewicz, Robert (12 August 2004). 10648: 7941: 4775:Relativistic composition of velocities 3262: 3050:. These events comprise the so-called 3039:, making timelike intervals positive. 16996: 16227: 15255: 14951: 14136: 13995: 13977:from the original on 25 February 2021 13883: 13865:from the original on 23 December 2020 13850: 13644: 13623: 13505: 13452: 13429: 13281: 13231: 13084:Ray d'Inverno; James Vickers (2022). 13058: 12677: 12530:from the original on 28 February 2008 12151: 12115: 11919: 11900: 11851: 11159:Privileged character of 3+1 spacetime 9754:{\displaystyle {\boldsymbol {p_{2}}}} 9721:{\displaystyle {\boldsymbol {p_{1}}}} 8601:{\displaystyle E=m_{\text{rel}}c^{2}} 7961:Extending momentum to four dimensions 7435:{\displaystyle f=(1-\beta _{r})f_{0}} 4895:axis. What is the composite velocity 2821:′ axis is tilted with respect to the 2801:′ = 0. But the points with 2510:can assume any real number value. If 1317:{\displaystyle \Delta {ct}=c\Delta t} 1128:Although two viewers may measure the 967: 166:Newton's law of universal gravitation 16208: 14749:The Anthropic Cosmological Principle 14595: 14531:"On the dimensionality of spacetime" 14449:Proceedings of the Amsterdam academy 14324: 13532: 13372: 13343: 13215:A first course in general relativity 12770:from the original on 6 November 2018 12454: 12443: 12401:from the original on 17 January 2023 12301:from the original on 17 January 2023 11995: 11977:from the original on 17 January 2023 10719:expressed in his inaugural lecture " 10107:is replaced by the highly nonlinear 9187: 8174: 6733:Deriving the Lorentz transformations 5251:adding citations to reliable sources 5222: 5208:regardless of the starting value of 4008:, the "moving" clock W′ runs slower. 3895:In the upper picture the place for W 3533:Time dilation and length contraction 3449:spacetime diagram, which are termed 3295:adding citations to reliable sources 3266: 2809:-direction of frame S with velocity 2681:adding citations to reliable sources 2652: 2111:, etc. We are always concerned with 1221:Then the squared spacetime interval 459:. A point in spacetime is called an 17587:Tolman–Oppenheimer–Volkoff equation 17540:Friedmann–Lemaître–Robertson–Walker 14938:Stanford Encyclopedia of Philosophy 14863:. Internet Archive: W. H. Freeman. 14837:. Oxford: Oxford University Press. 13752:but unaware of the large mass 13154:Stanford Encyclopedia of Philosophy 13009:Foundations of Hyperbolic Manifolds 12090: 11876:Stanford Encyclopedia of Philosophy 11841:would be inversely proportional to 11618:is the Galilean group and the group 11192:There are two kinds of dimensions: 11030:but obtain different descriptions. 10775: 10523: 4017:may be constructed as the event at 3464:speed—for all secants is less than 3023:of the event O. Likewise, the 2648: 24: 14732: 13886:General Relativity and Gravitation 13859:"Jules Henri Poincaré (1854–1912)" 13674:"Energetics of Charged Pion Decay" 13263:from the original on 27 April 2017 13194:from the original on 11 March 2017 13181: 12930:from the original on 23 March 2018 12360:The Universe of General Relativity 12332:from the original on 3 August 2022 12320:Williams, Matt (28 January 2022). 12103:from the original on 13 March 2017 11711:conformal group in four dimensions 10361:33.91 MeV − 29.79 MeV = 4.12 MeV 10121: 8326: 8125: 8082: 6645: 6537: 6525: 6505: 6462: 6450: 6430: 5318: 2874:{\displaystyle \tan(\theta )=v/c.} 2805:′ = 0 are moving in the 2093: 1679: 1656: 1625: 1598:the speed of light, converts time 1543: 1521: 1499: 1474: 1452: 1407: 1382: 1360: 1308: 1291: 1266: 1231: 1202: 1179: 1097: 1073: 1049: 1025: 990: 712:, with which he could explain the 604:History of Lorentz transformations 131:Introduction to general relativity 25: 17972: 17357:Hamilton–Jacobi–Einstein equation 16497:Discrete time and continuous time 14900: 14860:Spacetime Physics, Second Edition 13856: 13250: 13150:"Lightcones and Causal Structure" 13125:The Biggest Ideas in the Universe 12892:from the original on 28 July 2017 12659:from the original on 11 July 2017 12431:from the original on 23 June 2017 12421:"The Nobel Prize in Physics 1902" 11882:from the original on 16 July 2012 10761:, and spurred the development of 10727:differential geometry of surfaces 10591:The spacetime symmetry group for 10431:Figure 5–2. Equivalence principle 10033:Newtonian momenta, calculated as 9430:velocity of the original system, 9229: 7247: 6998:Substituting the expressions for 3800:Figure 2-10. Mutual time dilation 3703:additional, synchronized clocks W 3212:{\displaystyle {\sqrt {-s^{2}}}.} 2793:′ = 0 in frame S′. The 375:, wherein spacetime is curved by 216:Mathematics of general relativity 136:Mathematics of general relativity 17906: 17894: 17882: 17870: 17858: 17835: 17834: 16422: 16416: 16207: 16198: 16197: 15036: 14906: 14689:10.1103/PhysRevResearch.2.013217 14359:10.1111/j.1468-0068.2007.00645.x 13684:from the original on 21 May 2017 13671: 13160:from the original on 17 May 2019 11833:. More generally, in a space of 11573:Heisenberg uncertainty principle 11164:This section is an excerpt from 10654:This section is an excerpt from 10374:This section is an excerpt from 10367:Introduction to curved spacetime 10220: 10206: 9881:Energy and momentum conservation 9861: 9837: 9809:{\displaystyle m>m_{1}+m_{2}} 9745: 9712: 9594: 9586: 9581: 9573: 9501: 9476: 9359: 9334: 9301: 9276: 8924: 8052:, but the time component equals 6416:{\displaystyle x'=\gamma (x-vt)} 5765:. Event A has coordinates ( 5227: 4029:has equal spacetime interval as 3790: 3779: 3271: 3251:{\displaystyle {\sqrt {s^{2}}}.} 2657: 1252:{\displaystyle (\Delta {s})^{2}} 1154:four-dimensional Euclidean space 955:Spacetime in special relativity 767: 753: 737: 723: 623: 612: 251: 250: 237: 60: 14942:Space and Time: Inertial Frames 14703: 14646: 14589: 14467: 14454: 14407: 14257: 14192: 13735: 13665: 13617: 13583: 13526: 13471: 13309: 13275: 13148:Curiel, Erik; Bokulich, Peter. 13141: 13107: 12996: 12943: 12793:Einstein: His Life and Universe 12671: 12602: 12475: 12413: 12383: 12313: 12274: 12190: 11959:(Reprint ed.). Cambridge: 11801: 11791: 11778: 11759: 11749: 11732: 11703: 11600: 11583: 11553: 11540: 11526: 11498: 10376:Curved spacetime § Introduction 7267:right angles to the said line ( 6935:Substituting equation (4) into 6751:There have been many dozens of 5334:{\displaystyle \mathrm {T} =ct} 5238:needs additional citations for 5040:Substitute the expressions for 4571:with respect to frame S, since 4273: 4234:Basic mathematics of spacetime 4187: 4130: 4060:′-axis, and the worldline of W′ 3642: 3282:needs additional citations for 3102: 2668:needs additional citations for 1439:or for three space dimensions, 566:synchronize the clocks, in the 498:Mathematically, spacetime is a 432:provides an explanation of how 384: 32:Space and Time (disambiguation) 17164:Mass–energy equivalence (E=mc) 17022: 16570:History of timekeeping devices 14073:. W. W. Norton & Company. 13645:Idema, Timon (17 April 2019). 13512:. Cambridge University Press. 13257:The Physics and Relativity FAQ 12914:Cartan, É.; Fano, G. (1955) . 12217:(S1). College Park, Maryland: 12084: 11894: 11863: 11811:) follows from the concept of 11446:Einstein's thought experiments 11349:partial differential equations 11132: 11108: 10934: 10910: 10874: 10862: 10816: 10804: 10183: 10149: 10141: 10129: 9143: 9098: 9092: 9086: 9071: 9043: 8990: 8984: 8978: 8955: 8346: 8334: 8078: 8068: 8030: 8018: 7608: 7596: 7419: 7400: 6756:expressions of the underlying 6543: 6522: 6483: 6447: 6410: 6395: 6359: 6329: 5865: 5846: 5121: 5092: 4224: 4064:is the vertical through event 3910:′-axis) in the event labelled 3360: 3350: 3170:. The white line represents a 2851: 2845: 2789: = 0 in frame S and 2403: 2393: 2306: 2296: 2143: 2133: 2025:{\displaystyle ds^{2}=ds'^{2}} 1868: 1824: 1550: 1540: 1528: 1518: 1506: 1496: 1484: 1471: 1459: 1449: 1414: 1404: 1392: 1379: 1367: 1357: 1240: 1228: 1106: 1094: 1082: 1070: 1058: 1046: 1034: 1022: 839:equivalence of mass and energy 391: 13: 1: 14977: 14919:Albert Einstein on space–time 14538:Classical and Quantum Gravity 14266:Classical and Quantum Gravity 13533:Rose, H. H. (21 April 2008). 12552:. New York: Springer-Verlag. 11856: 11245:suggested that the world is " 10895:, as well as determining the 10021:{\displaystyle m=m_{1}+m_{2}} 9417:{\displaystyle m=m_{1}+m_{2}} 8454:. Einstein proposed that the 8312:in the rest frame, it equals 6423:etc., most often really mean 2996:(t>0), the other into the 2962: 1874:{\displaystyle (t',x',y',z')} 600:History of special relativity 14822:A Treatise on Time and Space 14781:and Ruth M. Williams (1992) 14632:10.1103/PhysRevD.106.L041501 14596:Feng, W.X. (3 August 2022). 14053:10.1016/0031-9163(64)90396-8 13590:Griffiths, David J. (2013). 13559:10.1088/0031-8949/9/1/014107 13316:Lerner, Lawrence S. (1997). 11607:(In the following, the group 11476:Philosophy of space and time 11255:law of universal gravitation 10887:. The metric determines the 10747:general theory of relativity 10474:and in Newton's second law, 8001:quantity, meaning that if a 7550:{\displaystyle w=\gamma w'.} 7520:. Because of time dilation, 7369:for a velocity parameter of 7287:for a velocity parameter of 5673:is greater than or equal to 4116:corresponds again to above. 1641:{\displaystyle \Delta s^{2}} 373:general theory of relativity 358:special theory of relativity 7: 17179:Relativistic Doppler effect 16517:Gravitational time dilation 16353:Barycentric Coordinate Time 16012:Geological history of Earth 14785:. Oxford University Press. 14783:Flat and curved space–times 14387:Leibniz, Gottfried (1880). 13213:Schutz, Bernard F. (1985). 12880:Minkowski, Hermann (1909). 12391:"George Francis FitzGerald" 12210:American Journal of Physics 11428: 10529:Is spacetime really curved? 7626:relativistic Doppler effect 7275:Longitudinal Doppler effect 7258:Relativistic Doppler effect 4219:gravitational time dilation 4202:from Stella's point of view 4108: = 0. The result 4096:is constructed (similar to 3886:in the upper pictures, and 3182:, i.e. the true distance = 679:Michelson–Morley experiment 534:Michelson–Morley experiment 524:(conventionally called the 445:Cartesian coordinate system 161:Introduction to gravitation 10: 17977: 17650:In computational physics: 17174:Relativity of simultaneity 16373:Geocentric Coordinate Time 16358:Barycentric Dynamical Time 16296:Coordinated Universal Time 15364:Orders of magnitude (time) 14568:10.1088/0264-9381/14/4/002 14389:"Discourse on metaphysics" 14296:10.1088/0264-9381/14/4/002 13282:Mould, Richard A. (1994). 13061:Introduction to Relativity 12884:[Space and Time]. 12548:Miller, Arthur I. (1998). 11961:Cambridge University Press 11461:Global spacetime structure 11313:spatial dimensions, where 11163: 10779: 10749:, made profound impact on 10689:at each point that varies 10653: 10584: 10394:Figure 5–1. Tidal effects. 10373: 9829:Choice of reference frames 9208:space translation symmetry 9206:in space they take place ( 9191: 9156:using the convention that 8379:In the low speed limit as 7945: 7454:with a velocity parameter 7251: 6920:Because of time dilation, 6736: 6629: = 600.00 s 5900: 5280: 5200:, then the formula yields 4824:{\displaystyle \beta =v/c} 4778: 4767:which is the common-sense 4724: or 4244: 3666: 3124:relativity of simultaneity 3075:Relativity of simultaneity 2966: 2104:{\displaystyle \Delta {x}} 1694:{\displaystyle \Delta s=0} 1277:{\displaystyle \Delta {x}} 1001:{\displaystyle \Delta {d}} 971: 958: 946:überflüssige Gelehrsamkeit 597: 591: 428:relative to the observer. 40:Spacetime (disambiguation) 36:Timespace (disambiguation) 29: 17832: 17664: 17529: 17501: 17487:Lense–Thirring precession 17370: 17319: 17281: 17260: 17249: 17207: 17151: 17135: 17077: 17069:Doubly special relativity 17041: 17030: 16927: 16818: 16785: 16759: 16640: 16540: 16527:Time-translation symmetry 16457: 16431: 16414: 16338:International Atomic Time 16288: 16265: 16193: 16135: 16088: 16075:Time translation symmetry 16020: 15960: 15950: 15872: 15799: 15719: 15660: 15544: 15462: 15372: 15334: 15320: 15289: 15238: 15217: 15153: 15091: 15045: 15034: 14985: 13003:Ratcliffe, J. G. (1994). 12789:Isaacson, Walter (2007). 12736:Einstein, Albert (1905). 12705:Walter, Scott A. (2007). 11901:Davis, Philip J. (2006). 11685:, but rather for a group 11358:, Tegmark maintains that 11239:Gottfried Wilhelm Leibniz 11138:{\displaystyle (x,y,z,t)} 10940:{\displaystyle (x,y,z,t)} 10587:Bondi–Metzner–Sachs group 10347:{\displaystyle E_{v}=pc,} 9220:time translation symmetry 7741:<!—end plainlist—> 7700:Transverse Doppler effect 7269:transverse Doppler effect 4781:Velocity addition formula 4206:But during the turnaround 4080:scales the time interval 3941:In the upper picture the 3890:′ in the lower pictures). 2737:Galilean reference frames 2600:{\displaystyle x=\pm ct.} 1881:, the spacetime interval 1214:{\displaystyle \Delta x.} 978:In three dimensions, the 690:George Francis FitzGerald 447:, these are often called 317:and the one dimension of 315:three dimensions of space 191:Derivations of relativity 17347:Post-Newtonian formalism 17337:Einstein field equations 17273:Mathematical formulation 17097:Hyperbolic orthogonality 15281: 14659:Physical Review Research 14443:10.1002/andp.19203660503 14159:10.1103/PhysRev.128.2851 12763:10.1002/andp.19053221004 12644:2027/uiug.30112063899089 12609:Poincare, Henri (1906). 11953:Schutz, Bernard (2004). 11548:Einstein synchronization 11492: 11243:Discourse on Metaphysics 10829:. This means the smooth 10739:differentiable manifolds 9875:center-of-momentum frame 9854:Center of Momentum Frame 9212:conservation of momentum 8912:{\displaystyle E=\pm pc} 8885:equals zero, it yields, 8371:Mass–energy relationship 6830:Lorentz transformation. 5912:addition of velocities. 5048:into the expression for 4241:Galilean transformations 4088:, which is shorter than 3992:. This yields the event 2155:{\displaystyle (ct)^{2}} 1974:{\displaystyle ds\neq 0} 1945:{\displaystyle ds=ds'=0} 1665:{\displaystyle \Delta t} 1188:{\displaystyle \Delta t} 892:The Relativity Principle 141:Einstein field equations 17058:Galilean transformation 17049:Principle of relativity 16787:Astronomical chronology 16760:Archaeology and geology 16467:Absolute space and time 16383:IERS Reference Meridian 16378:International Date Line 16289:International standards 16030:Absolute space and time 15681:Astronomical chronology 14923:Encyclopædia Britannica 14754:Oxford University Press 14462:Space, time, and matter 13405:Dynamics and Relativity 13059:Kogut, John B. (2001). 12888:. B. G. Teubner: 1–14. 12578:Galison, Peter (2003). 12511:10.1007/3-7643-7436-5_1 12281:Hughes, Stefan (2013). 12066:Collier, Peter (2017). 11829:has a surface area of 4 11188:-dimensional spacetimes 10790:Differentiable manifold 10501:should be equal to the 9961:{\displaystyle p'=p-mu} 9684:{\displaystyle E_{2}/c} 9645:{\displaystyle E_{1}/c} 7956:Mass–energy equivalence 7929:{\displaystyle \gamma } 7513:{\displaystyle \beta w} 7470:{\displaystyle \beta ,} 5897:Lorentz transformations 5708:In Fig. 3-3b, segments 5690:{\displaystyle \gamma } 5565:{\displaystyle \gamma } 5410:In Fig. 3-3a, segments 4757:{\displaystyle u=u'+v,} 4431:{\displaystyle x'=x-vt} 2766:as measured in frame S. 2733:standard configuration. 1195:and a spatial distance 896:Das Relativitätsprinzip 798:principle of relativity 106:Lorentz transformations 17143:Lorentz transformation 16080:Time reversal symmetry 15384:Italian six-hour clock 14219:"Lorentzian Manifolds" 14116:10.1098/rspa.1962.0161 14018:10.1103/PhysRev.57.147 13804:. Dover Publications. 13123:Carroll, Sean (2022). 12982:10.1002/andp.200810324 12455:Pais, Abraham (1982). 12395:The Linda Hall Library 12351:Stachel, John (2005). 12125:French, A. P. (1968). 12030:Lawden, D. F. (1982). 11523:), pure air, clear sky 11519:from the Greek αἰθήρ ( 11307: 11189: 11139: 11095: 11067: 11047: 11024: 11004: 10984: 10961: 10941: 10881: 10846: 10823: 10721: 10607:, the Poincaré group. 10432: 10395: 10348: 10190: 10099: 10098:{\displaystyle v'=v-u} 10058: 10022: 9962: 9918: 9917:{\displaystyle v'=v-u} 9810: 9755: 9722: 9685: 9646: 9604: 9554: 9418: 9368: 9310: 9239: 9224:conservation of energy 9176: 9150: 9050: 8913: 8869: 8763: 8696: 8602: 8555: 8492: 8434: 8359: 8270: 8232: 8184: 8152: 8103: 8037: 7971: 7930: 7907: 7826: 7709: 7690: 7618: 7551: 7514: 7491: 7471: 7450: 7436: 7353: 7113: 7061: 6748: 6711: 6550: 6493: 6417: 6369: 6269: 6084: 5903:Lorentz transformation 5885: 5691: 5660: 5566: 5544: 5407: 5393: 5364: 5335: 5296: 5134: 4825: 4790: 4758: 4718: 4717:{\displaystyle u'=u-v} 4549: 4531: 4496: 4464: 4432: 4210: 4164: 4123:affect the observer's 4000:-axis, and obviously: 3594: 3542: 3506: 3478: 3435: 3405: 3332: 3252: 3213: 3086: 3035:is greater than Δ 3008: 2980: 2913:′ = 0, 2875: 2758:Frame S′ moves in the 2724: 2716: 2636: 2601: 2562: 2531: 2504: 2455: 2358: 2261: 2233: 2213: 2212:{\displaystyle s^{2},} 2183: 2156: 2123: 2105: 2080: 2057: 2026: 1975: 1946: 1903: 1875: 1808: 1695: 1666: 1642: 1612: 1592: 1566: 1430: 1341: 1318: 1278: 1253: 1215: 1189: 1119: 1002: 887: 868: 794:operational definition 702:Lorentz transformation 550: 518: 354:Lorentz transformation 17611:Weyl−Lewis−Papapetrou 17352:Raychaudhuri equation 17291:Equivalence principle 16746:Weekday determination 16632:Sundial markup schema 15845:Time and fate deities 15790:The Unreality of Time 15729:A series and B series 14464:. Dover reprint: 284. 13506:Morin, David (2008). 13453:Morin, David (2017). 13373:Bais, Sander (2007). 13011:. New York. pp. 13005:"Hyperbolic geometry" 12482:Darrigol, O. (2005), 11419:cosmological constant 11308: 11175: 11140: 11096: 11068: 11048: 11025: 11005: 10985: 10962: 10942: 10891:geometry of spacetime 10882: 10880:{\displaystyle (3,1)} 10847: 10824: 10822:{\displaystyle (M,g)} 10767:differential topology 10755:representation theory 10713:local contributions. 10667:differential geometry 10581:Asymptotic symmetries 10438:equivalence principle 10430: 10393: 10349: 10191: 10100: 10059: 10023: 9963: 9919: 9811: 9756: 9723: 9686: 9647: 9605: 9560:. The total momentum 9555: 9419: 9369: 9311: 9237: 9177: 9151: 9051: 8914: 8870: 8764: 8697: 8603: 8556: 8493: 8435: 8360: 8271: 8269:{\displaystyle E/c=p} 8233: 8231:{\displaystyle E/p=c} 8182: 8153: 8104: 8038: 8036:{\displaystyle (x,t)} 7968: 7931: 7908: 7827: 7707: 7691: 7619: 7552: 7515: 7492: 7472: 7448: 7437: 7354: 7114: 7062: 6758:principle of locality 6746: 6712: 6551: 6494: 6418: 6370: 6290:is small enough, the 6270: 6085: 5886: 5692: 5677:, the expression for 5661: 5567: 5545: 5405: 5394: 5365: 5336: 5294: 5135: 4826: 4788: 4759: 4719: 4543: 4532: 4530:{\displaystyle t'=t.} 4497: 4465: 4433: 4378:coordinates and that 4214:Equivalence Principle 4183: 4162: 3592: 3540: 3507: 3479: 3436: 3434:{\displaystyle s^{2}} 3406: 3330: 3253: 3214: 3172:plane of simultaneity 3084: 3006: 2978: 2876: 2722: 2714: 2637: 2602: 2563: 2561:{\displaystyle s^{2}} 2532: 2530:{\displaystyle s^{2}} 2505: 2503:{\displaystyle s^{2}} 2456: 2359: 2262: 2260:{\displaystyle s^{2}} 2234: 2214: 2184: 2182:{\displaystyle x^{2}} 2157: 2121: 2106: 2081: 2058: 2027: 1981:, one can prove that 1976: 1947: 1904: 1876: 1809: 1696: 1667: 1643: 1613: 1593: 1567: 1431: 1342: 1319: 1279: 1254: 1216: 1190: 1120: 1003: 959:Further information: 881: 863: 843:equivalence principle 544: 519: 201:Differential geometry 101:Equivalence principle 17946:Theory of relativity 17652:Numerical relativity 17493:pulsar timing arrays 16767:Chronological dating 16507:Theory of relativity 16368:Daylight saving time 16096:Chronological dating 16065:Theory of relativity 15409:Daylight saving time 15154:Dimensions by number 14944:" by Robert DiSalle. 14915:at Wikimedia Commons 14818:Lucas, John Randolph 14715:technologyreview.com 14069:Thorne, Kip (1995). 12099:. Wolfram Research. 11768:In this linked image 11578:quantum entanglement 11306:{\displaystyle 5+2k} 11288: 11196:(bidirectional) and 11105: 11085: 11057: 11037: 11014: 10994: 10974: 10951: 10907: 10859: 10836: 10801: 10671:Riemannian manifolds 10319: 10113: 10072: 10057:{\displaystyle p=mv} 10039: 9986: 9932: 9891: 9774: 9736: 9703: 9660: 9621: 9566: 9434: 9382: 9324: 9266: 9175:{\displaystyle c=1.} 9160: 9062: 8946: 8891: 8801: 8705: 8627: 8569: 8509: 8466: 8403: 8318: 8246: 8208: 8117: 8065: 8015: 7920: 7847: 7766: 7635: 7561: 7524: 7501: 7481: 7458: 7391: 7376:, the wavelength is 7305: 7072: 7020: 6637: 6502: 6427: 6378: 6312: 6100: 5925: 5820: 5681: 5579: 5556: 5479: 5392:{\displaystyle c=1.} 5377: 5363:{\displaystyle w=ct} 5345: 5314: 5247:improve this article 5058: 4801: 4728: 4691: 4611:, can be written as 4507: 4495:{\displaystyle z'=z} 4475: 4463:{\displaystyle y'=y} 4443: 4402: 3674:Mutual time dilation 3496: 3468: 3451:invariant hyperbolae 3418: 3347: 3291:improve this article 3228: 3186: 2990:right circular cones 2836: 2773:= 0 for frame S and 2677:improve this article 2626: 2576: 2545: 2514: 2487: 2374: 2280: 2244: 2223: 2193: 2166: 2130: 2090: 2070: 2056:{\displaystyle s=s'} 2036: 1985: 1956: 1913: 1885: 1821: 1711: 1676: 1653: 1622: 1602: 1579: 1446: 1354: 1328: 1288: 1263: 1225: 1199: 1176: 1019: 987: 950:Minkowski spacetime. 935:conformal geometries 835:electromagnetic mass 508: 479:. Spacetime is thus 434:gravitational fields 307:space-time continuum 179:Relevant mathematics 17941:Theoretical physics 17936:Concepts in physics 17544:Friedmann equations 17438:Hulse–Taylor binary 17400:Gravitational waves 17296:Riemannian geometry 17122:Proper acceleration 17107:Maxwell's equations 17053:Galilean relativity 16977:Time value of money 16772:Geologic time scale 16627:History of sundials 16492:Cosmological decade 16444:Greenwich Mean Time 16275:Orders of magnitude 15389:Thai six-hour clock 14881:(1 December 2017). 14834:The Road to Reality 14681:2020PhRvR...2a3217S 14624:2022PhRvD.106d1501F 14560:1997CQGra..14L..69T 14492:1963NCim...27..636T 14435:1920AnP...366..440E 14288:1997CQGra..14L..69T 14151:1962PhRv..128.2851S 14108:1962RSPSA.269...21B 14045:1964PhL.....9..357W 14010:1940PhRv...57..147R 13955:1984CMaPh..94..379G 13908:1970GReGr...1....9D 13551:2008STAdM...9a4107R 12974:2008AnP...520..631K 12754:1905AnP...322..891E 12627:1906RCMP...21..129P 12503:2006eins.book....1D 12237:2001AmJPh..69S..24S 12131:Boca Raton, Florida 11629:the Lorentz group.) 11561:thought experiments 11451:Four-dimensionalism 10796:Lorentzian manifold 10786:Lorentzian manifold 10663:Riemannian geometry 10656:Riemannian geometry 10649:Riemannian geometry 10624:gravitational waves 10620:asymptotic symmetry 9823:elementary particle 8854: 7942:Energy and momentum 7759:light of frequency 3263:Invariant hyperbola 2955:Mercator projection 2777:′ = 0 for frame S′. 1902:{\displaystyle ds'} 1146:Lorentz contraction 1010:Pythagorean theorem 961:Minkowski spacetime 714:aberration of light 656:luminiferous aether 398:classical mechanics 48:Part of a series on 17593:Reissner–Nordström 17511:Brans–Dicke theory 17342:Linearized gravity 17169:Length contraction 17087:Frame of reference 17064:Special relativity 16952:Mental chronometry 16580:Marine chronometer 16432:Obsolete standards 15919:Rosy retrospection 15897:Mental chronometry 15721:Philosophy of time 15083:Degrees of freedom 14986:Dimensional spaces 14879:Arkani-Hamed, Nima 14824:. London: Methuen. 14804:Minkowski, Hermann 14500:10.1007/BF02784569 14423:Annalen der Physik 13963:10.1007/BF01224832 13916:10.1007/BF00759198 13651:LibreTexts Physics 13253:"The Twin Paradox" 13103:Extract of page 27 12952:Annalen der Physik 12742:Annalen der Physik 12635:10.1007/bf03013466 12491:Séminaire Poincaré 12137:. pp. 35–60. 12127:Special Relativity 11852:Additional details 11809:inverse-square law 11595:Annalen der Physik 11466:List of spacetimes 11303: 11230:differ from 3 and 11190: 11135: 11091: 11063: 11043: 11020: 11000: 10980: 10957: 10937: 10877: 10842: 10819: 10601:General Relativity 10593:Special Relativity 10493:gravitational mass 10433: 10396: 10344: 10186: 10095: 10054: 10018: 9958: 9914: 9806: 9751: 9718: 9681: 9642: 9600: 9550: 9414: 9364: 9306: 9240: 9172: 9146: 9046: 8909: 8865: 8840: 8759: 8692: 8598: 8551: 8488: 8430: 8355: 8295:massless particles 8266: 8228: 8185: 8148: 8099: 8033: 7972: 7926: 7903: 7822: 7710: 7686: 7614: 7547: 7510: 7487: 7467: 7451: 7432: 7349: 7239: = 0.666 7232: = 0.250 7221: = 0.500 7109: 7057: 6859:From the drawing, 6749: 6707: 6546: 6489: 6413: 6365: 6265: 6263: 6080: 6078: 5881: 5687: 5656: 5562: 5540: 5408: 5389: 5360: 5331: 5297: 5287:Length contraction 5130: 4821: 4791: 4754: 4714: 4550: 4527: 4492: 4460: 4428: 4165: 4145:thought experiment 3598:Length contraction 3595: 3543: 3502: 3474: 3431: 3401: 3333: 3248: 3209: 3087: 3029:timelike intervals 3009: 2981: 2871: 2725: 2717: 2632: 2597: 2558: 2527: 2500: 2451: 2354: 2257: 2229: 2209: 2179: 2152: 2124: 2101: 2076: 2053: 2022: 1971: 1942: 1899: 1871: 1804: 1691: 1662: 1638: 1608: 1591:{\displaystyle c,} 1588: 1562: 1426: 1340:{\displaystyle ct} 1337: 1314: 1274: 1249: 1211: 1185: 1170:spacetime interval 1166:length contraction 1115: 998: 968:Spacetime interval 916:invariant interval 888: 848:general relativity 684:stellar aberration 648:corpuscular theory 551: 547:frame of reference 514: 430:General relativity 422:special relativity 420:In the context of 330:Spacetime diagrams 311:mathematical model 305:, also called the 244:Physics portal 221:Spacetime topology 196:Spacetime diagrams 124:General relativity 96:Spacetime manifold 89:Spacetime concepts 78:General relativity 73:Special relativity 17961:Conceptual models 17846: 17845: 17660: 17659: 17639:Ozsváth–Schücking 17245: 17244: 17227:Minkowski diagram 17184:Thomas precession 17127:Relativistic mass 16990: 16989: 16800:Nuclear timescale 16482:Continuous signal 16221: 16220: 16131: 16130: 16106:Circadian rhythms 15924:Tense–aspect–mood 15779:Temporal finitism 15656: 15655: 15632:Grandfather clock 15249: 15248: 15058:Lebesgue covering 15023:Algebraic variety 14911:Media related to 14763:978-0-19-282147-8 14602:Physical Review D 14460:Weyl, H. (1922). 14368:on 24 August 2016 14235:978-3-642-02779-6 14080:978-0-393-31276-8 13833:Inside Relativity 13603:978-1-107-60217-5 13519:978-0-521-87622-3 13489:978-0-7506-2768-9 13464:978-1-5423-2351-2 13415:978-1-118-93329-9 13388:978-0-674-02611-7 13329:978-0-7637-0460-5 13295:978-0-387-95210-9 13097:978-0-19-886202-4 12958:(9–10): 631–690. 12804:978-0-7432-6473-0 12520:978-3-7643-7435-8 12379:on 15 April 2017. 12294:978-1-4675-7992-6 12245:10.1119/1.1371254 12198:Scherr, Rachel E. 12097:Wolfram Mathworld 12077:978-0-9573894-6-5 12043:978-0-486-42540-5 11996:Fock, V. (1966). 11912:978-1-4398-6432-6 11786:Kündig experiment 11744:hyperbolic metric 11727:hyperboloid model 11640:in the limit for 11591:Annals of Physics 11441:Complex spacetime 11094:{\displaystyle p} 11066:{\displaystyle q} 11046:{\displaystyle p} 11023:{\displaystyle p} 11003:{\displaystyle p} 10983:{\displaystyle p} 10960:{\displaystyle c} 10901:coordinate charts 10845:{\displaystyle g} 10679:Riemannian metric 10665:is the branch of 10637:supertranslations 10549:Such being said, 9426:traveling at the 9218:they take place ( 9188:Conservation laws 9089: 9056:or alternatively, 8981: 8847: 8735: 8690: 8689: 8620:at low velocity: 8585: 8531: 8476: 8457:relativistic mass 8393:approaches zero, 8305:is well-defined. 8198:A consequence of 8175:Momentum of light 7901: 7820: 7670: 7669: 7490:{\displaystyle w} 7337: 7242: 6728: 6705: 6701: 6161: 5981: 5879: 5654: 5653: 5626: 5625: 5535: 5279: 5278: 5271: 5125: 4009: 3956:(the reading of W 3937: 3891: 3767: 3753: 3752:running clock W′. 3517:(1+2)-dimensional 3505:{\displaystyle c} 3477:{\displaystyle c} 3337:(1+1)-dimensional 3323: 3322: 3315: 3243: 3204: 2897:′ = 0, 2825:axis by an angle 2709: 2708: 2701: 2635:{\displaystyle c} 2467:metric signatures 2232:{\displaystyle s} 2079:{\displaystyle x} 1611:{\displaystyle t} 855:Hermann Minkowski 775:Hermann Minkowski 704:. As a theory of 660:Fizeau experiment 530:Fizeau experiment 517:{\displaystyle c} 396:Non-relativistic 365:Hermann Minkowski 295: 294: 154:Classical gravity 16:(Redirected from 17968: 17911: 17910: 17909: 17899: 17898: 17887: 17886: 17885: 17875: 17874: 17863: 17862: 17854: 17838: 17837: 17621:van Stockum dust 17393:Two-body problem 17311:Mach's principle 17258: 17257: 17199:Terrell rotation 17039: 17038: 17017: 17010: 17003: 16994: 16993: 16691:Dominical letter 16622:Equation of time 16585:Marine sandglass 16426: 16420: 16398:Terrestrial Time 16255:Time measurement 16248: 16241: 16234: 16225: 16224: 16211: 16210: 16201: 16200: 16118:Glottochronology 15958: 15957: 15874:Human experience 15734:B-theory of time 15332: 15331: 15276: 15269: 15262: 15253: 15252: 15046:Other dimensions 15040: 15008:Projective space 14972: 14965: 14958: 14949: 14948: 14910: 14896: 14894: 14892: 14874: 14855:Wheeler, John A. 14848: 14800:Einstein, Albert 14775: 14752:(1st ed.). 14744:Tipler, Frank J. 14726: 14725: 14723: 14721: 14707: 14701: 14700: 14674: 14650: 14644: 14643: 14617: 14593: 14587: 14586: 14584: 14582: 14553: 14535: 14523: 14512: 14511: 14471: 14465: 14458: 14452: 14446: 14411: 14405: 14404: 14402: 14400: 14384: 14378: 14377: 14375: 14373: 14367: 14361:. Archived from 14352: 14334: 14322: 14316: 14315: 14281: 14261: 14255: 14254: 14252: 14250: 14245:on 15 April 2017 14244: 14238:. Archived from 14223: 14214: 14201: 14196: 14190: 14189: 14184: 14172: 14163: 14162: 14145:(6): 2851–2864. 14134: 14128: 14127: 14091: 14085: 14084: 14066: 14057: 14056: 14028: 14022: 14021: 13993: 13987: 13986: 13984: 13982: 13934: 13928: 13927: 13901: 13881: 13875: 13874: 13872: 13870: 13854: 13848: 13847: 13827: 13816: 13815: 13803: 13793: 13787: 13777: 13751: 13739: 13733: 13732: 13730: 13728: 13705: 13694: 13693: 13691: 13689: 13669: 13663: 13662: 13660: 13658: 13642: 13636: 13635: 13633: 13621: 13615: 13614: 13612: 13610: 13587: 13581: 13580: 13570: 13530: 13524: 13523: 13503: 13494: 13493: 13475: 13469: 13468: 13450: 13427: 13426: 13424: 13422: 13399: 13393: 13392: 13380: 13370: 13341: 13340: 13338: 13336: 13313: 13307: 13306: 13304: 13302: 13285:Basic Relativity 13279: 13273: 13272: 13270: 13268: 13251:Weiss, Michael. 13248: 13229: 13228: 13210: 13204: 13203: 13201: 13199: 13182:Savitt, Steven. 13179: 13170: 13169: 13167: 13165: 13145: 13139: 13138: 13120: 13114: 13111: 13105: 13101: 13081: 13075: 13074: 13056: 13041: 13040: 13034: 13026: 13000: 12994: 12993: 12967: 12947: 12941: 12939: 12937: 12935: 12911: 12902: 12901: 12899: 12897: 12877: 12868: 12867: 12856:10.2307/27757388 12839: 12833: 12832: 12830: 12818: 12809: 12808: 12796: 12786: 12780: 12779: 12777: 12775: 12765: 12733: 12727: 12726: 12724: 12722: 12702: 12693: 12692: 12675: 12669: 12668: 12666: 12664: 12646: 12606: 12600: 12599: 12575: 12564: 12563: 12545: 12539: 12538: 12537: 12535: 12529: 12488: 12479: 12473: 12472: 12452: 12441: 12440: 12438: 12436: 12417: 12411: 12410: 12408: 12406: 12387: 12381: 12380: 12378: 12372:. Archived from 12357: 12348: 12342: 12341: 12339: 12337: 12317: 12311: 12310: 12308: 12306: 12278: 12272: 12271: 12269: 12267: 12261: 12230: 12206: 12194: 12188: 12187: 12185: 12183: 12160: 12149: 12148: 12122: 12113: 12112: 12110: 12108: 12088: 12082: 12081: 12063: 12048: 12047: 12027: 12021: 12020: 12018: 12016: 11993: 11987: 11986: 11984: 11982: 11950: 11917: 11916: 11898: 11892: 11891: 11889: 11887: 11867: 11846: 11821: 11805: 11799: 11795: 11789: 11782: 11776: 11763: 11757: 11753: 11747: 11736: 11730: 11723:hyperbolic space 11707: 11701: 11649:, i.e. as group 11648: 11604: 11598: 11587: 11581: 11557: 11551: 11544: 11538: 11530: 11524: 11502: 11412: 11397: 11390: 11379: 11372: 11357: 11327:electromagnetism 11312: 11310: 11309: 11304: 11221: 11214: 11187: 11144: 11142: 11141: 11136: 11100: 11098: 11097: 11092: 11072: 11070: 11069: 11064: 11052: 11050: 11049: 11044: 11029: 11027: 11026: 11021: 11009: 11007: 11006: 11001: 10989: 10987: 10986: 10981: 10966: 10964: 10963: 10958: 10946: 10944: 10943: 10938: 10893: 10892: 10886: 10884: 10883: 10878: 10851: 10849: 10848: 10843: 10828: 10826: 10825: 10820: 10776:Curved manifolds 10724: 10717:Bernhard Riemann 10699:length of curves 10675:smooth manifolds 10571:bimetric gravity 10524:Technical topics 10486: 10464: 10362: 10355: 10353: 10351: 10350: 10345: 10331: 10330: 10308: 10306: 10305: 10298: 10295: 10294: 10285: 10283: 10282: 10275: 10274: 10265: 10264: 10263: 10256: 10255: 10244: 10224: 10210: 10197: 10195: 10193: 10192: 10187: 10182: 10181: 10180: 10170: 10165: 10148: 10125: 10124: 10106: 10104: 10102: 10101: 10096: 10082: 10065: 10063: 10061: 10060: 10055: 10029: 10027: 10025: 10024: 10019: 10017: 10016: 10004: 10003: 9979: 9969: 9967: 9965: 9964: 9959: 9942: 9925: 9923: 9921: 9920: 9915: 9901: 9865: 9841: 9817: 9815: 9813: 9812: 9807: 9805: 9804: 9792: 9791: 9762: 9760: 9758: 9757: 9752: 9750: 9749: 9748: 9729: 9727: 9725: 9724: 9719: 9717: 9716: 9715: 9693:add up to total 9692: 9690: 9688: 9687: 9682: 9677: 9672: 9671: 9653: 9651: 9649: 9648: 9643: 9638: 9633: 9632: 9611: 9609: 9607: 9606: 9601: 9599: 9598: 9597: 9585: 9584: 9559: 9557: 9556: 9551: 9549: 9545: 9544: 9543: 9531: 9530: 9516: 9511: 9507: 9506: 9505: 9504: 9494: 9493: 9481: 9480: 9479: 9469: 9468: 9451: 9450: 9449: 9425: 9423: 9421: 9420: 9415: 9413: 9412: 9400: 9399: 9375: 9373: 9371: 9370: 9365: 9363: 9362: 9357: 9351: 9350: 9338: 9337: 9332: 9317: 9315: 9313: 9312: 9307: 9305: 9304: 9299: 9293: 9292: 9280: 9279: 9274: 9194:Conservation law 9181: 9179: 9178: 9173: 9155: 9153: 9152: 9147: 9142: 9141: 9129: 9128: 9116: 9115: 9091: 9090: 9082: 9055: 9053: 9052: 9047: 9042: 9041: 9029: 9028: 9016: 9015: 9000: 8983: 8982: 8974: 8965: 8920: 8918: 8916: 8915: 8910: 8874: 8872: 8871: 8866: 8864: 8863: 8853: 8848: 8845: 8836: 8835: 8826: 8825: 8813: 8812: 8768: 8766: 8765: 8760: 8749: 8748: 8736: 8728: 8723: 8722: 8701: 8699: 8698: 8693: 8691: 8688: 8687: 8672: 8671: 8670: 8669: 8656: 8651: 8650: 8619: 8609: 8607: 8605: 8604: 8599: 8597: 8596: 8587: 8586: 8583: 8562: 8560: 8558: 8557: 8552: 8547: 8533: 8532: 8529: 8499: 8497: 8495: 8494: 8489: 8478: 8477: 8474: 8441: 8439: 8437: 8436: 8431: 8396: 8392: 8366: 8364: 8362: 8361: 8356: 8330: 8329: 8277: 8275: 8273: 8272: 8267: 8256: 8239: 8237: 8235: 8234: 8229: 8218: 8200:Maxwell's theory 8159: 8157: 8155: 8154: 8149: 8129: 8128: 8110: 8108: 8106: 8105: 8100: 8086: 8085: 8051: 8044: 8042: 8040: 8039: 8034: 7996: 7935: 7933: 7932: 7927: 7912: 7910: 7909: 7904: 7902: 7900: 7899: 7884: 7882: 7868: 7863: 7831: 7829: 7828: 7823: 7821: 7819: 7818: 7803: 7801: 7796: 7782: 7754: 7695: 7693: 7692: 7687: 7682: 7681: 7671: 7668: 7657: 7646: 7645: 7623: 7621: 7620: 7615: 7595: 7556: 7554: 7553: 7548: 7543: 7519: 7517: 7516: 7511: 7496: 7494: 7493: 7488: 7476: 7474: 7473: 7468: 7441: 7439: 7438: 7433: 7431: 7430: 7418: 7417: 7358: 7356: 7355: 7350: 7348: 7347: 7338: 7336: 7335: 7334: 7315: 7240: 7233: 7222: 7208: 7193: 7186: 7171: 7164: 7145: 7138: 7118: 7116: 7115: 7110: 7108: 7091: 7066: 7064: 7063: 7058: 7056: 7039: 6995: 6993: 6978: 6976: 6962: 6959: 6948: 6932: 6930: 6916: 6884: 6851: 6839: 6817: 6810: 6803: 6789: 6787: 6780: 6726: 6724: 6718: 6716: 6714: 6713: 6708: 6706: 6703: 6699: 6692: 6688: 6687: 6686: 6677: 6649: 6648: 6630: 6623: 6621: 6611: 6609: 6594: 6582: 6568:between events. 6555: 6553: 6552: 6547: 6515: 6498: 6496: 6495: 6490: 6482: 6481: 6472: 6440: 6422: 6420: 6419: 6414: 6388: 6374: 6372: 6371: 6366: 6358: 6357: 6348: 6322: 6274: 6272: 6271: 6266: 6264: 6260: 6238: 6216: 6212: 6211: 6197: 6167: 6163: 6162: 6160: 6159: 6150: 6149: 6137: 6132: 6089: 6087: 6086: 6081: 6079: 6065: 6043: 6031: 6027: 5999: 5987: 5983: 5982: 5980: 5979: 5970: 5962: 5939: 5890: 5888: 5887: 5882: 5880: 5872: 5864: 5863: 5833: 5744: 5743: 5742: 5741: 5696: 5694: 5693: 5688: 5665: 5663: 5662: 5657: 5655: 5652: 5651: 5636: 5632: 5627: 5624: 5623: 5614: 5609: 5608: 5593: 5589: 5571: 5569: 5568: 5563: 5549: 5547: 5546: 5541: 5536: 5534: 5533: 5524: 5519: 5518: 5503: 5501: 5446: 5445: 5444: 5443: 5398: 5396: 5395: 5390: 5369: 5367: 5366: 5361: 5340: 5338: 5337: 5332: 5321: 5274: 5267: 5263: 5260: 5254: 5231: 5223: 5196:is set equal to 5195: 5181: 5166: 5155: 5139: 5137: 5136: 5131: 5126: 5124: 5120: 5119: 5110: 5105: 5084: 5083: 5068: 5036: 5028: 5016: 5008: 4996: 4995: 4979: 4958: 4929: 4890: 4883: 4874: 4863: 4855: 4830: 4828: 4827: 4822: 4817: 4763: 4761: 4760: 4755: 4744: 4723: 4721: 4720: 4715: 4701: 4683: 4682: 4675: 4668: 4661:. This leads to 4660: 4659: 4642: 4627: 4618: 4610: 4609: 4597: 4588: 4580: 4566: 4536: 4534: 4533: 4528: 4517: 4501: 4499: 4498: 4493: 4485: 4469: 4467: 4466: 4461: 4453: 4437: 4435: 4434: 4429: 4412: 4394: 4392: 4384: 4377: 4361: 4359: 4352: 4345: 4338: 4329: 4309: 4307: 4300: 4293: 4286: 4271: 4112: > 4056:is the vertical 4004: < 3940: 3894: 3865: 3854: 3794: 3783: 3761: 3719: 3653:hyperbolic angle 3618: 3576: 3574: 3564: 3562: 3552: 3550: 3527:Fig. 2-7a. 3518: 3511: 3509: 3508: 3503: 3483: 3481: 3480: 3475: 3440: 3438: 3437: 3432: 3430: 3429: 3410: 3408: 3407: 3402: 3397: 3396: 3381: 3380: 3368: 3367: 3338: 3318: 3311: 3307: 3304: 3298: 3275: 3267: 3257: 3255: 3254: 3249: 3244: 3242: 3241: 3232: 3218: 3216: 3215: 3210: 3205: 3203: 3202: 3190: 3169: 3165: 3155: 3151: 3140: 3133: 3113: 3017:spatial distance 2880: 2878: 2877: 2872: 2864: 2729:reference frames 2704: 2697: 2693: 2690: 2684: 2661: 2653: 2649:Reference frames 2641: 2639: 2638: 2633: 2606: 2604: 2603: 2598: 2567: 2565: 2564: 2559: 2557: 2556: 2536: 2534: 2533: 2528: 2526: 2525: 2509: 2507: 2506: 2501: 2499: 2498: 2475: 2471: 2460: 2458: 2457: 2452: 2450: 2449: 2437: 2436: 2424: 2423: 2411: 2410: 2386: 2385: 2363: 2361: 2360: 2355: 2353: 2352: 2340: 2339: 2327: 2326: 2314: 2313: 2292: 2291: 2266: 2264: 2263: 2258: 2256: 2255: 2238: 2236: 2235: 2230: 2218: 2216: 2215: 2210: 2205: 2204: 2188: 2186: 2185: 2180: 2178: 2177: 2161: 2159: 2158: 2153: 2151: 2150: 2110: 2108: 2107: 2102: 2100: 2085: 2083: 2082: 2077: 2062: 2060: 2059: 2054: 2052: 2031: 2029: 2028: 2023: 2021: 2020: 2019: 2000: 1999: 1980: 1978: 1977: 1972: 1951: 1949: 1948: 1943: 1935: 1908: 1906: 1905: 1900: 1898: 1880: 1878: 1877: 1872: 1867: 1856: 1845: 1834: 1813: 1811: 1810: 1805: 1800: 1799: 1784: 1783: 1768: 1767: 1752: 1751: 1739: 1738: 1726: 1725: 1700: 1698: 1697: 1692: 1671: 1669: 1668: 1663: 1647: 1645: 1644: 1639: 1637: 1636: 1617: 1615: 1614: 1609: 1597: 1595: 1594: 1589: 1571: 1569: 1568: 1563: 1558: 1557: 1536: 1535: 1514: 1513: 1492: 1491: 1467: 1466: 1435: 1433: 1432: 1427: 1422: 1421: 1400: 1399: 1375: 1374: 1347:-coordinate is: 1346: 1344: 1343: 1338: 1323: 1321: 1320: 1315: 1301: 1284:in space and by 1283: 1281: 1280: 1275: 1273: 1258: 1256: 1255: 1250: 1248: 1247: 1238: 1220: 1218: 1217: 1212: 1194: 1192: 1191: 1186: 1124: 1122: 1121: 1116: 1114: 1113: 1104: 1090: 1089: 1080: 1066: 1065: 1056: 1042: 1041: 1032: 1007: 1005: 1004: 999: 997: 974:Causal structure 771: 757: 741: 727: 664:Hippolyte Fizeau 627: 616: 523: 521: 520: 515: 481:four-dimensional 323:four-dimensional 287: 280: 273: 259: 254: 253: 246: 242: 241: 211:Curved spacetime 64: 45: 44: 21: 17976: 17975: 17971: 17970: 17969: 17967: 17966: 17965: 17956:Time in physics 17921: 17920: 17917: 17907: 17905: 17893: 17883: 17881: 17869: 17857: 17849: 17847: 17842: 17828: 17656: 17560:BKL singularity 17550:Lemaître–Tolman 17525: 17521:Quantum gravity 17503: 17497: 17483:geodetic effect 17457:(together with 17427:LISA Pathfinder 17366: 17315: 17301:Penrose diagram 17283: 17277: 17252: 17241: 17237:Minkowski space 17203: 17147: 17131: 17079: 17073: 17033: 17026: 17021: 16991: 16986: 16923: 16814: 16781: 16755: 16636: 16536: 16487:Coordinate time 16459:Time in physics 16453: 16427: 16421: 16412: 16284: 16261: 16252: 16222: 16217: 16189: 16180:Time immemorial 16127: 16084: 16045:Coordinate time 16016: 15970:Geological time 15946: 15929:Time management 15892:Generation time 15876: 15868: 15813: 15795: 15715: 15674: 15652: 15540: 15458: 15375: 15368: 15324: 15316: 15285: 15280: 15250: 15245: 15234: 15213: 15149: 15087: 15041: 15032: 14998:Euclidean space 14981: 14976: 14934:Expert articles 14903: 14890: 14888: 14871: 14853:Taylor, E. F.; 14849:Chapters 17–18. 14845: 14779:George F. Ellis 14764: 14740:Barrow, John D. 14735: 14733:Further reading 14730: 14729: 14719: 14717: 14709: 14708: 14704: 14651: 14647: 14594: 14590: 14580: 14578: 14533: 14524: 14515: 14472: 14468: 14459: 14455: 14415:Ehrenfest, Paul 14412: 14408: 14398: 14396: 14385: 14381: 14371: 14369: 14365: 14350:10.1.1.404.7853 14332: 14323: 14319: 14262: 14258: 14248: 14246: 14242: 14236: 14221: 14215: 14204: 14197: 14193: 14173: 14166: 14139:Physical Review 14135: 14131: 14102:(1336): 21–52. 14092: 14088: 14081: 14067: 14060: 14033:Physics Letters 14029: 14025: 13998:Physical Review 13994: 13990: 13980: 13978: 13935: 13931: 13882: 13878: 13868: 13866: 13855: 13851: 13844: 13828: 13819: 13812: 13794: 13790: 13769: 13743: 13740: 13736: 13726: 13724: 13722: 13706: 13697: 13687: 13685: 13670: 13666: 13656: 13654: 13643: 13639: 13631:physics/9807044 13622: 13618: 13608: 13606: 13604: 13588: 13584: 13531: 13527: 13520: 13504: 13497: 13490: 13476: 13472: 13465: 13451: 13430: 13420: 13418: 13416: 13400: 13396: 13389: 13371: 13344: 13334: 13332: 13330: 13314: 13310: 13300: 13298: 13296: 13280: 13276: 13266: 13264: 13249: 13232: 13225: 13211: 13207: 13197: 13195: 13180: 13173: 13163: 13161: 13146: 13142: 13135: 13121: 13117: 13112: 13108: 13098: 13082: 13078: 13071: 13057: 13044: 13028: 13027: 13023: 13001: 12997: 12948: 12944: 12933: 12931: 12912: 12905: 12895: 12893: 12882:"Raum und Zeit" 12878: 12871: 12840: 12836: 12819: 12812: 12805: 12787: 12783: 12773: 12771: 12748:(10): 891–921. 12734: 12730: 12720: 12718: 12703: 12696: 12690: 12676: 12672: 12662: 12660: 12607: 12603: 12596: 12576: 12567: 12560: 12546: 12542: 12533: 12531: 12527: 12521: 12486: 12480: 12476: 12469: 12453: 12444: 12434: 12432: 12419: 12418: 12414: 12404: 12402: 12389: 12388: 12384: 12376: 12370: 12355: 12349: 12345: 12335: 12333: 12318: 12314: 12304: 12302: 12295: 12279: 12275: 12265: 12263: 12259: 12228:physics/0207109 12204: 12195: 12191: 12181: 12179: 12177: 12161: 12152: 12145: 12123: 12116: 12106: 12104: 12091:Rowland, Todd. 12089: 12085: 12078: 12064: 12051: 12044: 12028: 12024: 12014: 12012: 12010: 11994: 11990: 11980: 11978: 11971: 11951: 11920: 11913: 11899: 11895: 11885: 11883: 11868: 11864: 11859: 11854: 11849: 11816: 11806: 11802: 11796: 11792: 11783: 11779: 11764: 11760: 11754: 11750: 11740:Cartesian plane 11737: 11733: 11708: 11704: 11692: 11683: 11674: 11665: 11656: 11641: 11638: 11626: 11615: 11605: 11601: 11588: 11584: 11558: 11554: 11545: 11541: 11531: 11527: 11507:from the Latin 11503: 11499: 11495: 11490: 11431: 11426: 11425: 11410: 11392: 11385: 11374: 11367: 11352: 11289: 11286: 11285: 11216: 11209: 11177: 11169: 11161: 11106: 11103: 11102: 11086: 11083: 11082: 11058: 11055: 11054: 11038: 11035: 11034: 11015: 11012: 11011: 10995: 10992: 10991: 10975: 10972: 10971: 10967:is equal to 1. 10952: 10949: 10948: 10908: 10905: 10904: 10890: 10889: 10860: 10857: 10856: 10837: 10834: 10833: 10802: 10799: 10798: 10792: 10780:Main articles: 10778: 10771: 10770: 10659: 10651: 10618:addressed this 10616:Rainer K. Sachs 10589: 10583: 10560:more convenient 10540:more convenient 10535:conventionalist 10531: 10526: 10521: 10520: 10510: 10500: 10491:reason why the 10481: 10475: 10470: 10459: 10453: 10451: 10441: 10384:reference frame 10379: 10369: 10364: 10360: 10326: 10322: 10320: 10317: 10316: 10314: 10304: 10301: 10300: 10299: 10296: 10293: 10291: 10290: 10289: 10287: 10281: 10279: 10278: 10277: 10273: 10271: 10270: 10269: 10267: 10262: 10260: 10259: 10258: 10254: 10252: 10251: 10250: 10249: 10242: 10232: 10231: 10230: 10229: 10228: 10225: 10216: 10215: 10214: 10211: 10176: 10172: 10171: 10166: 10158: 10144: 10120: 10116: 10114: 10111: 10110: 10108: 10075: 10073: 10070: 10069: 10067: 10040: 10037: 10036: 10034: 10012: 10008: 9999: 9995: 9987: 9984: 9983: 9981: 9974: 9935: 9933: 9930: 9929: 9927: 9926:, the momentum 9894: 9892: 9889: 9888: 9886: 9883: 9870: 9869: 9868: 9867: 9866: 9858: 9857: 9851: 9845: 9842: 9831: 9800: 9796: 9787: 9783: 9775: 9772: 9771: 9769: 9763:add up to form 9744: 9740: 9739: 9737: 9734: 9733: 9731: 9711: 9707: 9706: 9704: 9701: 9700: 9698: 9673: 9667: 9663: 9661: 9658: 9657: 9655: 9634: 9628: 9624: 9622: 9619: 9618: 9616: 9593: 9589: 9580: 9576: 9569: 9567: 9564: 9563: 9561: 9539: 9535: 9526: 9522: 9521: 9517: 9512: 9500: 9496: 9495: 9489: 9485: 9475: 9471: 9470: 9464: 9460: 9459: 9455: 9442: 9438: 9437: 9435: 9432: 9431: 9408: 9404: 9395: 9391: 9383: 9380: 9379: 9377: 9358: 9353: 9352: 9346: 9342: 9333: 9328: 9327: 9325: 9322: 9321: 9319: 9300: 9295: 9294: 9288: 9284: 9275: 9270: 9269: 9267: 9264: 9263: 9261: 9232: 9196: 9190: 9161: 9158: 9157: 9137: 9133: 9124: 9120: 9111: 9107: 9081: 9080: 9063: 9060: 9059: 9037: 9033: 9024: 9020: 9011: 9007: 8996: 8973: 8972: 8961: 8947: 8944: 8943: 8927: 8892: 8889: 8888: 8886: 8884: 8859: 8855: 8849: 8844: 8831: 8827: 8821: 8817: 8808: 8804: 8802: 8799: 8798: 8781: 8744: 8740: 8727: 8718: 8714: 8706: 8703: 8702: 8683: 8679: 8665: 8661: 8657: 8655: 8646: 8642: 8628: 8625: 8624: 8615: 8592: 8588: 8582: 8578: 8570: 8567: 8566: 8564: 8543: 8528: 8524: 8510: 8507: 8506: 8504: 8473: 8469: 8467: 8464: 8463: 8461: 8404: 8401: 8400: 8398: 8394: 8380: 8373: 8325: 8321: 8319: 8316: 8315: 8313: 8252: 8247: 8244: 8243: 8241: 8240:. Rearranging, 8214: 8209: 8206: 8205: 8203: 8177: 8124: 8120: 8118: 8115: 8114: 8112: 8081: 8077: 8066: 8063: 8062: 8060: 8046: 8016: 8013: 8012: 8010: 7983: 7976:Linear momentum 7963: 7958: 7946:Main articles: 7944: 7921: 7918: 7917: 7895: 7891: 7883: 7875: 7864: 7856: 7848: 7845: 7844: 7840:with frequency 7814: 7810: 7802: 7797: 7789: 7775: 7767: 7764: 7763: 7752: 7739: 7702: 7677: 7673: 7658: 7647: 7644: 7636: 7633: 7632: 7588: 7562: 7559: 7558: 7536: 7525: 7522: 7521: 7502: 7499: 7498: 7482: 7479: 7478: 7459: 7456: 7455: 7426: 7422: 7413: 7409: 7392: 7389: 7388: 7374: 7367: 7343: 7339: 7330: 7326: 7319: 7314: 7306: 7303: 7302: 7292: 7285: 7277: 7260: 7252:Main articles: 7250: 7235: 7228: 7217: 7191: 7184: 7178: 7169: 7162: 7143: 7136: 7101: 7084: 7073: 7070: 7069: 7049: 7032: 7021: 7018: 7017: 6991: 6980: 6974: 6967: 6957: 6950: 6936: 6928: 6921: 6888: 6872: 6849: 6837: 6815: 6808: 6804:. To determine 6791: 6785: 6778: 6773: 6741: 6735: 6722: 6720: 6702: 6682: 6678: 6673: 6660: 6656: 6644: 6640: 6638: 6635: 6634: 6632: 6625: 6619: 6617: 6607: 6599: 6580: 6508: 6503: 6500: 6499: 6477: 6473: 6468: 6433: 6428: 6425: 6424: 6381: 6379: 6376: 6375: 6353: 6349: 6344: 6315: 6313: 6310: 6309: 6262: 6261: 6253: 6246: 6240: 6239: 6231: 6224: 6218: 6217: 6204: 6190: 6189: 6185: 6175: 6169: 6168: 6155: 6151: 6142: 6138: 6136: 6125: 6124: 6120: 6110: 6103: 6101: 6098: 6097: 6077: 6076: 6066: 6058: 6055: 6054: 6044: 6036: 6033: 6032: 6014: 6010: 6000: 5992: 5989: 5988: 5975: 5971: 5963: 5961: 5954: 5950: 5940: 5932: 5928: 5926: 5923: 5922: 5909: 5901:Main articles: 5899: 5871: 5859: 5855: 5829: 5821: 5818: 5817: 5773:) = ( 5733: 5731: 5726: 5725: 5682: 5679: 5678: 5647: 5643: 5631: 5619: 5615: 5610: 5604: 5600: 5588: 5580: 5577: 5576: 5557: 5554: 5553: 5529: 5525: 5520: 5514: 5510: 5502: 5497: 5480: 5477: 5476: 5435: 5433: 5428: 5427: 5378: 5375: 5374: 5346: 5343: 5342: 5317: 5315: 5312: 5311: 5289: 5281:Main articles: 5275: 5264: 5258: 5255: 5244: 5232: 5221: 5193: 5179: 5164: 5153: 5115: 5111: 5106: 5098: 5085: 5076: 5069: 5067: 5059: 5056: 5055: 5026: 5018: 5006: 4998: 4993: 4981: 4963: 4938: 4904: 4888: 4872: 4861: 4857: 4835: 4813: 4802: 4799: 4798: 4783: 4777: 4737: 4729: 4726: 4725: 4694: 4692: 4689: 4688: 4684:and ultimately 4680: 4673: 4666: 4662: 4657: 4644: 4629: 4616: 4612: 4607: 4599: 4586: 4582: 4572: 4564: 4510: 4508: 4505: 4504: 4478: 4476: 4473: 4472: 4446: 4444: 4441: 4440: 4405: 4403: 4400: 4399: 4390: 4386: 4379: 4363: 4357: 4350: 4343: 4336: 4331: 4311: 4305: 4298: 4291: 4284: 4279: 4253: 4249: 4243: 4236: 4227: 4133: 4127:of the object. 4063: 4055: 4037:is longer than 3971: 3967: 3959: 3950: 3931: 3927: 3918:is taken to be 3917: 3904: 3899:is taken to be 3898: 3881: 3877: 3863: 3804: 3803: 3802: 3801: 3797: 3796: 3795: 3786: 3785: 3784: 3735: 3731: 3727: 3723: 3710: 3706: 3676: 3671: 3665: 3616: 3572: 3570: 3560: 3558: 3548: 3546: 3535: 3516: 3497: 3494: 3493: 3469: 3466: 3465: 3425: 3421: 3419: 3416: 3415: 3392: 3388: 3376: 3372: 3363: 3359: 3348: 3345: 3344: 3336: 3319: 3308: 3302: 3299: 3288: 3276: 3265: 3237: 3233: 3231: 3229: 3226: 3225: 3198: 3194: 3189: 3187: 3184: 3183: 3180:proper distance 3167: 3157: 3153: 3142: 3135: 3131: 3108: 3099:reference frame 3077: 3021:timelike future 2971: 2965: 2951:Cartesian plane 2901:′ = − 2860: 2837: 2834: 2833: 2705: 2694: 2688: 2685: 2674: 2662: 2651: 2627: 2624: 2623: 2577: 2574: 2573: 2552: 2548: 2546: 2543: 2542: 2521: 2517: 2515: 2512: 2511: 2494: 2490: 2488: 2485: 2484: 2473: 2469: 2445: 2441: 2432: 2428: 2419: 2415: 2406: 2402: 2381: 2377: 2375: 2372: 2371: 2348: 2344: 2335: 2331: 2322: 2318: 2309: 2305: 2287: 2283: 2281: 2278: 2277: 2251: 2247: 2245: 2242: 2241: 2224: 2221: 2220: 2200: 2196: 2194: 2191: 2190: 2173: 2169: 2167: 2164: 2163: 2146: 2142: 2131: 2128: 2127: 2096: 2091: 2088: 2087: 2071: 2068: 2067: 2045: 2037: 2034: 2033: 2015: 2011: 2007: 1995: 1991: 1986: 1983: 1982: 1957: 1954: 1953: 1928: 1914: 1911: 1910: 1891: 1886: 1883: 1882: 1860: 1849: 1838: 1827: 1822: 1819: 1818: 1795: 1791: 1779: 1775: 1763: 1759: 1747: 1743: 1734: 1730: 1721: 1717: 1712: 1709: 1708: 1677: 1674: 1673: 1654: 1651: 1650: 1632: 1628: 1623: 1620: 1619: 1603: 1600: 1599: 1580: 1577: 1576: 1553: 1549: 1531: 1527: 1509: 1505: 1487: 1483: 1462: 1458: 1447: 1444: 1443: 1417: 1413: 1395: 1391: 1370: 1366: 1355: 1352: 1351: 1329: 1326: 1325: 1294: 1289: 1286: 1285: 1269: 1264: 1261: 1260: 1243: 1239: 1234: 1226: 1223: 1222: 1200: 1197: 1196: 1177: 1174: 1173: 1109: 1105: 1100: 1085: 1081: 1076: 1061: 1057: 1052: 1037: 1033: 1028: 1020: 1017: 1016: 993: 988: 985: 984: 976: 970: 963: 957: 920:discussed below 824:Albert Einstein 786: 785: 784: 783: 779: 778: 777: 772: 764: 763: 761:Albert Einstein 758: 749: 748: 747: 742: 734: 733: 731:Hendrik Lorentz 728: 694:Hendrik Lorentz 671:aether-dragging 636: 635: 634: 633: 632: 628: 619: 618: 617: 606: 598:Main articles: 594: 509: 506: 505: 394: 387: 377:mass and energy 369:Minkowski space 313:that fuses the 291: 262: 249: 236: 235: 227: 226: 225: 180: 172: 171: 170: 155: 147: 146: 145: 125: 117: 116: 115: 111:Minkowski space 90: 82: 43: 28: 23: 22: 15: 12: 11: 5: 17974: 17964: 17963: 17958: 17953: 17948: 17943: 17938: 17933: 17916: 17915: 17903: 17891: 17879: 17867: 17844: 17843: 17833: 17830: 17829: 17827: 17826: 17819: 17814: 17809: 17804: 17799: 17794: 17789: 17784: 17779: 17774: 17769: 17764: 17759: 17754: 17749: 17747:Choquet-Bruhat 17744: 17739: 17734: 17729: 17724: 17719: 17714: 17709: 17704: 17699: 17694: 17689: 17684: 17679: 17674: 17668: 17666: 17662: 17661: 17658: 17657: 17655: 17654: 17647: 17646: 17641: 17636: 17629: 17628: 17623: 17618: 17613: 17608: 17599:Axisymmetric: 17596: 17595: 17590: 17584: 17573: 17572: 17567: 17562: 17557: 17552: 17547: 17538:Cosmological: 17535: 17533: 17527: 17526: 17524: 17523: 17518: 17513: 17507: 17505: 17499: 17498: 17496: 17495: 17490: 17479:frame-dragging 17476: 17471: 17466: 17463:Einstein rings 17459:Einstein cross 17452: 17441: 17440: 17435: 17429: 17424: 17419: 17406: 17396: 17395: 17390: 17385: 17380: 17374: 17372: 17368: 17367: 17365: 17364: 17362:Ernst equation 17359: 17354: 17349: 17344: 17339: 17334: 17332:BSSN formalism 17329: 17323: 17321: 17317: 17316: 17314: 17313: 17308: 17303: 17298: 17293: 17287: 17285: 17279: 17278: 17276: 17275: 17270: 17264: 17262: 17255: 17247: 17246: 17243: 17242: 17240: 17239: 17234: 17229: 17224: 17219: 17213: 17211: 17205: 17204: 17202: 17201: 17196: 17191: 17189:Ladder paradox 17186: 17181: 17176: 17171: 17166: 17161: 17155: 17153: 17149: 17148: 17146: 17145: 17139: 17137: 17133: 17132: 17130: 17129: 17124: 17119: 17114: 17109: 17104: 17099: 17094: 17092:Speed of light 17089: 17083: 17081: 17075: 17074: 17072: 17071: 17066: 17061: 17055: 17045: 17043: 17036: 17028: 17027: 17020: 17019: 17012: 17005: 16997: 16988: 16987: 16985: 16984: 16979: 16974: 16972:Time metrology 16969: 16964: 16959: 16954: 16949: 16948: 16947: 16937: 16931: 16929: 16928:Related topics 16925: 16924: 16922: 16921: 16916: 16911: 16906: 16901: 16896: 16891: 16886: 16881: 16876: 16871: 16866: 16861: 16856: 16851: 16846: 16841: 16836: 16831: 16825: 16823: 16816: 16815: 16813: 16812: 16807: 16802: 16797: 16791: 16789: 16783: 16782: 16780: 16779: 16774: 16769: 16763: 16761: 16757: 16756: 16754: 16753: 16748: 16743: 16738: 16733: 16728: 16723: 16718: 16713: 16708: 16703: 16698: 16693: 16688: 16683: 16678: 16673: 16667: 16662: 16657: 16652: 16646: 16644: 16638: 16637: 16635: 16634: 16629: 16624: 16619: 16617:Dialing scales 16614: 16609: 16604: 16603: 16602: 16592: 16587: 16582: 16577: 16572: 16567: 16562: 16557: 16552: 16546: 16544: 16538: 16537: 16535: 16534: 16529: 16524: 16519: 16514: 16509: 16504: 16499: 16494: 16489: 16484: 16479: 16474: 16469: 16463: 16461: 16455: 16454: 16452: 16451: 16449:Prime meridian 16446: 16441: 16439:Ephemeris time 16435: 16433: 16429: 16428: 16415: 16413: 16411: 16410: 16408:180th meridian 16405: 16400: 16395: 16390: 16385: 16380: 16375: 16370: 16365: 16360: 16355: 16350: 16345: 16340: 16335: 16330: 16325: 16320: 16315: 16310: 16305: 16304: 16303: 16292: 16290: 16286: 16285: 16283: 16282: 16277: 16272: 16266: 16263: 16262: 16251: 16250: 16243: 16236: 16228: 16219: 16218: 16216: 16215: 16205: 16194: 16191: 16190: 16188: 16187: 16182: 16177: 16172: 16165: 16160: 16155: 16150: 16145: 16139: 16137: 16133: 16132: 16129: 16128: 16126: 16125: 16123:Time geography 16120: 16115: 16113:Clock reaction 16110: 16109: 16108: 16098: 16092: 16090: 16086: 16085: 16083: 16082: 16077: 16072: 16067: 16062: 16057: 16052: 16047: 16042: 16037: 16032: 16026: 16024: 16018: 16017: 16015: 16014: 16009: 16004: 16003: 16002: 15997: 15992: 15987: 15982: 15977: 15966: 15964: 15955: 15948: 15947: 15945: 15944: 15931: 15926: 15921: 15916: 15915: 15914: 15912:time signature 15909: 15899: 15894: 15889: 15883: 15881: 15870: 15869: 15867: 15866: 15865: 15864: 15854: 15853: 15852: 15842: 15837: 15832: 15827: 15822: 15816: 15814: 15812: 15811: 15806: 15800: 15797: 15796: 15794: 15793: 15786: 15784:Temporal parts 15781: 15776: 15771: 15766: 15761: 15756: 15754:Eternal return 15751: 15746: 15741: 15739:Chronocentrism 15736: 15731: 15725: 15723: 15717: 15716: 15714: 15713: 15708: 15703: 15698: 15693: 15688: 15683: 15677: 15675: 15673: 15672: 15667: 15661: 15658: 15657: 15654: 15653: 15651: 15650: 15649: 15648: 15634: 15629: 15624: 15619: 15618: 15617: 15612: 15611: 15610: 15605: 15595: 15590: 15585: 15580: 15579: 15578: 15568: 15567: 15566: 15550: 15548: 15542: 15541: 15539: 15538: 15531: 15526: 15524:Hindu Panchang 15521: 15516: 15511: 15506: 15501: 15496: 15491: 15490: 15489: 15484: 15479: 15468: 15466: 15460: 15459: 15457: 15456: 15451: 15446: 15441: 15436: 15431: 15426: 15421: 15416: 15411: 15406: 15401: 15396: 15391: 15386: 15380: 15378: 15370: 15369: 15367: 15366: 15361: 15356: 15351: 15346: 15340: 15338: 15329: 15318: 15317: 15315: 15314: 15309: 15304: 15299: 15293: 15291: 15287: 15286: 15279: 15278: 15271: 15264: 15256: 15247: 15246: 15239: 15236: 15235: 15233: 15232: 15227: 15221: 15219: 15215: 15214: 15212: 15211: 15203: 15198: 15193: 15188: 15183: 15178: 15173: 15168: 15163: 15157: 15155: 15151: 15150: 15148: 15147: 15142: 15137: 15135:Cross-polytope 15132: 15127: 15122: 15120:Hyperrectangle 15117: 15112: 15107: 15101: 15099: 15089: 15088: 15086: 15085: 15080: 15075: 15070: 15065: 15060: 15055: 15049: 15047: 15043: 15042: 15035: 15033: 15031: 15030: 15025: 15020: 15015: 15010: 15005: 15000: 14995: 14989: 14987: 14983: 14982: 14975: 14974: 14967: 14960: 14952: 14946: 14945: 14935: 14926: 14916: 14902: 14901:External links 14899: 14898: 14897: 14875: 14869: 14850: 14843: 14829:Penrose, Roger 14825: 14815: 14796:Lorentz, H. A. 14793: 14776: 14762: 14734: 14731: 14728: 14727: 14702: 14645: 14608:(4): L041501. 14588: 14544:(4): L69–L75. 14529:(April 1997). 14513: 14486:(3): 636–651. 14466: 14453: 14429:(5): 440–446. 14406: 14379: 14343:(2): 227–252. 14326:Skow, Bradford 14317: 14272:(4): L69–L75. 14256: 14234: 14202: 14191: 14164: 14129: 14086: 14079: 14058: 14039:(4): 357–359. 14023: 14004:(2): 147–150. 13988: 13949:(3): 379–396. 13929: 13876: 13857:Murzi, Mauro. 13849: 13842: 13817: 13810: 13788: 13734: 13720: 13695: 13664: 13637: 13616: 13602: 13582: 13525: 13518: 13495: 13488: 13470: 13463: 13428: 13414: 13394: 13387: 13342: 13328: 13308: 13294: 13274: 13230: 13223: 13205: 13171: 13140: 13133: 13115: 13106: 13096: 13076: 13069: 13042: 13021: 12995: 12942: 12903: 12869: 12834: 12810: 12803: 12781: 12728: 12717:on 28 May 2024 12694: 12688: 12670: 12601: 12594: 12565: 12558: 12540: 12519: 12474: 12467: 12442: 12425:NobelPrize.org 12412: 12382: 12368: 12343: 12326:Universe Today 12312: 12293: 12273: 12189: 12175: 12150: 12143: 12114: 12083: 12076: 12049: 12042: 12022: 12008: 11988: 11969: 11918: 11911: 11893: 11861: 11860: 11858: 11855: 11853: 11850: 11848: 11847: 11800: 11790: 11777: 11758: 11748: 11731: 11715:Laguerre group 11702: 11690: 11681: 11672: 11663: 11654: 11636: 11624: 11613: 11599: 11582: 11565:Solvay Debates 11552: 11539: 11525: 11496: 11494: 11491: 11489: 11488: 11486:Time geography 11483: 11478: 11473: 11468: 11463: 11458: 11453: 11448: 11443: 11438: 11432: 11430: 11427: 11415:dimensionality 11405:from an ideal 11302: 11299: 11296: 11293: 11266:Paul Ehrenfest 11259:John D. Barrow 11176:Properties of 11170: 11162: 11160: 11157: 11134: 11131: 11128: 11125: 11122: 11119: 11116: 11113: 11110: 11090: 11062: 11042: 11019: 10999: 10979: 10956: 10936: 10933: 10930: 10927: 10924: 10921: 10918: 10915: 10912: 10876: 10873: 10870: 10867: 10864: 10841: 10831:Lorentz metric 10818: 10815: 10812: 10809: 10806: 10777: 10774: 10660: 10652: 10650: 10647: 10597:Poincaré group 10585:Main article: 10582: 10579: 10566: 10565: 10564: 10563: 10556: 10533:In Poincaré's 10530: 10527: 10525: 10522: 10513: 10512: 10508: 10498: 10479: 10468: 10457: 10425: 10424: 10380: 10372: 10368: 10365: 10343: 10340: 10337: 10334: 10329: 10325: 10310: 10309: 10302: 10292: 10280: 10272: 10261: 10253: 10237: 10226: 10219: 10218: 10217: 10212: 10205: 10204: 10203: 10202: 10201: 10185: 10179: 10175: 10169: 10164: 10161: 10157: 10154: 10151: 10147: 10143: 10140: 10137: 10134: 10131: 10128: 10123: 10119: 10094: 10091: 10088: 10085: 10081: 10078: 10053: 10050: 10047: 10044: 10015: 10011: 10007: 10002: 9998: 9994: 9991: 9978: = 0 9957: 9954: 9951: 9948: 9945: 9941: 9938: 9913: 9910: 9907: 9904: 9900: 9897: 9882: 9879: 9860: 9859: 9843: 9836: 9835: 9834: 9833: 9832: 9830: 9827: 9803: 9799: 9795: 9790: 9786: 9782: 9779: 9747: 9743: 9714: 9710: 9680: 9676: 9670: 9666: 9641: 9637: 9631: 9627: 9612:is conserved. 9596: 9592: 9588: 9583: 9579: 9575: 9572: 9548: 9542: 9538: 9534: 9529: 9525: 9520: 9515: 9510: 9503: 9499: 9492: 9488: 9484: 9478: 9474: 9467: 9463: 9458: 9454: 9448: 9445: 9441: 9428:center of mass 9411: 9407: 9403: 9398: 9394: 9390: 9387: 9361: 9356: 9349: 9345: 9341: 9336: 9331: 9303: 9298: 9291: 9287: 9283: 9278: 9273: 9254: 9253: 9250: 9231: 9230:Total momentum 9228: 9192:Main article: 9189: 9186: 9183: 9182: 9171: 9168: 9165: 9145: 9140: 9136: 9132: 9127: 9123: 9119: 9114: 9110: 9106: 9103: 9100: 9097: 9094: 9088: 9085: 9079: 9076: 9073: 9070: 9067: 9057: 9045: 9040: 9036: 9032: 9027: 9023: 9019: 9014: 9010: 9006: 9003: 8999: 8995: 8992: 8989: 8986: 8980: 8977: 8971: 8968: 8964: 8960: 8957: 8954: 8951: 8926: 8923: 8908: 8905: 8902: 8899: 8896: 8882: 8876: 8875: 8862: 8858: 8852: 8843: 8839: 8834: 8830: 8824: 8820: 8816: 8811: 8807: 8792:invariant mass 8779: 8770: 8769: 8758: 8755: 8752: 8747: 8743: 8739: 8734: 8731: 8726: 8721: 8717: 8713: 8710: 8686: 8682: 8678: 8675: 8668: 8664: 8660: 8654: 8649: 8645: 8641: 8638: 8635: 8632: 8612: 8611: 8595: 8591: 8581: 8577: 8574: 8550: 8546: 8542: 8539: 8536: 8527: 8523: 8520: 8517: 8514: 8501: 8487: 8484: 8481: 8472: 8429: 8426: 8423: 8420: 8417: 8414: 8411: 8408: 8372: 8369: 8354: 8351: 8348: 8345: 8342: 8339: 8336: 8333: 8328: 8324: 8265: 8262: 8259: 8255: 8251: 8227: 8224: 8221: 8217: 8213: 8193:speed of light 8176: 8173: 8147: 8144: 8141: 8138: 8135: 8132: 8127: 8123: 8098: 8095: 8092: 8089: 8084: 8080: 8076: 8073: 8070: 8050: = 0 8032: 8029: 8026: 8023: 8020: 7962: 7959: 7943: 7940: 7925: 7914: 7913: 7898: 7894: 7890: 7887: 7881: 7878: 7874: 7871: 7867: 7862: 7859: 7855: 7852: 7833: 7832: 7817: 7813: 7809: 7806: 7800: 7795: 7792: 7788: 7785: 7781: 7778: 7774: 7771: 7738: 7737: 7734: 7727: 7726: 7719: 7715: 7701: 7698: 7697: 7696: 7685: 7680: 7676: 7667: 7664: 7661: 7656: 7653: 7650: 7643: 7640: 7613: 7610: 7607: 7604: 7601: 7598: 7594: 7591: 7587: 7584: 7581: 7578: 7575: 7572: 7569: 7566: 7546: 7542: 7539: 7535: 7532: 7529: 7509: 7506: 7486: 7466: 7463: 7443: 7442: 7429: 7425: 7421: 7416: 7412: 7408: 7405: 7402: 7399: 7396: 7372: 7365: 7360: 7359: 7346: 7342: 7333: 7329: 7325: 7322: 7318: 7313: 7310: 7290: 7283: 7276: 7273: 7264:Doppler effect 7254:Doppler effect 7249: 7248:Doppler effect 7246: 7177: 7174: 7122: 7121: 7120: 7119: 7107: 7104: 7100: 7097: 7094: 7090: 7087: 7083: 7080: 7077: 7067: 7055: 7052: 7048: 7045: 7042: 7038: 7035: 7031: 7028: 7025: 6996: 6964: 6933: 6918: 6885: 6857: 6737:Main article: 6734: 6731: 6698: 6695: 6691: 6685: 6681: 6676: 6672: 6669: 6666: 6663: 6659: 6655: 6652: 6647: 6643: 6545: 6542: 6539: 6536: 6533: 6530: 6527: 6524: 6521: 6518: 6514: 6511: 6507: 6488: 6485: 6480: 6476: 6471: 6467: 6464: 6461: 6458: 6455: 6452: 6449: 6446: 6443: 6439: 6436: 6432: 6412: 6409: 6406: 6403: 6400: 6397: 6394: 6391: 6387: 6384: 6364: 6361: 6356: 6352: 6347: 6343: 6340: 6337: 6334: 6331: 6328: 6325: 6321: 6318: 6276: 6275: 6259: 6256: 6252: 6249: 6247: 6245: 6242: 6241: 6237: 6234: 6230: 6227: 6225: 6223: 6220: 6219: 6215: 6210: 6207: 6203: 6200: 6196: 6193: 6188: 6184: 6181: 6178: 6176: 6174: 6171: 6170: 6166: 6158: 6154: 6148: 6145: 6141: 6135: 6131: 6128: 6123: 6119: 6116: 6113: 6111: 6109: 6106: 6105: 6091: 6090: 6075: 6072: 6069: 6067: 6064: 6061: 6057: 6056: 6053: 6050: 6047: 6045: 6042: 6039: 6035: 6034: 6030: 6026: 6023: 6020: 6017: 6013: 6009: 6006: 6003: 6001: 5998: 5995: 5991: 5990: 5986: 5978: 5974: 5969: 5966: 5960: 5957: 5953: 5949: 5946: 5943: 5941: 5938: 5935: 5931: 5930: 5898: 5895: 5892: 5891: 5878: 5875: 5870: 5867: 5862: 5858: 5854: 5851: 5848: 5845: 5842: 5839: 5836: 5832: 5828: 5825: 5801: = ( 5785: = ( 5686: 5667: 5666: 5650: 5646: 5642: 5639: 5635: 5630: 5622: 5618: 5613: 5607: 5603: 5599: 5596: 5592: 5587: 5584: 5561: 5539: 5532: 5528: 5523: 5517: 5513: 5509: 5506: 5500: 5496: 5493: 5490: 5487: 5484: 5400: 5399: 5388: 5385: 5382: 5371: 5359: 5356: 5353: 5350: 5330: 5327: 5324: 5320: 5277: 5276: 5235: 5233: 5226: 5220: 5217: 5214: 5213: 5187: 5143: 5142: 5141: 5140: 5129: 5123: 5118: 5114: 5109: 5104: 5101: 5097: 5094: 5091: 5088: 5082: 5079: 5075: 5072: 5066: 5063: 5038: 4960: 4931: 4832: 4831: 4820: 4816: 4812: 4809: 4806: 4779:Main article: 4776: 4773: 4765: 4764: 4753: 4750: 4747: 4743: 4740: 4736: 4733: 4713: 4710: 4707: 4704: 4700: 4697: 4538: 4537: 4526: 4523: 4520: 4516: 4513: 4502: 4491: 4488: 4484: 4481: 4470: 4459: 4456: 4452: 4449: 4438: 4427: 4424: 4421: 4418: 4415: 4411: 4408: 4247:Galilean group 4245:Main article: 4242: 4239: 4235: 4232: 4226: 4223: 4189:Figs. 2-6 4132: 4129: 4061: 4053: 4011: 4010: 3969: 3965: 3957: 3948: 3938: 3929: 3922: 3915: 3902: 3896: 3892: 3879: 3875: 3799: 3798: 3789: 3788: 3787: 3778: 3777: 3776: 3775: 3774: 3769: 3768: 3755: 3754: 3733: 3729: 3725: 3721: 3708: 3704: 3675: 3672: 3667:Main article: 3664: 3661: 3534: 3531: 3501: 3473: 3428: 3424: 3412: 3411: 3400: 3395: 3391: 3387: 3384: 3379: 3375: 3371: 3366: 3362: 3358: 3355: 3352: 3321: 3320: 3279: 3277: 3270: 3264: 3261: 3247: 3240: 3236: 3208: 3201: 3197: 3193: 3076: 3073: 3060:null separated 2967:Main article: 2964: 2961: 2917:′ = 2882: 2881: 2870: 2867: 2863: 2859: 2856: 2853: 2850: 2847: 2844: 2841: 2779: 2778: 2767: 2756: 2707: 2706: 2665: 2663: 2656: 2650: 2647: 2631: 2596: 2593: 2590: 2587: 2584: 2581: 2555: 2551: 2524: 2520: 2497: 2493: 2481: 2480: 2477: 2463: 2462: 2461: 2448: 2444: 2440: 2435: 2431: 2427: 2422: 2418: 2414: 2409: 2405: 2401: 2398: 2395: 2392: 2389: 2384: 2380: 2366: 2365: 2364: 2351: 2347: 2343: 2338: 2334: 2330: 2325: 2321: 2317: 2312: 2308: 2304: 2301: 2298: 2295: 2290: 2286: 2254: 2250: 2228: 2208: 2203: 2199: 2176: 2172: 2149: 2145: 2141: 2138: 2135: 2099: 2095: 2075: 2051: 2048: 2044: 2041: 2018: 2014: 2010: 2006: 2003: 1998: 1994: 1990: 1970: 1967: 1964: 1961: 1941: 1938: 1934: 1931: 1927: 1924: 1921: 1918: 1897: 1894: 1890: 1870: 1866: 1863: 1859: 1855: 1852: 1848: 1844: 1841: 1837: 1833: 1830: 1826: 1815: 1814: 1803: 1798: 1794: 1790: 1787: 1782: 1778: 1774: 1771: 1766: 1762: 1758: 1755: 1750: 1746: 1742: 1737: 1733: 1729: 1724: 1720: 1716: 1690: 1687: 1684: 1681: 1661: 1658: 1635: 1631: 1627: 1607: 1587: 1584: 1573: 1572: 1561: 1556: 1552: 1548: 1545: 1542: 1539: 1534: 1530: 1526: 1523: 1520: 1517: 1512: 1508: 1504: 1501: 1498: 1495: 1490: 1486: 1482: 1479: 1476: 1473: 1470: 1465: 1461: 1457: 1454: 1451: 1437: 1436: 1425: 1420: 1416: 1412: 1409: 1406: 1403: 1398: 1394: 1390: 1387: 1384: 1381: 1378: 1373: 1369: 1365: 1362: 1359: 1336: 1333: 1313: 1310: 1307: 1304: 1300: 1297: 1293: 1272: 1268: 1246: 1242: 1237: 1233: 1230: 1210: 1207: 1204: 1184: 1181: 1126: 1125: 1112: 1108: 1103: 1099: 1096: 1093: 1088: 1084: 1079: 1075: 1072: 1069: 1064: 1060: 1055: 1051: 1048: 1045: 1040: 1036: 1031: 1027: 1024: 996: 992: 969: 966: 956: 953: 912:Space and Time 908:Space and Time 900:Space and Time 789:Henri Poincaré 781: 780: 773: 766: 765: 759: 752: 751: 750: 745:Henri Poincaré 743: 736: 735: 729: 722: 721: 720: 719: 718: 675:simultaneously 629: 622: 621: 620: 611: 610: 609: 608: 607: 593: 590: 568:data reduction 526:speed-of-light 513: 393: 390: 386: 383: 344:events occur. 321:into a single 293: 292: 290: 289: 282: 275: 267: 264: 263: 261: 260: 247: 232: 229: 228: 224: 223: 218: 213: 208: 203: 198: 193: 188: 182: 181: 178: 177: 174: 173: 169: 168: 163: 157: 156: 153: 152: 149: 148: 144: 143: 138: 133: 127: 126: 123: 122: 119: 118: 114: 113: 108: 103: 98: 92: 91: 88: 87: 84: 83: 81: 80: 75: 69: 66: 65: 57: 56: 50: 49: 26: 18:Spatiotemporal 9: 6: 4: 3: 2: 17973: 17962: 17959: 17957: 17954: 17952: 17949: 17947: 17944: 17942: 17939: 17937: 17934: 17932: 17929: 17928: 17926: 17919: 17914: 17904: 17902: 17897: 17892: 17890: 17880: 17878: 17873: 17868: 17866: 17861: 17856: 17855: 17852: 17841: 17831: 17825: 17824: 17820: 17818: 17815: 17813: 17810: 17808: 17805: 17803: 17800: 17798: 17795: 17793: 17790: 17788: 17785: 17783: 17780: 17778: 17775: 17773: 17770: 17768: 17765: 17763: 17760: 17758: 17755: 17753: 17750: 17748: 17745: 17743: 17740: 17738: 17735: 17733: 17732:Chandrasekhar 17730: 17728: 17725: 17723: 17720: 17718: 17715: 17713: 17710: 17708: 17705: 17703: 17700: 17698: 17695: 17693: 17692:Schwarzschild 17690: 17688: 17685: 17683: 17680: 17678: 17675: 17673: 17670: 17669: 17667: 17663: 17653: 17649: 17648: 17645: 17642: 17640: 17637: 17635: 17631: 17630: 17627: 17624: 17622: 17619: 17617: 17614: 17612: 17609: 17606: 17602: 17598: 17597: 17594: 17591: 17588: 17585: 17583: 17579: 17578:Schwarzschild 17575: 17574: 17571: 17568: 17566: 17563: 17561: 17558: 17556: 17553: 17551: 17548: 17545: 17541: 17537: 17536: 17534: 17532: 17528: 17522: 17519: 17517: 17514: 17512: 17509: 17508: 17506: 17500: 17494: 17491: 17488: 17484: 17480: 17477: 17475: 17474:Shapiro delay 17472: 17470: 17467: 17464: 17460: 17456: 17453: 17450: 17446: 17443: 17442: 17439: 17436: 17433: 17430: 17428: 17425: 17423: 17420: 17418: 17417:collaboration 17414: 17410: 17407: 17405: 17401: 17398: 17397: 17394: 17391: 17389: 17386: 17384: 17383:Event horizon 17381: 17379: 17376: 17375: 17373: 17369: 17363: 17360: 17358: 17355: 17353: 17350: 17348: 17345: 17343: 17340: 17338: 17335: 17333: 17330: 17328: 17327:ADM formalism 17325: 17324: 17322: 17318: 17312: 17309: 17307: 17304: 17302: 17299: 17297: 17294: 17292: 17289: 17288: 17286: 17280: 17274: 17271: 17269: 17266: 17265: 17263: 17259: 17256: 17254: 17248: 17238: 17235: 17233: 17232:Biquaternions 17230: 17228: 17225: 17223: 17220: 17218: 17215: 17214: 17212: 17210: 17206: 17200: 17197: 17195: 17192: 17190: 17187: 17185: 17182: 17180: 17177: 17175: 17172: 17170: 17167: 17165: 17162: 17160: 17159:Time dilation 17157: 17156: 17154: 17150: 17144: 17141: 17140: 17138: 17134: 17128: 17125: 17123: 17120: 17118: 17115: 17113: 17112:Proper length 17110: 17108: 17105: 17103: 17100: 17098: 17095: 17093: 17090: 17088: 17085: 17084: 17082: 17076: 17070: 17067: 17065: 17062: 17059: 17056: 17054: 17050: 17047: 17046: 17044: 17040: 17037: 17035: 17029: 17025: 17018: 17013: 17011: 17006: 17004: 16999: 16998: 16995: 16983: 16980: 16978: 16975: 16973: 16970: 16968: 16965: 16963: 16960: 16958: 16955: 16953: 16950: 16946: 16943: 16942: 16941: 16938: 16936: 16933: 16932: 16930: 16926: 16920: 16917: 16915: 16912: 16910: 16907: 16905: 16902: 16900: 16897: 16895: 16892: 16890: 16887: 16885: 16882: 16880: 16877: 16875: 16872: 16870: 16867: 16865: 16862: 16860: 16857: 16855: 16852: 16850: 16847: 16845: 16842: 16840: 16837: 16835: 16832: 16830: 16827: 16826: 16824: 16822: 16821:units of time 16817: 16811: 16810:Sidereal time 16808: 16806: 16803: 16801: 16798: 16796: 16795:Galactic year 16793: 16792: 16790: 16788: 16784: 16778: 16775: 16773: 16770: 16768: 16765: 16764: 16762: 16758: 16752: 16751:Weekday names 16749: 16747: 16744: 16742: 16741:Tropical year 16739: 16737: 16734: 16732: 16729: 16727: 16724: 16722: 16719: 16717: 16714: 16712: 16709: 16707: 16706:Intercalation 16704: 16702: 16699: 16697: 16694: 16692: 16689: 16687: 16684: 16682: 16679: 16677: 16674: 16672:(lunar Hijri) 16671: 16668: 16666: 16663: 16661: 16658: 16656: 16653: 16651: 16648: 16647: 16645: 16643: 16639: 16633: 16630: 16628: 16625: 16623: 16620: 16618: 16615: 16613: 16610: 16608: 16605: 16601: 16598: 16597: 16596: 16593: 16591: 16588: 16586: 16583: 16581: 16578: 16576: 16573: 16571: 16568: 16566: 16563: 16561: 16558: 16556: 16553: 16551: 16548: 16547: 16545: 16543: 16539: 16533: 16530: 16528: 16525: 16523: 16520: 16518: 16515: 16513: 16512:Time dilation 16510: 16508: 16505: 16503: 16500: 16498: 16495: 16493: 16490: 16488: 16485: 16483: 16480: 16478: 16475: 16473: 16470: 16468: 16465: 16464: 16462: 16460: 16456: 16450: 16447: 16445: 16442: 16440: 16437: 16436: 16434: 16430: 16425: 16419: 16409: 16406: 16404: 16401: 16399: 16396: 16394: 16391: 16389: 16386: 16384: 16381: 16379: 16376: 16374: 16371: 16369: 16366: 16364: 16361: 16359: 16356: 16354: 16351: 16349: 16348:24-hour clock 16346: 16344: 16343:12-hour clock 16341: 16339: 16336: 16334: 16331: 16329: 16326: 16324: 16321: 16319: 16316: 16314: 16311: 16309: 16306: 16302: 16299: 16298: 16297: 16294: 16293: 16291: 16287: 16281: 16278: 16276: 16273: 16271: 16268: 16267: 16264: 16260: 16256: 16249: 16244: 16242: 16237: 16235: 16230: 16229: 16226: 16214: 16206: 16204: 16196: 16195: 16192: 16186: 16183: 16181: 16178: 16176: 16173: 16171: 16170: 16166: 16164: 16161: 16159: 16156: 16154: 16151: 16149: 16146: 16144: 16141: 16140: 16138: 16134: 16124: 16121: 16119: 16116: 16114: 16111: 16107: 16104: 16103: 16102: 16101:Chronobiology 16099: 16097: 16094: 16093: 16091: 16087: 16081: 16078: 16076: 16073: 16071: 16068: 16066: 16063: 16061: 16058: 16056: 16053: 16051: 16048: 16046: 16043: 16041: 16038: 16036: 16035:Arrow of time 16033: 16031: 16028: 16027: 16025: 16023: 16019: 16013: 16010: 16008: 16007:Geochronology 16005: 16001: 15998: 15996: 15993: 15991: 15988: 15986: 15983: 15981: 15978: 15976: 15973: 15972: 15971: 15968: 15967: 15965: 15963: 15959: 15956: 15954: 15949: 15943: 15939: 15935: 15932: 15930: 15927: 15925: 15922: 15920: 15917: 15913: 15910: 15908: 15905: 15904: 15903: 15900: 15898: 15895: 15893: 15890: 15888: 15885: 15884: 15882: 15880: 15875: 15871: 15863: 15860: 15859: 15858: 15857:Wheel of time 15855: 15851: 15848: 15847: 15846: 15843: 15841: 15838: 15836: 15833: 15831: 15828: 15826: 15823: 15821: 15818: 15817: 15815: 15810: 15807: 15805: 15802: 15801: 15798: 15792: 15791: 15787: 15785: 15782: 15780: 15777: 15775: 15772: 15770: 15767: 15765: 15762: 15760: 15757: 15755: 15752: 15750: 15747: 15745: 15742: 15740: 15737: 15735: 15732: 15730: 15727: 15726: 15724: 15722: 15718: 15712: 15709: 15707: 15704: 15702: 15701:Periodization 15699: 15697: 15694: 15692: 15689: 15687: 15684: 15682: 15679: 15678: 15676: 15671: 15668: 15666: 15663: 15662: 15659: 15647: 15646: 15642: 15641: 15640: 15639: 15635: 15633: 15630: 15628: 15627:Digital clock 15625: 15623: 15620: 15616: 15613: 15609: 15606: 15604: 15601: 15600: 15599: 15596: 15594: 15591: 15589: 15586: 15584: 15581: 15577: 15574: 15573: 15572: 15569: 15565: 15562: 15561: 15560: 15557: 15556: 15555: 15552: 15551: 15549: 15547: 15543: 15537: 15536: 15532: 15530: 15527: 15525: 15522: 15520: 15517: 15515: 15512: 15510: 15507: 15505: 15502: 15500: 15497: 15495: 15492: 15488: 15485: 15483: 15480: 15478: 15475: 15474: 15473: 15470: 15469: 15467: 15465: 15461: 15455: 15452: 15450: 15447: 15445: 15442: 15440: 15437: 15435: 15432: 15430: 15427: 15425: 15422: 15420: 15417: 15415: 15412: 15410: 15407: 15405: 15404:Relative hour 15402: 15400: 15399:24-hour clock 15397: 15395: 15394:12-hour clock 15392: 15390: 15387: 15385: 15382: 15381: 15379: 15377: 15371: 15365: 15362: 15360: 15357: 15355: 15352: 15350: 15347: 15345: 15342: 15341: 15339: 15337: 15333: 15330: 15328: 15323: 15319: 15313: 15310: 15308: 15305: 15303: 15300: 15298: 15295: 15294: 15292: 15288: 15284: 15277: 15272: 15270: 15265: 15263: 15258: 15257: 15254: 15244: 15243: 15237: 15231: 15228: 15226: 15223: 15222: 15220: 15216: 15210: 15208: 15204: 15202: 15199: 15197: 15194: 15192: 15189: 15187: 15184: 15182: 15179: 15177: 15174: 15172: 15169: 15167: 15164: 15162: 15159: 15158: 15156: 15152: 15146: 15143: 15141: 15138: 15136: 15133: 15131: 15128: 15126: 15125:Demihypercube 15123: 15121: 15118: 15116: 15113: 15111: 15108: 15106: 15103: 15102: 15100: 15098: 15094: 15090: 15084: 15081: 15079: 15076: 15074: 15071: 15069: 15066: 15064: 15061: 15059: 15056: 15054: 15051: 15050: 15048: 15044: 15039: 15029: 15026: 15024: 15021: 15019: 15016: 15014: 15011: 15009: 15006: 15004: 15001: 14999: 14996: 14994: 14991: 14990: 14988: 14984: 14980: 14973: 14968: 14966: 14961: 14959: 14954: 14953: 14950: 14943: 14939: 14936: 14933: 14930: 14927: 14924: 14921:13th edition 14920: 14917: 14914: 14909: 14905: 14904: 14886: 14885: 14880: 14876: 14872: 14870:0-7167-2327-1 14866: 14862: 14861: 14856: 14851: 14846: 14844:0-679-45443-8 14840: 14836: 14835: 14830: 14826: 14823: 14819: 14816: 14813: 14809: 14808:Weyl, Hermann 14805: 14801: 14797: 14794: 14792: 14791:0-19-851164-7 14788: 14784: 14780: 14777: 14773: 14769: 14765: 14759: 14755: 14751: 14750: 14745: 14741: 14737: 14736: 14716: 14712: 14706: 14698: 14694: 14690: 14686: 14682: 14678: 14673: 14668: 14665:(1): 013217. 14664: 14660: 14656: 14649: 14641: 14637: 14633: 14629: 14625: 14621: 14616: 14611: 14607: 14603: 14599: 14592: 14577: 14573: 14569: 14565: 14561: 14557: 14552: 14551:gr-qc/9702052 14547: 14543: 14539: 14532: 14528: 14522: 14520: 14518: 14509: 14505: 14501: 14497: 14493: 14489: 14485: 14481: 14480:Nuovo Cimento 14477: 14470: 14463: 14457: 14450: 14444: 14440: 14436: 14432: 14428: 14424: 14420: 14416: 14410: 14394: 14390: 14383: 14364: 14360: 14356: 14351: 14346: 14342: 14338: 14331: 14327: 14321: 14313: 14309: 14305: 14301: 14297: 14293: 14289: 14285: 14280: 14279:gr-qc/9702052 14275: 14271: 14267: 14260: 14241: 14237: 14231: 14227: 14220: 14213: 14211: 14209: 14207: 14200: 14195: 14188: 14183: 14178: 14171: 14169: 14160: 14156: 14152: 14148: 14144: 14140: 14133: 14125: 14121: 14117: 14113: 14109: 14105: 14101: 14097: 14090: 14082: 14076: 14072: 14065: 14063: 14054: 14050: 14046: 14042: 14038: 14034: 14027: 14019: 14015: 14011: 14007: 14003: 13999: 13992: 13976: 13972: 13968: 13964: 13960: 13956: 13952: 13948: 13944: 13940: 13933: 13925: 13921: 13917: 13913: 13909: 13905: 13900: 13899:gr-qc/0411023 13895: 13891: 13887: 13880: 13864: 13860: 13853: 13845: 13843:0-691-08472-6 13839: 13835: 13834: 13826: 13824: 13822: 13813: 13811:0-486-60081-5 13807: 13802: 13801: 13792: 13786:in spacetime. 13785: 13781: 13776: 13772: 13767: 13763: 13759: 13755: 13750: 13746: 13738: 13723: 13721:0-7167-0336-X 13717: 13713: 13712: 13704: 13702: 13700: 13683: 13679: 13675: 13668: 13652: 13648: 13641: 13632: 13627: 13620: 13605: 13599: 13595: 13594: 13586: 13578: 13574: 13569: 13564: 13560: 13556: 13552: 13548: 13545:(1): 014107. 13544: 13540: 13536: 13529: 13521: 13515: 13511: 13510: 13502: 13500: 13491: 13485: 13481: 13474: 13466: 13460: 13456: 13449: 13447: 13445: 13443: 13441: 13439: 13437: 13435: 13433: 13417: 13411: 13407: 13406: 13398: 13390: 13384: 13379: 13378: 13369: 13367: 13365: 13363: 13361: 13359: 13357: 13355: 13353: 13351: 13349: 13347: 13331: 13325: 13321: 13320: 13312: 13297: 13291: 13287: 13286: 13278: 13262: 13258: 13254: 13247: 13245: 13243: 13241: 13239: 13237: 13235: 13226: 13224:0-521-27703-5 13220: 13216: 13209: 13193: 13189: 13185: 13178: 13176: 13159: 13155: 13151: 13144: 13136: 13134:9780593186589 13130: 13126: 13119: 13110: 13104: 13099: 13093: 13089: 13088: 13080: 13072: 13070:0-12-417561-9 13066: 13062: 13055: 13053: 13051: 13049: 13047: 13038: 13032: 13024: 13022:0-387-94348-X 13018: 13014: 13010: 13006: 12999: 12991: 12987: 12983: 12979: 12975: 12971: 12966: 12961: 12957: 12953: 12946: 12929: 12925: 12921: 12917: 12910: 12908: 12891: 12887: 12883: 12876: 12874: 12865: 12861: 12857: 12853: 12849: 12845: 12838: 12829: 12824: 12817: 12815: 12806: 12800: 12795: 12794: 12785: 12769: 12764: 12759: 12755: 12751: 12747: 12743: 12739: 12732: 12716: 12712: 12708: 12701: 12699: 12691: 12689:0-8126-9067-2 12685: 12681: 12674: 12658: 12654: 12650: 12645: 12640: 12636: 12632: 12628: 12624: 12620: 12616: 12612: 12605: 12597: 12595:0-393-02001-0 12591: 12587: 12583: 12582: 12574: 12572: 12570: 12561: 12559:0-387-94870-8 12555: 12551: 12544: 12526: 12522: 12516: 12512: 12508: 12504: 12500: 12496: 12492: 12485: 12478: 12470: 12468:0-19-853907-X 12464: 12460: 12459: 12451: 12449: 12447: 12430: 12426: 12422: 12416: 12400: 12396: 12392: 12386: 12375: 12371: 12369:0-8176-4380-X 12365: 12361: 12354: 12347: 12331: 12327: 12323: 12316: 12300: 12296: 12290: 12286: 12285: 12277: 12258: 12254: 12250: 12246: 12242: 12238: 12234: 12229: 12224: 12220: 12216: 12212: 12211: 12203: 12199: 12193: 12178: 12176:0-7167-0336-X 12172: 12168: 12167: 12159: 12157: 12155: 12146: 12144:0-7487-6422-4 12140: 12136: 12132: 12128: 12121: 12119: 12102: 12098: 12094: 12087: 12079: 12073: 12069: 12062: 12060: 12058: 12056: 12054: 12045: 12039: 12035: 12034: 12026: 12011: 12009:0-08-010061-9 12005: 12001: 12000: 11992: 11976: 11972: 11970:0-521-45506-5 11966: 11962: 11958: 11957: 11949: 11947: 11945: 11943: 11941: 11939: 11937: 11935: 11933: 11931: 11929: 11927: 11925: 11923: 11914: 11908: 11904: 11897: 11881: 11877: 11873: 11866: 11862: 11844: 11840: 11836: 11832: 11828: 11825: 11819: 11814: 11810: 11804: 11794: 11787: 11781: 11774: 11770: 11769: 11762: 11752: 11745: 11741: 11735: 11728: 11724: 11720: 11716: 11712: 11706: 11699: 11698: 11693: 11689: 11684: 11680: 11675: 11671: 11666: 11662: 11657: 11653: 11647: 11645: 11639: 11635: 11630: 11627: 11623: 11619: 11616: 11612: 11608: 11603: 11596: 11592: 11586: 11579: 11574: 11570: 11566: 11562: 11556: 11549: 11543: 11536: 11535:Henri Bergson 11529: 11522: 11518: 11514: 11510: 11506: 11501: 11497: 11487: 11484: 11482: 11479: 11477: 11474: 11472: 11469: 11467: 11464: 11462: 11459: 11457: 11454: 11452: 11449: 11447: 11444: 11442: 11439: 11437: 11434: 11433: 11422: 11420: 11416: 11408: 11407:monatomic gas 11404: 11399: 11395: 11388: 11383: 11377: 11370: 11365: 11361: 11355: 11350: 11346: 11342: 11338: 11337:or disperse. 11336: 11332: 11328: 11325:'s theory of 11324: 11321:claimed that 11320: 11316: 11300: 11297: 11294: 11291: 11283: 11279: 11275: 11271: 11267: 11262: 11260: 11256: 11252: 11251:Immanuel Kant 11248: 11244: 11241:, who in the 11240: 11235: 11233: 11229: 11225: 11224:string theory 11219: 11212: 11207: 11203: 11199: 11195: 11185: 11181: 11174: 11167: 11156: 11154: 11150: 11148: 11129: 11126: 11123: 11120: 11117: 11114: 11111: 11088: 11079: 11076: 11060: 11040: 11031: 11017: 10997: 10977: 10968: 10954: 10931: 10928: 10925: 10922: 10919: 10916: 10913: 10902: 10898: 10894: 10871: 10868: 10865: 10855: 10839: 10832: 10813: 10810: 10807: 10797: 10791: 10787: 10783: 10773: 10768: 10764: 10760: 10757:, as well as 10756: 10752: 10748: 10744: 10740: 10736: 10732: 10728: 10723: 10718: 10714: 10712: 10708: 10704: 10700: 10696: 10692: 10688: 10687:tangent space 10684: 10683:inner product 10680: 10676: 10673:, defined as 10672: 10669:that studies 10668: 10664: 10657: 10646: 10644: 10642: 10638: 10632: 10629: 10625: 10621: 10617: 10613: 10612:Hermann Bondi 10608: 10606: 10602: 10598: 10594: 10588: 10578: 10574: 10572: 10561: 10557: 10554: 10553: 10552: 10551: 10550: 10547: 10545: 10541: 10536: 10518: 10507: 10504: 10503:inertial mass 10497: 10494: 10490: 10489:a priori 10484: 10478: 10473: 10467: 10462: 10456: 10449: 10445: 10439: 10435: 10434: 10429: 10421: 10420: 10419: 10416: 10414: 10409: 10404: 10402: 10392: 10388: 10385: 10377: 10371: 10357: 10341: 10338: 10335: 10332: 10327: 10323: 10307: 10284: 10248: 10247: 10240: 10236: 10223: 10209: 10200: 10177: 10173: 10167: 10162: 10159: 10155: 10152: 10145: 10138: 10135: 10132: 10126: 10117: 10092: 10089: 10086: 10083: 10079: 10076: 10051: 10048: 10045: 10042: 10031: 10013: 10009: 10005: 10000: 9996: 9992: 9989: 9977: 9971: 9955: 9952: 9949: 9946: 9943: 9939: 9936: 9911: 9908: 9905: 9902: 9898: 9895: 9878: 9876: 9864: 9855: 9849: 9844:Figure 3-11. 9840: 9826: 9824: 9819: 9801: 9797: 9793: 9788: 9784: 9780: 9777: 9766: 9696: 9678: 9674: 9668: 9664: 9639: 9635: 9629: 9625: 9613: 9546: 9540: 9536: 9532: 9527: 9523: 9518: 9513: 9508: 9490: 9486: 9482: 9465: 9461: 9456: 9452: 9429: 9409: 9405: 9401: 9396: 9392: 9388: 9385: 9347: 9343: 9339: 9289: 9285: 9281: 9258: 9251: 9248: 9247: 9246: 9243: 9236: 9227: 9225: 9221: 9217: 9213: 9209: 9205: 9201: 9195: 9185: 9169: 9166: 9163: 9138: 9134: 9130: 9125: 9121: 9117: 9112: 9108: 9104: 9101: 9095: 9083: 9077: 9074: 9068: 9065: 9058: 9038: 9034: 9030: 9025: 9021: 9017: 9012: 9008: 9004: 9001: 8997: 8993: 8987: 8975: 8969: 8966: 8962: 8958: 8952: 8949: 8942: 8941: 8940: 8938: 8937: 8932: 8925:Four-momentum 8922: 8906: 8903: 8900: 8897: 8894: 8881: 8860: 8856: 8850: 8841: 8837: 8832: 8828: 8822: 8818: 8814: 8809: 8805: 8797: 8796: 8795: 8793: 8787: 8785: 8778: 8773: 8756: 8753: 8750: 8745: 8741: 8737: 8732: 8729: 8724: 8719: 8715: 8711: 8708: 8684: 8680: 8676: 8673: 8666: 8662: 8658: 8652: 8647: 8643: 8639: 8636: 8633: 8630: 8623: 8622: 8621: 8618: 8593: 8589: 8579: 8575: 8572: 8548: 8544: 8540: 8537: 8534: 8525: 8521: 8518: 8515: 8512: 8502: 8485: 8482: 8479: 8470: 8459: 8458: 8453: 8449: 8445: 8427: 8424: 8421: 8418: 8415: 8412: 8409: 8406: 8391: 8387: 8383: 8378: 8377: 8376: 8368: 8352: 8349: 8343: 8340: 8337: 8331: 8322: 8311: 8306: 8304: 8300: 8296: 8292: 8287: 8285: 8281: 8263: 8260: 8257: 8253: 8249: 8225: 8222: 8219: 8215: 8211: 8201: 8196: 8194: 8190: 8181: 8172: 8170: 8169:four-momentum 8165: 8163: 8145: 8142: 8139: 8136: 8133: 8130: 8121: 8096: 8093: 8090: 8087: 8074: 8071: 8057: 8055: 8049: 8027: 8024: 8021: 8006: 8004: 8003:closed system 8000: 7995: 7994: 7989: = 7988: 7987: 7981: 7977: 7967: 7957: 7953: 7949: 7948:Four-momentum 7939: 7937: 7923: 7896: 7892: 7888: 7885: 7879: 7876: 7872: 7869: 7865: 7860: 7857: 7853: 7850: 7843: 7842: 7841: 7839: 7815: 7811: 7807: 7804: 7798: 7793: 7790: 7786: 7783: 7779: 7776: 7772: 7769: 7762: 7761: 7760: 7758: 7751: 7747: 7742: 7735: 7732: 7731: 7730: 7724: 7720: 7717: 7716: 7714: 7706: 7683: 7678: 7674: 7665: 7662: 7659: 7654: 7651: 7648: 7641: 7638: 7631: 7630: 7629: 7627: 7611: 7605: 7602: 7599: 7592: 7589: 7585: 7582: 7579: 7576: 7573: 7570: 7567: 7564: 7544: 7540: 7537: 7533: 7530: 7527: 7507: 7504: 7484: 7464: 7461: 7447: 7427: 7423: 7414: 7410: 7406: 7403: 7397: 7394: 7387: 7386: 7385: 7383: 7379: 7375: 7368: 7344: 7340: 7331: 7327: 7323: 7320: 7316: 7311: 7308: 7301: 7300: 7299: 7297: 7293: 7286: 7272: 7270: 7265: 7259: 7255: 7245: 7243: 7238: 7231: 7226: 7220: 7215: 7211: 7206: 7203: 7201: 7197: 7190: 7183: 7173: 7168: 7161: 7157: 7153: 7149: 7142: 7135: 7131: 7127: 7105: 7102: 7098: 7095: 7092: 7088: 7085: 7081: 7078: 7075: 7068: 7053: 7050: 7046: 7043: 7040: 7036: 7033: 7029: 7026: 7023: 7016: 7015: 7013: 7009: 7005: 7001: 6997: 6994: 6987: 6984: = 6983: 6977: 6971: = 6970: 6965: 6961: 6954: = 6953: 6947: 6944: = 6943: 6939: 6934: 6931: 6925: = 6924: 6919: 6915: 6912: = 6911: 6907: 6903: 6899: 6896: = 6895: 6891: 6886: 6883: 6880: + 6879: 6876: = 6875: 6870: 6866: 6862: 6858: 6855: 6848: 6844: 6840: 6833: 6832: 6831: 6829: 6825: 6821: 6814: 6807: 6802: 6798: 6795: = 6794: 6784: 6777: 6771: 6767: 6761: 6759: 6754: 6745: 6740: 6730: 6696: 6693: 6689: 6683: 6679: 6674: 6670: 6667: 6664: 6661: 6657: 6653: 6650: 6641: 6628: 6615: 6606: 6602: 6597: 6592: 6590: 6586: 6579: 6575: 6569: 6567: 6563: 6559: 6540: 6534: 6531: 6528: 6519: 6516: 6512: 6509: 6486: 6478: 6474: 6469: 6465: 6459: 6456: 6453: 6444: 6441: 6437: 6434: 6407: 6404: 6401: 6398: 6392: 6389: 6385: 6382: 6362: 6354: 6350: 6345: 6341: 6338: 6335: 6332: 6326: 6323: 6319: 6316: 6307: 6305: 6301: 6297: 6293: 6289: 6285: 6282: ≪ 6281: 6257: 6254: 6250: 6248: 6243: 6235: 6232: 6228: 6226: 6221: 6213: 6208: 6205: 6201: 6198: 6194: 6191: 6186: 6182: 6179: 6177: 6172: 6164: 6156: 6152: 6146: 6143: 6139: 6133: 6129: 6126: 6121: 6117: 6114: 6112: 6107: 6096: 6095: 6094: 6073: 6070: 6068: 6062: 6059: 6051: 6048: 6046: 6040: 6037: 6028: 6024: 6021: 6018: 6015: 6011: 6007: 6004: 6002: 5996: 5993: 5984: 5976: 5972: 5967: 5964: 5958: 5955: 5951: 5947: 5944: 5942: 5936: 5933: 5921: 5920: 5919: 5916: 5913: 5908: 5907:Lorentz group 5904: 5894: 5876: 5873: 5868: 5860: 5856: 5852: 5849: 5843: 5840: 5837: 5834: 5830: 5826: 5823: 5816: 5815: 5814: 5812: 5808: 5805: − 5804: 5800: 5796: 5792: 5789: − 5788: 5784: 5780: 5776: 5772: 5768: 5764: 5760: 5757: = 5756: 5752: 5749: = 5748: 5740: 5736: 5729: 5723: 5719: 5715: 5711: 5706: 5704: 5700: 5684: 5676: 5672: 5648: 5644: 5640: 5637: 5633: 5628: 5620: 5616: 5611: 5605: 5601: 5597: 5594: 5590: 5585: 5582: 5575: 5574: 5573: 5559: 5550: 5537: 5530: 5526: 5521: 5515: 5511: 5507: 5504: 5498: 5494: 5491: 5488: 5485: 5482: 5474: 5470: 5467: = 5466: 5462: 5459: = 5458: 5454: 5451: = 5450: 5442: 5438: 5431: 5425: 5421: 5417: 5413: 5404: 5386: 5383: 5380: 5372: 5357: 5354: 5351: 5348: 5328: 5325: 5322: 5310: 5309: 5308: 5306: 5301: 5293: 5288: 5284: 5283:Time dilation 5273: 5270: 5262: 5252: 5248: 5242: 5241: 5236:This section 5234: 5230: 5225: 5224: 5216: 5211: 5207: 5204: = 5203: 5199: 5192: 5188: 5185: 5182: + 5178: 5175: = 5174: 5170: 5163: 5159: 5152: 5148: 5147: 5146: 5127: 5116: 5112: 5107: 5102: 5099: 5095: 5089: 5086: 5080: 5077: 5073: 5070: 5064: 5061: 5054: 5053: 5051: 5047: 5043: 5039: 5035: 5031: 5025: 5021: 5015: 5011: 5005: 5001: 4992: 4988: 4984: 4978: 4974: 4970: 4966: 4961: 4957: 4953: 4949: 4945: 4941: 4936: 4932: 4927: 4923: 4919: 4915: 4911: 4907: 4902: 4901: 4900: 4898: 4894: 4887: 4882: 4878: 4871: 4867: 4860: 4854: 4850: 4846: 4842: 4838: 4818: 4814: 4810: 4807: 4804: 4797: 4796: 4795: 4787: 4782: 4772: 4770: 4751: 4748: 4745: 4741: 4738: 4734: 4731: 4711: 4708: 4705: 4702: 4698: 4695: 4687: 4686: 4685: 4679: 4672: 4665: 4656: 4652: 4648: 4641: 4637: 4633: 4626: 4622: 4615: 4606: 4602: 4596: 4592: 4585: 4579: 4575: 4570: 4563: 4559: 4554: 4547: 4542: 4524: 4521: 4518: 4514: 4511: 4503: 4489: 4486: 4482: 4479: 4471: 4457: 4454: 4450: 4447: 4439: 4425: 4422: 4419: 4416: 4413: 4409: 4406: 4398: 4397: 4396: 4389: 4382: 4375: 4371: 4367: 4356: 4349: 4342: 4335: 4327: 4323: 4319: 4315: 4304: 4297: 4290: 4283: 4277: 4276: 4275:Fig. 1-1 4269: 4265: 4261: 4257: 4248: 4238: 4231: 4222: 4220: 4215: 4209: 4207: 4203: 4199: 4195: 4191: 4190: 4182: 4179: 4176: 4173: 4171: 4170:path integral 4161: 4157: 4153: 4151: 4146: 4142: 4137: 4128: 4126: 4122: 4117: 4115: 4111: 4107: 4103: 4099: 4095: 4091: 4087: 4083: 4079: 4075: 4072:′-coordinate 4071: 4067: 4059: 4051: 4047: 4042: 4040: 4036: 4032: 4028: 4024: 4020: 4016: 4007: 4003: 3999: 3995: 3991: 3987: 3983: 3979: 3975: 3963: 3960:) is labeled 3955: 3952:of the event 3951: 3944: 3939: 3935: 3932:in the event 3925: 3921: 3913: 3909: 3905: 3893: 3889: 3885: 3873: 3869: 3862: 3858: 3853: 3852: 3851: 3849: 3845: 3841: 3837: 3833: 3829: 3825: 3821: 3816: 3814: 3810: 3793: 3782: 3773: 3765: 3760: 3759: 3758: 3751: 3747: 3743: 3739: 3718: 3717: 3716: 3714: 3702: 3697: 3696:only apparent 3692: 3690: 3686: 3682: 3670: 3660: 3658: 3654: 3650: 3646: 3645: 3644:Fig. 2-9 3639: 3637: 3632: 3630: 3626: 3622: 3615: 3610: 3606: 3601: 3599: 3591: 3587: 3585: 3583: 3582:time dilation 3578: 3568: 3556: 3539: 3530: 3528: 3524: 3522: 3513: 3499: 3491: 3487: 3471: 3462: 3457: 3454: 3452: 3448: 3444: 3426: 3422: 3398: 3393: 3389: 3385: 3382: 3377: 3373: 3369: 3364: 3356: 3353: 3343: 3342: 3341: 3329: 3325: 3317: 3314: 3306: 3296: 3292: 3286: 3285: 3280:This section 3278: 3274: 3269: 3268: 3260: 3258: 3245: 3238: 3234: 3223: 3206: 3199: 3195: 3191: 3181: 3175: 3173: 3164: 3160: 3149: 3145: 3138: 3127: 3125: 3121: 3115: 3111: 3106: 3105: 3104:Fig. 2-4 3100: 3096: 3092: 3083: 3079: 3072: 3070: 3068: 3063: 3061: 3057: 3053: 3049: 3045: 3040: 3038: 3034: 3030: 3026: 3025:timelike past 3022: 3018: 3014: 3005: 3001: 2999: 2995: 2991: 2987: 2977: 2973: 2970: 2960: 2958: 2956: 2952: 2946: 2944: 2941:must also be 2940: 2936: 2932: 2928: 2922: 2920: 2916: 2912: 2908: 2904: 2900: 2896: 2891: 2887: 2868: 2865: 2861: 2857: 2854: 2848: 2842: 2839: 2832: 2831: 2830: 2828: 2824: 2820: 2816: 2812: 2808: 2804: 2800: 2796: 2792: 2788: 2784: 2776: 2772: 2768: 2765: 2761: 2757: 2754: 2750: 2746: 2742: 2741: 2740: 2738: 2734: 2730: 2721: 2713: 2703: 2700: 2692: 2682: 2678: 2672: 2671: 2666:This section 2664: 2660: 2655: 2654: 2646: 2644: 2629: 2621: 2616: 2614: 2610: 2594: 2591: 2588: 2585: 2582: 2579: 2571: 2553: 2549: 2540: 2522: 2518: 2495: 2491: 2478: 2468: 2464: 2446: 2442: 2438: 2433: 2429: 2425: 2420: 2416: 2412: 2407: 2399: 2396: 2390: 2387: 2382: 2378: 2370: 2369: 2367: 2349: 2345: 2341: 2336: 2332: 2328: 2323: 2319: 2315: 2310: 2302: 2299: 2293: 2288: 2284: 2276: 2275: 2273: 2270: 2269: 2268: 2252: 2248: 2226: 2206: 2201: 2197: 2174: 2170: 2147: 2139: 2136: 2120: 2116: 2114: 2097: 2073: 2064: 2049: 2046: 2042: 2039: 2016: 2012: 2008: 2004: 2001: 1996: 1992: 1988: 1968: 1965: 1962: 1959: 1939: 1936: 1932: 1929: 1925: 1922: 1919: 1916: 1895: 1892: 1888: 1864: 1861: 1857: 1853: 1850: 1846: 1842: 1839: 1835: 1831: 1828: 1801: 1796: 1792: 1788: 1785: 1780: 1776: 1772: 1769: 1764: 1760: 1756: 1753: 1748: 1744: 1740: 1735: 1731: 1727: 1722: 1718: 1714: 1707: 1706: 1705: 1702: 1688: 1685: 1682: 1659: 1633: 1629: 1605: 1585: 1582: 1575:The constant 1559: 1554: 1546: 1537: 1532: 1524: 1515: 1510: 1502: 1493: 1488: 1480: 1477: 1468: 1463: 1455: 1442: 1441: 1440: 1423: 1418: 1410: 1401: 1396: 1388: 1385: 1376: 1371: 1363: 1350: 1349: 1348: 1334: 1331: 1311: 1305: 1302: 1298: 1295: 1270: 1244: 1235: 1208: 1205: 1182: 1171: 1167: 1163: 1162:time dilation 1159: 1155: 1149: 1147: 1142: 1139: 1135: 1131: 1110: 1101: 1091: 1086: 1077: 1067: 1062: 1053: 1043: 1038: 1029: 1015: 1014: 1013: 1011: 994: 983: 982: 975: 965: 962: 952: 951: 947: 942: 940: 936: 932: 928: 923: 921: 917: 913: 909: 905: 904:Raum und Zeit 901: 897: 893: 885: 884:Raum und Zeit 880: 876: 873: 872:David Hilbert 867: 862: 860: 856: 851: 849: 844: 840: 836: 831: 829: 825: 820: 818: 814: 813:four-velocity 810: 809:four-position 806: 801: 799: 795: 790: 776: 770: 762: 756: 746: 740: 732: 726: 717: 715: 711: 707: 703: 698: 695: 692:in 1889, and 691: 687: 685: 680: 676: 672: 667: 665: 661: 657: 653: 649: 645: 641: 626: 615: 605: 601: 596: 589: 587: 583: 579: 574: 571: 569: 563: 561: 555: 548: 543: 539: 537: 535: 531: 527: 511: 503: 502: 496: 494: 488: 484: 482: 478: 474: 470: 466: 462: 458: 454: 450: 446: 442: 437: 435: 431: 427: 423: 418: 416: 412: 408: 403: 399: 389: 382: 380: 378: 374: 370: 366: 361: 359: 355: 351: 345: 343: 339: 335: 331: 327: 324: 320: 316: 312: 308: 304: 300: 288: 283: 281: 276: 274: 269: 268: 266: 265: 258: 248: 245: 240: 234: 233: 231: 230: 222: 219: 217: 214: 212: 209: 207: 204: 202: 199: 197: 194: 192: 189: 187: 184: 183: 176: 175: 167: 164: 162: 159: 158: 151: 150: 142: 139: 137: 134: 132: 129: 128: 121: 120: 112: 109: 107: 104: 102: 99: 97: 94: 93: 86: 85: 79: 76: 74: 71: 70: 68: 67: 63: 59: 58: 55: 52: 51: 47: 46: 41: 37: 33: 19: 17918: 17822: 17516:Kaluza–Klein 17268:Introduction 17208: 17194:Twin paradox 16957:Decimal time 16686:Astronomical 16565:Complication 16560:Atomic clock 16471: 16175:Time capsule 16169:Tempus fugit 16167: 16089:Other fields 16059: 15788: 15769:Perdurantism 15691:Calendar era 15643: 15636: 15622:Cuckoo clock 15559:astronomical 15533: 15359:Unit of time 15290:Key concepts 15240: 15206: 15145:Hyperpyramid 15110:Hypersurface 15027: 15003:Affine space 14993:Vector space 14932:Scholarpedia 14889:. Retrieved 14883: 14859: 14833: 14821: 14811: 14782: 14747: 14718:. Retrieved 14714: 14705: 14662: 14658: 14648: 14605: 14601: 14591: 14579:. Retrieved 14541: 14537: 14527:Tegmark, Max 14483: 14479: 14475: 14469: 14461: 14456: 14448: 14426: 14422: 14409: 14397:. Retrieved 14392: 14382: 14370:. Retrieved 14363:the original 14340: 14336: 14320: 14269: 14265: 14259: 14247:. Retrieved 14240:the original 14225: 14199:maths.tcd.ie 14194: 14186: 14142: 14138: 14132: 14099: 14095: 14089: 14070: 14036: 14032: 14026: 14001: 13997: 13991: 13979:. Retrieved 13946: 13942: 13932: 13889: 13885: 13879: 13867:. Retrieved 13852: 13832: 13799: 13791: 13779: 13774: 13770: 13765: 13761: 13757: 13753: 13748: 13744: 13737: 13725:. Retrieved 13710: 13686:. Retrieved 13678:Hyperphysics 13677: 13667: 13655:. Retrieved 13650: 13640: 13619: 13607:. Retrieved 13592: 13585: 13542: 13538: 13528: 13508: 13479: 13473: 13454: 13419:. Retrieved 13404: 13397: 13376: 13333:. Retrieved 13318: 13311: 13299:. Retrieved 13284: 13277: 13265:. Retrieved 13256: 13214: 13208: 13196:. Retrieved 13187: 13162:. Retrieved 13153: 13143: 13124: 13118: 13109: 13086: 13079: 13060: 13008: 12998: 12955: 12951: 12945: 12932:. Retrieved 12926:(1): 39–43. 12923: 12919: 12894:. Retrieved 12885: 12847: 12843: 12837: 12792: 12784: 12772:. Retrieved 12745: 12741: 12731: 12719:. Retrieved 12715:the original 12710: 12679: 12673: 12661:. Retrieved 12618: 12614: 12604: 12580: 12549: 12543: 12532:, retrieved 12494: 12490: 12477: 12457: 12433:. Retrieved 12424: 12415: 12403:. Retrieved 12394: 12385: 12374:the original 12359: 12346: 12334:. Retrieved 12325: 12315: 12303:. Retrieved 12283: 12276: 12264:. Retrieved 12214: 12208: 12192: 12180:. Retrieved 12165: 12126: 12105:. Retrieved 12096: 12086: 12067: 12032: 12025: 12013:. Retrieved 11998: 11991: 11979:. Retrieved 11955: 11902: 11896: 11884:. Retrieved 11875: 11865: 11842: 11838: 11834: 11830: 11826: 11817: 11803: 11793: 11780: 11772: 11766: 11761: 11751: 11734: 11719:Möbius group 11705: 11696: 11695: 11687: 11686: 11678: 11677: 11669: 11668: 11660: 11659: 11651: 11650: 11643: 11642: 11633: 11632: 11628: 11621: 11620: 11617: 11610: 11609: 11606: 11602: 11594: 11590: 11585: 11555: 11542: 11528: 11520: 11516: 11515:, carrying; 11512: 11508: 11505:luminiferous 11504: 11500: 11471:Metric space 11400: 11393: 11386: 11375: 11368: 11353: 11344: 11339: 11319:Hermann Weyl 11314: 11263: 11236: 11231: 11227: 11217: 11210: 11205: 11201: 11191: 11183: 11179: 11155: 11151: 11080: 11075:non-singular 11032: 10969: 10888: 10793: 10772: 10751:group theory 10703:surface area 10678: 10661: 10645: 10640: 10636: 10633: 10627: 10609: 10604: 10590: 10575: 10567: 10559: 10548: 10543: 10539: 10532: 10516: 10505: 10495: 10488: 10487:there is no 10482: 10476: 10471: 10465: 10460: 10454: 10448:Lilliputians 10443: 10417: 10412: 10407: 10405: 10397: 10381: 10370: 10311: 10238: 10233: 10032: 9975: 9972: 9884: 9871: 9853: 9847: 9820: 9764: 9694: 9614: 9259: 9255: 9244: 9241: 9215: 9203: 9200:Emmy Noether 9197: 9184: 8935: 8934: 8930: 8928: 8879: 8877: 8788: 8783: 8776: 8774: 8771: 8616: 8613: 8455: 8451: 8447: 8443: 8389: 8385: 8381: 8374: 8309: 8307: 8302: 8298: 8290: 8288: 8283: 8279: 8197: 8192: 8188: 8186: 8166: 8161: 8058: 8053: 8047: 8007: 7998: 7992: 7990: 7985: 7984: 7973: 7938: 7915: 7834: 7749: 7745: 7743: 7740: 7728: 7722: 7711: 7628:is given by 7452: 7384:is given by 7381: 7377: 7370: 7363: 7361: 7298:is given by 7295: 7288: 7281: 7278: 7261: 7244: 7236: 7229: 7224: 7218: 7213: 7209: 7207: 7204: 7199: 7195: 7188: 7181: 7179: 7166: 7159: 7155: 7151: 7147: 7140: 7133: 7129: 7125: 7123: 7011: 7007: 7003: 6999: 6989: 6985: 6981: 6972: 6968: 6955: 6951: 6945: 6941: 6937: 6926: 6922: 6913: 6909: 6905: 6901: 6897: 6893: 6889: 6881: 6877: 6873: 6868: 6864: 6860: 6853: 6846: 6842: 6835: 6827: 6823: 6819: 6818:in terms of 6812: 6805: 6800: 6796: 6792: 6782: 6775: 6769: 6765: 6762: 6750: 6626: 6613: 6604: 6600: 6595: 6593: 6588: 6584: 6577: 6573: 6570: 6565: 6561: 6560:means Δ 6557: 6308: 6303: 6299: 6295: 6291: 6287: 6283: 6279: 6277: 6092: 5917: 5914: 5910: 5893: 5810: 5806: 5802: 5798: 5794: 5790: 5786: 5782: 5778: 5774: 5770: 5766: 5762: 5758: 5754: 5750: 5746: 5738: 5734: 5727: 5721: 5717: 5713: 5709: 5707: 5702: 5698: 5674: 5670: 5668: 5551: 5472: 5468: 5464: 5460: 5456: 5452: 5448: 5440: 5436: 5429: 5423: 5419: 5415: 5411: 5409: 5304: 5302: 5298: 5265: 5256: 5245:Please help 5240:verification 5237: 5215: 5209: 5205: 5201: 5197: 5190: 5183: 5176: 5172: 5168: 5161: 5157: 5150: 5144: 5049: 5045: 5041: 5033: 5029: 5023: 5019: 5013: 5009: 5003: 4999: 4990: 4986: 4982: 4976: 4972: 4968: 4964: 4955: 4951: 4947: 4943: 4939: 4934: 4925: 4921: 4917: 4913: 4909: 4905: 4896: 4892: 4885: 4880: 4876: 4869: 4865: 4858: 4852: 4848: 4844: 4840: 4836: 4833: 4792: 4768: 4766: 4677: 4670: 4663: 4654: 4650: 4646: 4639: 4635: 4631: 4624: 4620: 4613: 4604: 4600: 4594: 4590: 4583: 4577: 4573: 4568: 4561: 4557: 4555: 4551: 4545: 4544:Figure 3–1. 4387: 4380: 4373: 4369: 4365: 4354: 4347: 4340: 4333: 4325: 4321: 4317: 4313: 4302: 4295: 4288: 4281: 4274: 4267: 4263: 4259: 4255: 4250: 4237: 4228: 4211: 4205: 4201: 4197: 4193: 4188: 4184: 4180: 4177: 4174: 4166: 4154: 4150:twin paradox 4149: 4141:twin paradox 4138: 4134: 4131:Twin paradox 4125:observations 4124: 4120: 4118: 4113: 4109: 4105: 4101: 4097: 4093: 4089: 4085: 4081: 4077: 4073: 4069: 4065: 4057: 4049: 4045: 4043: 4038: 4034: 4030: 4026: 4022: 4018: 4014: 4012: 4005: 4001: 3997: 3993: 3989: 3985: 3981: 3977: 3973: 3961: 3953: 3946: 3945:-coordinate 3942: 3933: 3923: 3919: 3911: 3907: 3900: 3887: 3883: 3871: 3867: 3860: 3856: 3847: 3843: 3839: 3835: 3831: 3827: 3823: 3819: 3817: 3812: 3808: 3805: 3770: 3763: 3756: 3749: 3745: 3741: 3737: 3712: 3700: 3695: 3693: 3688: 3684: 3680: 3677: 3669:Twin paradox 3656: 3643: 3640: 3635: 3633: 3628: 3624: 3620: 3613: 3608: 3604: 3602: 3596: 3586: 3579: 3566: 3554: 3544: 3529: 3525: 3521:hyperboloids 3514: 3489: 3485: 3460: 3458: 3455: 3450: 3446: 3442: 3413: 3334: 3324: 3309: 3300: 3289:Please help 3284:verification 3281: 3259: 3221: 3179: 3176: 3171: 3162: 3158: 3147: 3143: 3136: 3128: 3119: 3116: 3109: 3103: 3094: 3090: 3088: 3078: 3071: 3064: 3059: 3055: 3051: 3047: 3043: 3041: 3036: 3032: 3028: 3024: 3020: 3016: 3012: 3010: 2985: 2982: 2972: 2959: 2947: 2942: 2938: 2934: 2930: 2926: 2923: 2918: 2914: 2910: 2906: 2902: 2898: 2894: 2889: 2885: 2883: 2826: 2822: 2818: 2814: 2810: 2806: 2802: 2798: 2794: 2790: 2786: 2783:relativistic 2782: 2780: 2774: 2770: 2763: 2759: 2752: 2748: 2744: 2732: 2726: 2695: 2686: 2675:Please help 2670:verification 2667: 2645: 2617: 2612: 2608: 2569: 2538: 2482: 2271: 2125: 2112: 2065: 1816: 1703: 1574: 1438: 1169: 1160:(because of 1150: 1143: 1137: 1133: 1129: 1127: 979: 977: 964: 949: 945: 943: 924: 915: 911: 907: 903: 899: 895: 891: 889: 883: 869: 864: 852: 832: 821: 805:four vectors 802: 793: 787: 709: 699: 688: 674: 668: 651: 637: 595: 581: 577: 575: 572: 564: 559: 556: 552: 538: 525: 499: 497: 492: 489: 485: 476: 472: 468: 464: 460: 456: 452: 448: 438: 419: 395: 388: 385:Fundamentals 381: 362: 346: 341: 337: 334:relativistic 306: 302: 296: 206:Curved space 95: 53: 17913:Outer space 17877:Mathematics 17605:Kerr–Newman 17576:Spherical: 17445:Other tests 17388:Singularity 17320:Formulation 17282:Fundamental 17136:Formulation 17117:Proper time 17078:Fundamental 16967:System time 16962:Metric time 16681:Solar Hijri 16607:Water clock 16590:Radio clock 16522:Time domain 16502:Proper time 16388:Leap second 16270:Chronometry 16185:Time travel 16163:System time 16070:Time domain 16055:Proper time 15879:use of time 15850:Father Time 15830:Immortality 15820:Ages of Man 15749:Endurantism 15706:Regnal year 15686:Big History 15615:water-based 15514:Solar Hijri 15424:Hexadecimal 15374:Measurement 15336:Chronometry 15322:Measurement 15230:Codimension 15209:-dimensions 15130:Hypersphere 15013:Free module 14581:16 December 13892:(18): 9–8. 12621:: 129–176. 12221:: S24–S35. 11511:, light, + 11403:black holes 11341:Max Tegmark 10711:integrating 8442:approaches 7757:blueshifted 7624:Hence, the 6852:must equal 6841:must equal 6566:differences 5370:are common. 4567:. Velocity 4225:Gravitation 3689:both slower 3625:world sheet 3222:proper time 3120:necessarily 2921:(event R). 2620:world lines 2483:In general 2113:differences 782:Figure 1-3. 631:aberration. 392:Definitions 186:Four-vector 17925:Categories 17757:Zel'dovich 17665:Scientists 17644:Alcubierre 17451:of Mercury 17449:precession 17378:Black hole 17261:Background 17253:relativity 17222:World line 17217:Light cone 17042:Background 17034:relativity 17024:Relativity 16982:Timekeeper 16935:Chronology 16919:Millennium 16805:Precession 16711:Julian day 16532:T-symmetry 16393:Solar time 16363:Civil time 15887:Chronemics 15862:Kalachakra 15774:Presentism 15759:Eternalism 15665:Chronology 15603:mechanical 15554:Main types 15472:Main types 15225:Hyperspace 15105:Hyperplane 14672:1906.05336 14615:2207.14317 14182:1703.05448 12850:: 85–121. 12093:"Manifold" 12015:14 October 11857:References 10413:appearance 10243:1 MeV = 10 7997:. It is a 7838:redshifted 6725:10 km 6622:10 km 5797:, we have 5259:March 2024 3607:along the 3303:March 2024 3000:(t<0). 2986:light cone 2969:light cone 2963:Light cone 2689:March 2024 972:See also: 941:are used. 931:hyperbolic 828:kinematics 817:four-force 710:local time 640:Arago spot 493:world line 441:dimensions 407:observer's 17931:Spacetime 17889:Astronomy 17727:Robertson 17712:Friedmann 17707:Eddington 17697:de Sitter 17531:Solutions 17409:detectors 17404:astronomy 17371:Phenomena 17306:Geodesics 17209:Spacetime 17152:Phenomena 16879:Fortnight 16726:Lunisolar 16716:Leap year 16650:Gregorian 16600:stopwatch 16575:Hourglass 16555:Astrarium 16472:Spacetime 16403:Time zone 16280:Metrology 16259:standards 16143:Leap year 16060:Spacetime 15934:Yesterday 15835:Dreamtime 15809:Mythology 15696:Deep time 15608:stopwatch 15583:hourglass 15564:astrarium 15494:Gregorian 15487:Lunisolar 15464:Calendars 15454:Time zone 15327:standards 15115:Hypercube 15093:Polytopes 15073:Minkowski 15068:Hausdorff 15063:Inductive 15028:Spacetime 14979:Dimension 14913:Spacetime 14697:211734117 14640:251196731 14508:119683293 14345:CiteSeerX 14312:250904081 14304:0264-9381 14124:120125096 13971:120021772 13784:geodesics 13672:Nave, R. 13031:cite book 12965:0808.2730 12828:1210.6929 12653:120211823 12435:13 August 12405:13 August 12336:13 August 12135:CRC Press 11456:Geography 11424:networks. 11364:electrons 11264:In 1920, 10897:geodesics 10854:signature 10763:algebraic 10735:geodesics 10156:− 10136:− 10122:′ 10090:− 9950:− 9909:− 9848:Lab Frame 9222:) yields 9210:) yields 9087:→ 9069:≡ 8979:→ 8953:≡ 8901:± 8815:− 8709:≈ 8681:β 8677:− 8637:γ 8513:γ 8483:γ 8422:γ 8410:γ 8407:β 8350:γ 8344:β 8341:− 8327:′ 8140:γ 8137:β 8134:− 8126:′ 8091:γ 8083:′ 7999:conserved 7924:γ 7893:β 7889:− 7870:γ 7812:β 7808:− 7784:γ 7666:β 7655:β 7652:− 7606:β 7586:γ 7577:β 7534:γ 7505:β 7462:β 7411:β 7407:− 7328:β 7099:γ 7096:β 7082:γ 7047:γ 7044:β 7030:γ 6665:− 6654:γ 6646:′ 6538:Δ 6532:− 6526:Δ 6520:γ 6506:Δ 6463:Δ 6457:− 6451:Δ 6445:γ 6431:Δ 6402:− 6393:γ 6336:− 6327:γ 6183:γ 6118:γ 6019:− 6008:γ 5959:− 5948:γ 5877:γ 5857:β 5853:− 5844:γ 5685:γ 5645:β 5641:− 5598:− 5583:γ 5560:γ 5508:− 4805:β 4709:− 4420:− 4104:in S, at 3627:onto the 3575:10 s 3563:10 s 3551:10 s 3386:± 3370:− 3192:− 3132:(A, B, C) 3067:causality 3056:lightlike 3052:spacelike 2849:θ 2843:⁡ 2829:given by 2609:lightlike 2586:± 2570:spacelike 2391:− 2342:− 2329:− 2316:− 2094:Δ 1966:≠ 1786:− 1770:− 1754:− 1680:Δ 1657:Δ 1626:Δ 1544:Δ 1538:− 1522:Δ 1516:− 1500:Δ 1494:− 1475:Δ 1453:Δ 1408:Δ 1402:− 1383:Δ 1361:Δ 1309:Δ 1292:Δ 1267:Δ 1232:Δ 1203:Δ 1180:Δ 1098:Δ 1074:Δ 1050:Δ 1026:Δ 991:Δ 822:In 1905, 807:, namely 443:. In the 415:Euclidean 409:state of 363:In 1908, 326:continuum 303:spacetime 54:Spacetime 17840:Category 17717:Lemaître 17682:Einstein 17672:Poincaré 17632:Others: 17616:Taub–NUT 17582:interior 17504:theories 17502:Advanced 17469:redshift 17284:concepts 17102:Rapidity 17080:concepts 16940:Duration 16914:Saeculum 16894:Olympiad 16736:Solstice 16665:Holocene 16642:Calendar 16542:Horology 16333:ISO 8601 16328:ISO 31-1 16203:Category 15951:Time in 15942:Tomorrow 15804:Religion 15744:Duration 15711:Timeline 15645:Timeline 15444:Sidereal 15312:Eternity 15242:Category 15218:See also 15018:Manifold 14857:(1992). 14831:(2004). 14814:. Dover. 14772:87028148 14746:(1986). 14576:15694111 14417:(1920). 14399:13 April 14372:13 April 14328:(2007). 14249:14 April 13975:Archived 13924:14295121 13863:Archived 13727:14 April 13682:Archived 13577:27877933 13421:24 April 13335:22 April 13301:22 April 13267:10 April 13261:Archived 13198:26 March 13192:Archived 13164:26 March 13158:Archived 12990:12020510 12928:Archived 12890:Archived 12864:27757388 12768:Archived 12657:Archived 12525:archived 12497:: 1–22, 12429:Archived 12399:Archived 12330:Archived 12299:Archived 12266:11 April 12257:Archived 12182:14 April 12107:24 March 12101:Archived 11975:Archived 11886:24 March 11880:Archived 11694:, where 11429:See also 11331:orbitals 11198:temporal 10782:Manifold 10759:analysis 10743:Einstein 10691:smoothly 10628:a priori 10610:In 1962 10401:geodesic 10239:Example: 10080:′ 9940:′ 9899:′ 9852:(right) 9846:(above) 8784:fruitful 8617:γmc 7952:Momentum 7880:′ 7861:′ 7794:′ 7780:′ 7593:′ 7541:′ 7210:Example: 7106:′ 7089:′ 7054:′ 7037:′ 6596:Example: 6513:′ 6438:′ 6386:′ 6320:′ 6258:′ 6236:′ 6209:′ 6195:′ 6147:′ 6130:′ 6063:′ 6041:′ 5997:′ 5937:′ 5799:γβk 5779:γβk 5745:, where 5103:′ 5081:′ 4742:′ 4699:′ 4546:Galilean 4515:′ 4483:′ 4451:′ 4410:′ 4092:; also, 3154:C, B, A. 2539:timelike 2162:and the 2050:′ 2013:′ 1933:′ 1896:′ 1865:′ 1854:′ 1843:′ 1832:′ 981:distance 859:Max Born 706:dynamics 582:observes 578:measures 560:observer 532:and the 501:manifold 426:velocity 257:Category 17865:Physics 17851:Portals 17782:Hawking 17777:Penrose 17762:Novikov 17742:Wheeler 17687:Hilbert 17677:Lorentz 17634:pp-wave 17455:lensing 17251:General 17032:Special 16909:Century 16899:Lustrum 16829:Instant 16701:Equinox 16670:Islamic 16612:Sundial 16477:Chronon 16213:Commons 16136:Related 16050:Instant 16040:Chronon 16022:Physics 15962:Geology 15953:science 15825:Destiny 15670:History 15638:History 15593:sundial 15576:quantum 15519:Chinese 15509:Islamic 15419:Decimal 15414:Chinese 15376:systems 15302:Present 15140:Simplex 15078:Fractal 14891:16 July 14820:(1973) 14810:(1952) 14720:16 June 14677:Bibcode 14620:Bibcode 14556:Bibcode 14488:Bibcode 14451:20:200. 14431:Bibcode 14284:Bibcode 14147:Bibcode 14104:Bibcode 14041:Bibcode 14006:Bibcode 13981:9 April 13951:Bibcode 13904:Bibcode 13869:9 April 13568:5099802 13547:Bibcode 12970:Bibcode 12934:6 April 12896:17 July 12774:7 April 12750:Bibcode 12721:15 July 12663:15 July 12623:Bibcode 12534:17 July 12499:Bibcode 12305:7 April 12253:8146369 12233:Bibcode 11481:Present 11411:(3 + 1) 11360:protons 11335:nucleus 11323:Maxwell 11208:. 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So 4385:when 4272:(see 4143:is a 3972:, as 3968:and W 3732:and W 3724:and W 3707:and W 3651:by a 3490:along 3414:with 3146:= 0.3 3091:after 3044:space 2272:Note: 933:, or 461:event 350:space 338:where 17951:Time 17752:Kerr 17702:Weyl 17601:Kerr 17461:and 17415:and 17413:LIGO 16889:Year 16874:Week 16864:Hour 16318:DUT1 16257:and 15877:and 15840:Kāla 15535:List 15529:Maya 15325:and 15297:Past 15283:Time 15186:Five 15181:Four 15161:Zero 15095:and 14893:2022 14865:ISBN 14839:ISBN 14787:ISBN 14768:LCCN 14758:ISBN 14722:2021 14583:2006 14401:2018 14374:2018 14337:Noûs 14300:ISSN 14251:2017 14230:ISBN 14100:A269 14075:ISBN 13983:2018 13871:2018 13838:ISBN 13806:ISBN 13773:and 13747:and 13729:2017 13716:ISBN 13690:2017 13659:2024 13611:2017 13598:ISBN 13573:PMID 13514:ISBN 13484:ISBN 13459:ISBN 13423:2017 13410:ISBN 13383:ISBN 13337:2017 13324:ISBN 13303:2017 13290:ISBN 13269:2017 13219:ISBN 13200:2017 13166:2017 13129:ISBN 13092:ISBN 13065:ISBN 13037:link 13017:ISBN 12936:2018 12898:2017 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8526:m 8522:= 8519:c 8516:m 8500:. 8486:m 8480:= 8471:m 8452:m 8428:v 8425:m 8419:= 8416:c 8413:m 8390:c 8388:/ 8386:v 8382:β 8353:E 8347:) 8338:1 8335:( 8332:= 8323:E 8310:E 8303:c 8301:/ 8299:E 8284:c 8282:/ 8280:E 8264:p 8261:= 8258:c 8254:/ 8250:E 8226:c 8223:= 8220:p 8216:/ 8212:E 8189:c 8162:c 8146:c 8143:m 8131:= 8122:p 8097:c 8094:m 8088:= 8079:) 8075:c 8072:m 8069:( 8048:p 8031:) 8028:t 8025:, 8022:x 8019:( 7993:v 7991:m 7986:p 7897:2 7886:1 7877:f 7873:= 7866:/ 7858:f 7854:= 7851:f 7816:2 7805:1 7799:/ 7791:f 7787:= 7777:f 7773:= 7770:f 7753:′ 7750:f 7746:r 7684:. 7679:0 7675:f 7663:+ 7660:1 7649:1 7642:= 7639:f 7612:. 7609:) 7603:+ 7600:1 7597:( 7590:w 7583:= 7580:w 7574:+ 7571:w 7568:= 7565:T 7545:. 7538:w 7531:= 7528:w 7508:w 7485:w 7465:, 7428:0 7424:f 7420:) 7415:r 7404:1 7401:( 7398:= 7395:f 7382:f 7373:r 7371:β 7366:r 7364:v 7345:0 7341:f 7332:s 7324:+ 7321:1 7317:1 7312:= 7309:f 7296:f 7291:s 7289:β 7284:s 7282:v 7237:β 7230:β 7225:c 7219:β 7214:c 7200:t 7196:x 7192:′ 7189:t 7185:′ 7182:x 7170:′ 7167:x 7163:′ 7160:t 7156:β 7152:c 7150:/ 7148:v 7144:′ 7141:w 7137:′ 7130:w 7103:w 7093:+ 7086:x 7079:= 7076:x 7051:x 7041:+ 7034:w 7027:= 7024:w 7012:b 7008:r 7004:a 7000:s 6992:′ 6982:b 6975:′ 6969:r 6963:. 6960:β 6958:′ 6952:s 6946:β 6942:a 6940:/ 6938:s 6929:′ 6923:a 6917:. 6914:β 6910:c 6908:/ 6906:v 6902:r 6900:/ 6898:b 6894:a 6892:/ 6890:s 6882:s 6878:r 6874:x 6869:b 6865:a 6861:w 6854:z 6850:′ 6847:z 6843:y 6836:y 6824:x 6820:w 6816:′ 6813:x 6809:′ 6806:w 6801:c 6799:/ 6797:v 6793:β 6788:) 6786:′ 6783:x 6779:′ 6776:w 6774:( 6770:x 6766:w 6764:( 6723:× 6704:s 6694:= 6690:) 6684:2 6680:c 6675:/ 6671:x 6668:v 6662:t 6658:( 6651:= 6642:t 6627:t 6620:× 6614:c 6608:′ 6605:t 6601:t 6589:v 6585:v 6581:′ 6578:S 6574:S 6562:x 6558:x 6544:) 6541:t 6535:v 6529:x 6523:( 6517:= 6510:x 6487:, 6484:) 6479:2 6475:c 6470:/ 6466:x 6460:v 6454:t 6448:( 6442:= 6435:t 6411:) 6408:t 6405:v 6399:x 6396:( 6390:= 6383:x 6363:, 6360:) 6355:2 6351:c 6346:/ 6342:x 6339:v 6333:t 6330:( 6324:= 6317:t 6304:c 6302:/ 6296:c 6294:/ 6292:v 6288:x 6284:c 6280:v 6255:z 6251:= 6244:z 6233:y 6229:= 6222:y 6214:) 6206:t 6202:v 6199:+ 6192:x 6187:( 6180:= 6173:x 6165:) 6157:2 6153:c 6144:x 6140:v 6134:+ 6127:t 6122:( 6115:= 6108:t 6074:z 6071:= 6060:z 6052:y 6049:= 6038:y 6029:) 6025:t 6022:v 6016:x 6012:( 6005:= 5994:x 5985:) 5977:2 5973:c 5968:x 5965:v 5956:t 5952:( 5945:= 5934:t 5874:1 5869:= 5866:) 5861:2 5850:1 5847:( 5841:= 5838:K 5835:O 5831:/ 5827:B 5824:O 5811:β 5795:β 5787:x 5783:w 5771:w 5767:x 5763:v 5761:/ 5759:c 5755:β 5747:k 5739:k 5735:w 5728:x 5720:/ 5703:v 5699:c 5675:c 5671:v 5649:2 5638:1 5634:1 5629:= 5621:2 5617:c 5612:/ 5606:2 5602:v 5595:1 5591:1 5586:= 5538:. 5531:2 5527:c 5522:/ 5516:2 5512:v 5505:1 5499:/ 5495:K 5492:O 5489:= 5486:B 5483:O 5473:v 5471:/ 5465:β 5463:/ 5461:x 5457:w 5449:k 5441:k 5437:x 5430:w 5422:/ 5384:= 5381:c 5358:t 5355:c 5352:= 5349:w 5329:t 5326:c 5323:= 5319:T 5272:) 5266:( 5261:) 5257:( 5243:. 5210:v 5206:c 5202:u 5198:c 5194:′ 5191:u 5184:v 5180:′ 5177:u 5173:u 5169:c 5167:/ 5165:′ 5158:v 5154:′ 5151:u 5128:. 5122:) 5117:2 5113:c 5108:/ 5100:u 5096:v 5093:( 5090:+ 5087:1 5078:u 5074:+ 5071:v 5065:= 5062:u 5050:u 5046:r 5042:b 5037:. 5034:c 5032:/ 5030:a 5027:′ 5024:u 5020:r 5014:c 5012:/ 5010:s 5007:′ 5004:u 5000:b 4994:′ 4991:β 4987:m 4985:/ 4983:n 4977:s 4975:/ 4973:b 4969:a 4967:/ 4965:r 4959:. 4956:β 4952:c 4950:/ 4948:v 4944:a 4942:/ 4940:s 4935:α 4930:. 4928:) 4926:b 4922:a 4918:r 4914:s 4912:( 4910:c 4906:u 4897:u 4893:x 4889:′ 4886:x 4881:m 4879:/ 4877:n 4873:′ 4870:β 4866:c 4864:/ 4862:′ 4859:u 4853:a 4851:/ 4849:s 4845:β 4841:c 4839:/ 4837:v 4819:c 4815:/ 4811:v 4808:= 4752:, 4749:v 4746:+ 4739:u 4735:= 4732:u 4712:v 4706:u 4703:= 4696:u 4681:′ 4678:t 4676:/ 4674:′ 4671:x 4667:′ 4664:u 4658:′ 4655:t 4653:) 4651:v 4647:u 4645:( 4640:t 4638:) 4636:v 4632:u 4630:( 4617:′ 4614:x 4608:′ 4605:t 4601:t 4591:x 4587:′ 4584:x 4574:x 4569:u 4565:′ 4562:u 4558:v 4525:. 4522:t 4519:= 4512:t 4490:z 4487:= 4480:z 4458:y 4455:= 4448:y 4426:t 4423:v 4417:x 4414:= 4407:x 4391:′ 4388:t 4381:t 4376:) 4374:z 4370:y 4366:x 4364:( 4360:) 4358:′ 4355:t 4351:′ 4348:z 4344:′ 4341:y 4337:′ 4334:x 4332:( 4328:) 4326:t 4322:z 4318:y 4314:x 4312:( 4308:) 4306:′ 4303:t 4299:′ 4296:z 4292:′ 4289:y 4285:′ 4282:x 4280:( 4270:) 4268:t 4264:z 4260:y 4256:x 4254:( 4194:x 4106:x 4102:A 4098:D 4094:B 4078:C 4074:D 4066:C 4062:2 4054:1 4046:v 4025:( 4023:C 4019:x 4015:D 3994:C 3990:A 3986:A 3970:2 3966:1 3962:B 3958:2 3954:A 3949:t 3947:A 3936:. 3934:C 3930:2 3926:′ 3924:x 3920:C 3916:2 3912:A 3903:x 3901:A 3897:2 3880:1 3876:1 3864:′ 3861:O 3857:O 3848:x 3844:x 3824:x 3820:x 3813:v 3809:v 3764:v 3742:d 3738:d 3734:2 3730:1 3726:2 3722:1 3713:d 3709:2 3705:1 3685:v 3681:v 3657:x 3636:x 3629:x 3621:x 3617:′ 3614:t 3609:x 3605:c 3573:× 3567:c 3561:× 3555:c 3549:× 3500:c 3472:c 3461:x 3445:– 3443:x 3427:2 3423:s 3399:, 3394:2 3390:s 3383:= 3378:2 3374:x 3365:2 3361:) 3357:t 3354:c 3351:( 3316:) 3310:( 3305:) 3301:( 3287:. 3246:. 3239:2 3235:s 3207:. 3200:2 3196:s 3163:c 3159:v 3150:, 3148:c 3144:v 3137:v 3110:v 3037:x 2943:θ 2939:x 2935:x 2927:x 2919:a 2911:x 2907:a 2903:a 2895:x 2890:x 2886:x 2869:. 2866:c 2862:/ 2858:v 2855:= 2852:) 2846:( 2827:θ 2811:v 2807:x 2803:x 2799:x 2791:t 2787:t 2775:t 2771:t 2764:v 2760:x 2753:z 2749:y 2745:x 2702:) 2696:( 2691:) 2687:( 2673:. 2630:c 2595:. 2592:t 2589:c 2583:= 2580:x 2554:2 2550:s 2523:2 2519:s 2496:2 2492:s 2447:2 2443:z 2439:+ 2434:2 2430:y 2426:+ 2421:2 2417:x 2413:+ 2408:2 2404:) 2400:t 2397:c 2394:( 2388:= 2383:2 2379:s 2350:2 2346:z 2337:2 2333:y 2324:2 2320:x 2311:2 2307:) 2303:t 2300:c 2297:( 2294:= 2289:2 2285:s 2253:2 2249:s 2227:s 2207:, 2202:2 2198:s 2175:2 2171:x 2148:2 2144:) 2140:t 2137:c 2134:( 2098:x 2074:x 2047:s 2043:= 2040:s 2017:2 2009:s 2005:d 2002:= 1997:2 1993:s 1989:d 1969:0 1963:s 1960:d 1940:0 1937:= 1930:s 1926:d 1923:= 1920:s 1917:d 1893:s 1889:d 1869:) 1862:z 1858:, 1851:y 1847:, 1840:x 1836:, 1829:t 1825:( 1802:. 1797:2 1793:z 1789:d 1781:2 1777:y 1773:d 1765:2 1761:x 1757:d 1749:2 1745:t 1741:d 1736:2 1732:c 1728:= 1723:2 1719:s 1715:d 1689:0 1686:= 1683:s 1660:t 1634:2 1630:s 1606:t 1586:, 1583:c 1560:. 1555:2 1551:) 1547:z 1541:( 1533:2 1529:) 1525:y 1519:( 1511:2 1507:) 1503:x 1497:( 1489:2 1485:) 1481:t 1478:c 1472:( 1469:= 1464:2 1460:) 1456:s 1450:( 1424:, 1419:2 1415:) 1411:x 1405:( 1397:2 1393:) 1389:t 1386:c 1380:( 1377:= 1372:2 1368:) 1364:s 1358:( 1335:t 1332:c 1312:t 1306:c 1303:= 1299:t 1296:c 1271:x 1245:2 1241:) 1236:s 1229:( 1209:. 1206:x 1183:t 1138:z 1134:y 1130:x 1111:2 1107:) 1102:z 1095:( 1092:+ 1087:2 1083:) 1078:y 1071:( 1068:+ 1063:2 1059:) 1054:x 1047:( 1044:= 1039:2 1035:) 1030:d 1023:( 995:d 918:( 902:( 894:( 512:c 477:t 473:z 469:y 465:x 457:z 453:y 449:x 286:e 279:t 272:v 42:. 20:)

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