Special relativity

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therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations. The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905. Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong. Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.

4512: 15129: 25671: 36082: 425: 5366:{\displaystyle {\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}} 20789: 25277: 15208: 23048: 21513: 1212:

to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelson–Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist. Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be

13521: 6781:, an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass-energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909) which he used to perform a novel derivation of the Lorentz transformation. 25666:{\displaystyle {\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}.} 11613: 854:"Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results ... How, then, could such a universal principle be found?" 988: 16776: 13190: 19938: 13532: 11622: 2924: 2915: 2904: 2895: 12445: 36130: 26245: 6741: 7177: 13313:

shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig. 5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen. This illusion has come to be known as

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journey, people on Earth will experience more time than they do - since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years. A full 40-year trip at 1

16822: 20863: 13238: 13272: 64: 36142: 31: 34876: 34016: 438: 23598: 16869: 14318: 13301: 23747: 6771: 25984: 36094: 36118: 11597:, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect). 36106: 9919: 7169:, articulated in its present form by Langevin in 1911. Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older. 13947: 11061: 10765: 19950: 18800: 22408:. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10) and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors. 24057:{\displaystyle {\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}.} 17216: 15105:. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. ( 2942:) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario. 18153: 21317: 18317: 26240:{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }} 27032: 33555: 611:). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible. Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth. 16525: 25797: 9637: 28307:

Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited

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The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the

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Fig. 7-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation

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Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating,

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The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and

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Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean

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For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a

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The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or

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in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are

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equations. These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything

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The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events ... The universal principle of the

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against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance

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Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in vacuum and the independence of physical laws (especially

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In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity,

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Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905, and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite

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Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins

16322: 14313:{\displaystyle {\begin{aligned}E_{0}&=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L\\H_{0}&=H_{1}+{\tfrac {1}{2}}L{\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\frac {1+(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}=H_{1}+{\frac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}} 13469:

The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these do not talk about matter or radiation, they only talk about space and time. The derivation

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Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip,

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This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in

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play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference

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Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes "The word 'special' is meant to intimate that the principle is restricted to the case ...". See p. 111 of The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, Dover

13369:, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself. 12473:

tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope. A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag, but was incompatible with

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Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation

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as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the

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Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at

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It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is

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Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident

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particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.

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Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless

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The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in

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Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation and induction of double refraction, and the induction of torques on charged

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Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics", 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the

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simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the

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after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth. During their

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Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in

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In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form

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faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light. For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is

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in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal

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Fig. 7-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity

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The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is

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Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls will subtend a right angle. (a) In the

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Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems which exist within the limit of 100 light years from Earth.

1026:. Since the speed of light is constant in relativity irrespective of the reference frame, pulses of light can be used to unambiguously measure distances and refer back to the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. 923:

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations. But the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the

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Consider the elastic collision scenario in Fig. 6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and

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A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two

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Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations.

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In Fig. 7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same

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There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's

1831: 748:, which introduced a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case". Some of the work of Albert Einstein in special relativity is built on the earlier work by 26813: 1228:

that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in a vacuum.

905:. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment. In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance. 13268:. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye. 12481:

Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include

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The interval AC in the diagram is 'space-like'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. But no frames are accessible by a Lorentz

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be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different

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with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which

13493:, Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time. 11658:

Fig. 4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the

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special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws ...

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interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light.

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elongated, and a passing object would have a skew appearance that has been likened to a rotation. A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although the surface of the sphere and the images on it will appear distorted.

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The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810,

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Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.

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In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being

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Fig. 4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the

13398:. Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies. 20800:

When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.

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Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore

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spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

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independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would

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Since there is no absolute reference frame in relativity theory, a concept of "moving" does not strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be

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their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.

27823:, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components. 13208:

Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments. In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of

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For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame

28419:(W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length." 32992: 26889: 15770: 14902:

to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that

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On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called

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distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of

29678:, footnote page 10–11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morley experiment had no role in the foundation of the theory." and "... the theory of relativity was not founded to explain its outcome at all". 17937: 27034:

similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one does not need to perform Lorentz transformations to determine the invariants.

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in the past. For this reason the null dual cone is also known as the "light cone". (The point in the lower left of the Fig. 10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

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so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally. Now the transformation of the contravariant components of the position 4-vector can be compactly written as:

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streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5-4 has been stretched out.

27662: 24083:, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other 18001: 12453:

of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the

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The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.

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While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a

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Special and General Theory, 1916; The Principal Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949.

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Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver.

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The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;

12457:. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light. (2) If the source is in motion, the displacement would be the consequence of 6892:

Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same

15872: 17812: 16345: 15303: 7122:

In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of

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In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig,

27815:

The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.

26405: 26701: 20466: 13434:: it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes 12112:, that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies. 19013:, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 17615: 7184:

Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the

27760: 22157:

at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she

14739: 22730: 21126: 19882: 13731: 8442: 7873: 15584: 6491: 2533: 6392:, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the 16161: 6334:

in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special

6400:

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of

28380:

in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions,

22468:. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples: 12580: 2359: 19778: 15977: 9914:{\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.} 24270: 22323:

in his honour, that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called

20911:(acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity 12076: 6128:

changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that

20631: 17515: 10262: 603:, which adjust distances and times for moving objects. Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as 14842: 27125: 7147:

reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with

1007:. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another. 882:– the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other. 27921:. In general, though, the components of the four-force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, 27467: 22422:

Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See

20734:

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving

2440: 27666:

We can work out what this invariant is by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

27116: 25884: 25255:-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of 2879: 24525: 24350: 23487: 12209: 2271: 22885: 12424: 26344:, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order 5859: 19216: 12676: 6736:

created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.

2187: 1369: 15951: 15470:

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc) equals the sum of the total energies of the emergent particles:

9492: 27917:" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times 20970:

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 7-5, the acceleration is finished, the ships will remain at a constant offset in some frame

20131: 15321:

Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles:

11638:

Consider the spacetime diagrams in Fig. 4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The

2705: 1663: 26882: 21322:

The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame

20695: 20376: 17028: 9222:

between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are

5565: 4505: 836:. Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the 27909: 17298: 23593: 15022: 13952: 1207:", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an 27473:. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by 12939: 7350: 4244: 23265: 22229:

are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of

9135: 7173:

order for the two observers to compare their proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.

1596: 23154: 21789: 13167: 19970:

An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the

13279:

For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be

10068: 6756:

at a moving clock would, in general, not at all be the same thing, because the time that would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the

24569: 22049: 11056:{\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .} 2786: 26582: 23692: 18921: 18476: 14632:

represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence,

7276: 28415:

Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in

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In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving

10760:{\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .} 8805: 8513: 7945: 4402:

of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. The invariance of this interval is a property of the

3973: 32896:(1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format). 21131: 18567: 11604:

For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams). A variety of causal paradoxes could then be constructed.

11601:

transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result.

8219: 7650: 4517: 2617: 1668: 1374: 23350: 20231:

The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum

7107: 1254:

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:

21844: 11537:

Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the

8273: 7704: 1224:

From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity)

27555: 26327: 26286: 7585: 6660: 29074: 24724: 25966: 25925: 24415: 21505:, the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just 15668: 14911:

mass with energy required an act of aesthetic faith, very characteristic of Einstein." Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.

12461:. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver. 8949: 7472: 20547: 20315: 18374:, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such 14930:

However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant

21905: 17849: 13275:

Figure 5–5. Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross.

19925: 17669: 17309: 12978: 5985:). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, 3397: 794:

is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak

17721: 17211:{\displaystyle {\begin{pmatrix}x'\\y'\\\end{pmatrix}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}.} 8724: 8157: 5681:). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, 1037:

In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called

32884:

is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.

27764:

We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.

11593:

In Fig. 4-6, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; that is, there is a frame of reference in which events A and B occur at the

9392: 5780: 5500: 3875: 3789: 13422:

in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is

8847: 8555: 5983: 27811: 21703: 18795:{\displaystyle {\begin{aligned}A_{0}'&=\gamma \left(A_{0}-(v/c)A_{1}\right)\\A_{1}'&=\gamma \left(A_{1}-(v/c)A_{0}\right)\\A_{2}'&=A_{2}\\A_{3}'&=A_{3}\end{aligned}}} 11363:

applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See

6961: 6098: 3720: 3608: 1271:

Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler and by Callahan. This is also the approach followed by the Knowledge articles

19601: 16610: 11204: 6282: 6191: 27027:{\displaystyle T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},} 26566: 15330: 11718:

axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality.

6023: 21450: 20942: 20871:

A spacetime diagram (Fig. 7-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude

549:

is the same for all observers, regardless of the motion of light source or observer. This is known as the principle of light constancy, or the principle of light speed invariance.

22464:

accelerate and measure the properties of particles moving at near the speed of light, where their behavior is consistent with relativity theory and inconsistent with the earlier

18148:{\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}},} 11165: 9184: 7527: 5679: 779:) is negligible. To correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate 24912: 22439:(1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities. 9315: 8344: 7987: 6858: 3653: 3541: 32769:-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics. 27670: 19502: 18312:{\displaystyle {\begin{pmatrix}ct\\x\end{pmatrix}}={\begin{pmatrix}\cosh \phi &\sinh \phi \\\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct'\\x'\end{pmatrix}}.} 16626: 14331: 8993: 7418: 28281:

For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation

28218: 22617: 20229: 19449: 12685: 10285: 8886: 5933: 5639: 19108: 15065:

will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1

9016:

The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the

21397: 20182: 12277: 6126: 893:(a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source. 28233:

of a moving rigid object is the spatial distance between the ends of the object measured at the same time. In the rest frame of the object the simultaneity is not required.

21070: 17968: 17843: 16567: 9288: 8307: 6701: 4396: 21635: 11135: 11113: 10106: 7775: 5436: 5403: 4238: 4022: 22164:

After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to

22150: 19399: 19331: 14630: 14590: 11944: 11907:

the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If

5713: 3434: 3172: 28454: 28439: 22083: 19073: 19044: 18537: 16520:{\displaystyle \quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}} 13117: 8559: 7991: 6417: 4106: 4084: 3491: 3147: 24560: 22985:(1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take 13346:

of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel. In Fig. 5-6, galaxy

11532: 4062: 24452: 23616:

and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance

21950: 20997: 20004: 15793: 11991: 11793: 11685: 9564: 9536: 4200: 4177: 3903: 3748: 3321: 3259: 3231: 700: 24131: 22103: 21588: 21475: 21118: 20909: 18987: 17735: 15227: 13227: 11996: 11818: 11745: 9627: 7738: 7192:

Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences.

6220: 3834: 3346: 3203: 27948:

to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/

21545: 20500: 20268: 19658: 19635: 17239: 15646: 10116: 6312: 3293: 874:

the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:

34579: 31630: 21024: 18329:

The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.

13939: 13905: 13844: 13817: 13758: 13589: 12966: 9984: 3685: 3573: 24374: 20380: 3125: 33622: 21312:{\displaystyle {\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}=\gamma L\end{aligned}}} 17519: 14510: 13786: 12838: 3460: 28755:

Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern)

27387: 21568: 21364: 21093: 20965: 11716: 6890: 3996: 3075: 3052: 21417: 21341: 20889: 15626: 14640: 14550: 14530: 13874: 13620: 13569: 12858: 12102: 11964: 11765: 11481: 11458: 11434: 9955: 9602: 7141: 6911: 6810: 6569: 6332: 6147: 6051: 5899: 5879: 5605: 5585: 3809: 3095: 3029: 3009: 2989: 2969: 1263:

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of

720: 19782: 8348: 7779: 25792:{\displaystyle (\partial _{\mu '}\phi )=\Lambda _{\mu '}{}^{\nu }(\partial _{\nu }\phi )\qquad \partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}.} 16317:{\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}} 15479: 13193:

Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.

35163: 33926: 6593:

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames.

1225: 11820:

axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only

3097:

so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the

33951: 25167:{\displaystyle P^{\mu }=mU^{\mu }=m\gamma (v)(c,v_{x},v_{y},v_{z})=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)=\left({\frac {E}{c}},\mathbf {p} \right).} 23367: 469: 28402:

in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to

22770: 16133:{\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}} 13628: 22185: 646:. It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and 28308:

by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the

28220:

with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..."

18355:, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are 33537: 32983:

from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.

29007: 26531:{\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}} 22233:

fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.

19673: 19345:

are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike

15044:

is the coordinate time as measured by people on Earth. Therefore, after one year of accelerating at 9.81 m/s, the spaceship will be travelling at

27830:, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the 26808:{\displaystyle T^{\alpha }S_{\alpha }=T^{\alpha }\eta _{\alpha \beta }S^{\beta }=T_{\alpha }\eta ^{\alpha \beta }S_{\beta }={\text{invariant scalar}}} 32573:

Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region".

26828: 22181: 20551: 17437: 11767:

axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the

4436: 737:. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into 28406:'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å). 27841: 4362:

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

2444: 31801: 29782: 27046: 25803:

which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.

23492: 14940: 14765: 12488: 2275: 32811: 31432: 25817: 12871: 24461: 24286: 22457:(1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle. 20838:

be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing.

6760:. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out. 2945:

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1.

23166: 4433:

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:

1538: 575:

Special relativity was described by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".

33562: 33250: 23058: 19401:

while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval

6813:

interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds

4509:

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:

28749: 18386:

one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.

18359:

relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of

6869:

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of

35330: 33020: 32296: 28550:

The Electromagnetic Mass and Momentum of a Spinning Electron, G. Breit, Proceedings of the National Academy of Sciences, Vol. 12, p.451, 1926

28032: 20785:) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues. 4414:

of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is,

33879: 12084:, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since 1970: 810:

to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as

34910: 32036: 30941:

Ferraro, Rafael; Sforza, Daniel M. (2005). "European Physical Society logo Arago (1810): the first experimental result against the ether".

25192: 13418:

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a

17: 35377: 31609: 27480: 13245:

Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of

12363: 11722:

It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the

1057:. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two 34626: 33339: 22168:

Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an

889:

which is independent of the state of motion of the emitting body" (from the preface). That is, light in vacuum propagates with the speed

818:, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this 31224: 27952:

times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.

23605:

As illustrated in Fig. 10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii =

23270: 22194:

led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the

19112: 2093: 35294: 35196: 34861: 33956: 33482: 28521: 15895: 1249: 1015:

frame has the ability to determine measurements of the time of events using a "clock" (any reference device with uniform periodicity).

32775: 30368: 20022: 18347:, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, 2202: 35356: 35325: 35156: 34650: 34396: 33512: 33372: 27316:{\displaystyle d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(c\,dt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}} 22472: 22213:

of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the

20635: 20319: 16967: 3498:. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that 2369: 1022:

is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in

35201: 17255: 9993: 5785: 35793: 34001: 31463: 31261: 28158: 26372: 24963: 24119: 22239:

in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a

21497:

Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as

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acceleration could reach 2,000,000 light-years to the Andromeda Galaxy. This same time dilation is why a muon travelling close to

11647:), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards ( 2790: 35262: 33885: 32799: 26391:

of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the

22925:

are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a

12127: 11993:

is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then

6551:

can be considered a manifestation of the relativity of simultaneity. Since relativity of simultaneity is a first order effect in

2621: 898: 13458:. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to 5505: 63: 34432: 34386: 33740: 33218: 31405: 28503: 22929:

of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1). Just as Euclidean space uses a

21708: 13124: 9335: 908:

The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (

35942: 28769:

Lin, Shih-Chun; Giallorenzi, Thomas G. (1979). "Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer".

22411:

Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably

17973:

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called

13177:

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the

11834:

is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel

3436:

The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that

1003:, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be − 35299: 35257: 35232: 34570: 33961: 32830: 32501: 32482: 32435: 32403: 32377: 32271: 32170: 32143: 32118: 31952: 31925: 31894: 31743: 31715: 31566: 31541: 31507: 31071: 30925: 30898: 30864: 30794: 30764: 30571: 30505: 30478: 30447: 30416: 30272: 30156: 30060: 29766: 29599: 29524: 29494: 29444: 29392: 29296: 28609: 27770: 22502: 22484: 22375: 21954: 21593:

The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:

13249:, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer. 11899:

According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed

9218:

Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring

21516:

Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed

20859:

correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.

18867: 18399: 18363:

in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the

35372: 35149: 34054: 30085: 29351: 23619: 22278: 19220: 15128: 12213: 9094: 1528:{\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}} 24920: 22763:

are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate

8738: 8446: 7878: 6966: 35398: 32390:

Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press,

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In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in

13622:

is given by the relativistic Doppler shift formula which Einstein had developed in his 1905 paper on special relativity:

11687:

axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives

8171: 7602: 3908: 1826:{\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}} 455: 32727: 31691:

Halbert, M.L. (1972). "Review of Experiments on Nucleon-Nucleon Bremsstrahlung". In Austin, S.M.; Crawley, G.M. (eds.).

30815: 28191:

Wald, General Relativity, p. 60: "... the special theory of relativity asserts that spacetime is the manifold

20010:

cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.

15141:

strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±

1660:

Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:

35977: 35656: 34307: 33946: 33769: 33611: 33452: 32952: 32933: 32454: 30915: 30005: 26817:

Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no

24095:

The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component

22443:

source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.

22289:", which require a unification including general relativity too, are active and ongoing areas in theoretical research. 21795: 19943:

Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.

15765:{\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }} 7282: 7040: 1876:

Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates

885:

The principle of invariant light speed – "... light is always propagated in empty space with a definite velocity

83: 32989:

An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.

16958:

is likewise interpreted as twice the tinted area. Fig. 7-2 presents plots of the sinh, cosh, and tanh functions.

13308:

sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.

7208: 35436: 35267: 35247: 34678: 34312: 33630: 33507: 33013: 32559: 32544: 32522: 32395: 32385: 32353: 32312: 32248: 32225: 32194: 31388: 31176: 30702: 30540: 30302: 30032: 29979: 29955: 29880: 29829: 29805: 29675: 29563: 29361: 29331: 28650: 28633: 28269: 28051: 22454: 20752: 17932:{\displaystyle \gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}} 780: 22203: 19967:

is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.

17420:{\displaystyle \tanh \phi ={\frac {\sinh \phi }{\cosh \phi }}={\frac {e^{\phi }-e^{-\phi }}{e^{\phi }+e^{-\phi }}}.} 15211:

Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle

13387:

increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.

11344:, then the intuitive Galilean transformations are recovered from the transformation equations for special relativity 4353:{\displaystyle \Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})} 2548: 35858: 35206: 35191: 34903: 33745: 32899: 29321: 28386: 22490: 22400:. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the 22172:

event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.

20506: 20274: 15132:

Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle

2715: 924:

principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

868: 507: 103: 36081: 35309: 34632: 21848: 8230: 7661: 6913:

that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as

4202:) are independent invariants, the values of which do not change when observed from different frames of reference. 503: 424: 35186: 34345: 33790: 32786: 31823:

Franklin, Jerrold (2010). "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity".

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Figure 10–3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.

21517: 19955:

Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).

19886: 17619: 7536: 6624: 4205:

In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an

3351: 588: 29126:

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

24714:{\displaystyle U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).} 17677: 14937:, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by: 11654:), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground. 8679: 8112: 3261:

axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure,

35252: 35216: 34448: 33600: 33472: 33166: 32639:

Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System".

31773: 29317: 26676:{\displaystyle \eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }} 25930: 25889: 24379: 22344: 22263: 16920:), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that 12808:

away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a

11881:

independently established that light travels more slowly in water than in air, thus validating a prediction of

8906: 7429: 5732: 5452: 3839: 3753: 3127:

The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events,

20788: 15447:{\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}} 12109: 8809: 6860:

A time interval measured using a single clock which is motionless in a particular reference frame is called a

5938: 35709: 35641: 35304: 34951: 34879: 34621: 34526: 34456: 33365: 29758:

The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity

28007: 26337: 21642: 6056: 3998:

units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as

3690: 3578: 968:

Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.

614:

Special relativity has a wide range of consequences that have been experimentally verified. They include the

570: 280: 34589: 32023: 30618:

Debs, Talal A.; Redhead, Michael L.G. (1996). "The twin "paradox" and the conventionality of simultaneity".

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require the use of tensors with appropriate rank, which themselves can be thought of as being built up from

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to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized

6916: 6744:

Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame

6234: 6152: 35734: 35289: 35242: 35237: 34443: 33578: 33467: 33462: 33006: 32940:

The Australian National University. Relativistic visual effects experienced through an interactive program.

31326:

Müller, Thomas; Boblest, Sebastian (2014). "Visual appearance of wireframe objects in special relativity".

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condensers moving at an angle with respect to the aether. Lorentz was perplexed by experiments such as the

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One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:

26687: 26568:, in those frames. Throughout we use the signs as above, different authors use different conventions – see 26541: 26352: 22607: 22509: 22496: 22478: 22447: 15061:

will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1

13326:

Another example where visual appearance is at odds with measurement comes from the observation of apparent

11206:

which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the

654:. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of 16731:{\displaystyle \quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}} 14463:{\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)} 13880:. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let 7165:

The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called

5988: 1061:(i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer 36162: 35972: 35172: 35064: 35049: 35034: 34993: 34896: 34484: 34218: 34203: 33735: 33694: 33638: 33487: 33412: 33303: 31586: 30719: 29679: 28153: 28037: 27657:{\displaystyle \mathbf {P} ^{2}=\eta ^{\mu \nu }P_{\mu }P_{\nu }=-\left({\frac {E}{c}}\right)^{2}+p^{2}.} 22371: 20797:

direction along the line between them with the same constant proper acceleration. Will the string break?

20782: 20776: 16885: 15207: 13378: 12797: 12781:{\displaystyle \tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}} 11889: 7488: 5644: 1050: 663: 639: 523: 442: 185: 170: 160: 108: 31793: 29798:

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

27929:

while the four-force is defined by the rate of change of momentum with respect to proper time, that is,

26690:

on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector

25971:

The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.

24869: 24854:{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.} 17988:

into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the

6816: 36072: 35783: 35603: 35044: 35029: 34771: 34213: 34166: 33662: 33477: 33134: 33107: 32974: 32805: 31440: 31088: 31018: 30439: 28302:. Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable. 28258:

of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.

24864: 22316: 22191: 19454: 16881: 15145:. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ± 13497: 12354:{\displaystyle c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx } 11538: 11170: 9020:

involves a long ladder traveling near the speed of light and being contained within a smaller garage).

8960: 7385: 7143:

to the left, in which case observer A would perceive B's clock as running slower than her local clock.

6345: 815: 772: 631: 615: 165: 155: 30319: 28194: 22202:

fields required certain behaviors on charged particles. The general study of moving charges forms the

20854:

matches her every move. So the spaceships stay the same distance apart, and the string does not break.

20187: 19404: 8856: 4407: 3877:

as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between

36134: 35455: 34801: 34427: 34108: 33570: 33517: 33497: 30995: 30671: 30053:

Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)

28243: 28115: 28012: 23612:

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the

22450:(1903) showed that the torque on a capacitor is independent of position and inertial reference frame. 19084: 11892:. The speed of light was measured in still water. What would be the speed of light in flowing water? 11364: 9245: 1200: 635: 347: 35937: 29640: 29615: 29471: 29091:"Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?" 27536:

So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

22945:. All equations and effects of special relativity can be derived from this rotational symmetry (the 21509:

the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

20146: 13075:{\displaystyle f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.} 11140: 9149: 36167: 36039: 35957: 35911: 35618: 35403: 34826: 34376: 34136: 34047: 34020: 33971: 33966: 33941: 33931: 33891: 33586: 33442: 33358: 33240: 29470:

Sean Carroll, Lecture Notes on General Relativity, ch. 1, "Special relativity and flat spacetime",

29122:"On some close collisions of fast β-particles with electrons, photographed by the expansion method" 28022: 26368: 21029: 17941: 17816: 16877: 16546: 15116:

Elastic scattering of charged elementary particles deviates from ideality due to the production of

15098: 13505: 12840:

as measured by an observer on the receiver or the source (The sign convention adopted here is that

9293: 9268: 8670:{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}} 8102:{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}} 6671: 4423: 4368: 1192: 1058: 32323:(transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow. 31977:

Sidney Coleman; Sheldon L. Glashow (1997). "Cosmic Ray and Neutrino Tests of Special Relativity".

23047: 21600: 16896:

This nonlinearity is an artifact of our choice of parameters. We have previously noted that in an

11118: 11096: 10089: 6732:). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed 5725:

In this case, the two events are separated by more space than time, and they are hence said to be

5445:

In this case, the two events are separated by more time than space, and they are hence said to be

5411: 5378: 4213: 4001: 3174:

have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

991:

Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity

36009: 35696: 35613: 35583: 34967: 34933: 34555: 34477: 34097: 34088: 33976: 32959:

The Australian National University. Relativistic visual effects explained with movies and images.

32756: 30470: 29693:

Jeroen van Dongen (2009). "On the role of the Michelson–Morley experiment: Einstein in Chicago".

22389: 22267: 22108: 21512: 19369: 19304: 15867:{\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }} 14595: 14555: 13241:

Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.

12115:

From the point of view of special relativity, Fizeau's result is nothing but an approximation to

11577: 11360: 5904: 5684: 5610: 3402: 3152: 1866: 1216:, even when measured by multiple systems that are moving at different (but constant) velocities. 1188: 1000: 977: 958: 879: 730: 596: 535: 299: 73: 28398:

The refractive index dependence of the presumed partial aether-drag was eventually confirmed by

22056: 19049: 19020: 18495: 17807:{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}} 15298:{\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}} 13088: 11846:

turned rapidly (although this does not violate causality or any other relativistic phenomenon).

11416:

Thomas rotation results from the relativity of simultaneity. In Fig. 4-5a, a rod of length

9937:

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing

6358:

Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer

4089: 4067: 3130: 741:. Events that occur at the same time for one observer can occur at different times for another. 35967: 35823: 35778: 35393: 35008: 35003: 34781: 34644: 34550: 34508: 34182: 34146: 33903: 33811: 33783: 33686: 33545: 33324: 33186: 32532: 32341: 31606: 28446:

algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo

24535: 24455: 22516: 22236: 22226: 18360: 15102: 13509: 13501: 13490: 12081: 11496: 8522: 6785:

mirrors. (Langevin himself made use of a light-clock oriented parallel to its line of motion.)

6103: 4427: 4419: 4140: 4134: 1859: 1363: 1238: 909: 807: 768: 734: 576: 123: 118: 38: 32743: 32472: 31959: 31917: 31911: 31524: 31497: 30802: 30771: 30433: 30402: 30264: 30258: 29756: 29589: 29531: 29501: 29090: 28645:

Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited,

28325:

This concept is counterintuitive at least for the fact that, in contrast to usual concepts of

24430: 23364:

If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:

22960:

coordinate. To make the time coordinate look like the space coordinates, it can be treated as

21912: 19980: 12469:

used this expected phenomenon in a failed attempt to measure the speed of light, and in 1870,

11969: 8314: 7954: 4182: 4159: 3613: 3501: 669: 36049: 36004: 35484: 35429: 34786: 34391: 34330: 33981: 33936: 33776: 33725: 33720: 33646: 33457: 33437: 33427: 33329: 33298: 33161: 33082: 32980: 32946:

A variety of visualizations of relativistic effects, from relativistic motion to black holes.

31235: 30823:

Proceedings of the Fifth International Conference for History of Science in Science Education

30557: 30464: 29745:

For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990

29434: 28517: 27767:

The rest energy is related to the mass according to the celebrated equation discussed above:

24915: 24072: 23055:

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space

22088: 21573: 20894: 20461:{\displaystyle \gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)} 18954: 18364: 13212: 12458: 12439: 11882: 7745: 6576: 6402: 6196: 1199:, and was incorporated into Newtonian physics. But in the late 19th century the existence of 917: 32995:

showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.

30888: 30852: 29005:

Penrose, Roger (24 October 2008). "The Apparent Shape of a Relativistically Moving Sphere".

22161:

receives any communications from Terence after 100 hours on his clock (dashed green lines).

21530: 20482: 20250: 19640: 19617: 17610:{\displaystyle {\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=} 17224: 15631: 13520: 11910: 6287: 3264: 36024: 35952: 35838: 35704: 35666: 35598: 35112: 34938: 34691: 34617: 34461: 34063: 33855: 33804: 33797: 33750: 33422: 33395: 33288: 33112: 33072: 32793: 32782:

Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.

32772: 32694: 32648: 32582: 32423: 32075: 31996: 31842: 31665: 31345: 31300: 31100: 31030: 30960: 30752: 30627: 30211: 30117: 29912: 29754: 29712: 29641:"November 1887: Michelson and Morley report their failure to detect the luminiferous ether" 29240: 29188: 29133: 29016: 28979: 28950: 28919: 28872: 28829: 28778: 28723: 28709:"Electromagnetic phenomena in a system moving with any velocity smaller than that of light" 28679: 28372:

under standard Lorentz transformation in analogous to the invariance of squared distances Δ

28066: 26345: 26333: 25800: 24418: 24076: 22461: 22431:

claims he made in his later years that it played no role in his development of the theory.

22412: 22354: 22286: 22240: 21422: 21002: 20914: 19974:(MCRF), enables application of special relativity to the analysis of accelerated particles. 16905: 15094: 13917: 13883: 13822: 13795: 13743: 13574: 12971:

For light, and with the receiver moving at relativistic speeds, clocks on the receiver are

12944: 11885: 9960: 7056: 6991: 6862: 6572: 3658: 3546: 3465: 965:

of symmetry transformations. Einstein later derived these transformations from his axioms.

650:

position. Rather than an invariant time interval between two events, there is an invariant

244: 78: 30654: 27913:

In the rest frame of the object, the time component of the four-force is zero unless the "

27755:{\displaystyle \mathbf {P} ^{2}=-\left({\frac {E_{\text{rest}}}{c}}\right)^{2}=-(mc)^{2}.} 24359: 21369: 14915:

shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.

14734:{\displaystyle \quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)} 11564:

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".

6612:. To find the relation between the times between these ticks as measured in both systems, 4036: 3104: 1203:

led some physicists to suggest that the universe was filled with a substance they called "

935:

is chosen so that, in relation to it, physical laws hold good in their simplest form, the

726:

in a vacuum. It also explains how the phenomena of electricity and magnetism are related.

8: 35901: 35724: 35714: 35563: 35548: 35504: 34962: 34766: 34761: 34751: 34583: 34532: 34350: 34335: 34161: 34092: 34040: 33263: 33122: 33117: 33102: 33077: 33054: 33030: 32924: 32160: 30853:"Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies" 30588: 29224: 28346: 24084: 23024:

are included in the metric tensor. These numerous conventions can be superseded by using

22926: 22725:{\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},} 22573: 22465: 22424: 22381: 21502: 21492: 20974: 19877:{\displaystyle \gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=} 18323: 16873: 14489: 13763: 13740:. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle 13383:

As an object's speed approaches the speed of light from an observer's point of view, its

13327: 12975:

relative to clocks at the source. The receiver will measure the received frequency to be

12822: 12454: 12435: 11770: 11662: 11573: 10470:{\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .} 9541: 9513: 9011: 8437:{\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}} 8287: 7868:{\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}} 3880: 3725: 3439: 3298: 3236: 3208: 2932:

Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.

1646: 1264: 1204: 987: 833: 795: 791: 784: 592: 584: 558: 519: 337: 294: 34756: 32698: 32652: 32586: 32427: 32079: 32000: 31846: 31669: 31471: 31357: 31349: 31304: 31269: 31104: 31034: 30964: 30756: 30631: 30215: 30121: 29916: 29820:

Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift",

29716: 29459: 29244: 29192: 29137: 29020: 28983: 28954: 28923: 28876: 28833: 28782: 28727: 28708: 28683: 28664: 23738:

The Lorentz transformation in standard configuration above, that is, for a boost in the

21550: 21455: 21346: 21098: 21075: 20947: 17725:

The Lorentz transformations take a simple form when expressed in terms of rapidity. The

15579:{\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}} 13189: 11798: 11725: 11698: 9607: 7718: 6872: 3978: 3814: 3326: 3183: 3057: 3034: 847: 36122: 36034: 35891: 35744: 35558: 35494: 35281: 35039: 34831: 34731: 34655: 34513: 34494: 34488: 34439: 34381: 34290: 34208: 34126: 33818: 33702: 33678: 33522: 33405: 33319: 33196: 33144: 32733: 32627: 32091: 32012: 31986: 31883: 31858: 31832: 31361: 31335: 31152: 30976: 30950: 30339: 30234: 30199: 30181: 29994: 29928: 29902: 29776: 29728: 29702: 29282: 29206: 29151: 29032: 28888: 28845: 28802: 28513: 28251: 28163: 28056: 23718: 22982: 22397: 22271: 21402: 21326: 20874: 20762:

or rotating. To accommodate this difference, general relativity uses curved spacetime.

19937: 15611: 15110: 14535: 14515: 13859: 13605: 13554: 12843: 12087: 11949: 11750: 11466: 11463:

In Fig. 4-5b, the same rod is observed from the frame of a rocket moving at speed

11443: 11419: 9940: 9587: 7126: 6896: 6795: 6554: 6354: 6317: 6132: 6036: 5884: 5864: 5590: 5570: 3794: 3080: 3014: 2994: 2974: 2954: 1053:

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

1019: 1011: 902: 745: 705: 651: 619: 527: 150: 32008: 31559:

Einstein's Mass-Energy Equation, Volume I: Early History and Philosophical Foundations

31201: 28938: 22595:

Special relativity uses a "flat" 4-dimensional Minkowski space – an example of a

22342:

could only be fully explained with special relativity. It was the first foundation of

20765:

In this section, we analyze several scenarios involving accelerated reference frames.

1049:

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

36029: 35798: 35773: 35588: 35499: 35479: 35346: 35211: 35097: 35069: 35059: 35054: 34841: 34673: 34665: 34266: 34223: 33909: 33334: 33283: 33228: 33208: 33094: 32871: 32824:

A good introduction to special relativity at the undergraduate level, using calculus.

32667: 32619: 32594: 32555: 32540: 32518: 32497: 32478: 32450: 32431: 32399: 32391: 32381: 32373: 32349: 32308: 32267: 32244: 32221: 32205: 32190: 32166: 32139: 32114: 31948: 31921: 31890: 31854: 31769: 31739: 31711: 31677: 31562: 31537: 31503: 31384: 31365: 31182: 31172: 31067: 31056: 30980: 30972: 30921: 30894: 30860: 30790: 30760: 30698: 30567: 30536: 30501: 30474: 30443: 30412: 30298: 30268: 30239: 30152: 30081: 30056: 30028: 30001: 29975: 29951: 29893:

Miller, D. J. (2010). "A constructive approach to the special theory of relativity".

29876: 29825: 29801: 29762: 29732: 29671: 29595: 29559: 29520: 29490: 29440: 29388: 29380: 29357: 29347: 29327: 29292: 29155: 29054: 29036: 28907: 28892: 28863:

G. A. Benford; D. L. Book & W. A. Newcomb (1970). "The Tachyonic Antitelephone".

28849: 28794: 28646: 28629: 28605: 28148: 28086: 27983: 27973: 25186: 24353: 23714: 22436: 22305: 22259: 14898:

Rindler has observed that Einstein's heuristic argument suggested merely that energy

13531: 13384: 13343: 11860: 11835: 11542: 11487:, relativity of simultaneity causes the observer in the rocket frame to observe (not 4144:

most humans encounter that some of the effects predicted by relativity are initially

3054:) axis is vertical, which is the opposite of the usual convention in kinematics. The 2939: 1276: 999:-axis, from the perspective of an observer stationary in the unprimed system. By the 757: 643: 402: 367: 332: 219: 190: 180: 175: 34711: 32881: 32631: 32095: 32063: 32016: 31862: 31439:. School of Physics and Astronomy, Rochester Institute of Technology. Archived from 30495: 29932: 26576: 24080: 22946: 22419:, and general relativity (in the limiting case of negligible gravitational fields). 15113:. Radioactive decay may be considered a sort of time-reversed inelastic collision.) 13736:

Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame

13726:{\displaystyle {\frac {f'}{f}}={\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}} 12466: 11221:

being aligned with each other (although not necessarily in standard configuration).

9578:, followed by straightforward mathematical manipulations and back-substitution from 1288: 962: 954: 920:

of space and the independence of measuring rods and clocks from their past history.

828:

had already postulated that there is no absolute and well-defined state of rest (no

753: 362: 36098: 36044: 35719: 35686: 35671: 35553: 35422: 34746: 34736: 34683: 34660: 34238: 33867: 33849: 33527: 33273: 33213: 33139: 32702: 32676: 32656: 32611: 32590: 32419: 32365: 32083: 32004: 31850: 31673: 31353: 31308: 30968: 30635: 30331: 30229: 30219: 29920: 29720: 29594:(illustrated ed.). Springer Science & Business Media. p. §1,11 p. 7. 29439:(Reprint of 1920 translation by Robert W. Lawson ed.). Routledge. p. 48. 29248: 29196: 29141: 29024: 28987: 28880: 28837: 28806: 28786: 28356: 28096: 28017: 26579:

is the most general group of transformations which preserves the Minkowski metric:

26569: 26398:

has components (valid with suitably chosen coordinates) which can be arranged in a

26392: 26341: 22961: 22934: 22930: 22577: 22293: 20809:

apart and remain the same distance apart during acceleration. During acceleration,

16775: 15219:

The relativistic momentum and total relativistic energy of a particle are given by

14931: 13508:, and the relationship between the frequency of light and its energy as implied by 13315: 13237: 11874: 11554: 11257:, then it would also be moving at the speed of light in any other frame, moving at 6486:{\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)} 4145: 819: 580: 372: 342: 34806: 32827: 23729: 19949: 12444: 11878: 7114:

a proper time interval because he did not measure it with a single resting clock.

2923: 2914: 2903: 2894: 1854:-axis, or indeed in any direction parallel to the motion (which are warped by the 36014: 35962: 35906: 35886: 35788: 35676: 35543: 35514: 35087: 34972: 34821: 34796: 34721: 34716: 34599: 34560: 34522: 34466: 34340: 34276: 33897: 33829: 33594: 33502: 33447: 33417: 33381: 33293: 33203: 33149: 33049: 32956: 32937: 32927:

Software visualizing several scenarios under the influence of special relativity.

32876: 32834: 32815: 32779: 32572: 32512: 32413: 32288: 32261: 32213: 31942: 31613: 31607:

The special theory of relativity shows that time and space are affected by motion

31148: 31130: 30784: 30746: 30408: 30022: 29663: 29553: 29514: 29484: 28970: 28499: 28382: 28247: 28134: 27374: 24122: 24075:, as this has numerous advantages. The notation makes it clear the equations are 22600: 22558: 22529: 22405: 22282: 19773:{\displaystyle V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=} 14860:

Comparing the above expression with the classical expression for kinetic energy,

14512:

seen in the above equation have a straightforward physical interpretation. Since

11376: 6740: 4118: 1232: 1208: 1196: 841: 837: 829: 825: 811: 749: 600: 554: 515: 499: 357: 352: 327: 304: 214: 34: 34604: 33837: 32965:

A computer program to show the effects of traveling close to the speed of light.

32893: 31291:

Boas, Mary L. (1961). "Apparent Shape of Large Objects at Relativistic Speeds".

31160: 27993: 24265:{\displaystyle X^{\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf {x} ).} 23706:

region is the information that the point is "receiving", while the cone in the +

22512:

demonstrated that the speed of light is independent of the speed of the emitter.

15216:

smaller as the velocity of the incident particle approaches the speed of light:

14757:

Taking a Taylor series expansion and neglecting higher order terms, he obtained

12071:{\displaystyle u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .} 7176: 4108:

are related to the unprimed coordinates through the Lorentz transformations and

36146: 36086: 36054: 36019: 35916: 35749: 35739: 35729: 35651: 35623: 35608: 35593: 35509: 35125: 34998: 34846: 34518: 34502: 34498: 34401: 34371: 34228: 34131: 33654: 33432: 33223: 33178: 33156: 33044: 32333: 32236: 32064:"Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905" 31168: 29460:

The Feynman Lectures on Physics Vol. I Ch. 15-9: Equivalence of mass and energy

27914: 26364: 26360: 25974:

More generally, most physical quantities are best described as (components of)

25179: 22599:. Minkowski spacetime appears to be very similar to the standard 3-dimensional 22554:

Figure 10–1. Orthogonality and rotation of coordinate systems compared between

22385: 22339: 22326: 22320: 22297: 22214: 22210: 21498: 20626:{\displaystyle \gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=} 17510:{\displaystyle \beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=} 16943: 16828: 15117: 13335: 13331: 13198: 13170: 11546: 10257:{\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .} 9017: 6757: 6349: 1870: 1650: 1607: 1599: 723: 627: 542: 429: 397: 258: 113: 35999: 35141: 32968: 32757:

Special relativity for a general audience (no mathematical knowledge required)

32707: 32680: 32087: 31944:

Einstein's general theory of relativity: with modern applications in cosmology

29724: 29028: 28968:

Terrell, James (15 November 1959). "Invisibility of the Lorentz Contraction".

22539: 11865: 2528:{\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .} 36156: 35991: 35896: 35808: 35681: 35351: 35277: 35130: 35024: 34836: 34816: 34811: 34726: 34594: 34422: 34366: 34198: 34151: 33873: 33843: 33730: 33233: 32660: 32602:

Darrigol, Olivier (2004). "The Mystery of the Poincaré–Einstein Connection".

32360:

Formal Structure of Electromagnetics: General Covariance and Electromagnetics

32209: 31736:

Formal Structure of Electromagnetics: General Covariance and Electromagnetics

31156: 30826: 29486:

Explorations in Mathematical Physics: The Concepts Behind an Elegant Language

28991: 28525: 28399: 28110: 28091: 27545: 26388: 26384: 25267: 24966: 24530: 23157: 23025: 22978: 22591:

of a light signal, a vector is orthogonal to itself if it lies on this line).

22550: 22416: 22401: 20823:

in the frame of the accelerating spaceships. After a sufficiently long time,

14837:{\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}} 13284: 13246: 12972: 12809: 12575:{\displaystyle \cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}} 11437: 9024: 6778: 6588: 6548: 4415: 2354:{\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .} 1095:

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

623: 263: 145: 32998: 32949: 32930: 32762: 31706:

Thomas, George B.; Weir, Maurice D.; Hass, Joel; Giordano, Frank R. (2008).

31470:. Department of Physics and Astronomy, University of Alabama. Archived from 31383:(1st ed.). Cambridge, New York: Cambridge University Press. p. 3. 29948:

The Geometry of Spacetime: An Introduction to Special and General Relativity

29489:(illustrated ed.). Springer Science & Business Media. p. 234. 14552:

are the energies of the arbitrary body in the moving and stationary frames,

13454:. The momentum is equal to the energy multiplied by the velocity divided by 13405:

arguments for the equivalence (and transmutability) of mass and energy, for

11582: 9053:

to the end points of the rod must be measured simultaneously in that system

7189:

article. We will examine in the following one such solution to the paradox.

6604:. The location of the clock on two different ticks is then characterized by 36110: 36059: 35863: 35848: 35813: 35661: 35646: 34856: 34776: 34741: 34271: 34233: 33986: 33861: 33258: 32685: 32670: 32623: 32409: 31134: 30243: 29253: 29228: 29201: 29176: 29172: 29146: 29121: 28884: 28798: 28580:

Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920

27978: 27462:{\displaystyle \mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,} 27324: 26356: 25271: 22545: 22359: 22331: 22301: 13551:

Fig. 6-1 (top). Consider a system of plane waves of light having frequency

12106:

the aether must be capable of sustaining different motions at the same time

12080:

Fizeau's results, although consistent with Fresnel's earlier hypothesis of

11397: 11386: 11351:

like Einstein considers) requires special treatment of the transformations.

7186: 7166: 7160: 6774:

Figure 4–3. Thought experiment using a light-clock to explain time dilation

6713:) between the two ticks as seen in the frame in which the clock is moving ( 4130: 4113:

x-coordinates in the unprimed frame are now at the same position in space.

764: 531: 392: 32914:

The problem of superluminal motion is discussed in an entertaining manner.

32887: 28376:

under rotations in Euclidean space. Although space and time have an equal

23016:

elsewhere to compensate; for instance, spatial coordinates are divided by

20758:

only when gravitation is significant that general relativity is required.

20139:

As indicated before, there are varying treatments for the energy-momentum

19341:

are not necessarily at right angles with each other. The rule is that two

18370:

On the other hand, general relativity, from the outset, relies heavily on

15161:|. (c) Reverting to the stationary frame, the rebound velocities are 13496:

Einstein took as starting assumptions his recently discovered formula for

13271: 13201:

is one of the main novel predictions of the special theory of relativity.

12474:

the results of the Michelson–Morley experiment, which apparently demanded

957:

provided the mathematical framework for relativity theory by proving that

35947: 35921: 35843: 35532: 35471: 35120: 34640: 34609: 34156: 33991: 33670: 33492: 30401:

Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002).

28790: 28716:

Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW)

28672:

Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW)

28559:

Kinematics of an electron with an axis. Phil. Mag. 3:1-22. L. H. Thomas.]

28443: 28076: 27838:

with respect to proper time. The covariant version of the four-force is:

27835: 27349: 25263:-acceleration are simpler by means of the Lorentz transformation matrix. 24962:

of an object are respectively the timelike and spacelike components of a

24563: 24425: 24125: 24068: 22404:

and below, quantum effects must be taken into consideration resulting in

22396:

in the region of interest. In a strong gravitational field, one must use

20862: 20727:

Einstein preferred to change the definition of momentum to one involving

18343:

Four‑vectors have been mentioned above in context of the energy–momentum

18338: 16909: 16821: 16782: 13419: 13360: 12470: 11640: 11558: 2884: 799: 377: 253: 32911: 32850: 32794:

Special relativity explained (using simple or more advanced mathematics)

32319: 30955: 30343: 30224: 30102: 20827:

will increase to a sufficiently large factor that the string must break.

11621: 11612: 11567: 6414:(the forward Lorentz transformation in terms of coordinate differences) 6374:, the events occur in the order C, B, A. In a reference frame moving at 30: 35828: 35102: 35092: 34988: 34851: 34417: 34261: 34256: 33996: 33278: 33268: 32337: 31991: 31493: 30748:

Applications of Electrodynamics in Theoretical Physics and Astrophysics

27968: 27831: 24071:", in Minkowski spacetime. The components of vectors are written using 22312: 21343:. As shown in Fig. 7-5, Bell's example asserts the moving lengths 20944:

turns out to be greater than the length along the line of simultaneity

13366: 13347: 13305: 11587: 6770: 6393: 4124: 407: 387: 229: 224: 36141: 31312: 29924: 28841: 27118:

so the squared length of the differential of the position four-vector

13788:. Because of conservation of momentum, the body remains stationary in 12868:

each other). Assume that the source is stationary in the medium. Then

6571:, instruments based on the Sagnac effect for their operation, such as 2078:{\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .} 35803: 35754: 35079: 34888: 34248: 33129: 32986: 30563: 30335: 29210: 28939:"Fresnel's coefficient for light of different colours. (Second part)" 28403: 28129: 26821:

appears in its trivial transformation. The magnitude of the 4-vector

23597: 23482:{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}} 22596: 22588: 22304:

atoms using both special relativity and the preliminary knowledge on

16868: 15078: 13595:. The frequency (and hence energy) of the waves as measured in frame 13402: 11831: 11550: 11493:) the clock at the right end of the rod as being advanced in time by 11314:|) are small with respect to the speed of light (that is, e.g., 1272: 1072:(pronounced "S prime" or "S dash") belongs to a second observer 1023: 803: 738: 495: 205: 32866: 32853:

A complete online book on relativity with an extensive bibliography.

30639: 29755:

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

28908:"Fresnel's coefficient for light of different colours. (First part)" 28862: 23163:

lie along a dual-cone (see Fig. 10-2) defined by the equation;

22880:{\displaystyle ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},} 16931:, but as twice the area of the sector swept out by the ray from the 12419:{\displaystyle {\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)} 6284:

and this relationship is frame independent due to the invariance of

35833: 35818: 34791: 34141: 32615: 31637:. California State University Affordable Learning Solutions Program 30617: 30080:. Copernicus Books (imprint of Springer Science + Business Media). 28326: 27988: 25978:. So to transform from one frame to another, we use the well-known 24562:

defined as the derivative of the position 4-vector with respect to

23160: 22335: 22221:

field generated by a moving charge disappears and becomes a purely

18858:

must be a standard vector in three-dimensional space. Therefore, a

16942:

measures for the unit circle are identical. Fig. 7-1b shows a

16765: 15093:

In Newtonian mechanics, analysis of collisions involves use of the

11824:

faster than the speed of light will result in causality violation.

4411: 1654: 1366:

specifies that these coordinates are related in the following way:

913: 729:

A defining feature of special relativity is the replacement of the

382: 33350: 32943: 31837: 31710:(Eleventh ed.). Boston: Pearson Education, Inc. p. 533. 31656:

Nakel, Werner (1994). "The elementary process of bremsstrahlung".

31340: 31186: 30497:

The feynman lectures on physics; vol I: The new millennium edition

29907: 29707: 22941:(that is the 4D distance between any two events) when viewed from 19211:{\displaystyle A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv } 18487:

transform between frames according to the Lorentz transformation.

11869:

Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.

6728:) between these ticks as measured in the rest frame of the clock ( 2182:{\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .} 1645:

coordinates are transformed. These Lorentz transformations form a

35527: 35489: 32962: 32890:

The theory of special relativity in an easily understandable way.

32859:

An introduction to special relativity at the undergraduate level.

31913:

Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles

30494:

Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2011).

29875:, Cambridge University Press, Cambridge, chapter 11, Appendix I, 29472:

http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html

28604:. Progress in Mathematical Physics. Vol. 40. pp. 1–21. 26683:

and this is the physical symmetry underlying special relativity.

24088: 18352: 16904:

The natural functions for expressing these relationships are the

15946:{\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}} 13339: 13229:. In Fig. 5-3b, the light is redshifted by the same factor. 12108:. A variety of theoretical explanations were proposed to explain 9236:

from each other when seen from another moving coordinate system.

776: 483: 32905: 32856: 32821: 32108: 31976: 31412:. Department of Mathematics, University of California, Riverside 30726:. Department of Mathematics, University of California, Riverside 29350:(1980). "Chapter 7: Special Relativity in Classical Mechanics". 22767:

derived from time, such that the distance differential fulfills

21489:

Event horizon § Apparent horizon of an accelerated particle

20805:

To observers in the rest frame, the spaceships start a distance

20126:{\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})} 13819:

be the energy of the body before emission of the two pulses and

2266:{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ } 1835:

This shows that the unprimed frame is moving with the velocity −

35853: 35445: 34471: 31631:"Mechanics and Relativity. Chapter 14: Relativistic Collisions" 30697:. CreateSpace Independent Publishing Platform. pp. 90–92. 29822:

Einstein's generation. The origins of the relativity revolution

29065::639, 1905 (English translation by W. Perrett and G.B. Jeffery) 29008:

Mathematical Proceedings of the Cambridge Philosophical Society

26877:{\displaystyle |\mathbf {T} |={\sqrt {T^{\alpha }T_{\alpha }}}} 25979: 25975: 23730:

Transformations of physical quantities between reference frames

23035:. Then space and time have equivalent units, and no factors of 22250: 22244: 20690:{\displaystyle \gamma \left({\frac {dE}{dt}},{\vec {f}}\right)} 20371:{\displaystyle {\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=} 17023:{\displaystyle {\text{slope}}=\tan a={\frac {\sin a}{\cos a}}.} 11273:

The resultant speed of two velocities with magnitude less than

11229: 9228:

not co-local, but are separated by distance (space), they will

6495:

It is clear that the two events that are simultaneous in frame

4500:{\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}} 4139:

The consequences of special relativity can be derived from the

2538:

If we take differentials instead of taking differences, we get

2435:{\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ } 1243: 1233:

Lorentz invariance as the essential core of special relativity

659: 626:, the relativistic velocity addition formula, the relativistic 546: 32289:

Texts by Einstein and text about history of special relativity

27904:{\displaystyle F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }} 26825:

is the positive square root of the inner product with itself:

17293:{\displaystyle \beta \equiv \tanh \phi \equiv {\frac {v}{c}},} 13300: 6507:), are not necessarily simultaneous in another inertial frame 6388:, the events occur in the order A, B, C. The white lines, the 5854:{\displaystyle \Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2}),} 939:

laws hold good in relation to any other system of coordinates

494:

for short, is a scientific theory of the relationship between

34032: 32263:

Spacetime and Geometry: An Introduction to General Relativity

31234:. Michigan State University, East Lansing, MI. Archived from 30920:. ill Valley, Calif.: University Science Books. p. 393. 30256: 29076:

On the Inertia of Energy Required by the Relativity Principle

27111:{\displaystyle dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }} 27043:

The coordinate differentials transform also contravariantly:

23588:{\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.} 22562: 19451:

and the invariant length of the relativistic momentum vector

15017:{\displaystyle v(t)={\frac {at}{\sqrt {1+a^{2}t^{2}/c^{2}}}}} 13941:

after their emission. We obtain the following relationships:

12671:{\displaystyle \sin \theta '={\frac {\sin \theta }{\gamma }}} 9196:) of the rod as measured in the frame in which it is moving ( 6597: 1191:, which states that physical laws have the same form in each 647: 591:

failed to detect the Earth's motion against the hypothesized

32802:– A simple introduction to the special theory of relativity. 31268:. Institut für Physik Universität Hildesheim. Archived from 31054: 25879:{\displaystyle T_{\mu '}=\Lambda _{\mu '}{}^{\nu }T_{\nu },} 25810:

variant components of a 4-vector transform according to the

22487:– relativistic effects on a fast-moving particle's half-life 12934:{\displaystyle f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}} 4064:

still plot as 45° diagonal lines. The primed coordinates of

2874:{\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .} 848:

Traditional "two postulates" approach to special relativity

34452: 32737: 32037:"Experiments that Apparently are NOT Consistent with SR/GR" 28750:"Investigation of oscillations excited by oscillating ions" 28665:"The rotation of the plane of polarization in moving media" 28626:

The Special Theory of Relativity, A Mathematical Exposition

24520:{\displaystyle T^{\mu '}=\Lambda ^{\mu '}{}_{\nu }T^{\nu }} 24345:{\displaystyle X^{\mu '}=\Lambda ^{\mu '}{}_{\nu }X^{\nu }} 23613: 22540:

Comparison between flat Euclidean space and Minkowski space

22175: 13477: 12802: 12204:{\displaystyle u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=} 11747:

axis (and the signal from A to B slightly steeper than the

11549:

of an elementary particle or the rotation of a macroscopic

10493:

The forward and inverse transformations for this case are:

6733: 6525:). Only if these events are additionally co-local in frame 5405:

is hence independent of the frame in which it is measured.

2700:{\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .} 655: 36105: 35414: 32908:– An overview with visualizations and minimal mathematics. 32902:

by J D Cresser Department of Physics Macquarie University.

31940: 30320:"The Principle of Relativity, and Non-Newtonian Mechanics" 29407: 29055:

Does the inertia of a body depend upon its energy content?

27469:

which means all velocity four-vectors have a magnitude of

23260:{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}} 22603:, but there is a crucial difference with respect to time. 22493:– time dilation in accordance with Lorentz transformations 16857:

is twice the area between the ray, the hyperbola, and the

14907:

inertial mass represented a reserve of energy. "To equate

12121:, the relativistic formula for composition of velocities. 11888:

and invalidating the corresponding prediction of Newton's

5560:{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t),} 3811:

is an integer. Likewise, draw gridlines parallel with the

1591:{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}} 1219: 31768:. Cambridge, UK: Cambridge University Press. p. 26. 30187:(1st ed.). San Francisco: W. H. Freeman and Company. 28702: 28700: 28254:. Lorentz believed that length contraction represented a 27038: 23149:{\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},} 21784:{\displaystyle \beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}.} 16924:

is interpreted, not as the angle between the ray and the

14918: 13162:{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}} 11488: 11460:) rises vertically along the y-axis in the ground frame. 7196:

illustrates a scenario where the traveling twin flies at

3722:

in the unprimed system. Draw gridlines parallel with the

1282: 31204:. Mathematics Department, University of British Columbia 30720:"Is Faster-Than-Light Travel or Communication Possible?" 30324:

Proceedings of the American Academy of Arts and Sciences

29591:

Essential Relativity: Special, General, and Cosmological

28520:

and Wilfrid Perrett (1923); Another English translation

27834:. It is the rate of change of the above energy momentum 23710:

section is the information that the point is "sending".

20243:

This results from taking the derivative of the velocity

11630:

Figure 4-7. Causality violation by the use of fictitious

11208:

relativistic velocity addition (or composition) formulas

9035:. In this system, the length of this rod is written as Δ 30659:. Berkeley: University of California Press. p. 54. 30400: 29410:"What is the experimental basis of Special Relativity?" 28438:

Rapidity arises naturally as a coordinates on the pure

27539: 24092:

earlier equations, Cartesian coordinates will be used.

23742:-direction, can be recast into matrix form as follows: 22956:

above depends on the metric and on the choices for the

22481:– testing relativistic Doppler effect and time dilation 22044:{\displaystyle dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct).} 16811:

is twice the area between the ray, the circle, and the

14883:, Einstein then noted: "If a body gives off the energy 13232: 9239: 982: 763:

The theory is "special" in that it only applies in the

756:. The theory became essentially complete in 1907, with 32133: 31889:. Cambridge, Massachusetts: Harvard University Press. 31705: 29668:

Personal Knowledge: Towards a Post-Critical Philosophy

28697: 28272:

which failed to validate his theoretical expectations.

26430: 25553: 25431: 25286: 25274:φ transforms covariantly rather than contravariantly: 23980: 23934: 23822: 23756: 23713:

The geometry of Minkowski space can be depicted using

23687:{\textstyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}} 23622: 21503:

Section "Invariant hyperbola" of the article Spacetime

18916:{\displaystyle (c\Delta t,\Delta x,\Delta y,\Delta z)} 18471:{\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)} 18272: 18212: 18180: 18118: 18052: 18010: 17184: 17121: 17082: 16961:

For the unit circle, the slope of the ray is given by

14163: 14072: 14004: 13986: 13907:

be the energy of the body measured in reference frame

9060:. In other words, the measurement is characterized by 7285: 7211: 7043: 6969: 6919: 5991: 3911: 2885:

Graphical representation of the Lorentz transformation

1182: 27:

Theory of interwoven space and time by Albert Einstein

36070: 33623:

Die Grundlagen der Einsteinschen Relativitäts-Theorie

32041:

What is the experimental basis of Special Relativity?

30535:. New York: John Wiley & sons. pp. 958–959. 30318:

Lewis, Gilbert Newton; Tolman, Richard Chase (1909).

28197: 27844: 27773: 27673: 27558: 27483: 27390: 27128: 27049: 26892: 26831: 26704: 26585: 26544: 26408: 26294: 26253: 25987: 25933: 25892: 25820: 25679: 25519: 25495: 25471: 25447: 25435: 25389: 25360: 25331: 25302: 25290: 25280: 25242:{\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}.} 25195: 24975: 24923: 24872: 24727: 24572: 24538: 24464: 24433: 24382: 24362: 24289: 24134: 23750: 23717:, which are useful also in understanding many of the 23495: 23370: 23273: 23169: 23061: 22937:. Basically, special relativity can be stated as the 22773: 22620: 22111: 22091: 22059: 21957: 21915: 21851: 21798: 21711: 21645: 21603: 21576: 21553: 21533: 21458: 21425: 21405: 21372: 21349: 21329: 21129: 21101: 21078: 21032: 21005: 20977: 20950: 20917: 20897: 20877: 20770: 20638: 20554: 20509: 20485: 20383: 20322: 20277: 20253: 20190: 20149: 20025: 19983: 19889: 19785: 19676: 19660:

is the proper time between the two events that yield

19643: 19620: 19544: 19457: 19407: 19372: 19307: 19277:{\displaystyle A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}} 19223: 19115: 19087: 19052: 19023: 18957: 18870: 18565: 18498: 18402: 18174: 18004: 17944: 17852: 17819: 17738: 17680: 17622: 17522: 17440: 17312: 17258: 17227: 17076: 16970: 16629: 16575: 16549: 16348: 16164: 15980: 15898: 15796: 15671: 15634: 15614: 15482: 15333: 15230: 14943: 14768: 14643: 14598: 14558: 14538: 14518: 14492: 14334: 13950: 13920: 13886: 13862: 13825: 13798: 13766: 13746: 13631: 13608: 13577: 13557: 13215: 13127: 13091: 12981: 12947: 12874: 12846: 12825: 12688: 12592: 12491: 12366: 12280: 12268:{\displaystyle {\frac {c/n\pm v}{1\pm v/cn}}\approx } 12216: 12130: 12090: 11999: 11972: 11952: 11913: 11801: 11773: 11753: 11728: 11701: 11665: 11568:

Causality and prohibition of motion faster than light

11561:

orbit to the angular velocity of the orbital motion.

11499: 11469: 11446: 11422: 11173: 11143: 11121: 11099: 10791: 10504: 10288: 10119: 10092: 9996: 9963: 9943: 9882: 9834: 9814: 9778: 9736: 9640: 9610: 9590: 9544: 9516: 9418: 9338: 9296: 9271: 9152: 9130:{\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}} 9097: 8963: 8909: 8859: 8812: 8741: 8682: 8562: 8525: 8449: 8351: 8317: 8290: 8233: 8174: 8115: 7994: 7957: 7881: 7782: 7748: 7721: 7664: 7605: 7539: 7491: 7432: 7388: 7129: 7031:{\textstyle t_{B}=2L{\big /}{\sqrt {c^{2}-v^{2}}}={}} 6899: 6875: 6819: 6798: 6674: 6627: 6557: 6420: 6320: 6290: 6237: 6199: 6155: 6135: 6106: 6059: 6039: 5941: 5907: 5887: 5867: 5788: 5735: 5687: 5647: 5613: 5593: 5573: 5508: 5455: 5414: 5381: 5268: 5194: 5087: 4795: 4607: 4515: 4439: 4371: 4247: 4216: 4185: 4162: 4092: 4070: 4039: 4004: 3981: 3883: 3842: 3817: 3797: 3756: 3728: 3693: 3661: 3616: 3581: 3549: 3504: 3468: 3442: 3405: 3354: 3329: 3301: 3267: 3239: 3211: 3186: 3155: 3133: 3107: 3083: 3060: 3037: 3017: 2997: 2977: 2957: 2793: 2718: 2624: 2551: 2447: 2372: 2278: 2205: 2096: 1973: 1666: 1541: 1372: 897:

The constancy of the speed of light was motivated by

798:, that is, at a sufficiently small scale (e.g., when 708: 672: 32837:– An algebraic and integral calculus derivation for 31909: 31147: 30493: 29079:, A. Einstein, Annalen der Physik 23 (1907): 371–384 28355:

is also an invariant, generated by a variant of the

28182:

reprint of 1923 translation by Methuen and Company.]

27529:{\displaystyle 2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.} 26340:, which has six components: three are the classical 24955:{\displaystyle \mathbf {p} =\gamma (v)m\mathbf {v} } 22523: 21482: 19361:

The magnitude of a vector is the inner product of a

18925:

under Lorentz transformations as well as rotations.

13362:

Consequences derived from the Lorentz transformation

8800:{\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}} 8508:{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}} 7940:{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}} 6396:

with respect to the origin of all considered frames.

4125:

Consequences derived from the Lorentz transformation

3968:{\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}} 32894:

An Introduction to the Special Theory of Relativity

31093:

Journal of the Royal Astronomical Society of Canada

29356:(2nd ed.). Addison-Wesley Publishing Company. 29326:(3rd ed.). John Wiley & Sons. Chapter 11. 28656: 28242:The results of the Michelson–Morley experiment led 22995:, rather than a "disguised" Euclidean metric using 22186:

Covariant formulation of classical electromagnetism

21419:to be fixed, thereby forcing the rest frame length 15137:

stationary frame, an incident sphere traveling at 2

14323:From the above equations, we obtain the following: 13849:Next, consider the same system observed from frame 12429: 11534:and the rod is correspondingly observed as tilted. 11359:direction in the standard configuration. The above 8214:{\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}} 7645:{\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}} 6963:so that the total round-trip time of the pulses is 3348:axes are tilted from the unprimed axes by an angle 32748:English Translation as published in the 1923 book 31882: 31616:. Library.thinkquest.org. Retrieved on 2013-04-24. 30917:Explanatory Supplement to the Astronomical Almanac 30487: 30180: 30179:Taylor, Edwin F.; Wheeler, John Archibald (1966). 29993: 29552:Taylor, Edwin F.; Wheeler, John Archibald (1992). 29408:Tom Roberts & Siegmar Schleif (October 2007). 28212: 27903: 27805: 27754: 27656: 27528: 27461: 27315: 27110: 27026: 26876: 26807: 26675: 26560: 26530: 26321: 26280: 26239: 25960: 25919: 25878: 25791: 25665: 25241: 25166: 24954: 24906: 24853: 24713: 24554: 24519: 24446: 24424:More generally, all contravariant components of a 24409: 24368: 24344: 24264: 24056: 23686: 23587: 23481: 23345:{\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},} 23344: 23259: 23148: 22879: 22724: 22144: 22097: 22077: 22043: 21944: 21899: 21838: 21783: 21697: 21629: 21582: 21562: 21539: 21469: 21444: 21411: 21391: 21358: 21335: 21311: 21112: 21087: 21064: 21018: 20991: 20959: 20936: 20903: 20883: 20689: 20625: 20541: 20494: 20460: 20370: 20309: 20262: 20223: 20176: 20125: 19998: 19919: 19876: 19772: 19652: 19629: 19595: 19496: 19443: 19393: 19325: 19276: 19210: 19102: 19067: 19038: 18981: 18915: 18794: 18531: 18470: 18311: 18147: 17962: 17931: 17837: 17806: 17715: 17663: 17609: 17509: 17419: 17292: 17233: 17210: 17022: 16730: 16604: 16561: 16519: 16316: 16132: 15969:From these we obtain the following relationships: 15945: 15866: 15764: 15640: 15620: 15578: 15446: 15297: 15016: 14836: 14733: 14624: 14584: 14544: 14524: 14504: 14462: 14312: 13933: 13899: 13868: 13838: 13811: 13780: 13760:with respect to the x-axis. Each pulse has energy 13752: 13725: 13614: 13583: 13563: 13390:The energy content of an object at rest with mass 13252:Scientists make a fundamental distinction between 13221: 13161: 13111: 13074: 12960: 12933: 12852: 12832: 12780: 12670: 12574: 12418: 12353: 12267: 12203: 12096: 12070: 11985: 11958: 11938: 11812: 11787: 11759: 11739: 11710: 11679: 11526: 11475: 11452: 11428: 11198: 11159: 11129: 11107: 11055: 10759: 10469: 10256: 10100: 10062: 9978: 9949: 9913: 9621: 9596: 9558: 9530: 9486: 9386: 9309: 9282: 9178: 9129: 9039:. To measure the length of this rod in the system 8987: 8943: 8880: 8841: 8799: 8718: 8669: 8549: 8507: 8436: 8338: 8301: 8267: 8213: 8151: 8101: 7981: 7939: 7867: 7769: 7732: 7698: 7644: 7579: 7521: 7466: 7412: 7345:{\textstyle f''=f{\sqrt {(1+\beta )/(1-\beta )}}.} 7344: 7270: 7135: 7102:{\textstyle t_{A}{\big /}{\sqrt {1-v^{2}/c^{2}}}.} 7101: 7030: 6955: 6905: 6884: 6852: 6804: 6752:The measurements that we would get if we actually 6695: 6654: 6563: 6485: 6326: 6306: 6276: 6214: 6185: 6141: 6120: 6092: 6045: 6017: 5977: 5927: 5893: 5873: 5853: 5774: 5707: 5673: 5633: 5599: 5579: 5559: 5494: 5430: 5397: 5365: 4499: 4390: 4352: 4232: 4194: 4179:) and the temporal separation between two events ( 4171: 4100: 4078: 4056: 4033:of two photons passing through the origin at time 4016: 3990: 3967: 3897: 3869: 3828: 3803: 3783: 3742: 3714: 3679: 3647: 3602: 3567: 3535: 3485: 3454: 3428: 3391: 3340: 3315: 3287: 3253: 3225: 3197: 3166: 3141: 3119: 3101:of two photons passing through the origin at time 3089: 3069: 3046: 3023: 3003: 2983: 2963: 2873: 2780: 2699: 2611: 2527: 2434: 2353: 2265: 2181: 2077: 1825: 1590: 1527: 1112:moves, for simplicity, in a single direction: the 767:where the spacetime is "flat", that is, where the 744:Until several years later when Einstein developed 714: 694: 33556:Investigations on the Theory of Brownian Movement 32415:Relativity: An Introduction to the Special Theory 31822: 31464:"Jets, Superluminal Motion, and Gamma-Ray Bursts" 31057:"The Optics and Electrodynamics of Moving Bodies" 30559:Relativity: An introduction to space-time physics 30311: 29692: 29519:(illustrated ed.). OUP Oxford. p. 226. 29115: 29113: 29111: 28567: 28565: 28316:remained those defined with respect to the ether. 23352:which is the equation of a circle of radius 22182:Classical electromagnetism and special relativity 21839:{\displaystyle \gamma =1/{\sqrt {1-\beta ^{2}}}=} 20731:rather than give up on conservation of momentum. 16906:hyperbolic analogs of the trigonometric functions 14887:in the form of radiation, its mass diminishes by 13372: 13283:as length contracted. In 1959, James Terrell and 11545:is a relativistic correction that applies to the 11232:) were moving at the speed of light in one frame 9434: 9421: 9349: 9341: 7271:{\textstyle f'=f{\sqrt {(1-\beta )/(1+\beta )}}.} 6314:From this, we observe that the speed of light is 506:, the theory is presented as being based on just 36154: 32447:Special Relativity (M.I.T. Introductory Physics) 31916:(2nd ed.). John Wiley & Sons. pp. 31584: 31550: 31502:. Courier Dover Publications. pp. 177–178. 31499:Concepts of Mass in Classical and Modern Physics 31433:""Superluminal" motions in astronomical sources" 30886: 30782: 30751:(illustrated ed.). CRC Press. p. 206. 30695:Special Relativity for the Enthusiastic Beginner 30526: 30524: 29277: 29275: 29273: 20813:is a length contracted distance of the distance 12812:term, and that is the treatment described here. 6579:, are capable of extreme levels of sensitivity. 4029:. Since the speed of light is an invariant, the 35171: 32510: 32259: 31522: 30816:"Measuring Speed of Light: Why? Speed of what?" 30672:"Special Relativity Lecture Notes – Section 10" 30530: 30463:Ellis, George F. R.; Williams, Ruth M. (2000). 29432: 28820:Shaw, R. (1962). "Length Contraction Paradox". 28599: 24454:transform from one frame to another frame by a 22358:, becomes necessary; in which particles can be 20702:As expected, the final components of the above 17221:In a spacetime diagram, the velocity parameter 16888:). Sinh is red, cosh is blue and tanh is green. 9584:yields the Lorentz transformation of the speed 1116:-direction of frame S with a constant velocity 33563:Relativity: The Special and the General Theory 32491: 32470: 32061: 31787: 31785: 31759: 31757: 31755: 31624: 31622: 31379:Zensus, J. Anton; Pearson, Timothy J. (1987). 30288: 30286: 30284: 30142: 30140: 30138: 30020: 29547: 29545: 29543: 29541: 29539: 29436:Relativity: The Special and the General Theory 29383:(1970). "Chapter IX: Relativistic Mechanics". 29375: 29373: 29312: 29310: 29308: 29132:(830). The Royal Society Publishing: 630–637. 29108: 28936: 28905: 28768: 28562: 26332:An example of a four-dimensional second order 22206:, which is a step towards special relativity. 20792:Figure 7–4. Dewan–Beran–Bell spaceship paradox 15199:, indicating that the vectors are orthogonal. 12815:Assume the receiver and the source are moving 12791: 12448:Figure 5–2. Illustration of stellar aberration 11277:is always a velocity with magnitude less than 10063:{\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.} 9265:moves in the x direction with velocity vector 6537:), will they be simultaneous in another frame 2612:{\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ } 840:, including both the laws of mechanics and of 814:. As long as the universe can be modeled as a 35430: 35157: 34904: 34048: 33366: 33028: 33014: 32109:J.A. Wheeler; C. Misner; K.S. Thorne (1973). 32028: 31885:Very Special Relativity: An Illustrated Guide 31794:"Can Special Relativity Handle Acceleration?" 31729: 31727: 31580: 31578: 31378: 31325: 31129:. London: Logos Press Limited. pp. 6–9. 30940: 30580: 30521: 30462: 30362: 30360: 30257:Daniel Kleppner & David Kolenkow (1973). 30200:"Relativity in the Global Positioning System" 30178: 30174: 30172: 30170: 30168: 30069: 30046: 30044: 29867: 29865: 29863: 29861: 29859: 29857: 29551: 29270: 29223: 28033:Special relativity (alternative formulations) 25189:is the proper time derivative of 4-velocity: 20721: 20542:{\displaystyle F\equiv {\frac {dP}{d\tau }}=} 20310:{\displaystyle A\equiv {\frac {dV}{d\tau }}=} 9487:{\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.} 7279:she receives signals at the enhanced rate of 6339: 3687:in the primed coordinate system transform to 3610:in the unprimed coordinate system. Likewise, 3575:in the primed coordinate system transform to 2781:{\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ } 463: 32449:(1st ed.). W. W. Norton & Company. 31974:The number of works is vast, see as example: 31468:Galaxies and the Universe - WWW Course Notes 31406:"Apparent Superluminal Velocity of Galaxies" 30807: 30611: 30369:"Paul Langeyin and the Theory of Relativity" 30094: 29855: 29853: 29851: 29849: 29847: 29845: 29843: 29841: 29839: 29837: 29781:: CS1 maint: multiple names: authors list ( 29171: 28537:"Science and Common Sense", P. W. Bridgman, 22251:Theories of relativity and quantum mechanics 21900:{\displaystyle {\sqrt {c^{2}t^{2}-s^{2}}}/s} 18828: 16708: 16659: 16497: 16426: 16294: 16206: 16110: 16022: 13602:that is moving along the x-axis at velocity 13342:, and other astronomical objects that eject 13184: 9046:, in which the rod is moving, the distances 8268:{\displaystyle t_{2}'={\frac {L}{\gamma v}}} 7699:{\displaystyle t_{1}'={\frac {L}{\gamma v}}} 7180:Figure 4-4. Doppler analysis of twin paradox 6227:In this case, the two events are said to be 5408:In considering the physical significance of 4156:In Galilean relativity, an object's length ( 1244:Alternative approaches to special relativity 946:moving in uniform translation relatively to 553:The first postulate was first formulated by 32550:Tipler, Paul, and Llewellyn, Ralph (2002). 31782: 31752: 31649: 31619: 31319: 31141: 30844: 30533:Fundamental Physics: Extended Third Edition 30317: 30281: 30151:(2nd ed.). New York: Springer-Verlag. 30135: 29761:. Courier Dover Publications. p. 111. 29583: 29581: 29579: 29577: 29575: 29536: 29370: 29305: 29050: 29048: 29046: 28741: 28345:lead to, now not broadly used, concepts of 28312:states of the field for a moving observer. 28250:independently to propose the phenomenon of 26322:{\displaystyle \Lambda ^{\chi '}{}_{\psi }} 26281:{\displaystyle \Lambda _{\chi '}{}^{\psi }} 20846:has the same acceleration that he has, and 19920:{\displaystyle (\gamma ,\gamma {\vec {v}})} 17664:{\displaystyle \tanh(\phi _{1}+\phi _{2}),} 7580:{\displaystyle fT'={\frac {2fL}{\gamma v}}} 7117: 6655:{\displaystyle \Delta t'=\gamma \,\Delta t} 3392:{\displaystyle \alpha =\tan ^{-1}(\beta ),} 1858:factor) and perpendicular; see the article 1846:-axis. The transformation can apply to the 564: 35437: 35423: 35164: 35150: 34911: 34897: 34055: 34041: 33373: 33359: 33021: 33007: 32867:Relativity: the Special and General Theory 32297:Relativity: The Special and General Theory 32185:Jean-Bernard Zuber & Claude Itzykson, 32158: 32034: 31941:Øyvind Grøn & Sigbjørn Hervik (2007). 31733: 31724: 31575: 31556: 31492: 30913: 30907: 30376:Japanese Studies in the History of Science 30357: 30165: 30041: 29985: 29886: 29616:"James Clerk Maxwell: a force for physics" 29285:(2013). "Electrodynamics and Relativity". 28602:Physical Applications of Homogeneous Balls 26371:for the electromagnetic field, namely the 18928: 18389: 17716:{\displaystyle \phi =\phi _{1}+\phi _{2}.} 17033:In the Cartesian plane, rotation of point 15149:. Energy conservation requires that | 13591:relative to the x-axis of reference frame 13537:Figure 6-1. Einstein's 1905 derivation of 12864:if the receiver and the source are moving 11946:is the speed of light in still water, and 9077:to find the relation between the lengths Δ 8719:{\displaystyle ={\frac {fL}{v}}(1+\beta )} 8152:{\displaystyle ={\frac {fL}{v}}(1-\beta )} 4272: 4261: 4121:, but the frames are actually equivalent. 1347:in a reference frame moving at a velocity 1250:Derivations of the Lorentz transformations 971: 470: 456: 35326:Tests of relativistic energy and momentum 32828:Relativity Calculator: Special Relativity 32789:. Einstein's Theory of Special Relativity 32706: 32638: 31990: 31836: 31816: 31339: 31284: 31165:The Feynman Lectures on Physics: Volume 1 30954: 30789:. University of Pittsburgh. p. 107. 30686: 30531:Halliday, David; Resnick, Robert (1988). 30233: 30223: 30191: 30103:"The Genesis of the Theory of Relativity" 30075: 29906: 29834: 29706: 29688: 29686: 29426: 29346: 29281: 29252: 29200: 29167: 29165: 29145: 28758:. Leiden: E. J. Brill. (subsection § 31). 28495: 28493: 28491: 28489: 28487: 28485: 28483: 28337:for non-coinciding events), and that the 28200: 27455: 27230: 27204: 27190: 27160: 25961:{\displaystyle \Lambda ^{\mu '}{}_{\nu }} 25920:{\displaystyle \Lambda _{\mu '}{}^{\nu }} 25799:only in Cartesian coordinates. It is the 24410:{\displaystyle \Lambda ^{\mu '}{}_{\nu }} 23051:Figure 10–2. Three-dimensional dual-cone. 22475:– testing the limiting speed of particles 22473:Tests of relativistic energy and momentum 20143:so that one may also see it expressed as 18322:In other words, Lorentz boosts represent 15040:is the acceleration of the spaceship and 13856:that is moving along the x-axis at speed 13058: 12829: 9832: 9742: 9706: 9480: 9387:{\displaystyle \mathbf {|u|} =u=dx/dt\,.} 9380: 9261:in standard configuration. A particle in 8944:{\displaystyle T'={\frac {2L}{\gamma v}}} 7467:{\displaystyle T'={\frac {2L}{\gamma v}}} 6792:Observer A holds a light-clock of length 6763: 6645: 6458: 5775:{\displaystyle |\Delta x/\Delta t|>c,} 5544: 5495:{\displaystyle |\Delta x/\Delta t|<c,} 5352: 5330: 5249: 5175: 5112: 5066: 5034: 4992: 4940: 4877: 4820: 4753: 4457: 4453: 3870:{\displaystyle (k\beta \gamma ,k\gamma )} 3791:as measured in the unprimed frame, where 3784:{\displaystyle (k\gamma ,k\beta \gamma )} 2754: 2587: 2408: 2241: 1044: 35794:Covariance and contravariance of vectors 34002:Emergency Committee of Atomic Scientists 32601: 32320:Henri Poincaré and the Relativity Theory 31876: 31874: 31872: 31516: 31266:Space Time Travel: Relativity visualized 30813: 30744: 30663: 30586: 30297:. New York: W. W. Norton & Company. 30100: 29972:Introduction to the Theory of Relativity 29945: 29939: 29824:, Chicago: University of Chicago Press, 29800:(3rd ed.). Incomprehensible Books. 29748: 29572: 29320:(1999). "Special Theory of Relativity". 29119: 29043: 28593: 28541:, Vol. 79, No. 1 (Jul. 1954), pp. 32–39. 28351:. It is immediate that the negative of Δ 24529:Examples of other 4-vectors include the 23596: 23046: 22549: 22534: 22277:and quantum mechanics can be unified is 22225:field in a comoving frame of reference. 22176:Relativity and unifying electromagnetism 21511: 20861: 20787: 20781:The Dewan–Beran–Bell spaceship paradox ( 19507: 19359:Invariance of the magnitude of a vector: 16867: 15206: 15202: 15127: 15090:particles, requires special relativity. 15073:is observed to travel much farther than 14925:Space travel under constant acceleration 13299: 13292:contracted, an approaching object would 13270: 13236: 13188: 12803:Relativistic longitudinal Doppler effect 12443: 11864: 11581: 10108:not aligned along the x-axis, we write: 8842:{\displaystyle ={\frac {2fL}{\gamma v}}} 7175: 6769: 6739: 6353: 5978:{\displaystyle v=c^{2}\Delta t/\Delta x} 986: 48:, the paper founding special relativity. 29: 35263:Lorentz-violating neutrino oscillations 32977:visualizing the Lorentz transformation. 32745:On the Electrodynamics of Moving Bodies 32675: 32241:An Introduction to Quantum Field Theory 32043:. University of California at Riverside 31708:Thomas' Calculus: Early Transcendentals 31690: 31486: 31055:Janssen, Michel; Stachel, John (2004). 30934: 30855:. In Kox, A.J.; Eisenstaedt, J (eds.). 30850: 30146: 29795: 29789: 29587: 29385:The Variational Principles of Mechanics 29379: 29316: 29068: 29004: 28998: 28967: 28961: 28762: 28747: 28706: 28662: 28571:Einstein, Autobiographical Notes, 1949. 28522:On the Electrodynamics of Moving Bodies 28514:On the Electrodynamics of Moving Bodies 27806:{\displaystyle E_{\text{rest}}=mc^{2}.} 23724: 22330:, it also led to the prediction of the 22279:one of the unsolved problems in physics 21698:{\displaystyle c^{2}t^{2}-x^{2}=s^{2},} 20475:This is the derivative of the momentum 15648:, and factoring out common terms from ( 6956:{\textstyle \pm {\sqrt {c^{2}-v^{2}}},} 6093:{\displaystyle c^{2}\Delta t/\Delta x,} 3715:{\displaystyle (\gamma ,\beta \gamma )} 3603:{\displaystyle (\beta \gamma ,\gamma )} 1220:Relativity without the second postulate 739:a single continuum known as "spacetime" 504:On the Electrodynamics of Moving Bodies 14: 36155: 34918: 33741:Albert Einstein World Award of Science 33219:Atomic, molecular, and optical physics 32444: 31960:Extract of page 195 (with units where 31763: 31016: 31010: 30786:Four Decades of Scientific Explanation 30656:The Theory of the Relativity of Motion 30652: 30431: 30292: 30078:Secrets of the Old One: Einstein, 1905 30050: 29991: 29892: 29683: 29512: 29453: 29162: 29088: 28480: 27323:is an invariant. Notice that when the 27039:Relativistic kinematics and invariance 26355:is another second order antisymmetric 26329:. All tensors transform by this rule. 19596:{\displaystyle dS\equiv (dt,dx,dy,dz)} 16605:{\displaystyle \beta _{2}=\beta _{3},} 14919:How far can you travel from the Earth? 13914:before emission of the two pulses and 13792:after emission of the two pulses. Let 12819:from each other with a relative speed 11557:of the spin of a particle following a 8895:Twin's calculation as to how much the 6277:{\displaystyle |\Delta x/\Delta t|=c,} 6186:{\displaystyle c^{2}\Delta t/\Delta x} 4430:between any Cartesian inertial frame. 1283:Lorentz transformation and its inverse 910:made in almost all theories of physics 806:. But general relativity incorporates 595:. These led to the development of the 35418: 35331:Kaufmann–Bucherer–Neumann experiments 35300:Experimental testing of time dilation 35258:Antimatter tests of Lorentz violation 35233:Modern searches for Lorentz violation 35145: 34892: 34036: 33354: 33002: 32981:Original interactive FLASH Animations 32851:MathPages – Reflections on Relativity 32408: 32113:. W.H. Freeman & Co. p. 58. 32068:Archive for History of Exact Sciences 31869: 31655: 31628: 30987: 30893:(2nd ed.). Springer. p. 8. 30880: 30859:. Boston: Birkhäuser. pp. 1–13. 30692: 30589:"L'Évolution de l'espace et du temps" 30555: 30366: 30197: 29974:, Dover edition, Chapter IV, page 36 29695:Archive for History of Exact Sciences 29291:(4th ed.). Pearson. Chapter 12. 26561:{\displaystyle \eta ^{\alpha \beta }} 26359:, with six components: three for the 22503:Modern searches for Lorentz violation 22485:Experimental testing of time dilation 22376:Criticism of the theory of relativity 18158:and the inverse Lorentz boost in the 15123: 15084: 9005: 7715:Number of pulses received at initial 6788:Consider the scenario illustrated in 6018:{\textstyle {\sqrt {-\Delta s^{2}}},} 5782:and given the Lorentz transformation 5502:and given the Lorentz transformation 4418:, in the x-direction) with all other 4151: 1960:, and the differences are defined as 1637:coordinates are unaffected; only the 802:are negligible) and in conditions of 32822:The Hogg Notes on Special Relativity 32732:Einstein's original work in German, 32372:(2nd ed.), Oxford University Press. 31880: 31766:A first course in general relativity 31430: 31290: 31225:"Appearances at Relativistic Speeds" 31124: 31118: 31086: 30914:Seidelmann, P. Kenneth, ed. (1992). 30669: 29996:Space and Time in Special Relativity 29482: 29387:(4th ed.). Dover Publications. 28930: 28899: 28819: 28813: 27540:Relativistic dynamics and invariance 26373:electromagnetic stress–energy tensor 22939:invariance of any spacetime interval 22499:– testing isotropy of space and mass 19972:momentarily comoving reference frame 16872:Figure 7–2. Plot of the three basic 16754: 16620: 16339: 16155: 15971: 15889: 15787: 15662: 15473: 15324: 15221: 14759: 14634: 14325: 13233:Measurement versus visual appearance 10782: 10495: 10279: 10110: 9987: 9631: 9409: 9329: 9240:Lorentz transformation of velocities 8284:Number of signals received at final 983:Reference frames and relative motion 899:Maxwell's theory of electromagnetism 35399:Test theories of special relativity 34627:Tolman–Oppenheimer–Volkoff equation 34580:Friedmann–Lemaître–Robertson–Walker 33380: 32900:Lecture Notes on Special Relativity 32566: 32554:(4th ed.). W. H. Freeman & Co. 32539:(2nd ed.). W. H. Freeman & Co. 31222: 31161:"Relativistic Effects in Radiation" 29873:Introduction to Classical Mechanics 29177:"A Theory of Electrons and Protons" 28072:Moving magnet and conductor problem 22085:from step 2 and the expression for 19610:This results when the displacement 18827:in the earlier presentation of the 11854: 11355:There is nothing special about the 7594:Time when traveler's turnaround is 7522:{\displaystyle fT={\frac {2fL}{v}}} 5674:{\displaystyle v=\Delta x/\Delta t} 4406:Lorentz transform (also called the 1842:There is nothing special about the 1839:, as measured in the primed frame. 1183:Lack of an absolute reference frame 24: 35657:Tensors in curvilinear coordinates 33770:Albert Einstein: Creator and Rebel 33483:Einstein–Infeld–Hoffmann equations 33453:Einstein relation (kinetic theory) 32729:Zur Elektrodynamik bewegter Körper 32477:. University of California Press. 32370:Introduction to Special Relativity 32283: 30857:The universe of general relativity 30024:Introduction to special relativity 28505:Zur Elektrodynamik bewegter Körper 28474: 27826:If a particle is not traveling at 27072: 26650: 26626: 26538:which is equal to its reciprocal, 26296: 26255: 26181: 26154: 26130: 26106: 26079: 26055: 25935: 25894: 25840: 25770: 25766: 25752: 25735: 25708: 25684: 25530: 25522: 25506: 25498: 25482: 25474: 25458: 25450: 25400: 25392: 25371: 25363: 25342: 25334: 25313: 25305: 24907:{\displaystyle E=\gamma (v)mc^{2}} 24484: 24384: 24309: 22243:form, that is, in the language of 22209:The Lorentz transformation of the 20771:Dewan–Beran–Bell spaceship paradox 19977:Since photons move on null lines, 19528:or for infinitesimal separations, 19524:Δt, Δx, Δy, Δz 18904: 18895: 18886: 18877: 18805:which comes from simply replacing 16543:For the symmetrical case in which 11849: 11436:in its rest frame (i.e., having a 11406:Figure 4-5. Thomas–Wigner rotation 11370: 9153: 9115: 9098: 6853:{\displaystyle t_{\text{A}}=2L/c.} 6675: 6646: 6628: 6600:is at rest in the unprimed system 6459: 6443: 6421: 6254: 6243: 6177: 6166: 6107: 6081: 6070: 5997: 5969: 5958: 5908: 5824: 5812: 5789: 5752: 5741: 5688: 5665: 5654: 5614: 5567:it is evident that there exists a 5545: 5532: 5509: 5472: 5461: 5415: 5382: 5340: 5318: 5237: 5163: 5100: 5054: 5022: 4980: 4928: 4904: 4893: 4865: 4808: 4780: 4769: 4741: 4685: 4668: 4613: 4592: 4546: 4530: 4484: 4468: 4440: 4372: 4334: 4318: 4302: 4283: 4248: 4217: 4186: 4163: 2488: 2468: 2448: 2409: 2391: 2373: 2319: 2304: 2279: 2242: 2229: 2206: 2141: 2097: 2026: 1974: 1355:-axis with respect to that frame, 46:Zur Elektrodynamik bewegter Körper 25: 36179: 35357:Michelson–Gale–Pearson experiment 35268:Lorentz-violating electrodynamics 35248:Experiments of Rayleigh and Brace 34397:Hamilton–Jacobi–Einstein equation 33962:Einstein–Oppenheimer relationship 33631:The Einstein Theory of Relativity 32963:Warp Special Relativity Simulator 32721: 32716: 32134:J.R. Forshaw; A.G. Smith (2009). 31791: 31403: 31259: 31089:"The Relativistic Doppler Effect" 30998:. The Royal Observatory Greenwich 30993: 30717: 30435:Astronomy, A Physical Perspective 28270:experiments of Rayleigh and Brace 22524:Technical discussion of spacetime 22455:Experiments of Rayleigh and Brace 21909:From the relativistic force law, 21483:Accelerated observer with horizon 20753:Acceleration (special relativity) 19497:{\displaystyle E^{2}-p^{2}c^{2}.} 18935:Closure under linear combination: 18480:is a "4-vector" if its component 18382:spacetime, and not just within a 15608:) into its components, replacing 13478:Einstein's 1905 demonstration of 11199:{\displaystyle \mathbf {u=u'+v} } 11090:can be interpreted as giving the 9027:is at rest and aligned along the 8988:{\displaystyle T={\frac {2L}{v}}} 7413:{\displaystyle T={\frac {2L}{v}}} 7151:A's and B's measurement of time. 6362:. In a reference frame moving at 5438:, there are three cases to note: 901:and the lack of evidence for the 834:Galileo's principle of relativity 642:, the speed of causality and the 583:appeared to be incompatible with 44:" were published. These included 36140: 36128: 36116: 36104: 36092: 36080: 35310:Length contraction confirmations 35207:de Sitter double star experiment 34875: 34874: 34015: 34014: 33746:Einstein Prize for Laser Science 32918: 32888:Understanding Special Relativity 32882:Special Relativity Lecture Notes 32681:"Special relativity: Kinematics" 32253: 32230: 32199: 32179: 32152: 32127: 32102: 32062:John D. Norton, John D. (2004). 32055: 32035:Roberts, Tom; Schleif, Siegmar. 31968: 31934: 31903: 31804:from the original on 7 June 2017 31699: 31684: 31600: 31461: 31199: 29217: 29181:Proceedings of the Royal Society 29096:. Physics Department, ENS Cachan 29082: 28461: 28432: 28422: 28387:four-dimensional Euclidean space 28341:-denotation is misleading. This 28213:{\displaystyle \mathbb {R} ^{4}} 27676: 27561: 27393: 27134: 26838: 25152: 24948: 24925: 24701: 24252: 22652: 22641: 22427:for a more detailed discussion. 22258:relativity can be combined with 20224:{\displaystyle (E,{\vec {p}}c).} 19948: 19936: 19444:{\displaystyle c^{2}t^{2}-x^{2}} 16820: 16781:Figure 7-1a. A ray through the 16774: 13530: 13519: 12430:Relativistic aberration of light 11620: 11611: 11396: 11385: 11192: 11189: 11182: 11178: 11175: 11146: 11123: 11101: 10291: 10121: 10094: 9426: 9345: 9299: 9273: 8881:{\displaystyle {\frac {2fL}{v}}} 8733:Total number of received pulses 6582: 2922: 2913: 2902: 2893: 1918:, another event has coordinates 869:Postulates of special relativity 785:accelerating frames of reference 733:of Newtonian mechanics with the 437: 436: 423: 62: 33791:Einstein: His Life and Universe 33340:Timeline of physics discoveries 31910:R. Resnick; R. Eisberg (1985). 31455: 31424: 31410:The Original Usenet Physics FAQ 31397: 31372: 31253: 31216: 31193: 31080: 31048: 30996:"Airy's Water Telescope (1870)" 30776: 30738: 30711: 30646: 30549: 30456: 30425: 30394: 30250: 30014: 29964: 29814: 29739: 29657: 29633: 29608: 29558:(2nd ed.). W. H. Freeman. 29516:Relativity Made Relatively Easy 29506: 29476: 29464: 29288:Introduction to Electrodynamics 28856: 28639: 28618: 28583: 28574: 28553: 28544: 28409: 28392: 28362: 28319: 28275: 28236: 28223: 28185: 28175: 28116:Relativity in the APS formalism 27940:In a continuous medium, the 3D 27544:The invariant magnitude of the 25750: 24783: 23359: 23042: 22519:– no "aether flow obstruction". 22152:which is a constant expression. 20746: 19103:{\displaystyle A\cdot B\equiv } 18854:The last three components of a 16827:Figure 7-1b. A ray through the 16631: 16630: 16350: 16349: 16166: 16165: 15982: 15981: 15900: 15899: 15798: 15797: 15673: 15672: 15484: 15483: 15335: 15334: 15272: 15266: 15232: 15231: 14770: 14769: 14645: 14644: 14336: 14335: 11966:is the speed of the water, and 11347:Attaching a frame to a photon ( 10964: 10875: 10671: 10585: 7154: 929:Special principle of relativity 35378:Refutations of emission theory 35217:Measurements of neutrino speed 34204:Mass–energy equivalence (E=mc) 34062: 33413:Mass–energy equivalence (E=mc) 32266:. Addison Wesley. p. 22. 31798:The Physics and Relativity FAQ 31629:Idema, Timon (17 April 2019). 31557:Fernflores, Francisco (2018). 29401: 29340: 29120:Champion, Frank Clive (1932). 29089:Baglio, Julien (26 May 2007). 28531: 28052:Einstein's thought experiments 27740: 27730: 27304: 27294: 27282: 27272: 27260: 27250: 27238: 27224: 26843: 26833: 25747: 25731: 25701: 25680: 25065: 25020: 25017: 25011: 24941: 24935: 24888: 24882: 24737: 24731: 24705: 24691: 24688: 24682: 24673: 24628: 24625: 24619: 24256: 24239: 24233: 24206: 24200: 24148: 22345:relativistic quantum mechanics 22264:relativistic quantum mechanics 22072: 22063: 22035: 22026: 22015: 21993: 21984: 21975: 21724: 21715: 20676: 20598: 20439: 20433: 20421: 20362: 20356: 20338: 20215: 20206: 20191: 20177:{\displaystyle (E,{\vec {p}})} 20171: 20165: 20150: 20120: 20067: 20061: 20055: 20032: 19914: 19908: 19890: 19745: 19709: 19590: 19554: 19268: 19253: 19059: 19030: 18910: 18871: 18702: 18688: 18625: 18611: 18526: 18499: 17655: 17629: 16744: 16614: 16533: 16449: 16429: 16399: 16380: 16370: 16358: 16330: 16270: 16258: 16146: 16086: 16074: 15438: 15403: 15362: 15260: 15239: 14953: 14947: 14395: 14369: 14363: 14337: 14200: 14186: 14109: 14095: 13673: 13659: 13373:Equivalence of mass and energy 12662: 12650: 12636: 12627: 12557: 12543: 12110:Fresnel's dragging coefficient 11224:We note the following points: 11210:, valid for the three axes of 11160:{\displaystyle \mathbf {u'} ,} 10458: 10362: 10356: 10302: 10245: 10179: 10173: 10128: 9713: 9691: 9179:{\displaystyle \Delta t'=0\ .} 8780: 8760: 8713: 8701: 8601: 8581: 8488: 8468: 8382: 8370: 8146: 8134: 8033: 8013: 7920: 7900: 7813: 7801: 7334: 7322: 7314: 7302: 7260: 7248: 7240: 7228: 6261: 6239: 5845: 5809: 5759: 5737: 5551: 5529: 5479: 5457: 4947: 4862: 4697: 4665: 4347: 4299: 4008: 3960: 3941: 3933: 3914: 3864: 3843: 3778: 3757: 3709: 3694: 3674: 3662: 3642: 3617: 3597: 3582: 3562: 3550: 3530: 3505: 3383: 3377: 3077:axis is scaled by a factor of 2766: 2734: 2594: 2572: 2420: 2388: 2248: 2226: 1769: 1744: 1724: 1684: 1653:, that parameter being called 1471: 1456: 1428: 1398: 1291:to have spacetime coordinates 13: 1: 35710:Exterior covariant derivative 35642:Tensor (intrinsic definition) 32925:Raytracing Special Relativity 32420:World Scientific Publications 32009:10.1016/S0370-2693(97)00638-2 31585:Philip Gibbs & Don Koks. 31358:10.1088/0143-0807/35/6/065025 30500:. Basic Books. p. 15-5. 30055:. Mew York: Springer-Verlag. 29264: 28512:17: 891; English translation 28008:History of special relativity 26338:relativistic angular momentum 25251:The transformation rules for 24721:where the Lorentz factor is: 22190:Theoretical investigation in 21547:approaches a limit of one as 21065:{\displaystyle x_{B}=x_{A}+L} 20891:acceleration for proper time 20867:and after acceleration stops. 17963:{\displaystyle =\sinh \phi .} 17838:{\displaystyle =\cosh \phi ,} 16938:. Numerically, the angle and 16618:) takes on the simpler form: 16562:{\displaystyle \phi =\theta } 15959: 15880: 15778: 15656: 15650: 15604: 15592: 15460: 15311: 14850: 14747: 14476: 11632:"instantaneous communicators" 11167:and they replace the formula 9510:Substituting expressions for 9310:{\displaystyle \mathbf {u'} } 9283:{\displaystyle \mathbf {u} .} 9189:This shows that the length (Δ 9071:, which can be combined with 9031:-axis in the unprimed system 6696:{\displaystyle \Delta x=0\ .} 6231:separated. This implies that 5729:separated. This implies that 5449:separated. This implies that 4391:{\displaystyle \Delta s^{2},} 1127:The origins of frames S and S 931:: If a system of coordinates 571:History of special relativity 35735:Raising and lowering indices 35373:Refutations of aether theory 35295:Moessbauer rotor experiments 35197:Moessbauer rotor experiments 35192:Kennedy–Thorndike experiment 32595:10.1016/0031-9163(64)91095-9 32463:Silberstein, Ludwik (1914). 32327: 32317:Logunov, Anatoly A. (2005). 32022:An overview can be found on 31693:The Two-Body Force in Nuclei 31678:10.1016/0370-1573(94)00068-9 30432:Cutner, Mark Leslie (2003). 30260:An Introduction to Mechanics 30204:Living Reviews in Relativity 28229:In a spacetime setting, the 28159:Velocity composition paradox 28082:Relativistic heat conduction 28062:Relativistic Euler equations 27821:Newton's third law of motion 26688:raising and lowering indices 26353:electromagnetic field tensor 26288:is the reciprocal matrix of 25927:is the reciprocal matrix of 24067:and we have 4d vectors, or " 22943:any inertial reference frame 22491:Kennedy–Thorndike experiment 22294:Bohr–Sommerfeld atomic model 21630:{\displaystyle \beta =ct/x.} 21072:are the ships' positions in 18332: 17241:is the analog of slope. The 15109:collisions are discussed in 15032:) is the velocity at a time 14486:The two differences of form 12117: 11296:| (and then also | 11130:{\displaystyle \mathbf {v} } 11108:{\displaystyle \mathbf {u} } 11086: 11080: 11069: 10773: 10483: 10270: 10101:{\displaystyle \mathbf {u} } 10076: 7483:Total number of pulses sent 5431:{\displaystyle \Delta s^{2}} 5398:{\displaystyle \Delta s^{2}} 4233:{\displaystyle \Delta s^{2}} 4017:{\displaystyle \beta \to 1.} 1610:in vacuum, and the velocity 524:inertial frames of reference 488:special theory of relativity 18:Special Theory of Relativity 7: 35973:Gluon field strength tensor 35444: 35187:Michelson–Morley experiment 35173:Tests of special relativity 35050:Relativistic Doppler effect 34994:Inertial frame of reference 34219:Relativistic Doppler effect 33736:Albert Einstein Peace Prize 33513:Unsuccessful investigations 33304:Quantum information science 32750:The Principle of Relativity 31825:European Journal of Physics 31764:Schutz, Bernard F. (1985). 31561:. New York: Momentum Pres. 31328:European Journal of Physics 31293:American Journal of Physics 30943:European Journal of Physics 30653:Tolman, Richard C. (1917). 30620:American Journal of Physics 30466:Flat and Curved Space-times 29946:Callahan, James J. (2011). 29895:American Journal of Physics 28822:American Journal of Physics 28038:Relativity priority dispute 27955: 26686:The metric can be used for 25673:which is the transpose of: 22610:of distance (line element) 22372:Tests of special relativity 22362:throughout space and time. 22315:constructed an influential 22145:{\displaystyle F=mc^{2}/s,} 21501:. In the text accompanying 19394:{\displaystyle A\cdot A=0,} 19326:{\displaystyle A\cdot B=0.} 19297:Two vectors are said to be 16759: 15111:Spacetime#Conservation laws 14625:{\displaystyle H_{1}-E_{1}} 14585:{\displaystyle H_{0}-E_{0}} 13354: 12798:Relativistic Doppler effect 12792:Relativistic Doppler effect 9927: 9580: 9574: 9568: 9500: 9400: 9073: 8166:Time for remainder of trip 6706:This shows that the time (Δ 6614: 6410: 6053:greater than and less than 5928:{\displaystyle \Delta t'=0} 5708:{\displaystyle \Delta s/c,} 5634:{\displaystyle \Delta x'=0} 3429:{\displaystyle \beta =v/c.} 3167:{\displaystyle {\text{B}},} 3011:axis is horizontal and the 2709: 2542: 2363: 2196: 2087: 1964: 1865:A quantity invariant under 1629:-axis. For simplicity, the 830:privileged reference frames 589:Michelson–Morley experiment 171:Relativistic Doppler effect 109:Inertial frame of reference 37:around 1905, the year his " 10: 36184: 35784:Cartan formalism (physics) 35604:Penrose graphical notation 35253:Trouton–Rankine experiment 35045:Relativity of simultaneity 34690:In computational physics: 34214:Relativity of simultaneity 33663:Picasso at the Lapin Agile 33601:Russell–Einstein Manifesto 33473:Bose–Einstein correlations 33135:Classical electromagnetism 32348:. W. H. Freeman & Co. 31855:10.1088/0143-0807/31/2/006 31738:. Dover Publications Inc. 31381:Superluminal Radio Sources 31167:. Reading, Massachusetts: 31062:. In Stachel, John (ed.). 30973:10.1088/0143-0807/26/1/020 30440:Cambridge University Press 30147:Rindler, Wolfgang (1977). 30101:Darrigol, Olivier (2005). 30076:Bernstein, Jeremy (2006). 30051:Miller, Arthur I. (1998). 29588:Rindler, Wolfgang (1977). 29513:Steane, Andrew M. (2012). 28943:Proc. Kon. Acad. Van Weten 28912:Proc. Kon. Acad. Van Weten 26363:and another three for the 22983:Misner, Thorne and Wheeler 22949:) of Minkowski spacetime. 22543: 22527: 22384:is accurate only when the 22380:Special relativity in its 22369: 22317:relativistic wave equation 22204:Liénard–Wiechert potential 22192:classical electromagnetism 22179: 22078:{\displaystyle \beta (ct)} 21486: 20774: 20750: 20722:4-vectors and physical law 19068:{\displaystyle {\vec {B}}} 19039:{\displaystyle {\vec {A}}} 18532:{\displaystyle (ct,x,y,z)} 18336: 16884:) and hyperbolic tangent ( 16763: 15095:conservation laws for mass 14922: 13498:relativistic Doppler shift 13489:In his fourth of his 1905 13376: 13112:{\displaystyle \beta =v/c} 12795: 12433: 11903:the medium plus the speed 11858: 11571: 11374: 9243: 9009: 7158: 6586: 6346:Relativity of simultaneity 6343: 6340:Relativity of simultaneity 4128: 4101:{\displaystyle {\text{B}}} 4079:{\displaystyle {\text{A}}} 3975:times the spacing between 3142:{\displaystyle {\text{A}}} 1247: 1236: 975: 866: 832:), a principle now called 816:pseudo-Riemannian manifold 616:relativity of simultaneity 568: 166:Relativity of simultaneity 35990: 35930: 35879: 35872: 35764: 35695: 35632: 35576: 35523: 35470: 35463: 35456:Glossary of tensor theory 35452: 35386: 35365: 35339: 35318: 35305:Hafele–Keating experiment 35276: 35225: 35179: 35111: 35078: 35017: 34981: 34947: 34926: 34872: 34704: 34569: 34541: 34527:Lense–Thirring precession 34410: 34359: 34321: 34300: 34289: 34247: 34191: 34175: 34117: 34109:Doubly special relativity 34081: 34070: 34010: 33919: 33828: 33759: 33713: 33610: 33571:The Meaning of Relativity 33536: 33388: 33312: 33249: 33177: 33093: 33065: 33037: 32785:Audio: Cain/Gay (2006) – 32708:10.4249/scholarpedia.8520 32474:Space, Time and Spacetime 32305:The Meaning of Relativity 32303:Einstein, Albert (1996). 32294:Einstein, Albert (1920). 32088:10.1007/s00407-004-0085-6 31947:. Springer. p. 195. 31587:"The Relativistic Rocket" 31536:. Springer. p. 221. 31437:Physics 200 Lecture Notes 31202:"Relativistic Distortion" 30887:Richard A. Mould (2001). 30783:Wesley C. Salmon (2006). 30027:. Wiley. pp. 62–63. 29992:Mermin, N. David (1968). 29725:10.1007/s00407-009-0050-5 29323:Classical Electrodynamics 29029:10.1017/S0305004100033776 28748:Lorentz, Hendrik (1895). 28244:George Francis FitzGerald 28013:Doubly special relativity 26387:allows one to define the 26378: 25980:tensor transformation law 24555:{\displaystyle U^{\mu },} 22365: 20710:corresponding to spatial 20016:Energy–momentum 4-vector: 18999:Inner-product invariance: 18843:, etc. we generally mean 18549:if it transforms (in the 17729:factor can be written as 16908:. Fig. 7-1a shows a 13185:Transverse Doppler effect 11643:of A and B are vertical ( 11527:{\displaystyle Lv/c^{2},} 11365:Velocity-addition formula 9246:Velocity-addition formula 9211:) in its own rest frame ( 8550:{\displaystyle f''t_{2}'} 6121:{\displaystyle \Delta t'} 1131:are coincident when time 1059:Galilean reference frames 36040:Gregorio Ricci-Curbastro 35912:Riemann curvature tensor 35619:Van der Waerden notation 35404:Standard-Model Extension 35290:Ives–Stilwell experiment 35243:Trouton–Noble experiment 35238:Hughes–Drever experiment 35065:Bell's spaceship paradox 34387:Post-Newtonian formalism 34377:Einstein field equations 34313:Mathematical formulation 34137:Hyperbolic orthogonality 33967:Albert Einstein Archives 33892:Bernhard Caesar Einstein 33639:Relics: Einstein's Brain 33587:The Evolution of Physics 33468:Bose–Einstein statistics 33463:Bose–Einstein condensate 33443:Einstein field equations 33241:Condensed matter physics 32661:10.1103/PhysRevA.56.4405 32511:Sergey Stepanov (2018). 32465:The Theory of Relativity 32358:Post, E.J., 1997 (1962) 32260:Sean M. Carroll (2004). 31695:. Boston, MA.: Springer. 31523:John J. Stachel (2002). 31019:"Airy's water telescope" 30745:Ginsburg, David (1989). 30263:. McGraw-Hill. pp. 29433:Albert Einstein (2001). 28992:10.1103/PhysRev.116.1041 28600:Yaakov Friedman (2004). 28266:Trouton–Noble experiment 28169: 28154:Bell's spaceship paradox 28028:Rietdijk–Putnam argument 28023:Einstein synchronisation 27550:energy–momentum relation 25814:Lorentz transformation: 24447:{\displaystyle T^{\nu }} 24099:and spacelike component 22999:as the time coordinate. 22515:Experiments to test the 22497:Hughes–Drever experiment 22479:Ives–Stilwell experiment 22448:Trouton–Noble experiment 21945:{\displaystyle F=dp/dt=} 20783:Bell's spaceship paradox 20777:Bell's spaceship paradox 20235:is a conserved quantity. 19999:{\displaystyle d\tau =0} 18835:As usual, when we write 18326:in Minkowski spacetime. 15660:) yields the following: 13506:conservation of momentum 13260:on the one hand, versus 11986:{\displaystyle u_{\pm }} 11841:This is not to say that 11539:spin of moving particles 8339:{\displaystyle f''t_{2}} 7982:{\displaystyle f't_{1}'} 7118:Reciprocal time dilation 4195:{\displaystyle \Delta t} 4172:{\displaystyle \Delta r} 3648:{\displaystyle (x',ct')} 3536:{\displaystyle (x',ct')} 1209:absolute reference frame 1193:inertial reference frame 1039:transformation equations 760:'s papers on spacetime. 731:Galilean transformations 695:{\displaystyle E=mc^{2}} 565:Origins and significance 534:). This is known as the 186:Bell's spaceship paradox 36010:Elwin Bruno Christoffel 35943:Angular momentum tensor 35614:Tetrad (index notation) 35584:Abstract index notation 35035:Mass–energy equivalence 34968:Galilean transformation 34934:Principle of relativity 34098:Galilean transformation 34089:Principle of relativity 33977:Einstein Papers Project 33488:Einstein–de Haas effect 32950:Through Einstein's Eyes 32492:Lawrence Sklar (1992). 32471:Lawrence Sklar (1977). 32307:. Fine Communications. 32136:Dynamics and Relativity 30587:Langevin, Paul (1911). 30471:Oxford University Press 30367:Cuvaj, Camillo (1971). 30021:Robert Resnick (1968). 29796:Collier, Peter (2017). 29229:"The Positive Electron" 28937:Zeeman, Pieter (1915). 28906:Zeeman, Pieter (1914). 28707:Lorentz, H. A. (1904). 27384:has an invariant form: 27373:is differential of the 27348:is the differential of 23721:in special relativity. 22390:gravitational potential 22268:quantum electrodynamics 22098:{\displaystyle \gamma } 21583:{\displaystyle \gamma } 21095:the positions in frame 20904:{\displaystyle \sigma } 19516:Otherwise known as the 18982:{\displaystyle C=aA+aB} 18929:Properties of 4-vectors 18830:Lorentz transformation. 18390:Definition of 4-vectors 15628:with the dimensionless 13571:traveling in direction 13379:Mass–energy equivalence 13344:relativistic-speed jets 13222:{\displaystyle \gamma } 12968:is the speed of sound. 12104:depends on wavelength, 12082:partial aether dragging 11578:Tachyonic antitelephone 11485:in the frame of the rod 7770:{\displaystyle f't_{1}} 6215:{\displaystyle v>c.} 4408:Poincaré transformation 1867:Lorentz transformations 1189:principle of relativity 1001:principle of relativity 978:Principle of relativity 972:Principle of relativity 959:Lorentz transformations 880:principle of relativity 735:Lorentz transformations 664:mass–energy equivalence 640:mass–energy equivalence 636:a universal speed limit 607:relativistic velocities 597:Lorentz transformations 536:principle of relativity 300:Galilean transformation 161:Mass–energy equivalence 74:Principle of relativity 35824:Levi-Civita connection 35394:One-way speed of light 35009:Lorentz transformation 34183:Lorentz transformation 33904:Thomas Martin Einstein 33812:Introducing Relativity 33784:Einstein for Beginners 33687:Einstein and Eddington 33478:Einstein–Cartan theory 33325:Nobel Prize in Physics 33187:Relativistic mechanics 32533:John Archibald Wheeler 32445:French, A. P. (1968). 32342:John Archibald Wheeler 32138:. Wiley. p. 247. 31017:Hollis, H. P. (1937). 30293:French, A. P. (1968). 29970:P. G. Bergmann (1976) 29950:. New York: Springer. 29254:10.1103/PhysRev.43.491 29202:10.1098/rspa.1930.0013 29147:10.1098/rspa.1932.0108 28885:10.1103/PhysRevD.2.263 28663:Lorentz, H.A. (1902). 28539:The Scientific Monthly 28214: 27905: 27807: 27756: 27658: 27530: 27463: 27317: 27112: 27028: 26878: 26809: 26694:with another 4-vector 26677: 26562: 26532: 26323: 26282: 26241: 25962: 25921: 25880: 25793: 25667: 25243: 25168: 24956: 24908: 24855: 24715: 24556: 24521: 24456:Lorentz transformation 24448: 24411: 24370: 24346: 24266: 24058: 23688: 23602: 23589: 23483: 23346: 23261: 23150: 23052: 22933:, so spacetime uses a 22881: 22726: 22592: 22517:aether drag hypothesis 22505:– various modern tests 22146: 22099: 22079: 22045: 21946: 21901: 21840: 21785: 21699: 21631: 21584: 21564: 21541: 21540:{\displaystyle \beta } 21523: 21471: 21446: 21413: 21393: 21360: 21337: 21313: 21114: 21089: 21066: 21020: 20993: 20961: 20938: 20905: 20885: 20868: 20793: 20691: 20627: 20543: 20496: 20495:{\displaystyle \tau .} 20462: 20372: 20311: 20264: 20263:{\displaystyle \tau .} 20241:Acceleration 4-vector: 20225: 20178: 20127: 20000: 19921: 19878: 19774: 19654: 19653:{\displaystyle d\tau } 19631: 19630:{\displaystyle d\tau } 19597: 19514:Displacement 4-vector: 19498: 19445: 19395: 19327: 19278: 19212: 19104: 19069: 19040: 18983: 18917: 18796: 18533: 18472: 18313: 18149: 17964: 17933: 17839: 17808: 17717: 17665: 17611: 17511: 17421: 17294: 17235: 17234:{\displaystyle \beta } 17212: 17024: 16889: 16880:), hyperbolic cosine ( 16732: 16606: 16563: 16521: 16318: 16134: 15947: 15868: 15766: 15642: 15641:{\displaystyle \beta } 15622: 15580: 15448: 15299: 15212: 15133: 15018: 14838: 14735: 14626: 14586: 14546: 14526: 14506: 14464: 14314: 13935: 13901: 13870: 13846:after their emission. 13840: 13813: 13782: 13754: 13727: 13616: 13585: 13565: 13502:conservation of energy 13491:Annus mirabilis papers 13321:Terrell–Penrose effect 13309: 13276: 13242: 13223: 13194: 13181:with a moving source. 13163: 13113: 13076: 12962: 12935: 12854: 12834: 12782: 12672: 12576: 12449: 12420: 12355: 12269: 12205: 12098: 12072: 11987: 11960: 11940: 11939:{\displaystyle u'=c/n} 11870: 11814: 11789: 11761: 11741: 11712: 11681: 11595:same location in space 11590: 11528: 11477: 11454: 11430: 11228:If an object (e.g., a 11200: 11161: 11131: 11115:of the two velocities 11109: 11057: 10761: 10471: 10258: 10102: 10064: 9980: 9951: 9915: 9623: 9598: 9560: 9532: 9488: 9388: 9311: 9284: 9180: 9131: 8989: 8945: 8901:twin should have aged 8882: 8843: 8801: 8720: 8671: 8551: 8509: 8438: 8340: 8303: 8269: 8215: 8153: 8103: 7983: 7941: 7869: 7771: 7734: 7700: 7646: 7581: 7523: 7468: 7414: 7346: 7272: 7181: 7137: 7103: 7032: 6957: 6907: 6886: 6854: 6806: 6775: 6764:Langevin's light-clock 6745: 6697: 6656: 6577:fiber optic gyroscopes 6565: 6487: 6397: 6328: 6308: 6307:{\displaystyle s^{2}.} 6278: 6216: 6187: 6149:could be greater than 6143: 6122: 6094: 6047: 6019: 5979: 5929: 5895: 5875: 5855: 5776: 5709: 5675: 5635: 5601: 5581: 5561: 5496: 5432: 5399: 5367: 4501: 4392: 4354: 4234: 4196: 4173: 4141:Lorentz transformation 4135:Relativistic mechanics 4102: 4080: 4058: 4018: 3992: 3969: 3899: 3871: 3830: 3805: 3785: 3744: 3716: 3681: 3649: 3604: 3569: 3537: 3487: 3456: 3430: 3393: 3342: 3317: 3289: 3288:{\displaystyle v=c/2.} 3255: 3233:axes of frame S'. The 3227: 3199: 3168: 3143: 3121: 3091: 3071: 3048: 3025: 3005: 2985: 2965: 2875: 2782: 2701: 2613: 2529: 2436: 2355: 2267: 2183: 2079: 1860:Lorentz transformation 1827: 1592: 1529: 1364:Lorentz transformation 1261: 1239:Lorentz transformation 1055:standard configuration 1045:Standard configuration 1008: 952: 861:Autobiographical Notes 856: 808:non-Euclidean geometry 773:energy–momentum tensor 771:(a consequence of the 769:curvature of spacetime 716: 696: 662:, as expressed in the 124:Lorentz transformation 49: 36050:Jan Arnoldus Schouten 36005:Augustin-Louis Cauchy 35485:Differential geometry 35202:Resonator experiments 34651:Weyl−Lewis−Papapetrou 34392:Raychaudhuri equation 34331:Equivalence principle 33982:Einstein refrigerator 33972:Einstein's Blackboard 33846:(second wife; cousin) 33777:Einstein and Religion 33726:Albert Einstein Medal 33721:Albert Einstein Award 33579:The World as I See It 33518:Wave–particle duality 33498:Bohr–Einstein debates 33458:Cosmological constant 33438:Equivalence principle 33428:Einstein coefficients 33330:Philosophy of physics 32906:SpecialRelativity.net 32494:Philosophy of Physics 32362:. Dover Publications. 31881:Bais, Sander (2007). 30814:Lauginie, P. (2004). 30693:Morin, David (2017). 30556:Adams, Steve (1997). 30404:Understanding Physics 28518:George Barker Jeffery 28215: 27906: 27808: 27757: 27659: 27531: 27464: 27318: 27113: 27029: 26879: 26810: 26678: 26563: 26533: 26324: 26283: 26242: 25963: 25922: 25881: 25794: 25668: 25244: 25169: 24957: 24916:relativistic momentum 24909: 24856: 24716: 24557: 24522: 24449: 24412: 24371: 24347: 24267: 24073:tensor index notation 24059: 23689: 23600: 23590: 23484: 23347: 23262: 23151: 23050: 22882: 22727: 22553: 22535:Geometry of spacetime 22462:Particle accelerators 22360:created and destroyed 22334:of the electron (the 22147: 22100: 22080: 22046: 21947: 21902: 21841: 21786: 21700: 21632: 21590:approaches infinity. 21585: 21570:increases. Likewise, 21565: 21542: 21515: 21472: 21447: 21445:{\displaystyle A'B''} 21414: 21394: 21361: 21338: 21314: 21115: 21090: 21067: 21021: 21019:{\displaystyle x_{A}} 20994: 20962: 20939: 20937:{\displaystyle A'B''} 20906: 20886: 20865: 20791: 20751:Further information: 20692: 20628: 20544: 20497: 20463: 20373: 20312: 20265: 20226: 20179: 20128: 20001: 19922: 19879: 19775: 19655: 19632: 19598: 19508:Examples of 4-vectors 19499: 19446: 19396: 19333:Unlike the case with 19328: 19279: 19213: 19105: 19070: 19041: 18984: 18918: 18797: 18534: 18473: 18365:field strength tensor 18351:, and more generally 18314: 18150: 17965: 17934: 17840: 17809: 17718: 17666: 17612: 17512: 17422: 17295: 17236: 17213: 17025: 16871: 16733: 16607: 16564: 16522: 16319: 16135: 15948: 15869: 15767: 15643: 15623: 15581: 15449: 15300: 15210: 15203:Relativistic analysis 15131: 15019: 14839: 14736: 14627: 14587: 14547: 14527: 14507: 14465: 14315: 13936: 13934:{\displaystyle H_{1}} 13902: 13900:{\displaystyle H_{0}} 13871: 13841: 13839:{\displaystyle E_{1}} 13814: 13812:{\displaystyle E_{0}} 13783: 13755: 13753:{\displaystyle \phi } 13728: 13617: 13586: 13584:{\displaystyle \phi } 13566: 13303: 13274: 13240: 13224: 13192: 13164: 13114: 13077: 12963: 12961:{\displaystyle c_{s}} 12936: 12855: 12835: 12783: 12673: 12577: 12459:light-time correction 12447: 12440:Light-time correction 12421: 12356: 12270: 12206: 12099: 12073: 11988: 11961: 11941: 11868: 11815: 11790: 11762: 11742: 11713: 11682: 11585: 11529: 11478: 11455: 11431: 11201: 11162: 11132: 11110: 11058: 10762: 10472: 10259: 10103: 10065: 9981: 9979:{\displaystyle -v\ .} 9952: 9916: 9624: 9599: 9561: 9533: 9489: 9389: 9312: 9290:What is its velocity 9285: 9181: 9143:for events satisfying 9132: 9023:Similarly, suppose a 8990: 8946: 8883: 8844: 8802: 8721: 8672: 8552: 8510: 8439: 8341: 8304: 8270: 8216: 8154: 8104: 7984: 7942: 7870: 7772: 7735: 7701: 7647: 7582: 7524: 7469: 7415: 7347: 7273: 7179: 7138: 7104: 7033: 6958: 6908: 6887: 6855: 6807: 6773: 6743: 6698: 6665:for events satisfying 6657: 6618:can be used to find: 6573:ring laser gyroscopes 6566: 6488: 6403:absolute simultaneity 6390:lines of simultaneity 6357: 6329: 6309: 6279: 6217: 6188: 6144: 6123: 6095: 6048: 6020: 5980: 5930: 5896: 5876: 5856: 5777: 5710: 5676: 5636: 5602: 5582: 5562: 5497: 5433: 5400: 5368: 4502: 4393: 4355: 4235: 4197: 4174: 4103: 4081: 4059: 4019: 3993: 3970: 3900: 3872: 3831: 3806: 3786: 3745: 3717: 3682: 3680:{\displaystyle (1,0)} 3650: 3605: 3570: 3568:{\displaystyle (0,1)} 3538: 3488: 3486:{\displaystyle t'=0.} 3457: 3431: 3394: 3343: 3318: 3290: 3256: 3228: 3200: 3169: 3144: 3122: 3092: 3072: 3049: 3026: 3006: 2991:axes of frame S. The 2986: 2966: 2876: 2783: 2702: 2614: 2530: 2437: 2356: 2268: 2184: 2080: 1828: 1625:, is parallel to the 1593: 1530: 1256: 1201:electromagnetic waves 1120:as measured in frame 990: 926: 852: 717: 697: 104:Einstein's postulates 33: 36025:Carl Friedrich Gauss 35958:stress–energy tensor 35953:Cauchy stress tensor 35705:Covariant derivative 35667:Antisymmetric tensor 35599:Multi-index notation 34939:Theory of relativity 34692:Numerical relativity 34533:pulsar timing arrays 33856:Hans Albert Einstein 33805:I Am Albert Einstein 33751:Einstein Prize (APS) 33423:Photoelectric effect 33396:Theory of relativity 33289:Mathematical physics 32931:Real Time Relativity 32187:Quantum Field Theory 31125:Gill, T. P. (1965). 30851:Stachel, J. (2005). 30803:Section 3.7 page 107 30198:Ashby, Neil (2003). 30149:Essential Relativity 28791:10.1364/AO.18.000915 28256:physical contraction 28195: 28067:Lorentz ether theory 27842: 27771: 27671: 27556: 27481: 27388: 27126: 27047: 26890: 26829: 26702: 26583: 26542: 26406: 26369:stress–energy tensor 26367:. There is also the 26346:antisymmetric tensor 26334:antisymmetric tensor 26292: 26251: 25985: 25931: 25890: 25818: 25806:More generally, the 25801:covariant derivative 25677: 25278: 25193: 24973: 24921: 24870: 24725: 24570: 24536: 24462: 24431: 24380: 24369:{\displaystyle \nu } 24360: 24287: 24132: 24077:manifestly covariant 23748: 23725:Physics in spacetime 23620: 23493: 23368: 23271: 23167: 23059: 22771: 22618: 22583:(red lines labelled 22508:Experiments to test 22413:quantum field theory 22355:quantum field theory 22287:theory of everything 22241:manifestly covariant 22109: 22089: 22057: 21955: 21913: 21849: 21796: 21709: 21643: 21601: 21574: 21551: 21531: 21456: 21423: 21403: 21392:{\displaystyle A'B'} 21370: 21347: 21327: 21127: 21099: 21076: 21030: 21003: 20975: 20948: 20915: 20895: 20875: 20636: 20552: 20507: 20483: 20381: 20320: 20275: 20251: 20188: 20147: 20023: 20006:for a photon, and a 19981: 19887: 19783: 19674: 19641: 19618: 19542: 19518:spacetime separation 19455: 19405: 19370: 19305: 19221: 19113: 19085: 19050: 19021: 19017:. In the following, 18955: 18868: 18862:must transform like 18563: 18496: 18400: 18324:hyperbolic rotations 18172: 18002: 17942: 17850: 17817: 17736: 17678: 17620: 17520: 17438: 17310: 17256: 17225: 17074: 16968: 16874:Hyperbolic functions 16627: 16573: 16547: 16346: 16162: 15978: 15896: 15794: 15669: 15632: 15612: 15480: 15331: 15228: 14941: 14766: 14641: 14596: 14556: 14536: 14516: 14490: 14332: 13948: 13918: 13884: 13860: 13823: 13796: 13764: 13744: 13629: 13606: 13575: 13555: 13365:dealt strictly with 13213: 13125: 13089: 12979: 12945: 12872: 12844: 12823: 12686: 12590: 12489: 12364: 12278: 12214: 12128: 12088: 11997: 11970: 11950: 11911: 11886:wave theory of light 11799: 11771: 11751: 11726: 11699: 11663: 11497: 11467: 11444: 11420: 11171: 11141: 11119: 11097: 10789: 10502: 10286: 10117: 10090: 9994: 9961: 9941: 9638: 9608: 9588: 9542: 9514: 9416: 9336: 9294: 9269: 9250:Consider two frames 9150: 9095: 8961: 8907: 8857: 8810: 8739: 8680: 8560: 8523: 8447: 8349: 8315: 8288: 8231: 8172: 8113: 7992: 7955: 7879: 7780: 7746: 7719: 7662: 7603: 7537: 7489: 7430: 7386: 7283: 7209: 7127: 7041: 6967: 6917: 6897: 6873: 6863:proper time interval 6817: 6796: 6672: 6625: 6555: 6418: 6318: 6288: 6235: 6197: 6153: 6133: 6104: 6057: 6037: 5989: 5939: 5905: 5885: 5865: 5786: 5733: 5685: 5645: 5611: 5591: 5571: 5506: 5453: 5412: 5379: 4513: 4437: 4369: 4245: 4214: 4183: 4160: 4090: 4068: 4057:{\displaystyle t'=0} 4037: 4002: 3979: 3909: 3881: 3840: 3815: 3795: 3754: 3750:axis through points 3726: 3691: 3659: 3614: 3579: 3547: 3502: 3466: 3440: 3403: 3352: 3327: 3299: 3265: 3237: 3209: 3184: 3153: 3131: 3120:{\displaystyle t=0.} 3105: 3081: 3058: 3035: 3015: 2995: 2975: 2955: 2938:Spacetime diagrams ( 2791: 2716: 2622: 2549: 2445: 2370: 2276: 2203: 2094: 1971: 1664: 1539: 1370: 961:are a subset of his 796:gravitational fields 706: 670: 79:Theory of relativity 35902:Nonmetricity tensor 35757:(2nd-order tensors) 35725:Hodge star operator 35715:Exterior derivative 35564:Transport phenomena 35549:Continuum mechanics 35505:Multilinear algebra 35319:Relativistic energy 35088:Minkowski spacetime 35004:Maxwell's equations 34963:Galilean relativity 34584:Friedmann equations 34478:Hulse–Taylor binary 34440:Gravitational waves 34336:Riemannian geometry 34162:Proper acceleration 34147:Maxwell's equations 34093:Galilean relativity 33508:Thought experiments 33264:Atmospheric physics 33103:Classical mechanics 33031:branches of physics 32699:2011SchpJ...6.8520R 32653:1997PhRvA..56.4405W 32587:1964PhL....12..260A 32531:Taylor, Edwin, and 32428:1989rist.book.....Q 32162:The Road to Reality 32159:R. Penrose (2007). 32080:2004AHES...59...45N 32001:1997PhLB..405..249C 31847:2010EJPh...31..291F 31734:E. J. Post (1962). 31670:1994PhR...243..317N 31443:on 16 February 2017 31431:Richmond, Michael. 31350:2014EJPh...35f5025M 31305:1961AmJPh..29..283B 31262:"The Ball is Round" 31171:. pp. 34–7 f. 31153:Leighton, Robert B. 31149:Feynman, Richard P. 31105:1968JRASC..62..105S 31035:1937Obs....60..103H 30965:2005EJPh...26..195F 30772:Extract of page 206 30757:1989aetp.book.....G 30632:1996AmJPh..64..384D 30225:10.12942/lrr-2003-1 30216:2003LRR.....6....1A 30122:2006eins.book....1D 29917:2010AmJPh..78..633M 29871:David Morin (2007) 29717:2009arXiv0908.1545V 29532:Extract of page 226 29502:Extract of page 234 29353:Classical Mechanics 29283:Griffiths, David J. 29245:1933PhRv...43..491A 29193:1930RSPSA.126..360D 29138:1932RSPSA.136..630C 29021:1959PCPS...55..137P 28984:1959PhRv..116.1041T 28955:1915KNAB...18..398Z 28924:1914KNAB...17..445Z 28877:1970PhRvD...2..263B 28834:1962AmJPh..30...72S 28783:1979ApOpt..18..915L 28728:1903KNAB....6..809L 28684:1901KNAB....4..669L 28455:rotation generators 28368:The invariance of Δ 26572:alternative signs. 26236: 26050: 24865:relativistic energy 24847: 24829: 24811: 24085:physical quantities 23719:thought experiments 23681: 23663: 23645: 23555: 23534: 23513: 23452: 23431: 23410: 23312: 23291: 23230: 23209: 23116: 23095: 22952:The actual form of 22927:rotational symmetry 22873: 22852: 22831: 22810: 22718: 22697: 22676: 22574:Minkowski spacetime 22466:Newtonian mechanics 22425:classical mechanics 22382:Minkowski spacetime 22319:, now known as the 22237:Maxwell's equations 22227:Maxwell's equations 22105:from step 3 yields 21493:Rindler coordinates 21295: 21279: 21202: 21146: 20992:{\displaystyle S'.} 19349:is orthogonal with 18770: 18736: 18659: 18582: 18361:Maxwell's equations 17674:or in other words, 16876:: hyperbolic sine ( 16700: 16292: 16223: 16108: 16039: 14505:{\displaystyle H-E} 13781:{\displaystyle L/2} 13510:Maxwell's equations 13328:superluminal motion 13304:Figure 5–6. Galaxy 12833:{\displaystyle v\,} 12455:aberration of light 12436:Aberration of light 11788:{\displaystyle ct'} 11680:{\displaystyle -x'} 11574:Causality (physics) 11349:riding a light beam 11020: 10992: 10931: 10903: 10847: 10820: 10684: 10598: 10517: 10355: 10336: 10317: 9559:{\displaystyle dt'} 9531:{\displaystyle dx'} 9012:Lorentz contraction 8546: 8302:{\displaystyle f''} 8246: 7978: 7677: 7380:Total time of trip 7200:to and from a star 3898:{\displaystyle ct'} 3743:{\displaystyle ct'} 3455:{\displaystyle t=0} 3316:{\displaystyle ct'} 3254:{\displaystyle ct'} 3226:{\displaystyle ct'} 2068: 2052: 2016: 2000: 1647:one-parameter group 1265:Minkowski spacetime 792:Galilean relativity 593:luminiferous aether 585:Newtonian mechanics 577:Maxwell's equations 559:Galilean invariance 528:frames of reference 522:(identical) in all 295:Galilean relativity 215:Minkowski spacetime 119:Maxwell's equations 36163:Special relativity 36135:History of science 36035:Tullio Levi-Civita 35978:Metric tensor (GR) 35892:Levi-Civita symbol 35745:Tensor contraction 35559:General relativity 35495:Euclidean geometry 35282:Length contraction 35226:Lorentz invariance 35098:Spacetime diagrams 35040:Length contraction 34920:Special relativity 34633:Reissner–Nordström 34551:Brans–Dicke theory 34382:Linearized gravity 34209:Length contraction 34127:Frame of reference 34104:Special relativity 33957:Things named after 33819:Subtle is the Lord 33642:(1994 documentary) 33634:(1923 documentary) 33626:(1922 documentary) 33523:Gravitational wave 33406:General relativity 33401:Special relativity 33320:History of physics 32955:2013-05-14 at the 32936:2013-05-08 at the 32912:Relativity 4-ever? 32857:Special Relativity 32833:2013-03-21 at the 32814:2013-04-25 at the 32778:2010-02-01 at the 32734:Annalen der Physik 32668:Special Relativity 32514:Relativistic World 32496:. Westview Press. 31635:LibreTexts Physics 31612:2012-10-21 at the 31127:The Doppler Effect 30473:. pp. 28–29. 30295:Special Relativity 30110:Séminaire Poincaré 29483:Koks, Don (2006). 29414:Usenet Physics FAQ 29381:Lanczos, Cornelius 29348:Goldstein, Herbert 29059:Annalen der Physik 28510:Annalen der Physik 28252:length contraction 28210: 28164:Lighthouse paradox 28057:physical cosmology 27944:combines with the 27901: 27803: 27752: 27654: 27526: 27459: 27313: 27122:constructed using 27108: 27024: 26874: 26805: 26673: 26558: 26528: 26522: 26319: 26278: 26237: 26204: 25988: 25958: 25917: 25876: 25789: 25663: 25654: 25542: 25538: 25514: 25490: 25466: 25444: 25417: 25413: 25384: 25355: 25326: 25299: 25239: 25164: 24952: 24904: 24851: 24833: 24815: 24797: 24711: 24552: 24517: 24444: 24407: 24366: 24352:where there is an 24342: 24262: 24054: 24045: 23966: 23923: 23808: 23715:Minkowski diagrams 23684: 23667: 23649: 23631: 23603: 23585: 23541: 23520: 23499: 23479: 23438: 23417: 23396: 23342: 23298: 23277: 23257: 23216: 23195: 23146: 23102: 23081: 23053: 23012:, with factors of 22977:(this is called a 22877: 22859: 22838: 22817: 22796: 22722: 22704: 22683: 22662: 22593: 22398:general relativity 22392:is much less than 22217:. Conversely, the 22142: 22095: 22075: 22041: 21942: 21897: 21836: 21781: 21695: 21627: 21580: 21563:{\displaystyle ct} 21560: 21537: 21524: 21470:{\displaystyle S'} 21467: 21442: 21409: 21399:measured in frame 21389: 21359:{\displaystyle AB} 21356: 21333: 21309: 21307: 21283: 21267: 21190: 21134: 21113:{\displaystyle S'} 21110: 21088:{\displaystyle S,} 21085: 21062: 21016: 20989: 20960:{\displaystyle AB} 20957: 20934: 20901: 20881: 20869: 20794: 20687: 20623: 20539: 20492: 20458: 20368: 20307: 20260: 20221: 20174: 20123: 19996: 19917: 19874: 19770: 19650: 19627: 19608:Velocity 4-vector: 19593: 19494: 19441: 19391: 19323: 19274: 19208: 19100: 19065: 19036: 18979: 18913: 18792: 18790: 18758: 18724: 18647: 18570: 18529: 18468: 18309: 18300: 18261: 18198: 18165:may be written as 18145: 18136: 18107: 18038: 17995:may be written as 17960: 17929: 17835: 17804: 17713: 17661: 17607: 17507: 17417: 17290: 17231: 17208: 17199: 17173: 17107: 17020: 16890: 16728: 16686: 16602: 16559: 16517: 16314: 16278: 16209: 16130: 16094: 16025: 15943: 15864: 15762: 15638: 15618: 15576: 15444: 15295: 15213: 15171:. The dot product 15134: 15124:Newtonian analysis 15085:Elastic collisions 15014: 14834: 14731: 14622: 14582: 14542: 14522: 14502: 14460: 14310: 14308: 14172: 14081: 14013: 13995: 13931: 13897: 13876:relative to frame 13866: 13836: 13809: 13778: 13750: 13723: 13612: 13581: 13561: 13310: 13277: 13243: 13219: 13195: 13159: 13109: 13072: 12958: 12931: 12850: 12830: 12778: 12668: 12572: 12450: 12416: 12351: 12265: 12201: 12094: 12068: 11983: 11956: 11936: 11890:corpuscular theory 11871: 11813:{\displaystyle x'} 11810: 11785: 11757: 11740:{\displaystyle x'} 11737: 11711:{\displaystyle +x} 11708: 11677: 11591: 11524: 11473: 11450: 11426: 11196: 11157: 11127: 11105: 11053: 11008: 10980: 10919: 10891: 10835: 10808: 10757: 10672: 10586: 10505: 10467: 10343: 10324: 10305: 10254: 10098: 10060: 9976: 9947: 9911: 9903: 9853: 9830: 9797: 9761: 9622:{\displaystyle u'} 9619: 9594: 9556: 9528: 9484: 9384: 9307: 9280: 9232:occur at the same 9207:than its length (Δ 9176: 9127: 9006:Length contraction 8985: 8941: 8878: 8839: 8797: 8716: 8667: 8547: 8534: 8505: 8434: 8336: 8299: 8265: 8234: 8211: 8149: 8099: 7979: 7966: 7937: 7865: 7767: 7733:{\displaystyle f'} 7730: 7696: 7665: 7642: 7577: 7519: 7464: 7410: 7342: 7268: 7182: 7133: 7099: 7028: 6953: 6903: 6885:{\displaystyle v.} 6882: 6850: 6802: 6776: 6746: 6693: 6652: 6561: 6483: 6398: 6324: 6304: 6274: 6212: 6183: 6139: 6118: 6090: 6043: 6015: 5975: 5925: 5891: 5871: 5851: 5772: 5705: 5671: 5631: 5597: 5577: 5557: 5492: 5428: 5395: 5363: 5361: 5291: 5217: 5131: 4839: 4636: 4497: 4388: 4350: 4230: 4207:invariant interval 4192: 4169: 4152:Invariant interval 4098: 4076: 4054: 4014: 3991:{\displaystyle ct} 3988: 3965: 3895: 3867: 3829:{\displaystyle x'} 3826: 3801: 3781: 3740: 3712: 3677: 3645: 3600: 3565: 3533: 3483: 3452: 3426: 3389: 3341:{\displaystyle x'} 3338: 3313: 3285: 3251: 3223: 3198:{\displaystyle x'} 3195: 3164: 3139: 3117: 3087: 3070:{\displaystyle ct} 3067: 3047:{\displaystyle ct} 3044: 3021: 3001: 2981: 2961: 2940:Minkowski diagrams 2871: 2778: 2697: 2609: 2525: 2432: 2351: 2263: 2179: 2075: 2056: 2040: 2004: 1988: 1823: 1821: 1588: 1525: 1523: 1009: 903:luminiferous ether 746:general relativity 712: 692: 652:spacetime interval 620:length contraction 492:special relativity 430:Physics portal 151:Length contraction 56:Special relativity 50: 36068: 36067: 36030:Hermann Grassmann 35986: 35985: 35938:Moment of inertia 35799:Differential form 35774:Affine connection 35589:Einstein notation 35572: 35571: 35500:Exterior calculus 35480:Coordinate system 35412: 35411: 35352:Sagnac experiment 35347:Fizeau experiment 35212:Hammar experiment 35139: 35138: 35070:Ehrenfest paradox 35060:Relativistic disk 35055:Thomas precession 35030:Relativistic mass 34886: 34885: 34700: 34699: 34679:Ozsváth–Schücking 34285: 34284: 34267:Minkowski diagram 34224:Thomas precession 34167:Relativistic mass 34030: 34029: 33927:Awards and honors 33910:Siegbert Einstein 33798:Einstein's Cosmos 33348: 33347: 33335:Physics education 33284:Materials science 33251:Interdisciplinary 33209:Quantum mechanics 32872:Project Gutenberg 32677:Rindler, Wolfgang 32641:Physical Review A 32537:Spacetime Physics 32503:978-0-8133-0625-4 32484:978-0-520-03174-6 32437:978-9971-5-0612-4 32404:978-0-19-927583-0 32378:978-0-19-853952-0 32273:978-0-8053-8732-2 32206:Charles W. Misner 32172:978-0-679-77631-4 32165:. Vintage books. 32145:978-0-470-01460-8 32120:978-0-7167-0344-0 31979:Physics Letters B 31954:978-0-387-69199-2 31927:978-0-471-87373-0 31896:978-0-674-02611-7 31745:978-0-486-65427-0 31717:978-0-321-49575-4 31568:978-1-60650-857-2 31543:978-0-8176-4143-6 31509:978-0-486-29998-3 31313:10.1119/1.1937751 31159:(February 1977). 31087:Sher, D. (1968). 31073:978-1-4020-1308-9 30927:978-0-935702-68-2 30900:978-0-387-95210-9 30866:978-0-8176-4380-5 30796:978-0-8229-5926-7 30766:978-2-88124-719-4 30670:Takeuchi, Tatsu. 30573:978-0-7484-0621-0 30507:978-0-465-02414-8 30480:978-0-19-850657-7 30449:978-0-521-82196-4 30418:978-0-387-98756-9 30274:978-0-07-035048-9 30183:Spacetime Physics 30158:978-0-387-10090-6 30062:978-0-387-94870-6 29925:10.1119/1.3298908 29768:978-0-486-60081-9 29601:978-3-540-07970-5 29555:Spacetime Physics 29526:978-0-19-966286-9 29496:978-0-387-32793-8 29446:978-0-415-25384-0 29394:978-0-486-65067-8 29298:978-0-321-85656-2 28865:Physical Review D 28842:10.1119/1.1941907 28611:978-0-8176-3339-4 28417:Einstein's Cosmos 28335:positive definite 28149:Ehrenfest paradox 28087:Relativistic disk 27984:Arnold Sommerfeld 27974:Hermann Minkowski 27883: 27781: 27712: 27706: 27626: 27546:momentum 4-vector 27333:is negative that 27019: 27018:invariant scalars 26872: 26803: 25784: 25537: 25513: 25489: 25465: 25443: 25412: 25383: 25354: 25325: 25298: 25234: 25187:four-acceleration 25146: 25084: 24781: 24780: 24611: 24376:from 0 to 3, and 24354:implied summation 24128:with components: 23702:The cone in the − 23682: 23039:appear anywhere. 23002:Some authors use 22606:In 3D space, the 22561:through circular 22437:Fizeau experiment 22306:quantum mechanics 22260:quantum mechanics 21887: 21831: 21776: 21705:we conclude that 21597:We remember that 21412:{\displaystyle S} 21336:{\displaystyle S} 20884:{\displaystyle k} 20706:are all standard 20679: 20665: 20613: 20601: 20581: 20534: 20451: 20436: 20410: 20359: 20336: 20302: 20209: 20168: 20058: 19911: 19864: 19841: 19818: 19765: 19701: 19271: 19256: 19062: 19033: 17927: 17926: 17885: 17884: 17802: 17801: 17768: 17767: 17602: 17502: 17412: 17349: 17285: 17015: 16974: 16755:Beyond the basics 16752: 16751: 16726: 16541: 16540: 16515: 16338: 16337: 16312: 16154: 16153: 16128: 15967: 15966: 15888: 15887: 15786: 15785: 15621:{\displaystyle v} 15600: 15599: 15468: 15467: 15441: 15406: 15365: 15319: 15318: 15270: 15263: 15242: 15012: 15011: 14858: 14857: 14822: 14805: 14755: 14754: 14718: 14717: 14545:{\displaystyle E} 14525:{\displaystyle H} 14484: 14483: 14447: 14446: 14304: 14303: 14248: 14247: 14171: 14157: 14156: 14080: 14012: 13994: 13869:{\displaystyle v} 13721: 13720: 13645: 13615:{\displaystyle v} 13564:{\displaystyle f} 13385:relativistic mass 13262:visual appearance 13157: 13156: 13056: 13055: 12914: 12853:{\displaystyle v} 12776: 12742: 12708: 12666: 12570: 12409: 12375: 12341: 12310: 12297: 12260: 12196: 12097:{\displaystyle n} 12064: 12055: 12021: 11959:{\displaystyle v} 11861:Fizeau experiment 11836:faster than light 11760:{\displaystyle x} 11691:than it was sent. 11543:Thomas precession 11476:{\displaystyle v} 11453:{\displaystyle L} 11429:{\displaystyle L} 11305:| and | 11290:| and | 11077: 11076: 11049: 11045: 10960: 10956: 10871: 10867: 10781: 10780: 10753: 10749: 10667: 10663: 10581: 10577: 10491: 10490: 10463: 10430: 10397: 10342: 10323: 10278: 10277: 10250: 10227: 10204: 10162: 10146: 10084: 10083: 10055: 9972: 9950:{\displaystyle v} 9935: 9934: 9906: 9902: 9856: 9852: 9829: 9796: 9769: 9760: 9680: 9597:{\displaystyle u} 9508: 9507: 9408: 9407: 9172: 9125: 9003: 9002: 8983: 8939: 8876: 8837: 8758: 8699: 8647: 8579: 8466: 8414: 8368: 8263: 8209: 8196: 8132: 8079: 8011: 7898: 7845: 7799: 7694: 7640: 7627: 7575: 7517: 7462: 7408: 7337: 7263: 7136:{\displaystyle v} 7094: 7021: 6948: 6906:{\displaystyle c} 6827: 6805:{\displaystyle L} 6689: 6564:{\displaystyle v} 6476: 6370:, as measured by 6327:{\displaystyle c} 6142:{\displaystyle v} 6046:{\displaystyle v} 6010: 5894:{\displaystyle c} 5874:{\displaystyle v} 5808: 5600:{\displaystyle c} 5580:{\displaystyle v} 5528: 5290: 5216: 5130: 4861: 4838: 4664: 4635: 4270: 4269: 4096: 4074: 3963: 3804:{\displaystyle k} 3159: 3137: 3090:{\displaystyle c} 3024:{\displaystyle t} 3004:{\displaystyle x} 2984:{\displaystyle t} 2964:{\displaystyle x} 2867: 2833: 2808: 2777: 2771: 2733: 2693: 2664: 2644: 2608: 2605: 2599: 2571: 2521: 2487: 2462: 2431: 2425: 2387: 2347: 2318: 2298: 2262: 2259: 2253: 2225: 2175: 2140: 2137: 2131: 2071: 2025: 2019: 1586: 1585: 1455: 1397: 1277:Minkowski diagram 912:), including the 859:Albert Einstein: 775:and representing 758:Hermann Minkowski 715:{\displaystyle c} 644:Thomas precession 632:relativistic mass 480: 479: 220:Spacetime diagram 191:Ehrenfest paradox 181:Relativistic disk 176:Thomas precession 156:Relativistic mass 16:(Redirected from 36175: 36145: 36144: 36133: 36132: 36131: 36121: 36120: 36119: 36109: 36108: 36097: 36096: 36095: 36085: 36084: 36076: 36045:Bernhard Riemann 35877: 35876: 35720:Exterior product 35687:Two-point tensor 35672:Symmetric tensor 35554:Electromagnetism 35468: 35467: 35439: 35432: 35425: 35416: 35415: 35166: 35159: 35152: 35143: 35142: 34913: 34906: 34899: 34890: 34889: 34878: 34877: 34661:van Stockum dust 34433:Two-body problem 34351:Mach's principle 34298: 34297: 34239:Terrell rotation 34079: 34078: 34057: 34050: 34043: 34034: 34033: 34018: 34017: 33912:(distant cousin) 33906:(great-grandson) 33868:Hermann Einstein 33850:Lieserl Einstein 33528:Tea leaf paradox 33375: 33368: 33361: 33352: 33351: 33274:Chemical physics 33214:Particle physics 33140:Classical optics 33023: 33016: 33009: 33000: 32999: 32971: 32944:Spacetime travel 32874: 32846: 32800:Bondi K-Calculus 32712: 32710: 32664: 32635: 32598: 32567:Journal articles 32528: 32507: 32488: 32460: 32441: 32366:Wolfgang Rindler 32278: 32277: 32257: 32251: 32234: 32228: 32203: 32197: 32183: 32177: 32176: 32156: 32150: 32149: 32131: 32125: 32124: 32106: 32100: 32099: 32059: 32053: 32052: 32050: 32048: 32032: 32026: 32020: 31994: 31985:(3–4): 249–252. 31972: 31966: 31958: 31938: 31932: 31931: 31907: 31901: 31900: 31888: 31878: 31867: 31866: 31840: 31820: 31814: 31813: 31811: 31809: 31800:. math.ucr.edu. 31789: 31780: 31779: 31761: 31750: 31749: 31731: 31722: 31721: 31703: 31697: 31696: 31688: 31682: 31681: 31653: 31647: 31646: 31644: 31642: 31626: 31617: 31604: 31598: 31597: 31595: 31593: 31582: 31573: 31572: 31554: 31548: 31547: 31520: 31514: 31513: 31490: 31484: 31483: 31481: 31479: 31459: 31453: 31452: 31450: 31448: 31428: 31422: 31421: 31419: 31417: 31404:Chase, Scott I. 31401: 31395: 31394: 31376: 31370: 31369: 31343: 31323: 31317: 31316: 31288: 31282: 31281: 31279: 31277: 31257: 31251: 31250: 31248: 31246: 31241:on 13 April 2017 31240: 31229: 31223:Signell, Peter. 31220: 31214: 31213: 31211: 31209: 31197: 31191: 31190: 31145: 31139: 31138: 31122: 31116: 31115: 31113: 31111: 31084: 31078: 31077: 31061: 31052: 31046: 31045: 31043: 31041: 31014: 31008: 31007: 31005: 31003: 30991: 30985: 30984: 30958: 30938: 30932: 30931: 30911: 30905: 30904: 30890:Basic Relativity 30884: 30878: 30877: 30875: 30873: 30848: 30842: 30841: 30839: 30837: 30831: 30825:. Archived from 30820: 30811: 30805: 30800: 30780: 30774: 30770: 30742: 30736: 30735: 30733: 30731: 30715: 30709: 30708: 30690: 30684: 30683: 30681: 30679: 30667: 30661: 30660: 30650: 30644: 30643: 30615: 30609: 30608: 30606: 30604: 30584: 30578: 30577: 30553: 30547: 30546: 30528: 30519: 30518: 30516: 30514: 30491: 30485: 30484: 30460: 30454: 30453: 30429: 30423: 30422: 30398: 30392: 30391: 30389: 30387: 30373: 30364: 30355: 30354: 30352: 30350: 30336:10.2307/20022495 30315: 30309: 30308: 30290: 30279: 30278: 30254: 30248: 30247: 30237: 30227: 30195: 30189: 30188: 30186: 30176: 30163: 30162: 30144: 30133: 30132: 30130: 30128: 30107: 30098: 30092: 30091: 30087:978-0387-26005-1 30073: 30067: 30066: 30048: 30039: 30038: 30018: 30012: 30011: 29999: 29989: 29983: 29968: 29962: 29961: 29943: 29937: 29936: 29910: 29890: 29884: 29869: 29832: 29818: 29812: 29811: 29793: 29787: 29786: 29780: 29772: 29752: 29746: 29743: 29737: 29736: 29710: 29690: 29681: 29661: 29655: 29654: 29652: 29651: 29637: 29631: 29630: 29628: 29627: 29612: 29606: 29605: 29585: 29570: 29569: 29549: 29534: 29530: 29510: 29504: 29500: 29480: 29474: 29468: 29462: 29457: 29451: 29450: 29430: 29424: 29423: 29421: 29420: 29405: 29399: 29398: 29377: 29368: 29367: 29344: 29338: 29337: 29318:Jackson, John D. 29314: 29303: 29302: 29279: 29259: 29258: 29256: 29221: 29215: 29214: 29204: 29187:(801): 360–365. 29169: 29160: 29159: 29149: 29117: 29106: 29105: 29103: 29101: 29095: 29086: 29080: 29072: 29066: 29052: 29041: 29040: 29002: 28996: 28995: 28978:(4): 1041–1045. 28965: 28959: 28958: 28934: 28928: 28927: 28903: 28897: 28896: 28860: 28854: 28853: 28817: 28811: 28810: 28766: 28760: 28759: 28745: 28739: 28738: 28736: 28734: 28713: 28704: 28695: 28694: 28692: 28690: 28669: 28660: 28654: 28643: 28637: 28622: 28616: 28615: 28597: 28591: 28587: 28581: 28578: 28572: 28569: 28560: 28557: 28551: 28548: 28542: 28535: 28529: 28497: 28469: 28465: 28459: 28452: 28451: 28440:boost generators 28436: 28430: 28426: 28420: 28413: 28407: 28396: 28390: 28366: 28360: 28357:metric signature 28329:, it may assume 28323: 28317: 28306: 28301: 28288: 28279: 28273: 28262: 28240: 28234: 28227: 28221: 28219: 28217: 28216: 28211: 28209: 28208: 28203: 28189: 28183: 28179: 28097:Born coordinates 28018:Bondi k-calculus 27946:density of power 27942:density of force 27910: 27908: 27907: 27902: 27900: 27899: 27884: 27882: 27874: 27873: 27872: 27859: 27854: 27853: 27812: 27810: 27809: 27804: 27799: 27798: 27783: 27782: 27779: 27761: 27759: 27758: 27753: 27748: 27747: 27723: 27722: 27717: 27713: 27708: 27707: 27704: 27698: 27685: 27684: 27679: 27663: 27661: 27660: 27655: 27650: 27649: 27637: 27636: 27631: 27627: 27619: 27606: 27605: 27596: 27595: 27586: 27585: 27570: 27569: 27564: 27535: 27533: 27532: 27527: 27519: 27518: 27509: 27508: 27499: 27498: 27468: 27466: 27465: 27460: 27454: 27453: 27438: 27437: 27428: 27427: 27418: 27417: 27402: 27401: 27396: 27372: 27371: 27370: 27347: 27346: 27345: 27322: 27320: 27319: 27314: 27312: 27311: 27290: 27289: 27268: 27267: 27246: 27245: 27217: 27216: 27203: 27202: 27189: 27188: 27173: 27172: 27159: 27158: 27143: 27142: 27137: 27117: 27115: 27114: 27109: 27107: 27106: 27094: 27093: 27088: 27085: 27084: 27083: 27067: 27066: 27065: 27033: 27031: 27030: 27025: 27020: 27017: 27012: 27011: 27006: 27003: 27002: 26993: 26992: 26987: 26984: 26983: 26974: 26973: 26968: 26965: 26964: 26952: 26951: 26946: 26943: 26942: 26933: 26932: 26927: 26924: 26923: 26911: 26910: 26905: 26902: 26901: 26883: 26881: 26880: 26875: 26873: 26871: 26870: 26861: 26860: 26851: 26846: 26841: 26836: 26820: 26814: 26812: 26811: 26806: 26804: 26802:invariant scalar 26801: 26796: 26795: 26786: 26785: 26773: 26772: 26760: 26759: 26750: 26749: 26737: 26736: 26724: 26723: 26714: 26713: 26682: 26680: 26679: 26674: 26672: 26671: 26666: 26663: 26662: 26661: 26648: 26647: 26642: 26639: 26638: 26637: 26624: 26623: 26622: 26614: 26598: 26597: 26570:Minkowski metric 26567: 26565: 26564: 26559: 26557: 26556: 26537: 26535: 26534: 26529: 26527: 26526: 26421: 26420: 26401: 26393:Minkowski metric 26342:angular momentum 26328: 26326: 26325: 26320: 26318: 26317: 26312: 26309: 26308: 26307: 26287: 26285: 26284: 26279: 26277: 26276: 26271: 26268: 26267: 26266: 26246: 26244: 26243: 26238: 26235: 26221: 26203: 26202: 26197: 26194: 26193: 26192: 26176: 26175: 26170: 26167: 26166: 26165: 26152: 26151: 26146: 26143: 26142: 26141: 26128: 26127: 26122: 26119: 26118: 26117: 26101: 26100: 26095: 26092: 26091: 26090: 26077: 26076: 26071: 26068: 26067: 26066: 26049: 26048: 26037: 26029: 26020: 26019: 26008: 26000: 25967: 25965: 25964: 25959: 25957: 25956: 25951: 25948: 25947: 25946: 25926: 25924: 25923: 25918: 25916: 25915: 25910: 25907: 25906: 25905: 25885: 25883: 25882: 25877: 25872: 25871: 25862: 25861: 25856: 25853: 25852: 25851: 25835: 25834: 25833: 25798: 25796: 25795: 25790: 25785: 25783: 25782: 25781: 25765: 25760: 25759: 25743: 25742: 25730: 25729: 25724: 25721: 25720: 25719: 25697: 25696: 25695: 25672: 25670: 25669: 25664: 25659: 25658: 25547: 25546: 25539: 25536: 25528: 25520: 25515: 25512: 25504: 25496: 25491: 25488: 25480: 25472: 25467: 25464: 25456: 25448: 25445: 25436: 25422: 25421: 25414: 25411: 25410: 25398: 25390: 25385: 25382: 25381: 25369: 25361: 25356: 25353: 25352: 25340: 25332: 25327: 25324: 25323: 25311: 25303: 25300: 25291: 25248: 25246: 25245: 25240: 25235: 25233: 25225: 25224: 25223: 25210: 25205: 25204: 25173: 25171: 25170: 25165: 25160: 25156: 25155: 25147: 25139: 25129: 25125: 25124: 25123: 25111: 25110: 25098: 25097: 25085: 25077: 25064: 25063: 25051: 25050: 25038: 25037: 25001: 25000: 24985: 24984: 24961: 24959: 24958: 24953: 24951: 24928: 24913: 24911: 24910: 24905: 24903: 24902: 24860: 24858: 24857: 24852: 24846: 24841: 24828: 24823: 24810: 24805: 24793: 24792: 24782: 24779: 24778: 24769: 24764: 24763: 24748: 24744: 24720: 24718: 24717: 24712: 24704: 24672: 24671: 24659: 24658: 24646: 24645: 24612: 24610: 24602: 24601: 24600: 24587: 24582: 24581: 24561: 24559: 24558: 24553: 24548: 24547: 24526: 24524: 24523: 24518: 24516: 24515: 24506: 24505: 24500: 24497: 24496: 24495: 24479: 24478: 24477: 24453: 24451: 24450: 24445: 24443: 24442: 24416: 24414: 24413: 24408: 24406: 24405: 24400: 24397: 24396: 24395: 24375: 24373: 24372: 24367: 24351: 24349: 24348: 24343: 24341: 24340: 24331: 24330: 24325: 24322: 24321: 24320: 24304: 24303: 24302: 24281: 24272:where we define 24271: 24269: 24268: 24263: 24255: 24199: 24198: 24186: 24185: 24173: 24172: 24160: 24159: 24144: 24143: 24117: 24063: 24061: 24060: 24055: 24050: 24049: 23971: 23970: 23928: 23927: 23813: 23812: 23805: 23793: 23781: 23769: 23694:away and a time 23693: 23691: 23690: 23685: 23683: 23680: 23675: 23662: 23657: 23644: 23639: 23630: 23594: 23592: 23591: 23586: 23581: 23580: 23568: 23567: 23554: 23549: 23533: 23528: 23512: 23507: 23488: 23486: 23485: 23480: 23478: 23477: 23465: 23464: 23451: 23446: 23430: 23425: 23409: 23404: 23383: 23382: 23351: 23349: 23348: 23343: 23338: 23337: 23325: 23324: 23311: 23306: 23290: 23285: 23266: 23264: 23263: 23258: 23256: 23255: 23243: 23242: 23229: 23224: 23208: 23203: 23182: 23181: 23156:we see that the 23155: 23153: 23152: 23147: 23142: 23141: 23129: 23128: 23115: 23110: 23094: 23089: 23074: 23073: 23034: 23011: 22994: 22981:). According to 22976: 22935:Minkowski metric 22931:Euclidean metric 22924: 22886: 22884: 22883: 22878: 22872: 22867: 22851: 22846: 22830: 22825: 22809: 22804: 22786: 22785: 22762: 22731: 22729: 22728: 22723: 22717: 22712: 22696: 22691: 22675: 22670: 22655: 22644: 22633: 22632: 22578:hyperbolic angle 22151: 22149: 22148: 22143: 22135: 22130: 22129: 22104: 22102: 22101: 22096: 22084: 22082: 22081: 22076: 22050: 22048: 22047: 22042: 22022: 22014: 22013: 21971: 21951: 21949: 21948: 21943: 21932: 21906: 21904: 21903: 21898: 21893: 21888: 21886: 21885: 21873: 21872: 21863: 21862: 21853: 21845: 21843: 21842: 21837: 21832: 21830: 21829: 21814: 21812: 21790: 21788: 21787: 21782: 21777: 21775: 21774: 21762: 21761: 21752: 21751: 21742: 21740: 21704: 21702: 21701: 21696: 21691: 21690: 21678: 21677: 21665: 21664: 21655: 21654: 21636: 21634: 21633: 21628: 21620: 21589: 21587: 21586: 21581: 21569: 21567: 21566: 21561: 21546: 21544: 21543: 21538: 21476: 21474: 21473: 21468: 21466: 21451: 21449: 21448: 21443: 21441: 21433: 21418: 21416: 21415: 21410: 21398: 21396: 21395: 21390: 21388: 21380: 21365: 21363: 21362: 21357: 21342: 21340: 21339: 21334: 21318: 21316: 21315: 21310: 21308: 21291: 21275: 21259: 21247: 21243: 21227: 21226: 21198: 21185: 21181: 21171: 21170: 21142: 21119: 21117: 21116: 21111: 21109: 21094: 21092: 21091: 21086: 21071: 21069: 21068: 21063: 21055: 21054: 21042: 21041: 21025: 21023: 21022: 21017: 21015: 21014: 20998: 20996: 20995: 20990: 20985: 20966: 20964: 20963: 20958: 20943: 20941: 20940: 20935: 20933: 20925: 20910: 20908: 20907: 20902: 20890: 20888: 20887: 20882: 20822: 20819: 20742: 20738: 20730: 20717: 20713: 20709: 20705: 20696: 20694: 20693: 20688: 20686: 20682: 20681: 20680: 20672: 20666: 20664: 20656: 20648: 20632: 20630: 20629: 20624: 20619: 20615: 20614: 20612: 20604: 20603: 20602: 20594: 20587: 20582: 20580: 20572: 20564: 20548: 20546: 20545: 20540: 20535: 20533: 20525: 20517: 20501: 20499: 20498: 20493: 20479:with respect to 20478: 20467: 20465: 20464: 20459: 20457: 20453: 20452: 20450: 20442: 20438: 20437: 20429: 20416: 20411: 20409: 20401: 20393: 20377: 20375: 20374: 20369: 20361: 20360: 20352: 20337: 20335: 20324: 20316: 20314: 20313: 20308: 20303: 20301: 20293: 20285: 20269: 20267: 20266: 20261: 20247:with respect to 20246: 20234: 20230: 20228: 20227: 20222: 20211: 20210: 20202: 20183: 20181: 20180: 20175: 20170: 20169: 20161: 20142: 20132: 20130: 20129: 20124: 20119: 20118: 20106: 20105: 20093: 20092: 20077: 20060: 20059: 20051: 20042: 20009: 20005: 20003: 20002: 19997: 19966: 19952: 19940: 19926: 19924: 19923: 19918: 19913: 19912: 19904: 19883: 19881: 19880: 19875: 19870: 19866: 19865: 19863: 19855: 19847: 19842: 19840: 19832: 19824: 19819: 19817: 19809: 19801: 19779: 19777: 19776: 19771: 19766: 19764: 19760: 19748: 19707: 19702: 19700: 19692: 19684: 19659: 19657: 19656: 19651: 19636: 19634: 19633: 19628: 19613: 19602: 19600: 19599: 19594: 19535: 19527: 19503: 19501: 19500: 19495: 19490: 19489: 19480: 19479: 19467: 19466: 19450: 19448: 19447: 19442: 19440: 19439: 19427: 19426: 19417: 19416: 19400: 19398: 19397: 19392: 19364: 19348: 19344: 19340: 19336: 19332: 19330: 19329: 19324: 19293: 19283: 19281: 19280: 19275: 19273: 19272: 19264: 19258: 19257: 19249: 19243: 19242: 19233: 19232: 19217: 19215: 19214: 19209: 19204: 19203: 19194: 19193: 19181: 19180: 19171: 19170: 19158: 19157: 19148: 19147: 19135: 19134: 19125: 19124: 19109: 19107: 19106: 19101: 19078: 19074: 19072: 19071: 19066: 19064: 19063: 19055: 19045: 19043: 19042: 19037: 19035: 19034: 19026: 19016: 19012: 18994: 18990: 18988: 18986: 18985: 18980: 18948: 18924: 18922: 18920: 18919: 18914: 18861: 18857: 18801: 18799: 18798: 18793: 18791: 18787: 18786: 18766: 18753: 18752: 18732: 18719: 18715: 18714: 18713: 18698: 18684: 18683: 18655: 18642: 18638: 18637: 18636: 18621: 18607: 18606: 18578: 18555: 18548: 18540: 18538: 18536: 18535: 18530: 18479: 18477: 18475: 18474: 18469: 18467: 18463: 18462: 18461: 18449: 18448: 18436: 18435: 18423: 18422: 18377: 18373: 18350: 18346: 18318: 18316: 18315: 18310: 18305: 18304: 18297: 18285: 18266: 18265: 18203: 18202: 18164: 18154: 18152: 18151: 18146: 18141: 18140: 18112: 18111: 18043: 18042: 18035: 18023: 17994: 17969: 17967: 17966: 17961: 17938: 17936: 17935: 17930: 17928: 17919: 17918: 17903: 17902: 17891: 17886: 17883: 17882: 17867: 17863: 17844: 17842: 17841: 17836: 17813: 17811: 17810: 17805: 17803: 17794: 17793: 17778: 17774: 17769: 17766: 17765: 17750: 17746: 17722: 17720: 17719: 17714: 17709: 17708: 17696: 17695: 17670: 17668: 17667: 17662: 17654: 17653: 17641: 17640: 17616: 17614: 17613: 17608: 17603: 17601: 17600: 17599: 17584: 17583: 17561: 17560: 17559: 17541: 17540: 17524: 17516: 17514: 17513: 17508: 17503: 17501: 17500: 17499: 17490: 17489: 17473: 17472: 17471: 17459: 17458: 17448: 17426: 17424: 17423: 17418: 17413: 17411: 17410: 17409: 17394: 17393: 17383: 17382: 17381: 17366: 17365: 17355: 17350: 17348: 17337: 17326: 17299: 17297: 17296: 17291: 17286: 17278: 17249:, is defined by 17240: 17238: 17237: 17232: 17217: 17215: 17214: 17209: 17204: 17203: 17178: 17177: 17112: 17111: 17104: 17092: 17062: 17060: 17053: 17044: 17029: 17027: 17026: 17021: 17016: 17014: 17003: 16992: 16975: 16972: 16941: 16937: 16930: 16899: 16852: 16840: 16824: 16806: 16794: 16778: 16746: 16737: 16735: 16734: 16729: 16727: 16725: 16724: 16723: 16719: 16699: 16694: 16679: 16678: 16669: 16657: 16656: 16647: 16642: 16621: 16611: 16609: 16608: 16603: 16598: 16597: 16585: 16584: 16568: 16566: 16565: 16560: 16535: 16526: 16524: 16523: 16518: 16516: 16514: 16513: 16512: 16508: 16489: 16488: 16467: 16466: 16457: 16456: 16441: 16440: 16424: 16423: 16412: 16392: 16391: 16378: 16373: 16340: 16332: 16323: 16321: 16320: 16315: 16313: 16311: 16310: 16309: 16305: 16291: 16286: 16277: 16254: 16253: 16241: 16233: 16232: 16222: 16217: 16204: 16203: 16192: 16191: 16181: 16176: 16175: 16156: 16148: 16139: 16137: 16136: 16131: 16129: 16127: 16126: 16125: 16121: 16107: 16102: 16093: 16070: 16069: 16057: 16049: 16048: 16038: 16033: 16020: 16019: 16008: 16007: 15997: 15992: 15991: 15972: 15961: 15952: 15950: 15949: 15944: 15942: 15941: 15929: 15928: 15910: 15909: 15890: 15882: 15873: 15871: 15870: 15865: 15863: 15852: 15851: 15842: 15841: 15829: 15818: 15817: 15808: 15807: 15788: 15780: 15771: 15769: 15768: 15763: 15761: 15750: 15749: 15740: 15739: 15727: 15716: 15715: 15706: 15705: 15693: 15692: 15683: 15682: 15663: 15647: 15645: 15644: 15639: 15627: 15625: 15624: 15619: 15594: 15585: 15583: 15582: 15577: 15575: 15574: 15562: 15561: 15549: 15548: 15536: 15535: 15523: 15522: 15507: 15506: 15494: 15493: 15474: 15462: 15453: 15451: 15450: 15445: 15443: 15442: 15437: 15436: 15427: 15421: 15420: 15408: 15407: 15402: 15401: 15392: 15386: 15385: 15367: 15366: 15361: 15360: 15351: 15345: 15344: 15325: 15313: 15304: 15302: 15301: 15296: 15294: 15293: 15271: 15268: 15265: 15264: 15256: 15244: 15243: 15235: 15222: 15198: 15170: 15160: 15154: 15081:(when at rest). 15023: 15021: 15020: 15015: 15013: 15010: 15009: 15000: 14995: 14994: 14985: 14984: 14969: 14968: 14960: 14879: 14877: 14876: 14873: 14870: 14852: 14843: 14841: 14840: 14835: 14833: 14832: 14823: 14821: 14820: 14808: 14806: 14798: 14793: 14792: 14780: 14779: 14760: 14749: 14740: 14738: 14737: 14732: 14730: 14726: 14719: 14716: 14715: 14706: 14701: 14700: 14685: 14681: 14668: 14667: 14655: 14654: 14635: 14631: 14629: 14628: 14623: 14621: 14620: 14608: 14607: 14591: 14589: 14588: 14583: 14581: 14580: 14568: 14567: 14551: 14549: 14548: 14543: 14531: 14529: 14528: 14523: 14511: 14509: 14508: 14503: 14478: 14469: 14467: 14466: 14461: 14459: 14455: 14448: 14445: 14444: 14435: 14430: 14429: 14414: 14410: 14394: 14393: 14381: 14380: 14362: 14361: 14349: 14348: 14326: 14319: 14317: 14316: 14311: 14309: 14305: 14302: 14301: 14292: 14287: 14286: 14271: 14267: 14262: 14261: 14249: 14246: 14245: 14236: 14231: 14230: 14215: 14214: 14213: 14196: 14178: 14173: 14164: 14158: 14155: 14154: 14145: 14140: 14139: 14124: 14123: 14122: 14105: 14087: 14082: 14073: 14067: 14066: 14050: 14049: 14030: 14029: 14014: 14005: 13996: 13987: 13981: 13980: 13964: 13963: 13940: 13938: 13937: 13932: 13930: 13929: 13913: 13906: 13904: 13903: 13898: 13896: 13895: 13875: 13873: 13872: 13867: 13855: 13845: 13843: 13842: 13837: 13835: 13834: 13818: 13816: 13815: 13810: 13808: 13807: 13787: 13785: 13784: 13779: 13774: 13759: 13757: 13756: 13751: 13732: 13730: 13729: 13724: 13722: 13719: 13718: 13709: 13704: 13703: 13688: 13687: 13686: 13669: 13651: 13646: 13641: 13633: 13621: 13619: 13618: 13613: 13601: 13590: 13588: 13587: 13582: 13570: 13568: 13567: 13562: 13534: 13523: 13453: 13433: 13414: 13316:Terrell rotation 13228: 13226: 13225: 13220: 13168: 13166: 13165: 13160: 13158: 13155: 13154: 13139: 13135: 13118: 13116: 13115: 13110: 13105: 13081: 13079: 13078: 13073: 13068: 13067: 13057: 13054: 13043: 13032: 13031: 13026: 13025: 13016: 13012: 12991: 12990: 12967: 12965: 12964: 12959: 12957: 12956: 12940: 12938: 12937: 12932: 12930: 12929: 12920: 12916: 12915: 12913: 12912: 12900: 12884: 12883: 12859: 12857: 12856: 12851: 12839: 12837: 12836: 12831: 12787: 12785: 12784: 12779: 12777: 12769: 12761: 12760: 12756: 12747: 12743: 12741: 12730: 12719: 12709: 12704: 12696: 12677: 12675: 12674: 12669: 12667: 12665: 12646: 12622: 12611: 12606: 12581: 12579: 12578: 12573: 12571: 12569: 12553: 12535: 12531: 12510: 12505: 12425: 12423: 12422: 12417: 12415: 12411: 12410: 12408: 12407: 12395: 12376: 12368: 12360: 12358: 12357: 12352: 12347: 12343: 12342: 12340: 12329: 12316: 12312: 12311: 12303: 12298: 12290: 12274: 12272: 12271: 12266: 12261: 12259: 12252: 12237: 12227: 12218: 12210: 12208: 12207: 12202: 12197: 12195: 12194: 12193: 12184: 12176: 12161: 12154: 12145: 12140: 12139: 12118:Equation 10 12103: 12101: 12100: 12095: 12077: 12075: 12074: 12069: 12062: 12061: 12057: 12056: 12054: 12053: 12041: 12022: 12014: 12009: 12008: 11992: 11990: 11989: 11984: 11982: 11981: 11965: 11963: 11962: 11957: 11945: 11943: 11942: 11937: 11932: 11921: 11875:Hippolyte Fizeau 11855:Dragging effects 11819: 11817: 11816: 11811: 11809: 11794: 11792: 11791: 11786: 11784: 11766: 11764: 11763: 11758: 11746: 11744: 11743: 11738: 11736: 11717: 11715: 11714: 11709: 11686: 11684: 11683: 11678: 11676: 11652: 11624: 11615: 11555:angular velocity 11533: 11531: 11530: 11525: 11520: 11519: 11510: 11482: 11480: 11479: 11474: 11459: 11457: 11456: 11451: 11435: 11433: 11432: 11427: 11400: 11389: 11343: 11341: 11337: 11336: 11334: 11333: 11328: 11325: 11313: 11312: 11304: 11303: 11295: 11289: 11269: 11264: 11256: 11250: 11242: 11220: 11205: 11203: 11202: 11197: 11195: 11188: 11166: 11164: 11163: 11158: 11153: 11152: 11136: 11134: 11133: 11128: 11126: 11114: 11112: 11111: 11106: 11104: 11087:Equation 14 11081:Equation 10 11071: 11062: 11060: 11059: 11054: 11047: 11046: 11044: 11043: 11039: 11038: 11037: 11028: 11016: 10988: 10979: 10974: 10973: 10958: 10957: 10955: 10954: 10950: 10949: 10948: 10939: 10927: 10899: 10890: 10885: 10884: 10869: 10868: 10866: 10865: 10864: 10855: 10843: 10827: 10816: 10806: 10801: 10800: 10783: 10775: 10766: 10764: 10763: 10758: 10751: 10750: 10748: 10747: 10743: 10742: 10741: 10732: 10724: 10723: 10699: 10698: 10689: 10680: 10665: 10664: 10662: 10661: 10657: 10656: 10655: 10646: 10638: 10637: 10613: 10612: 10603: 10594: 10579: 10578: 10576: 10575: 10574: 10565: 10557: 10556: 10540: 10533: 10532: 10522: 10513: 10496: 10485: 10476: 10474: 10473: 10468: 10461: 10457: 10446: 10441: 10428: 10424: 10413: 10408: 10395: 10391: 10380: 10375: 10351: 10340: 10332: 10321: 10313: 10298: 10297: 10280: 10272: 10263: 10261: 10260: 10255: 10248: 10238: 10225: 10215: 10202: 10192: 10172: 10171: 10160: 10156: 10155: 10144: 10140: 10139: 10124: 10111: 10107: 10105: 10104: 10099: 10097: 10078: 10069: 10067: 10066: 10061: 10056: 10054: 10053: 10052: 10043: 10035: 10020: 10013: 10004: 9988: 9985: 9983: 9982: 9977: 9970: 9956: 9954: 9953: 9948: 9929: 9920: 9918: 9917: 9912: 9907: 9905: 9904: 9901: 9900: 9891: 9883: 9873: 9862: 9857: 9855: 9854: 9851: 9843: 9835: 9831: 9828: 9827: 9815: 9805: 9798: 9795: 9787: 9779: 9775: 9770: 9768: 9767: 9763: 9762: 9759: 9758: 9749: 9737: 9716: 9686: 9681: 9679: 9678: 9666: 9665: 9653: 9648: 9632: 9628: 9626: 9625: 9620: 9618: 9603: 9601: 9600: 9595: 9565: 9563: 9562: 9557: 9555: 9537: 9535: 9534: 9529: 9527: 9502: 9493: 9491: 9490: 9485: 9479: 9468: 9463: 9449: 9438: 9437: 9432: 9424: 9410: 9402: 9393: 9391: 9390: 9385: 9373: 9353: 9352: 9344: 9330: 9323: 9316: 9314: 9313: 9308: 9306: 9305: 9289: 9287: 9286: 9281: 9276: 9260: 9234:spatial distance 9202: 9195: 9185: 9183: 9182: 9177: 9170: 9163: 9146: 9142: 9139: 9136: 9134: 9133: 9128: 9126: 9121: 9113: 9108: 9087: 9070: 9068: 9059: 9051: 9045: 8999: 8994: 8992: 8991: 8986: 8984: 8979: 8971: 8955: 8950: 8948: 8947: 8942: 8940: 8938: 8930: 8922: 8917: 8887: 8885: 8884: 8879: 8877: 8872: 8861: 8848: 8846: 8845: 8840: 8838: 8836: 8828: 8817: 8806: 8804: 8803: 8798: 8796: 8795: 8791: 8778: 8777: 8759: 8754: 8743: 8725: 8723: 8722: 8717: 8700: 8695: 8687: 8676: 8674: 8673: 8668: 8666: 8665: 8661: 8652: 8648: 8646: 8635: 8624: 8617: 8616: 8612: 8599: 8598: 8580: 8575: 8567: 8556: 8554: 8553: 8548: 8542: 8533: 8514: 8512: 8511: 8506: 8504: 8503: 8499: 8486: 8485: 8467: 8462: 8454: 8443: 8441: 8440: 8435: 8433: 8432: 8428: 8419: 8415: 8413: 8402: 8391: 8369: 8364: 8356: 8345: 8343: 8342: 8337: 8335: 8334: 8325: 8308: 8306: 8305: 8300: 8298: 8279: 8274: 8272: 8271: 8266: 8264: 8262: 8251: 8242: 8225: 8220: 8218: 8217: 8212: 8210: 8202: 8197: 8189: 8184: 8183: 8158: 8156: 8155: 8150: 8133: 8128: 8120: 8108: 8106: 8105: 8100: 8098: 8097: 8093: 8084: 8080: 8078: 8067: 8056: 8049: 8048: 8044: 8031: 8030: 8012: 8007: 7999: 7988: 7986: 7985: 7980: 7974: 7965: 7946: 7944: 7943: 7938: 7936: 7935: 7931: 7918: 7917: 7899: 7894: 7886: 7874: 7872: 7871: 7866: 7864: 7863: 7859: 7850: 7846: 7844: 7833: 7822: 7800: 7795: 7787: 7776: 7774: 7773: 7768: 7766: 7765: 7756: 7739: 7737: 7736: 7731: 7729: 7710: 7705: 7703: 7702: 7697: 7695: 7693: 7682: 7673: 7656: 7651: 7649: 7648: 7643: 7641: 7633: 7628: 7620: 7615: 7614: 7586: 7584: 7583: 7578: 7576: 7574: 7566: 7555: 7550: 7528: 7526: 7525: 7520: 7518: 7513: 7502: 7478: 7473: 7471: 7470: 7465: 7463: 7461: 7453: 7445: 7440: 7424: 7419: 7417: 7416: 7411: 7409: 7404: 7396: 7356: 7355: 7351: 7349: 7348: 7343: 7338: 7321: 7301: 7293: 7277: 7275: 7274: 7269: 7264: 7247: 7227: 7219: 7203: 7199: 7195: 7142: 7140: 7139: 7134: 7108: 7106: 7105: 7100: 7095: 7093: 7092: 7083: 7078: 7077: 7062: 7060: 7059: 7053: 7052: 7037: 7035: 7034: 7029: 7027: 7022: 7020: 7019: 7007: 7006: 6997: 6995: 6994: 6979: 6978: 6962: 6960: 6959: 6954: 6949: 6947: 6946: 6934: 6933: 6924: 6912: 6910: 6909: 6904: 6891: 6889: 6888: 6883: 6859: 6857: 6856: 6851: 6843: 6829: 6828: 6825: 6811: 6809: 6808: 6803: 6791: 6749:the same point. 6724:than the time (Δ 6719: 6712: 6702: 6700: 6699: 6694: 6687: 6668: 6664: 6661: 6659: 6658: 6653: 6638: 6611: 6570: 6568: 6567: 6562: 6543: 6536: 6524: 6522: 6513: 6506: 6492: 6490: 6489: 6484: 6482: 6478: 6477: 6475: 6474: 6465: 6453: 6431: 6384:with respect to 6383: 6333: 6331: 6330: 6325: 6313: 6311: 6310: 6305: 6300: 6299: 6283: 6281: 6280: 6275: 6264: 6253: 6242: 6221: 6219: 6218: 6213: 6192: 6190: 6189: 6184: 6176: 6165: 6164: 6148: 6146: 6145: 6140: 6127: 6125: 6124: 6119: 6117: 6099: 6097: 6096: 6091: 6080: 6069: 6068: 6052: 6050: 6049: 6044: 6033:. For values of 6024: 6022: 6021: 6016: 6011: 6009: 6008: 5993: 5984: 5982: 5981: 5976: 5968: 5957: 5956: 5935:(in particular, 5934: 5932: 5931: 5926: 5918: 5900: 5898: 5897: 5892: 5880: 5878: 5877: 5872: 5860: 5858: 5857: 5852: 5844: 5843: 5834: 5806: 5799: 5781: 5779: 5778: 5773: 5762: 5751: 5740: 5714: 5712: 5711: 5706: 5698: 5680: 5678: 5677: 5672: 5664: 5641:(in particular, 5640: 5638: 5637: 5632: 5624: 5606: 5604: 5603: 5598: 5586: 5584: 5583: 5578: 5566: 5564: 5563: 5558: 5526: 5519: 5501: 5499: 5498: 5493: 5482: 5471: 5460: 5437: 5435: 5434: 5429: 5427: 5426: 5404: 5402: 5401: 5396: 5394: 5393: 5372: 5370: 5369: 5364: 5362: 5358: 5357: 5356: 5336: 5335: 5334: 5317: 5316: 5301: 5297: 5293: 5292: 5289: 5288: 5279: 5278: 5269: 5255: 5254: 5253: 5236: 5235: 5223: 5219: 5218: 5215: 5214: 5205: 5204: 5195: 5181: 5180: 5179: 5162: 5161: 5152: 5151: 5136: 5132: 5129: 5128: 5119: 5118: 5117: 5116: 5099: 5098: 5088: 5085: 5084: 5072: 5071: 5070: 5053: 5052: 5040: 5039: 5038: 5021: 5020: 5011: 5010: 4998: 4997: 4996: 4979: 4978: 4969: 4968: 4953: 4946: 4945: 4944: 4927: 4926: 4914: 4903: 4883: 4882: 4881: 4859: 4858: 4857: 4845: 4841: 4840: 4837: 4836: 4827: 4826: 4825: 4824: 4807: 4806: 4796: 4790: 4779: 4759: 4758: 4757: 4740: 4739: 4725: 4724: 4709: 4705: 4704: 4695: 4678: 4662: 4661: 4660: 4648: 4647: 4642: 4638: 4637: 4634: 4633: 4624: 4623: 4608: 4602: 4585: 4584: 4575: 4574: 4558: 4557: 4542: 4541: 4529: 4528: 4506: 4504: 4503: 4498: 4496: 4495: 4480: 4479: 4467: 4466: 4452: 4451: 4410:), making it an 4397: 4395: 4394: 4389: 4384: 4383: 4359: 4357: 4356: 4351: 4346: 4345: 4330: 4329: 4314: 4313: 4295: 4294: 4282: 4281: 4271: 4267: 4263: 4260: 4259: 4241: 4239: 4237: 4236: 4231: 4229: 4228: 4201: 4199: 4198: 4193: 4178: 4176: 4175: 4170: 4146:counterintuitive 4107: 4105: 4104: 4099: 4097: 4094: 4085: 4083: 4082: 4077: 4075: 4072: 4063: 4061: 4060: 4055: 4047: 4023: 4021: 4020: 4015: 3997: 3995: 3994: 3989: 3974: 3972: 3971: 3966: 3964: 3959: 3958: 3940: 3932: 3931: 3913: 3904: 3902: 3901: 3896: 3894: 3876: 3874: 3873: 3868: 3835: 3833: 3832: 3827: 3825: 3810: 3808: 3807: 3802: 3790: 3788: 3787: 3782: 3749: 3747: 3746: 3741: 3739: 3721: 3719: 3718: 3713: 3686: 3684: 3683: 3678: 3654: 3652: 3651: 3646: 3641: 3627: 3609: 3607: 3606: 3601: 3574: 3572: 3571: 3566: 3542: 3540: 3539: 3534: 3529: 3515: 3492: 3490: 3489: 3484: 3476: 3461: 3459: 3458: 3453: 3435: 3433: 3432: 3427: 3419: 3398: 3396: 3395: 3390: 3373: 3372: 3347: 3345: 3344: 3339: 3337: 3322: 3320: 3319: 3314: 3312: 3294: 3292: 3291: 3286: 3281: 3260: 3258: 3257: 3252: 3250: 3232: 3230: 3229: 3224: 3222: 3204: 3202: 3201: 3196: 3194: 3173: 3171: 3170: 3165: 3160: 3157: 3148: 3146: 3145: 3140: 3138: 3135: 3126: 3124: 3123: 3118: 3096: 3094: 3093: 3088: 3076: 3074: 3073: 3068: 3053: 3051: 3050: 3045: 3030: 3028: 3027: 3022: 3010: 3008: 3007: 3002: 2990: 2988: 2987: 2982: 2970: 2968: 2967: 2962: 2926: 2917: 2906: 2897: 2880: 2878: 2877: 2872: 2865: 2864: 2860: 2859: 2858: 2849: 2844: 2831: 2824: 2806: 2787: 2785: 2784: 2779: 2775: 2769: 2765: 2747: 2731: 2711: 2706: 2704: 2703: 2698: 2691: 2690: 2686: 2685: 2684: 2675: 2662: 2642: 2635: 2618: 2616: 2615: 2610: 2606: 2603: 2597: 2569: 2562: 2544: 2534: 2532: 2531: 2526: 2519: 2518: 2514: 2513: 2512: 2503: 2498: 2485: 2478: 2460: 2441: 2439: 2438: 2433: 2429: 2423: 2419: 2401: 2385: 2365: 2360: 2358: 2357: 2352: 2345: 2344: 2340: 2339: 2338: 2329: 2316: 2296: 2289: 2272: 2270: 2269: 2264: 2260: 2257: 2251: 2223: 2216: 2198: 2188: 2186: 2185: 2180: 2173: 2172: 2171: 2159: 2158: 2138: 2135: 2129: 2128: 2127: 2115: 2114: 2089: 2084: 2082: 2081: 2076: 2069: 2064: 2048: 2036: 2023: 2017: 2012: 1996: 1984: 1966: 1959: 1954: 1944: 1935: 1917: 1912: 1902: 1893: 1832: 1830: 1829: 1824: 1822: 1815: 1793: 1768: 1754: 1723: 1722: 1713: 1708: 1694: 1620: 1597: 1595: 1594: 1589: 1587: 1584: 1583: 1574: 1569: 1568: 1553: 1549: 1534: 1532: 1531: 1526: 1524: 1507: 1485: 1453: 1442: 1427: 1426: 1417: 1395: 1384: 1361: 1346: 1344: 1337: 1330: 1323: 1310: 1195:, dates back to 1174: 1154: 1147: 1145: 1138:for frame S and 1137: 1130: 1111: 1101: 1078: 1071: 1051:reference frames 1012:Reference frames 945: 863: 820:curved spacetime 721: 719: 718: 713: 701: 699: 698: 693: 691: 690: 609: 608: 581:electromagnetism 472: 465: 458: 445: 440: 439: 432: 428: 427: 320: 247: 208: 139: 97: 66: 52: 51: 21: 36183: 36182: 36178: 36177: 36176: 36174: 36173: 36172: 36168:Albert Einstein 36153: 36152: 36151: 36139: 36129: 36127: 36117: 36115: 36103: 36093: 36091: 36079: 36071: 36069: 36064: 36015:Albert Einstein 35982: 35963:Einstein tensor 35926: 35907:Ricci curvature 35887:Kronecker delta 35873:Notable tensors 35868: 35789:Connection form 35766: 35760: 35691: 35677:Tensor operator 35634: 35628: 35568: 35544:Computer vision 35537: 35519: 35515:Tensor calculus 35459: 35448: 35443: 35413: 35408: 35382: 35361: 35335: 35314: 35280: 35272: 35221: 35175: 35170: 35140: 35135: 35107: 35074: 35013: 34989:Relative motion 34977: 34973:Aether theories 34956: 34943: 34922: 34917: 34887: 34882: 34868: 34696: 34600:BKL singularity 34590:Lemaître–Tolman 34565: 34561:Quantum gravity 34543: 34537: 34523:geodetic effect 34497:(together with 34467:LISA Pathfinder 34406: 34355: 34341:Penrose diagram 34323: 34317: 34292: 34281: 34277:Minkowski space 34243: 34187: 34171: 34119: 34113: 34073: 34066: 34061: 34031: 34026: 34006: 33952:Religious views 33947:Political views 33915: 33900:(granddaughter) 33898:Evelyn Einstein 33886:Robert Einstein 33880:Eduard Einstein 33824: 33761: 33755: 33709: 33679:Einstein's Gift 33613: 33606: 33547:Annus mirabilis 33532: 33503:Teleparallelism 33448:Einstein radius 33418:Brownian motion 33384: 33382:Albert Einstein 33379: 33349: 33344: 33308: 33294:Medical physics 33245: 33204:Nuclear physics 33173: 33167:Non-equilibrium 33089: 33061: 33033: 33027: 32969: 32957:Wayback Machine 32938:Wayback Machine 32921: 32877:Albert Einstein 32864: 32838: 32835:Wayback Machine 32816:Wayback Machine 32796: 32780:Wayback Machine 32773:Einstein Online 32759: 32724: 32719: 32575:Physics Letters 32569: 32525: 32504: 32485: 32457: 32438: 32422:. p. 128. 32330: 32291: 32286: 32284:Further reading 32281: 32274: 32258: 32254: 32235: 32231: 32214:John A. Wheeler 32204: 32200: 32184: 32180: 32173: 32157: 32153: 32146: 32132: 32128: 32121: 32107: 32103: 32060: 32056: 32046: 32044: 32033: 32029: 32021: 31975: 31973: 31969: 31955: 31939: 31935: 31928: 31908: 31904: 31897: 31879: 31870: 31821: 31817: 31807: 31805: 31792:Gibbs, Philip. 31790: 31783: 31776: 31762: 31753: 31746: 31732: 31725: 31718: 31704: 31700: 31689: 31685: 31658:Physics Reports 31654: 31650: 31640: 31638: 31627: 31620: 31614:Wayback Machine 31605: 31601: 31591: 31589: 31583: 31576: 31569: 31555: 31551: 31544: 31521: 31517: 31510: 31491: 31487: 31477: 31475: 31474:on 1 March 2017 31460: 31456: 31446: 31444: 31429: 31425: 31415: 31413: 31402: 31398: 31391: 31377: 31373: 31324: 31320: 31289: 31285: 31275: 31273: 31258: 31254: 31244: 31242: 31238: 31232:Project PHYSNET 31227: 31221: 31217: 31207: 31205: 31198: 31194: 31179: 31146: 31142: 31123: 31119: 31109: 31107: 31085: 31081: 31074: 31059: 31053: 31049: 31039: 31037: 31023:The Observatory 31015: 31011: 31001: 30999: 30994:Dolan, Graham. 30992: 30988: 30956:physics/0412055 30939: 30935: 30928: 30912: 30908: 30901: 30885: 30881: 30871: 30869: 30867: 30849: 30845: 30835: 30833: 30829: 30818: 30812: 30808: 30797: 30781: 30777: 30767: 30743: 30739: 30729: 30727: 30718:Gibbs, Philip. 30716: 30712: 30705: 30691: 30687: 30677: 30675: 30674:. Virginia Tech 30668: 30664: 30651: 30647: 30640:10.1119/1.18252 30616: 30612: 30602: 30600: 30585: 30581: 30574: 30554: 30550: 30543: 30529: 30522: 30512: 30510: 30508: 30492: 30488: 30481: 30469:(2n ed.). 30461: 30457: 30450: 30442:. p. 128. 30430: 30426: 30419: 30411:. p. 422. 30409:Springer-Verlag 30399: 30395: 30385: 30383: 30371: 30365: 30358: 30348: 30346: 30330:(25): 709–726. 30316: 30312: 30305: 30291: 30282: 30275: 30255: 30251: 30196: 30192: 30177: 30166: 30159: 30145: 30136: 30126: 30124: 30105: 30099: 30095: 30088: 30074: 30070: 30063: 30049: 30042: 30035: 30019: 30015: 30008: 30000:. McGraw-Hill. 29990: 29986: 29969: 29965: 29958: 29944: 29940: 29891: 29887: 29870: 29835: 29819: 29815: 29808: 29794: 29790: 29774: 29773: 29769: 29753: 29749: 29744: 29740: 29691: 29684: 29664:Michael Polanyi 29662: 29658: 29649: 29647: 29639: 29638: 29634: 29625: 29623: 29614: 29613: 29609: 29602: 29586: 29573: 29566: 29550: 29537: 29527: 29511: 29507: 29497: 29481: 29477: 29469: 29465: 29458: 29454: 29447: 29431: 29427: 29418: 29416: 29406: 29402: 29395: 29378: 29371: 29364: 29345: 29341: 29334: 29315: 29306: 29299: 29280: 29271: 29267: 29262: 29222: 29218: 29170: 29163: 29118: 29109: 29099: 29097: 29093: 29087: 29083: 29073: 29069: 29053: 29044: 29003: 28999: 28971:Physical Review 28966: 28962: 28935: 28931: 28904: 28900: 28861: 28857: 28818: 28814: 28767: 28763: 28746: 28742: 28732: 28730: 28711: 28705: 28698: 28688: 28686: 28667: 28661: 28657: 28644: 28640: 28624:Das, A. (1993) 28623: 28619: 28612: 28598: 28594: 28588: 28584: 28579: 28575: 28570: 28563: 28558: 28554: 28549: 28545: 28536: 28532: 28500:Albert Einstein 28498: 28481: 28477: 28475:Primary sources 28472: 28466: 28462: 28449: 28447: 28437: 28433: 28427: 28423: 28414: 28410: 28397: 28393: 28383:Minkowski space 28367: 28363: 28343:negative square 28333:values (is not 28324: 28320: 28304: 28303: 28286: 28282: 28280: 28276: 28260: 28259: 28248:Hendrik Lorentz 28241: 28237: 28228: 28224: 28204: 28199: 28198: 28196: 28193: 28192: 28190: 28186: 28180: 28176: 28172: 28135:conventionalism 27958: 27895: 27891: 27875: 27868: 27864: 27860: 27858: 27849: 27845: 27843: 27840: 27839: 27794: 27790: 27778: 27774: 27772: 27769: 27768: 27743: 27739: 27718: 27703: 27699: 27697: 27693: 27692: 27680: 27675: 27674: 27672: 27669: 27668: 27645: 27641: 27632: 27618: 27614: 27613: 27601: 27597: 27591: 27587: 27578: 27574: 27565: 27560: 27559: 27557: 27554: 27553: 27542: 27514: 27510: 27504: 27500: 27491: 27487: 27482: 27479: 27478: 27449: 27445: 27433: 27429: 27423: 27419: 27410: 27406: 27397: 27392: 27391: 27389: 27386: 27385: 27380:The 4-velocity 27375:proper distance 27363: 27361: 27360: 27337: 27335: 27334: 27307: 27303: 27285: 27281: 27263: 27259: 27241: 27237: 27212: 27208: 27198: 27194: 27181: 27177: 27168: 27164: 27154: 27150: 27138: 27133: 27132: 27127: 27124: 27123: 27102: 27098: 27089: 27087: 27086: 27076: 27075: 27071: 27058: 27057: 27053: 27048: 27045: 27044: 27041: 27016: 27007: 27005: 27004: 26998: 26994: 26988: 26986: 26985: 26979: 26975: 26969: 26967: 26966: 26960: 26956: 26947: 26945: 26944: 26938: 26934: 26928: 26926: 26925: 26919: 26915: 26906: 26904: 26903: 26897: 26893: 26891: 26888: 26887: 26866: 26862: 26856: 26852: 26850: 26842: 26837: 26832: 26830: 26827: 26826: 26818: 26800: 26791: 26787: 26778: 26774: 26768: 26764: 26755: 26751: 26742: 26738: 26732: 26728: 26719: 26715: 26709: 26705: 26703: 26700: 26699: 26667: 26665: 26664: 26654: 26653: 26649: 26643: 26641: 26640: 26630: 26629: 26625: 26615: 26607: 26606: 26602: 26590: 26586: 26584: 26581: 26580: 26549: 26545: 26543: 26540: 26539: 26521: 26520: 26515: 26510: 26505: 26499: 26498: 26493: 26488: 26483: 26477: 26476: 26471: 26466: 26461: 26455: 26454: 26449: 26444: 26439: 26426: 26425: 26413: 26409: 26407: 26404: 26403: 26399: 26381: 26313: 26311: 26310: 26300: 26299: 26295: 26293: 26290: 26289: 26272: 26270: 26269: 26259: 26258: 26254: 26252: 26249: 26248: 26222: 26208: 26198: 26196: 26195: 26185: 26184: 26180: 26171: 26169: 26168: 26158: 26157: 26153: 26147: 26145: 26144: 26134: 26133: 26129: 26123: 26121: 26120: 26110: 26109: 26105: 26096: 26094: 26093: 26083: 26082: 26078: 26072: 26070: 26069: 26059: 26058: 26054: 26041: 26030: 26022: 26021: 26012: 26001: 25993: 25992: 25986: 25983: 25982: 25952: 25950: 25949: 25939: 25938: 25934: 25932: 25929: 25928: 25911: 25909: 25908: 25898: 25897: 25893: 25891: 25888: 25887: 25867: 25863: 25857: 25855: 25854: 25844: 25843: 25839: 25826: 25825: 25821: 25819: 25816: 25815: 25777: 25773: 25769: 25764: 25755: 25751: 25738: 25734: 25725: 25723: 25722: 25712: 25711: 25707: 25688: 25687: 25683: 25678: 25675: 25674: 25653: 25652: 25647: 25642: 25637: 25631: 25630: 25625: 25620: 25615: 25609: 25608: 25603: 25598: 25593: 25581: 25580: 25575: 25570: 25559: 25549: 25548: 25541: 25540: 25529: 25521: 25518: 25516: 25505: 25497: 25494: 25492: 25481: 25473: 25470: 25468: 25457: 25449: 25446: 25434: 25427: 25426: 25416: 25415: 25403: 25399: 25391: 25388: 25386: 25374: 25370: 25362: 25359: 25357: 25345: 25341: 25333: 25330: 25328: 25316: 25312: 25304: 25301: 25289: 25282: 25281: 25279: 25276: 25275: 25226: 25219: 25215: 25211: 25209: 25200: 25196: 25194: 25191: 25190: 25151: 25138: 25137: 25133: 25119: 25115: 25106: 25102: 25093: 25089: 25076: 25075: 25071: 25059: 25055: 25046: 25042: 25033: 25029: 24996: 24992: 24980: 24976: 24974: 24971: 24970: 24947: 24924: 24922: 24919: 24918: 24898: 24894: 24871: 24868: 24867: 24842: 24837: 24824: 24819: 24806: 24801: 24788: 24784: 24774: 24770: 24765: 24759: 24755: 24743: 24726: 24723: 24722: 24700: 24667: 24663: 24654: 24650: 24641: 24637: 24603: 24596: 24592: 24588: 24586: 24577: 24573: 24571: 24568: 24567: 24543: 24539: 24537: 24534: 24533: 24511: 24507: 24501: 24499: 24498: 24488: 24487: 24483: 24470: 24469: 24465: 24463: 24460: 24459: 24438: 24434: 24432: 24429: 24428: 24401: 24399: 24398: 24388: 24387: 24383: 24381: 24378: 24377: 24361: 24358: 24357: 24336: 24332: 24326: 24324: 24323: 24313: 24312: 24308: 24295: 24294: 24290: 24288: 24285: 24284: 24273: 24251: 24194: 24190: 24181: 24177: 24168: 24164: 24155: 24151: 24139: 24135: 24133: 24130: 24129: 24100: 24044: 24043: 24037: 24036: 24030: 24029: 24005: 24004: 23976: 23975: 23965: 23964: 23958: 23957: 23951: 23950: 23944: 23943: 23930: 23929: 23922: 23921: 23916: 23911: 23906: 23900: 23899: 23894: 23889: 23884: 23878: 23877: 23872: 23867: 23862: 23850: 23849: 23844: 23839: 23828: 23818: 23817: 23807: 23806: 23798: 23795: 23794: 23786: 23783: 23782: 23774: 23771: 23770: 23762: 23752: 23751: 23749: 23746: 23745: 23732: 23727: 23676: 23671: 23658: 23653: 23640: 23635: 23629: 23621: 23618: 23617: 23576: 23572: 23563: 23559: 23550: 23545: 23529: 23524: 23508: 23503: 23494: 23491: 23490: 23473: 23469: 23460: 23456: 23447: 23442: 23426: 23421: 23405: 23400: 23378: 23374: 23369: 23366: 23365: 23362: 23333: 23329: 23320: 23316: 23307: 23302: 23286: 23281: 23272: 23269: 23268: 23251: 23247: 23238: 23234: 23225: 23220: 23204: 23199: 23177: 23173: 23168: 23165: 23164: 23137: 23133: 23124: 23120: 23111: 23106: 23090: 23085: 23069: 23065: 23060: 23057: 23056: 23045: 23029: 23003: 22986: 22971: 22965: 22922: 22915: 22908: 22901: 22888: 22868: 22863: 22847: 22842: 22826: 22821: 22805: 22800: 22781: 22777: 22772: 22769: 22768: 22760: 22753: 22746: 22733: 22713: 22708: 22692: 22687: 22671: 22666: 22651: 22640: 22628: 22624: 22619: 22616: 22615: 22601:Euclidean space 22559:Euclidean space 22548: 22542: 22537: 22532: 22530:Minkowski space 22526: 22510:emission theory 22406:quantum gravity 22378: 22370:Main articles: 22368: 22283:quantum gravity 22253: 22231:electromagnetic 22188: 22180:Main articles: 22178: 22131: 22125: 22121: 22110: 22107: 22106: 22090: 22087: 22086: 22058: 22055: 22054: 22018: 22009: 22005: 21967: 21956: 21953: 21952: 21928: 21914: 21911: 21910: 21889: 21881: 21877: 21868: 21864: 21858: 21854: 21852: 21850: 21847: 21846: 21825: 21821: 21813: 21808: 21797: 21794: 21793: 21770: 21766: 21757: 21753: 21747: 21743: 21741: 21736: 21710: 21707: 21706: 21686: 21682: 21673: 21669: 21660: 21656: 21650: 21646: 21644: 21641: 21640: 21616: 21602: 21599: 21598: 21575: 21572: 21571: 21552: 21549: 21548: 21532: 21529: 21528: 21495: 21487:Main articles: 21485: 21459: 21457: 21454: 21453: 21434: 21426: 21424: 21421: 21420: 21404: 21401: 21400: 21381: 21373: 21371: 21368: 21367: 21348: 21345: 21344: 21328: 21325: 21324: 21306: 21305: 21287: 21271: 21260: 21252: 21249: 21248: 21222: 21218: 21217: 21213: 21203: 21194: 21187: 21186: 21166: 21162: 21161: 21157: 21147: 21138: 21130: 21128: 21125: 21124: 21102: 21100: 21097: 21096: 21077: 21074: 21073: 21050: 21046: 21037: 21033: 21031: 21028: 21027: 21010: 21006: 21004: 21001: 21000: 20978: 20976: 20973: 20972: 20949: 20946: 20945: 20926: 20918: 20916: 20913: 20912: 20896: 20893: 20892: 20876: 20873: 20872: 20817: 20814: 20779: 20773: 20755: 20749: 20740: 20736: 20728: 20724: 20715: 20711: 20707: 20703: 20671: 20670: 20657: 20649: 20647: 20646: 20642: 20637: 20634: 20633: 20605: 20593: 20592: 20588: 20586: 20573: 20565: 20563: 20562: 20558: 20553: 20550: 20549: 20526: 20518: 20516: 20508: 20505: 20504: 20484: 20481: 20480: 20476: 20473:Force 4-vector: 20443: 20428: 20427: 20417: 20415: 20402: 20394: 20392: 20391: 20387: 20382: 20379: 20378: 20351: 20350: 20328: 20323: 20321: 20318: 20317: 20294: 20286: 20284: 20276: 20273: 20272: 20252: 20249: 20248: 20244: 20232: 20201: 20200: 20189: 20186: 20185: 20160: 20159: 20148: 20145: 20144: 20140: 20114: 20110: 20101: 20097: 20088: 20084: 20073: 20050: 20049: 20038: 20024: 20021: 20020: 20007: 19982: 19979: 19978: 19964: 19960: 19959: 19958: 19957: 19956: 19953: 19945: 19944: 19941: 19903: 19902: 19888: 19885: 19884: 19856: 19848: 19846: 19833: 19825: 19823: 19810: 19802: 19800: 19793: 19789: 19784: 19781: 19780: 19756: 19749: 19708: 19706: 19693: 19685: 19683: 19675: 19672: 19671: 19642: 19639: 19638: 19619: 19616: 19615: 19611: 19543: 19540: 19539: 19529: 19521: 19510: 19485: 19481: 19475: 19471: 19462: 19458: 19456: 19453: 19452: 19435: 19431: 19422: 19418: 19412: 19408: 19406: 19403: 19402: 19371: 19368: 19367: 19362: 19346: 19342: 19338: 19334: 19306: 19303: 19302: 19291: 19263: 19262: 19248: 19247: 19238: 19234: 19228: 19224: 19222: 19219: 19218: 19199: 19195: 19189: 19185: 19176: 19172: 19166: 19162: 19153: 19149: 19143: 19139: 19130: 19126: 19120: 19116: 19114: 19111: 19110: 19086: 19083: 19082: 19076: 19054: 19053: 19051: 19048: 19047: 19025: 19024: 19022: 19019: 19018: 19014: 19010: 18992: 18956: 18953: 18952: 18950: 18946: 18931: 18869: 18866: 18865: 18863: 18859: 18855: 18826: 18815: 18789: 18788: 18782: 18778: 18771: 18762: 18755: 18754: 18748: 18744: 18737: 18728: 18721: 18720: 18709: 18705: 18694: 18679: 18675: 18674: 18670: 18660: 18651: 18644: 18643: 18632: 18628: 18617: 18602: 18598: 18597: 18593: 18583: 18574: 18566: 18564: 18561: 18560: 18556:) according to 18550: 18546: 18497: 18494: 18493: 18491: 18485: 18457: 18453: 18444: 18440: 18431: 18427: 18418: 18414: 18413: 18409: 18401: 18398: 18397: 18395: 18392: 18375: 18371: 18348: 18344: 18341: 18335: 18299: 18298: 18290: 18287: 18286: 18278: 18268: 18267: 18260: 18259: 18248: 18236: 18235: 18224: 18208: 18207: 18197: 18196: 18190: 18189: 18176: 18175: 18173: 18170: 18169: 18159: 18135: 18134: 18128: 18127: 18114: 18113: 18106: 18105: 18094: 18079: 18078: 18064: 18048: 18047: 18037: 18036: 18028: 18025: 18024: 18016: 18006: 18005: 18003: 18000: 17999: 17989: 17943: 17940: 17939: 17914: 17910: 17892: 17890: 17878: 17874: 17862: 17851: 17848: 17847: 17818: 17815: 17814: 17789: 17785: 17773: 17761: 17757: 17745: 17737: 17734: 17733: 17704: 17700: 17691: 17687: 17679: 17676: 17675: 17649: 17645: 17636: 17632: 17621: 17618: 17617: 17595: 17591: 17579: 17575: 17562: 17555: 17551: 17536: 17532: 17525: 17523: 17521: 17518: 17517: 17495: 17491: 17485: 17481: 17474: 17467: 17463: 17454: 17450: 17449: 17447: 17439: 17436: 17435: 17402: 17398: 17389: 17385: 17384: 17374: 17370: 17361: 17357: 17356: 17354: 17338: 17327: 17325: 17311: 17308: 17307: 17277: 17257: 17254: 17253: 17226: 17223: 17222: 17198: 17197: 17191: 17190: 17180: 17179: 17172: 17171: 17160: 17148: 17147: 17133: 17117: 17116: 17106: 17105: 17097: 17094: 17093: 17085: 17078: 17077: 17075: 17072: 17071: 17058: 17051: 17046: 17034: 17004: 16993: 16991: 16971: 16969: 16966: 16965: 16939: 16932: 16925: 16897: 16866: 16865: 16864: 16863: 16862: 16842: 16831: 16825: 16817: 16816: 16796: 16785: 16779: 16768: 16762: 16757: 16715: 16711: 16707: 16695: 16690: 16674: 16670: 16665: 16658: 16652: 16648: 16646: 16638: 16628: 16625: 16624: 16593: 16589: 16580: 16576: 16574: 16571: 16570: 16548: 16545: 16544: 16504: 16500: 16496: 16484: 16480: 16462: 16458: 16452: 16448: 16436: 16432: 16425: 16419: 16408: 16387: 16383: 16379: 16377: 16357: 16347: 16344: 16343: 16301: 16297: 16293: 16287: 16282: 16273: 16249: 16245: 16237: 16228: 16224: 16218: 16213: 16205: 16199: 16187: 16183: 16182: 16180: 16171: 16167: 16163: 16160: 16159: 16117: 16113: 16109: 16103: 16098: 16089: 16065: 16061: 16053: 16044: 16040: 16034: 16029: 16021: 16015: 16003: 15999: 15998: 15996: 15987: 15983: 15979: 15976: 15975: 15937: 15933: 15924: 15920: 15905: 15901: 15897: 15894: 15893: 15859: 15847: 15843: 15837: 15833: 15825: 15813: 15809: 15803: 15799: 15795: 15792: 15791: 15757: 15745: 15741: 15735: 15731: 15723: 15711: 15707: 15701: 15697: 15688: 15684: 15678: 15674: 15670: 15667: 15666: 15633: 15630: 15629: 15613: 15610: 15609: 15602:Breaking down ( 15570: 15566: 15557: 15553: 15544: 15540: 15531: 15527: 15518: 15514: 15502: 15498: 15489: 15485: 15481: 15478: 15477: 15432: 15428: 15426: 15425: 15416: 15412: 15397: 15393: 15391: 15390: 15381: 15377: 15356: 15352: 15350: 15349: 15340: 15336: 15332: 15329: 15328: 15289: 15285: 15267: 15255: 15254: 15234: 15233: 15229: 15226: 15225: 15205: 15172: 15162: 15156: 15155:| = | 15150: 15126: 15087: 15005: 15001: 14996: 14990: 14986: 14980: 14976: 14961: 14959: 14942: 14939: 14938: 14927: 14921: 14874: 14871: 14868: 14867: 14865: 14828: 14824: 14816: 14812: 14807: 14797: 14788: 14784: 14775: 14771: 14767: 14764: 14763: 14711: 14707: 14702: 14696: 14692: 14680: 14679: 14675: 14663: 14659: 14650: 14646: 14642: 14639: 14638: 14616: 14612: 14603: 14599: 14597: 14594: 14593: 14576: 14572: 14563: 14559: 14557: 14554: 14553: 14537: 14534: 14533: 14517: 14514: 14513: 14491: 14488: 14487: 14440: 14436: 14431: 14425: 14421: 14409: 14408: 14404: 14389: 14385: 14376: 14372: 14357: 14353: 14344: 14340: 14333: 14330: 14329: 14307: 14306: 14297: 14293: 14288: 14282: 14278: 14266: 14257: 14253: 14241: 14237: 14232: 14226: 14222: 14209: 14192: 14179: 14177: 14162: 14150: 14146: 14141: 14135: 14131: 14118: 14101: 14088: 14086: 14071: 14062: 14058: 14051: 14045: 14041: 14038: 14037: 14025: 14021: 14003: 13985: 13976: 13972: 13965: 13959: 13955: 13951: 13949: 13946: 13945: 13925: 13921: 13919: 13916: 13915: 13911: 13891: 13887: 13885: 13882: 13881: 13861: 13858: 13857: 13853: 13830: 13826: 13824: 13821: 13820: 13803: 13799: 13797: 13794: 13793: 13770: 13765: 13762: 13761: 13745: 13742: 13741: 13714: 13710: 13705: 13699: 13695: 13682: 13665: 13652: 13650: 13634: 13632: 13630: 13627: 13626: 13607: 13604: 13603: 13599: 13576: 13573: 13572: 13556: 13553: 13552: 13549: 13548: 13547: 13546: 13545: 13535: 13526: 13525: 13524: 13487: 13435: 13423: 13406: 13381: 13375: 13357: 13235: 13214: 13211: 13210: 13197:The transverse 13187: 13150: 13146: 13134: 13126: 13123: 13122: 13101: 13090: 13087: 13086: 13063: 13059: 13044: 13033: 13030: 13021: 13017: 13002: 12998: 12986: 12982: 12980: 12977: 12976: 12952: 12948: 12946: 12943: 12942: 12925: 12921: 12908: 12904: 12899: 12892: 12888: 12879: 12875: 12873: 12870: 12869: 12845: 12842: 12841: 12824: 12821: 12820: 12805: 12800: 12794: 12768: 12752: 12748: 12731: 12720: 12718: 12714: 12713: 12697: 12695: 12687: 12684: 12683: 12642: 12623: 12612: 12610: 12599: 12591: 12588: 12587: 12549: 12536: 12527: 12511: 12509: 12498: 12490: 12487: 12486: 12442: 12434:Main articles: 12432: 12403: 12399: 12394: 12387: 12383: 12367: 12365: 12362: 12361: 12333: 12328: 12321: 12317: 12302: 12289: 12288: 12284: 12279: 12276: 12275: 12248: 12238: 12223: 12219: 12217: 12215: 12212: 12211: 12189: 12185: 12180: 12169: 12162: 12147: 12146: 12144: 12135: 12131: 12129: 12126: 12125: 12089: 12086: 12085: 12049: 12045: 12040: 12033: 12029: 12013: 12004: 12000: 11998: 11995: 11994: 11977: 11973: 11971: 11968: 11967: 11951: 11948: 11947: 11928: 11914: 11912: 11909: 11908: 11863: 11857: 11852: 11850:Optical effects 11802: 11800: 11797: 11796: 11777: 11772: 11769: 11768: 11752: 11749: 11748: 11729: 11727: 11724: 11723: 11700: 11697: 11696: 11669: 11664: 11661: 11660: 11650: 11636: 11635: 11634: 11633: 11631: 11627: 11626: 11625: 11617: 11616: 11580: 11570: 11553:, relating the 11515: 11511: 11506: 11498: 11495: 11494: 11468: 11465: 11464: 11445: 11442: 11441: 11421: 11418: 11417: 11410: 11409: 11408: 11407: 11403: 11402: 11401: 11392: 11391: 11390: 11379: 11377:Thomas rotation 11373: 11371:Thomas rotation 11339: 11329: 11326: 11321: 11320: 11318: 11317: 11315: 11310: 11306: 11301: 11297: 11291: 11285: 11260: 11258: 11248: 11243: 11233: 11218: 11181: 11174: 11172: 11169: 11168: 11145: 11144: 11142: 11139: 11138: 11122: 11120: 11117: 11116: 11100: 11098: 11095: 11094: 11033: 11029: 11024: 11012: 11001: 10997: 10993: 10984: 10978: 10969: 10965: 10944: 10940: 10935: 10923: 10912: 10908: 10904: 10895: 10889: 10880: 10876: 10860: 10856: 10851: 10839: 10828: 10812: 10807: 10805: 10796: 10792: 10790: 10787: 10786: 10737: 10733: 10728: 10719: 10715: 10708: 10704: 10700: 10694: 10690: 10688: 10676: 10651: 10647: 10642: 10633: 10629: 10622: 10618: 10614: 10608: 10604: 10602: 10590: 10570: 10566: 10561: 10552: 10548: 10541: 10528: 10524: 10523: 10521: 10509: 10503: 10500: 10499: 10450: 10442: 10434: 10417: 10409: 10401: 10384: 10376: 10368: 10347: 10328: 10309: 10290: 10289: 10287: 10284: 10283: 10234: 10211: 10188: 10167: 10163: 10151: 10147: 10135: 10131: 10120: 10118: 10115: 10114: 10093: 10091: 10088: 10087: 10048: 10044: 10039: 10028: 10021: 10006: 10005: 10003: 9995: 9992: 9991: 9962: 9959: 9958: 9942: 9939: 9938: 9896: 9892: 9884: 9881: 9874: 9863: 9861: 9844: 9836: 9833: 9823: 9819: 9813: 9806: 9788: 9780: 9777: 9776: 9774: 9754: 9750: 9738: 9735: 9725: 9721: 9717: 9687: 9685: 9671: 9667: 9658: 9654: 9652: 9641: 9639: 9636: 9635: 9611: 9609: 9606: 9605: 9589: 9586: 9585: 9581:Equation 7 9575:Equation 8 9569:Equation 5 9548: 9543: 9540: 9539: 9520: 9515: 9512: 9511: 9472: 9464: 9456: 9442: 9433: 9425: 9420: 9419: 9417: 9414: 9413: 9369: 9348: 9340: 9339: 9337: 9334: 9333: 9321: 9298: 9297: 9295: 9292: 9291: 9272: 9270: 9267: 9266: 9258: 9248: 9242: 9200: 9193: 9156: 9151: 9148: 9147: 9144: 9140: 9137: 9114: 9112: 9101: 9096: 9093: 9092: 9085: 9074:Equation 4 9066: 9061: 9057: 9049: 9043: 9014: 9008: 8997: 8972: 8970: 8962: 8959: 8958: 8953: 8931: 8923: 8921: 8910: 8908: 8905: 8904: 8862: 8860: 8858: 8855: 8854: 8829: 8818: 8816: 8811: 8808: 8807: 8787: 8783: 8779: 8773: 8769: 8744: 8742: 8740: 8737: 8736: 8688: 8686: 8681: 8678: 8677: 8657: 8653: 8636: 8625: 8623: 8619: 8618: 8608: 8604: 8600: 8594: 8590: 8568: 8566: 8561: 8558: 8557: 8538: 8526: 8524: 8521: 8520: 8495: 8491: 8487: 8481: 8477: 8455: 8453: 8448: 8445: 8444: 8424: 8420: 8403: 8392: 8390: 8386: 8385: 8357: 8355: 8350: 8347: 8346: 8330: 8326: 8318: 8316: 8313: 8312: 8291: 8289: 8286: 8285: 8277: 8255: 8250: 8238: 8232: 8229: 8228: 8223: 8201: 8188: 8179: 8175: 8173: 8170: 8169: 8121: 8119: 8114: 8111: 8110: 8109: 8089: 8085: 8068: 8057: 8055: 8051: 8050: 8040: 8036: 8032: 8026: 8022: 8000: 7998: 7993: 7990: 7989: 7970: 7958: 7956: 7953: 7952: 7927: 7923: 7919: 7913: 7909: 7887: 7885: 7880: 7877: 7876: 7875: 7855: 7851: 7834: 7823: 7821: 7817: 7816: 7788: 7786: 7781: 7778: 7777: 7761: 7757: 7749: 7747: 7744: 7743: 7722: 7720: 7717: 7716: 7708: 7686: 7681: 7669: 7663: 7660: 7659: 7654: 7632: 7619: 7610: 7606: 7604: 7601: 7600: 7567: 7556: 7554: 7543: 7538: 7535: 7534: 7503: 7501: 7490: 7487: 7486: 7476: 7454: 7446: 7444: 7433: 7431: 7428: 7427: 7422: 7397: 7395: 7387: 7384: 7383: 7371: 7363: 7362:Measured by the 7317: 7300: 7286: 7284: 7281: 7280: 7243: 7226: 7212: 7210: 7207: 7206: 7201: 7197: 7193: 7163: 7157: 7128: 7125: 7124: 7120: 7088: 7084: 7079: 7073: 7069: 7061: 7055: 7054: 7048: 7044: 7042: 7039: 7038: 7026: 7015: 7011: 7002: 6998: 6996: 6990: 6989: 6974: 6970: 6968: 6965: 6964: 6942: 6938: 6929: 6925: 6923: 6918: 6915: 6914: 6898: 6895: 6894: 6874: 6871: 6870: 6839: 6824: 6820: 6818: 6815: 6814: 6797: 6794: 6793: 6789: 6766: 6717: 6710: 6673: 6670: 6669: 6666: 6662: 6631: 6626: 6623: 6622: 6615:Equation 3 6605: 6591: 6585: 6556: 6553: 6552: 6541: 6530: 6520: 6515: 6511: 6500: 6470: 6466: 6454: 6452: 6442: 6438: 6424: 6419: 6416: 6415: 6411:Equation 3 6375: 6352: 6342: 6319: 6316: 6315: 6295: 6291: 6289: 6286: 6285: 6260: 6249: 6238: 6236: 6233: 6232: 6198: 6195: 6194: 6172: 6160: 6156: 6154: 6151: 6150: 6134: 6131: 6130: 6110: 6105: 6102: 6101: 6076: 6064: 6060: 6058: 6055: 6054: 6038: 6035: 6034: 6027:proper distance 6004: 6000: 5992: 5990: 5987: 5986: 5964: 5952: 5948: 5940: 5937: 5936: 5911: 5906: 5903: 5902: 5886: 5883: 5882: 5866: 5863: 5862: 5861:there exists a 5839: 5835: 5830: 5792: 5787: 5784: 5783: 5758: 5747: 5736: 5734: 5731: 5730: 5694: 5686: 5683: 5682: 5660: 5646: 5643: 5642: 5617: 5612: 5609: 5608: 5592: 5589: 5588: 5572: 5569: 5568: 5512: 5507: 5504: 5503: 5478: 5467: 5456: 5454: 5451: 5450: 5422: 5418: 5413: 5410: 5409: 5389: 5385: 5380: 5377: 5376: 5360: 5359: 5351: 5347: 5343: 5329: 5325: 5321: 5312: 5308: 5299: 5298: 5284: 5280: 5274: 5270: 5267: 5260: 5256: 5248: 5244: 5240: 5231: 5227: 5210: 5206: 5200: 5196: 5193: 5186: 5182: 5174: 5170: 5166: 5157: 5153: 5147: 5143: 5134: 5133: 5124: 5120: 5111: 5107: 5103: 5094: 5090: 5089: 5086: 5080: 5076: 5065: 5061: 5057: 5048: 5044: 5033: 5029: 5025: 5016: 5012: 5006: 5002: 4991: 4987: 4983: 4974: 4970: 4964: 4960: 4951: 4950: 4939: 4935: 4931: 4922: 4918: 4907: 4896: 4876: 4872: 4868: 4853: 4849: 4832: 4828: 4819: 4815: 4811: 4802: 4798: 4797: 4794: 4783: 4772: 4752: 4748: 4744: 4735: 4731: 4730: 4726: 4720: 4716: 4707: 4706: 4700: 4696: 4688: 4671: 4656: 4652: 4643: 4629: 4625: 4616: 4609: 4606: 4595: 4591: 4587: 4586: 4580: 4576: 4570: 4566: 4559: 4553: 4549: 4537: 4533: 4524: 4520: 4516: 4514: 4511: 4510: 4491: 4487: 4475: 4471: 4462: 4458: 4447: 4443: 4438: 4435: 4434: 4379: 4375: 4370: 4367: 4366: 4341: 4337: 4325: 4321: 4309: 4305: 4290: 4286: 4277: 4273: 4262: 4255: 4251: 4246: 4243: 4242: 4224: 4220: 4215: 4212: 4211: 4210: 4184: 4181: 4180: 4161: 4158: 4157: 4154: 4137: 4127: 4119:Cartesian plane 4093: 4091: 4088: 4087: 4071: 4069: 4066: 4065: 4040: 4038: 4035: 4034: 4003: 4000: 3999: 3980: 3977: 3976: 3954: 3950: 3936: 3927: 3923: 3912: 3910: 3907: 3906: 3887: 3882: 3879: 3878: 3841: 3838: 3837: 3818: 3816: 3813: 3812: 3796: 3793: 3792: 3755: 3752: 3751: 3732: 3727: 3724: 3723: 3692: 3689: 3688: 3660: 3657: 3656: 3655:coordinates of 3634: 3620: 3615: 3612: 3611: 3580: 3577: 3576: 3548: 3545: 3544: 3543:coordinates of 3522: 3508: 3503: 3500: 3499: 3469: 3467: 3464: 3463: 3441: 3438: 3437: 3415: 3404: 3401: 3400: 3365: 3361: 3353: 3350: 3349: 3330: 3328: 3325: 3324: 3305: 3300: 3297: 3296: 3277: 3266: 3263: 3262: 3243: 3238: 3235: 3234: 3215: 3210: 3207: 3206: 3187: 3185: 3182: 3181: 3156: 3154: 3151: 3150: 3134: 3132: 3129: 3128: 3106: 3103: 3102: 3082: 3079: 3078: 3059: 3056: 3055: 3036: 3033: 3032: 3016: 3013: 3012: 2996: 2993: 2992: 2976: 2973: 2972: 2956: 2953: 2952: 2936: 2935: 2934: 2933: 2929: 2928: 2927: 2919: 2918: 2909: 2908: 2907: 2899: 2898: 2887: 2854: 2850: 2845: 2837: 2817: 2813: 2809: 2792: 2789: 2788: 2758: 2740: 2717: 2714: 2713: 2680: 2676: 2671: 2649: 2645: 2628: 2623: 2620: 2619: 2555: 2550: 2547: 2546: 2508: 2504: 2499: 2491: 2471: 2467: 2463: 2446: 2443: 2442: 2412: 2394: 2371: 2368: 2367: 2334: 2330: 2325: 2303: 2299: 2282: 2277: 2274: 2273: 2209: 2204: 2201: 2200: 2167: 2163: 2154: 2150: 2123: 2119: 2110: 2106: 2095: 2092: 2091: 2060: 2044: 2029: 2008: 1992: 1977: 1972: 1969: 1968: 1957: 1952: 1947: 1942: 1937: 1933: 1926: 1919: 1915: 1910: 1905: 1900: 1895: 1891: 1884: 1877: 1820: 1819: 1808: 1801: 1795: 1794: 1786: 1779: 1773: 1772: 1761: 1747: 1734: 1728: 1727: 1718: 1714: 1709: 1701: 1687: 1674: 1667: 1665: 1662: 1661: 1651:linear mappings 1618: 1579: 1575: 1570: 1564: 1560: 1548: 1540: 1537: 1536: 1522: 1521: 1508: 1500: 1497: 1496: 1486: 1478: 1475: 1474: 1443: 1435: 1432: 1431: 1422: 1418: 1413: 1385: 1377: 1373: 1371: 1368: 1367: 1359: 1342: 1335: 1328: 1321: 1316: 1292: 1285: 1252: 1246: 1241: 1235: 1226:it can be shown 1222: 1185: 1172: 1152: 1143: 1139: 1132: 1128: 1109: 1099: 1076: 1069: 1047: 995:only along the 985: 980: 974: 943: 871: 865: 858: 850: 842:electrodynamics 838:laws of physics 826:Galileo Galilei 812:Minkowski space 750:Hendrik Lorentz 707: 704: 703: 686: 682: 671: 668: 667: 606: 605: 601:Hendrik Lorentz 573: 567: 555:Galileo Galilei 516:laws of physics 502:'s 1905 paper, 500:Albert Einstein 476: 435: 422: 421: 414: 413: 412: 322: 321: 318: 310: 309: 305:Aether theories 290: 289: 288: 269: 268: 249: 248: 243: 235: 234: 210: 209: 204: 196: 195: 141: 140: 137: 129: 128: 99: 98: 95: 40:Annus Mirabilis 35:Albert Einstein 28: 23: 22: 15: 12: 11: 5: 36181: 36171: 36170: 36165: 36150: 36149: 36137: 36125: 36113: 36101: 36089: 36066: 36065: 36063: 36062: 36057: 36055:Woldemar Voigt 36052: 36047: 36042: 36037: 36032: 36027: 36022: 36020:Leonhard Euler 36017: 36012: 36007: 36002: 35996: 35994: 35992:Mathematicians 35988: 35987: 35984: 35983: 35981: 35980: 35975: 35970: 35965: 35960: 35955: 35950: 35945: 35940: 35934: 35932: 35928: 35927: 35925: 35924: 35919: 35917:Torsion tensor 35914: 35909: 35904: 35899: 35894: 35889: 35883: 35881: 35874: 35870: 35869: 35867: 35866: 35861: 35856: 35851: 35846: 35841: 35836: 35831: 35826: 35821: 35816: 35811: 35806: 35801: 35796: 35791: 35786: 35781: 35776: 35770: 35768: 35762: 35761: 35759: 35758: 35752: 35750:Tensor product 35747: 35742: 35740:Symmetrization 35737: 35732: 35730:Lie derivative 35727: 35722: 35717: 35712: 35707: 35701: 35699: 35693: 35692: 35690: 35689: 35684: 35679: 35674: 35669: 35664: 35659: 35654: 35652:Tensor density 35649: 35644: 35638: 35636: 35630: 35629: 35627: 35626: 35624:Voigt notation 35621: 35616: 35611: 35609:Ricci calculus 35606: 35601: 35596: 35594:Index notation 35591: 35586: 35580: 35578: 35574: 35573: 35570: 35569: 35567: 35566: 35561: 35556: 35551: 35546: 35540: 35538: 35536: 35535: 35530: 35524: 35521: 35520: 35518: 35517: 35512: 35510:Tensor algebra 35507: 35502: 35497: 35492: 35490:Dyadic algebra 35487: 35482: 35476: 35474: 35465: 35461: 35460: 35453: 35450: 35449: 35442: 35441: 35434: 35427: 35419: 35410: 35409: 35407: 35406: 35401: 35396: 35390: 35388: 35384: 35383: 35381: 35380: 35375: 35369: 35367: 35363: 35362: 35360: 35359: 35354: 35349: 35343: 35341: 35337: 35336: 35334: 35333: 35328: 35322: 35320: 35316: 35315: 35313: 35312: 35307: 35302: 35297: 35292: 35286: 35284: 35274: 35273: 35271: 35270: 35265: 35260: 35255: 35250: 35245: 35240: 35235: 35229: 35227: 35223: 35222: 35220: 35219: 35214: 35209: 35204: 35199: 35194: 35189: 35183: 35181: 35180:Speed/isotropy 35177: 35176: 35169: 35168: 35161: 35154: 35146: 35137: 35136: 35134: 35133: 35128: 35123: 35117: 35115: 35109: 35108: 35106: 35105: 35100: 35095: 35090: 35084: 35082: 35076: 35075: 35073: 35072: 35067: 35062: 35057: 35052: 35047: 35042: 35037: 35032: 35027: 35021: 35019: 35015: 35014: 35012: 35011: 35006: 35001: 34999:Speed of light 34996: 34991: 34985: 34983: 34979: 34978: 34976: 34975: 34970: 34965: 34959: 34957: 34955: 34954: 34948: 34945: 34944: 34942: 34941: 34936: 34930: 34928: 34924: 34923: 34916: 34915: 34908: 34901: 34893: 34884: 34883: 34873: 34870: 34869: 34867: 34866: 34859: 34854: 34849: 34844: 34839: 34834: 34829: 34824: 34819: 34814: 34809: 34804: 34799: 34794: 34789: 34787:Choquet-Bruhat 34784: 34779: 34774: 34769: 34764: 34759: 34754: 34749: 34744: 34739: 34734: 34729: 34724: 34719: 34714: 34708: 34706: 34702: 34701: 34698: 34697: 34695: 34694: 34687: 34686: 34681: 34676: 34669: 34668: 34663: 34658: 34653: 34648: 34639:Axisymmetric: 34636: 34635: 34630: 34624: 34613: 34612: 34607: 34602: 34597: 34592: 34587: 34578:Cosmological: 34575: 34573: 34567: 34566: 34564: 34563: 34558: 34553: 34547: 34545: 34539: 34538: 34536: 34535: 34530: 34519:frame-dragging 34516: 34511: 34506: 34503:Einstein rings 34499:Einstein cross 34492: 34481: 34480: 34475: 34469: 34464: 34459: 34446: 34436: 34435: 34430: 34425: 34420: 34414: 34412: 34408: 34407: 34405: 34404: 34402:Ernst equation 34399: 34394: 34389: 34384: 34379: 34374: 34372:BSSN formalism 34369: 34363: 34361: 34357: 34356: 34354: 34353: 34348: 34343: 34338: 34333: 34327: 34325: 34319: 34318: 34316: 34315: 34310: 34304: 34302: 34295: 34287: 34286: 34283: 34282: 34280: 34279: 34274: 34269: 34264: 34259: 34253: 34251: 34245: 34244: 34242: 34241: 34236: 34231: 34229:Ladder paradox 34226: 34221: 34216: 34211: 34206: 34201: 34195: 34193: 34189: 34188: 34186: 34185: 34179: 34177: 34173: 34172: 34170: 34169: 34164: 34159: 34154: 34149: 34144: 34139: 34134: 34132:Speed of light 34129: 34123: 34121: 34115: 34114: 34112: 34111: 34106: 34101: 34095: 34085: 34083: 34076: 34068: 34067: 34060: 34059: 34052: 34045: 34037: 34028: 34027: 34025: 34024: 34011: 34008: 34007: 34005: 34004: 33999: 33994: 33989: 33984: 33979: 33974: 33969: 33964: 33959: 33954: 33949: 33944: 33939: 33934: 33929: 33923: 33921: 33917: 33916: 33914: 33913: 33907: 33901: 33895: 33889: 33883: 33877: 33871: 33865: 33859: 33853: 33847: 33841: 33834: 33832: 33826: 33825: 33823: 33822: 33815: 33808: 33801: 33794: 33787: 33780: 33773: 33765: 33763: 33757: 33756: 33754: 33753: 33748: 33743: 33738: 33733: 33728: 33723: 33717: 33715: 33711: 33710: 33708: 33707: 33699: 33691: 33690:(2008 TV film) 33683: 33675: 33667: 33659: 33655:Young Einstein 33651: 33647:Insignificance 33643: 33635: 33627: 33618: 33616: 33608: 33607: 33605: 33604: 33598: 33595:Why Socialism? 33591: 33583: 33575: 33567: 33559: 33552: 33542: 33540: 33534: 33533: 33531: 33530: 33525: 33520: 33515: 33510: 33505: 33500: 33495: 33490: 33485: 33480: 33475: 33470: 33465: 33460: 33455: 33450: 33445: 33440: 33435: 33433:Einstein solid 33430: 33425: 33420: 33415: 33410: 33409: 33408: 33403: 33392: 33390: 33386: 33385: 33378: 33377: 33370: 33363: 33355: 33346: 33345: 33343: 33342: 33337: 33332: 33327: 33322: 33316: 33314: 33310: 33309: 33307: 33306: 33301: 33296: 33291: 33286: 33281: 33276: 33271: 33266: 33261: 33255: 33253: 33247: 33246: 33244: 33243: 33238: 33237: 33236: 33231: 33226: 33216: 33211: 33206: 33201: 33200: 33199: 33194: 33183: 33181: 33175: 33174: 33172: 33171: 33170: 33169: 33164: 33157:Thermodynamics 33154: 33153: 33152: 33147: 33137: 33132: 33127: 33126: 33125: 33120: 33115: 33110: 33099: 33097: 33091: 33090: 33088: 33087: 33086: 33085: 33075: 33069: 33067: 33063: 33062: 33060: 33059: 33058: 33057: 33047: 33041: 33039: 33035: 33034: 33026: 33025: 33018: 33011: 33003: 32997: 32996: 32990: 32984: 32978: 32970:Animation clip 32966: 32960: 32947: 32941: 32928: 32920: 32917: 32916: 32915: 32909: 32903: 32897: 32891: 32885: 32879: 32862: 32860: 32854: 32848: 32825: 32819: 32803: 32795: 32792: 32791: 32790: 32787:Astronomy Cast 32783: 32770: 32763:Einstein Light 32758: 32755: 32754: 32753: 32741: 32723: 32722:Original works 32720: 32718: 32717:External links 32715: 32714: 32713: 32673: 32665: 32647:(6): 4405–09. 32636: 32616:10.1086/430652 32599: 32581:(3): 260–262. 32568: 32565: 32564: 32563: 32552:Modern Physics 32548: 32529: 32523: 32517:. De Gruyter. 32508: 32502: 32489: 32483: 32468: 32461: 32456:978-0393097931 32455: 32442: 32436: 32406: 32388: 32363: 32356: 32334:Charles Misner 32329: 32326: 32325: 32324: 32315: 32301: 32290: 32287: 32285: 32282: 32280: 32279: 32272: 32252: 32237:George Sterman 32229: 32198: 32178: 32171: 32151: 32144: 32126: 32119: 32101: 32054: 32027: 31992:hep-ph/9703240 31967: 31953: 31933: 31926: 31902: 31895: 31868: 31831:(2): 291–298. 31815: 31781: 31774: 31751: 31744: 31723: 31716: 31698: 31683: 31664:(6): 317–353. 31648: 31618: 31599: 31574: 31567: 31549: 31542: 31526:Einstein from 31515: 31508: 31485: 31454: 31423: 31396: 31389: 31371: 31318: 31283: 31272:on 12 May 2017 31252: 31215: 31192: 31177: 31169:Addison-Wesley 31157:Sands, Matthew 31140: 31117: 31079: 31072: 31064:Going Critical 31047: 31009: 30986: 30949:(1): 195–204. 30933: 30926: 30906: 30899: 30879: 30865: 30843: 30832:on 4 July 2015 30806: 30795: 30775: 30765: 30737: 30710: 30703: 30685: 30662: 30645: 30626:(4): 384–392. 30610: 30579: 30572: 30566:. p. 54. 30548: 30541: 30520: 30506: 30486: 30479: 30455: 30448: 30424: 30417: 30393: 30356: 30310: 30303: 30280: 30273: 30249: 30190: 30164: 30157: 30134: 30093: 30086: 30068: 30061: 30040: 30033: 30013: 30007:978-0881334203 30006: 29984: 29963: 29956: 29938: 29901:(6): 633–638. 29885: 29833: 29813: 29806: 29788: 29767: 29747: 29738: 29701:(6): 655–663. 29682: 29656: 29632: 29607: 29600: 29571: 29564: 29535: 29525: 29505: 29495: 29475: 29463: 29452: 29445: 29425: 29400: 29393: 29369: 29362: 29339: 29332: 29304: 29297: 29268: 29266: 29263: 29261: 29260: 29239:(6): 491–494. 29216: 29161: 29107: 29081: 29067: 29042: 29015:(1): 137–139. 28997: 28960: 28929: 28898: 28871:(2): 263–265. 28855: 28812: 28777:(6): 915–931. 28771:Applied Optics 28761: 28740: 28696: 28655: 28638: 28617: 28610: 28592: 28582: 28573: 28561: 28552: 28543: 28530: 28478: 28476: 28473: 28471: 28470: 28460: 28453:) on the pure 28431: 28421: 28408: 28391: 28361: 28348:imaginary time 28318: 28274: 28235: 28222: 28207: 28202: 28184: 28173: 28171: 28168: 28167: 28166: 28161: 28156: 28151: 28145: 28144: 28142: 28138: 28137: 28132: 28126: 28125: 28123: 28119: 28118: 28113: 28107: 28106: 28104: 28100: 28099: 28094: 28089: 28084: 28079: 28074: 28069: 28064: 28059: 28054: 28048: 28047: 28045: 28041: 28040: 28035: 28030: 28025: 28020: 28015: 28010: 28004: 28003: 28001: 27997: 27996: 27991: 27986: 27981: 27976: 27971: 27965: 27964: 27962: 27957: 27954: 27915:invariant mass 27898: 27894: 27890: 27887: 27881: 27878: 27871: 27867: 27863: 27857: 27852: 27848: 27802: 27797: 27793: 27789: 27786: 27777: 27751: 27746: 27742: 27738: 27735: 27732: 27729: 27726: 27721: 27716: 27711: 27702: 27696: 27691: 27688: 27683: 27678: 27653: 27648: 27644: 27640: 27635: 27630: 27625: 27622: 27617: 27612: 27609: 27604: 27600: 27594: 27590: 27584: 27581: 27577: 27573: 27568: 27563: 27548:generates the 27541: 27538: 27525: 27522: 27517: 27513: 27507: 27503: 27497: 27494: 27490: 27486: 27458: 27452: 27448: 27444: 27441: 27436: 27432: 27426: 27422: 27416: 27413: 27409: 27405: 27400: 27395: 27310: 27306: 27302: 27299: 27296: 27293: 27288: 27284: 27280: 27277: 27274: 27271: 27266: 27262: 27258: 27255: 27252: 27249: 27244: 27240: 27236: 27233: 27229: 27226: 27223: 27220: 27215: 27211: 27207: 27201: 27197: 27193: 27187: 27184: 27180: 27176: 27171: 27167: 27163: 27157: 27153: 27149: 27146: 27141: 27136: 27131: 27105: 27101: 27097: 27092: 27082: 27079: 27074: 27070: 27064: 27061: 27056: 27052: 27040: 27037: 27023: 27015: 27010: 27001: 26997: 26991: 26982: 26978: 26972: 26963: 26959: 26955: 26950: 26941: 26937: 26931: 26922: 26918: 26914: 26909: 26900: 26896: 26869: 26865: 26859: 26855: 26849: 26845: 26840: 26835: 26799: 26794: 26790: 26784: 26781: 26777: 26771: 26767: 26763: 26758: 26754: 26748: 26745: 26741: 26735: 26731: 26727: 26722: 26718: 26712: 26708: 26670: 26660: 26657: 26652: 26646: 26636: 26633: 26628: 26621: 26618: 26613: 26610: 26605: 26601: 26596: 26593: 26589: 26577:Poincaré group 26555: 26552: 26548: 26525: 26519: 26516: 26514: 26511: 26509: 26506: 26504: 26501: 26500: 26497: 26494: 26492: 26489: 26487: 26484: 26482: 26479: 26478: 26475: 26472: 26470: 26467: 26465: 26462: 26460: 26457: 26456: 26453: 26450: 26448: 26445: 26443: 26440: 26438: 26435: 26432: 26431: 26429: 26424: 26419: 26416: 26412: 26380: 26377: 26365:magnetic field 26361:electric field 26316: 26306: 26303: 26298: 26275: 26265: 26262: 26257: 26234: 26231: 26228: 26225: 26220: 26217: 26214: 26211: 26207: 26201: 26191: 26188: 26183: 26179: 26174: 26164: 26161: 26156: 26150: 26140: 26137: 26132: 26126: 26116: 26113: 26108: 26104: 26099: 26089: 26086: 26081: 26075: 26065: 26062: 26057: 26053: 26047: 26044: 26040: 26036: 26033: 26028: 26025: 26018: 26015: 26011: 26007: 26004: 25999: 25996: 25991: 25955: 25945: 25942: 25937: 25914: 25904: 25901: 25896: 25875: 25870: 25866: 25860: 25850: 25847: 25842: 25838: 25832: 25829: 25824: 25788: 25780: 25776: 25772: 25768: 25763: 25758: 25754: 25749: 25746: 25741: 25737: 25733: 25728: 25718: 25715: 25710: 25706: 25703: 25700: 25694: 25691: 25686: 25682: 25662: 25657: 25651: 25648: 25646: 25643: 25641: 25638: 25636: 25633: 25632: 25629: 25626: 25624: 25621: 25619: 25616: 25614: 25611: 25610: 25607: 25604: 25602: 25599: 25597: 25594: 25592: 25589: 25586: 25583: 25582: 25579: 25576: 25574: 25571: 25569: 25566: 25563: 25560: 25558: 25555: 25554: 25552: 25545: 25535: 25532: 25527: 25524: 25517: 25511: 25508: 25503: 25500: 25493: 25487: 25484: 25479: 25476: 25469: 25463: 25460: 25455: 25452: 25442: 25439: 25433: 25432: 25430: 25425: 25420: 25409: 25406: 25402: 25397: 25394: 25387: 25380: 25377: 25373: 25368: 25365: 25358: 25351: 25348: 25344: 25339: 25336: 25329: 25322: 25319: 25315: 25310: 25307: 25297: 25294: 25288: 25287: 25285: 25259:-velocity and 25238: 25232: 25229: 25222: 25218: 25214: 25208: 25203: 25199: 25180:invariant mass 25163: 25159: 25154: 25150: 25145: 25142: 25136: 25132: 25128: 25122: 25118: 25114: 25109: 25105: 25101: 25096: 25092: 25088: 25083: 25080: 25074: 25070: 25067: 25062: 25058: 25054: 25049: 25045: 25041: 25036: 25032: 25028: 25025: 25022: 25019: 25016: 25013: 25010: 25007: 25004: 24999: 24995: 24991: 24988: 24983: 24979: 24950: 24946: 24943: 24940: 24937: 24934: 24931: 24927: 24901: 24897: 24893: 24890: 24887: 24884: 24881: 24878: 24875: 24850: 24845: 24840: 24836: 24832: 24827: 24822: 24818: 24814: 24809: 24804: 24800: 24796: 24791: 24787: 24777: 24773: 24768: 24762: 24758: 24754: 24751: 24747: 24742: 24739: 24736: 24733: 24730: 24710: 24707: 24703: 24699: 24696: 24693: 24690: 24687: 24684: 24681: 24678: 24675: 24670: 24666: 24662: 24657: 24653: 24649: 24644: 24640: 24636: 24633: 24630: 24627: 24624: 24621: 24618: 24615: 24609: 24606: 24599: 24595: 24591: 24585: 24580: 24576: 24551: 24546: 24542: 24514: 24510: 24504: 24494: 24491: 24486: 24482: 24476: 24473: 24468: 24441: 24437: 24404: 24394: 24391: 24386: 24365: 24339: 24335: 24329: 24319: 24316: 24311: 24307: 24301: 24298: 24293: 24261: 24258: 24254: 24250: 24247: 24244: 24241: 24238: 24235: 24232: 24229: 24226: 24223: 24220: 24217: 24214: 24211: 24208: 24205: 24202: 24197: 24193: 24189: 24184: 24180: 24176: 24171: 24167: 24163: 24158: 24154: 24150: 24147: 24142: 24138: 24081:Poincaré group 24053: 24048: 24042: 24039: 24038: 24035: 24032: 24031: 24028: 24025: 24022: 24019: 24016: 24013: 24010: 24007: 24006: 24003: 24000: 23997: 23994: 23991: 23988: 23985: 23982: 23981: 23979: 23974: 23969: 23963: 23960: 23959: 23956: 23953: 23952: 23949: 23946: 23945: 23942: 23939: 23936: 23935: 23933: 23926: 23920: 23917: 23915: 23912: 23910: 23907: 23905: 23902: 23901: 23898: 23895: 23893: 23890: 23888: 23885: 23883: 23880: 23879: 23876: 23873: 23871: 23868: 23866: 23863: 23861: 23858: 23855: 23852: 23851: 23848: 23845: 23843: 23840: 23838: 23835: 23832: 23829: 23827: 23824: 23823: 23821: 23816: 23811: 23804: 23801: 23797: 23796: 23792: 23789: 23785: 23784: 23780: 23777: 23773: 23772: 23768: 23765: 23761: 23758: 23757: 23755: 23731: 23728: 23726: 23723: 23679: 23674: 23670: 23666: 23661: 23656: 23652: 23648: 23643: 23638: 23634: 23628: 23625: 23584: 23579: 23575: 23571: 23566: 23562: 23558: 23553: 23548: 23544: 23540: 23537: 23532: 23527: 23523: 23519: 23516: 23511: 23506: 23502: 23498: 23476: 23472: 23468: 23463: 23459: 23455: 23450: 23445: 23441: 23437: 23434: 23429: 23424: 23420: 23416: 23413: 23408: 23403: 23399: 23395: 23392: 23389: 23386: 23381: 23377: 23373: 23361: 23358: 23341: 23336: 23332: 23328: 23323: 23319: 23315: 23310: 23305: 23301: 23297: 23294: 23289: 23284: 23280: 23276: 23254: 23250: 23246: 23241: 23237: 23233: 23228: 23223: 23219: 23215: 23212: 23207: 23202: 23198: 23194: 23191: 23188: 23185: 23180: 23176: 23172: 23145: 23140: 23136: 23132: 23127: 23123: 23119: 23114: 23109: 23105: 23101: 23098: 23093: 23088: 23084: 23080: 23077: 23072: 23068: 23064: 23044: 23041: 23020:or factors of 22969: 22947:Poincaré group 22920: 22913: 22906: 22899: 22876: 22871: 22866: 22862: 22858: 22855: 22850: 22845: 22841: 22837: 22834: 22829: 22824: 22820: 22816: 22813: 22808: 22803: 22799: 22795: 22792: 22789: 22784: 22780: 22776: 22758: 22751: 22744: 22721: 22716: 22711: 22707: 22703: 22700: 22695: 22690: 22686: 22682: 22679: 22674: 22669: 22665: 22661: 22658: 22654: 22650: 22647: 22643: 22639: 22636: 22631: 22627: 22623: 22614:is defined by 22541: 22538: 22536: 22533: 22528:Main article: 22525: 22522: 22521: 22520: 22513: 22506: 22500: 22494: 22488: 22482: 22476: 22459: 22458: 22451: 22444: 22440: 22386:absolute value 22367: 22364: 22340:fine structure 22321:Dirac equation 22298:fine structure 22296:explained the 22252: 22249: 22215:magnetic field 22211:electric field 22177: 22174: 22154: 22153: 22141: 22138: 22134: 22128: 22124: 22120: 22117: 22114: 22094: 22074: 22071: 22068: 22065: 22062: 22051: 22040: 22037: 22034: 22031: 22028: 22025: 22021: 22017: 22012: 22008: 22004: 22001: 21998: 21995: 21992: 21989: 21986: 21983: 21980: 21977: 21974: 21970: 21966: 21963: 21960: 21941: 21938: 21935: 21931: 21927: 21924: 21921: 21918: 21907: 21896: 21892: 21884: 21880: 21876: 21871: 21867: 21861: 21857: 21835: 21828: 21824: 21820: 21817: 21811: 21807: 21804: 21801: 21791: 21780: 21773: 21769: 21765: 21760: 21756: 21750: 21746: 21739: 21735: 21732: 21729: 21726: 21723: 21720: 21717: 21714: 21694: 21689: 21685: 21681: 21676: 21672: 21668: 21663: 21659: 21653: 21649: 21637: 21626: 21623: 21619: 21615: 21612: 21609: 21606: 21579: 21559: 21556: 21536: 21499:event horizons 21484: 21481: 21465: 21462: 21440: 21437: 21432: 21429: 21408: 21387: 21384: 21379: 21376: 21355: 21352: 21332: 21320: 21319: 21304: 21301: 21298: 21294: 21290: 21286: 21282: 21278: 21274: 21270: 21266: 21263: 21261: 21258: 21255: 21251: 21250: 21246: 21242: 21239: 21236: 21233: 21230: 21225: 21221: 21216: 21212: 21209: 21206: 21204: 21201: 21197: 21193: 21189: 21188: 21184: 21180: 21177: 21174: 21169: 21165: 21160: 21156: 21153: 21150: 21148: 21145: 21141: 21137: 21133: 21132: 21108: 21105: 21084: 21081: 21061: 21058: 21053: 21049: 21045: 21040: 21036: 21013: 21009: 20988: 20984: 20981: 20956: 20953: 20932: 20929: 20924: 20921: 20900: 20880: 20856: 20855: 20828: 20775:Main article: 20772: 20769: 20748: 20745: 20723: 20720: 20700: 20699: 20698: 20697: 20685: 20678: 20675: 20669: 20663: 20660: 20655: 20652: 20645: 20641: 20622: 20618: 20611: 20608: 20600: 20597: 20591: 20585: 20579: 20576: 20571: 20568: 20561: 20557: 20538: 20532: 20529: 20524: 20521: 20515: 20512: 20491: 20488: 20470: 20469: 20468: 20456: 20449: 20446: 20441: 20435: 20432: 20426: 20423: 20420: 20414: 20408: 20405: 20400: 20397: 20390: 20386: 20367: 20364: 20358: 20355: 20349: 20346: 20343: 20340: 20334: 20331: 20327: 20306: 20300: 20297: 20292: 20289: 20283: 20280: 20259: 20256: 20237: 20236: 20220: 20217: 20214: 20208: 20205: 20199: 20196: 20193: 20173: 20167: 20164: 20158: 20155: 20152: 20136: 20135: 20134: 20133: 20122: 20117: 20113: 20109: 20104: 20100: 20096: 20091: 20087: 20083: 20080: 20076: 20072: 20069: 20066: 20063: 20057: 20054: 20048: 20045: 20041: 20037: 20034: 20031: 20028: 20012: 20011: 19995: 19992: 19989: 19986: 19975: 19968: 19954: 19947: 19946: 19942: 19935: 19934: 19933: 19932: 19931: 19930: 19929: 19928: 19927: 19916: 19910: 19907: 19901: 19898: 19895: 19892: 19873: 19869: 19862: 19859: 19854: 19851: 19845: 19839: 19836: 19831: 19828: 19822: 19816: 19813: 19808: 19805: 19799: 19796: 19792: 19788: 19769: 19763: 19759: 19755: 19752: 19747: 19744: 19741: 19738: 19735: 19732: 19729: 19726: 19723: 19720: 19717: 19714: 19711: 19705: 19699: 19696: 19691: 19688: 19682: 19679: 19649: 19646: 19626: 19623: 19614:is divided by 19605: 19604: 19603: 19592: 19589: 19586: 19583: 19580: 19577: 19574: 19571: 19568: 19565: 19562: 19559: 19556: 19553: 19550: 19547: 19532:dt, dx, dy, dz 19509: 19506: 19505: 19504: 19493: 19488: 19484: 19478: 19474: 19470: 19465: 19461: 19438: 19434: 19430: 19425: 19421: 19415: 19411: 19390: 19387: 19384: 19381: 19378: 19375: 19355: 19354: 19322: 19319: 19316: 19313: 19310: 19295: 19287: 19286: 19285: 19284: 19270: 19267: 19261: 19255: 19252: 19246: 19241: 19237: 19231: 19227: 19207: 19202: 19198: 19192: 19188: 19184: 19179: 19175: 19169: 19165: 19161: 19156: 19152: 19146: 19142: 19138: 19133: 19129: 19123: 19119: 19099: 19096: 19093: 19090: 19061: 19058: 19032: 19029: 18996: 18978: 18975: 18972: 18969: 18966: 18963: 18960: 18930: 18927: 18912: 18909: 18906: 18903: 18900: 18897: 18894: 18891: 18888: 18885: 18882: 18879: 18876: 18873: 18824: 18813: 18803: 18802: 18785: 18781: 18777: 18774: 18772: 18769: 18765: 18761: 18757: 18756: 18751: 18747: 18743: 18740: 18738: 18735: 18731: 18727: 18723: 18722: 18718: 18712: 18708: 18704: 18701: 18697: 18693: 18690: 18687: 18682: 18678: 18673: 18669: 18666: 18663: 18661: 18658: 18654: 18650: 18646: 18645: 18641: 18635: 18631: 18627: 18624: 18620: 18616: 18613: 18610: 18605: 18601: 18596: 18592: 18589: 18586: 18584: 18581: 18577: 18573: 18569: 18568: 18528: 18525: 18522: 18519: 18516: 18513: 18510: 18507: 18504: 18501: 18483: 18466: 18460: 18456: 18452: 18447: 18443: 18439: 18434: 18430: 18426: 18421: 18417: 18412: 18408: 18405: 18391: 18388: 18378:even within a 18337:Main article: 18334: 18331: 18320: 18319: 18308: 18303: 18296: 18293: 18289: 18288: 18284: 18281: 18277: 18274: 18273: 18271: 18264: 18258: 18255: 18252: 18249: 18247: 18244: 18241: 18238: 18237: 18234: 18231: 18228: 18225: 18223: 18220: 18217: 18214: 18213: 18211: 18206: 18201: 18195: 18192: 18191: 18188: 18185: 18182: 18181: 18179: 18156: 18155: 18144: 18139: 18133: 18130: 18129: 18126: 18123: 18120: 18119: 18117: 18110: 18104: 18101: 18098: 18095: 18093: 18090: 18087: 18084: 18081: 18080: 18077: 18074: 18071: 18068: 18065: 18063: 18060: 18057: 18054: 18053: 18051: 18046: 18041: 18034: 18031: 18027: 18026: 18022: 18019: 18015: 18012: 18011: 18009: 17971: 17970: 17959: 17956: 17953: 17950: 17947: 17925: 17922: 17917: 17913: 17909: 17906: 17901: 17898: 17895: 17889: 17881: 17877: 17873: 17870: 17866: 17861: 17858: 17855: 17845: 17834: 17831: 17828: 17825: 17822: 17800: 17797: 17792: 17788: 17784: 17781: 17777: 17772: 17764: 17760: 17756: 17753: 17749: 17744: 17741: 17712: 17707: 17703: 17699: 17694: 17690: 17686: 17683: 17672: 17671: 17660: 17657: 17652: 17648: 17644: 17639: 17635: 17631: 17628: 17625: 17606: 17598: 17594: 17590: 17587: 17582: 17578: 17574: 17571: 17568: 17565: 17558: 17554: 17550: 17547: 17544: 17539: 17535: 17531: 17528: 17506: 17498: 17494: 17488: 17484: 17480: 17477: 17470: 17466: 17462: 17457: 17453: 17446: 17443: 17428: 17427: 17416: 17408: 17405: 17401: 17397: 17392: 17388: 17380: 17377: 17373: 17369: 17364: 17360: 17353: 17347: 17344: 17341: 17336: 17333: 17330: 17324: 17321: 17318: 17315: 17301: 17300: 17289: 17284: 17281: 17276: 17273: 17270: 17267: 17264: 17261: 17230: 17219: 17218: 17207: 17202: 17196: 17193: 17192: 17189: 17186: 17185: 17183: 17176: 17170: 17167: 17164: 17161: 17159: 17156: 17153: 17150: 17149: 17146: 17143: 17140: 17137: 17134: 17132: 17129: 17126: 17123: 17122: 17120: 17115: 17110: 17103: 17100: 17096: 17095: 17091: 17088: 17084: 17083: 17081: 17031: 17030: 17019: 17013: 17010: 17007: 17002: 16999: 16996: 16990: 16987: 16984: 16981: 16978: 16944:unit hyperbola 16829:unit hyperbola 16826: 16819: 16818: 16780: 16773: 16772: 16771: 16770: 16769: 16764:Main article: 16761: 16758: 16756: 16753: 16750: 16749: 16740: 16738: 16722: 16718: 16714: 16710: 16706: 16703: 16698: 16693: 16689: 16685: 16682: 16677: 16673: 16668: 16664: 16661: 16655: 16651: 16645: 16641: 16637: 16634: 16601: 16596: 16592: 16588: 16583: 16579: 16558: 16555: 16552: 16539: 16538: 16529: 16527: 16511: 16507: 16503: 16499: 16495: 16492: 16487: 16483: 16479: 16476: 16473: 16470: 16465: 16461: 16455: 16451: 16447: 16444: 16439: 16435: 16431: 16428: 16422: 16418: 16415: 16411: 16407: 16404: 16401: 16398: 16395: 16390: 16386: 16382: 16376: 16372: 16369: 16366: 16363: 16360: 16356: 16353: 16336: 16335: 16326: 16324: 16308: 16304: 16300: 16296: 16290: 16285: 16281: 16276: 16272: 16269: 16266: 16263: 16260: 16257: 16252: 16248: 16244: 16240: 16236: 16231: 16227: 16221: 16216: 16212: 16208: 16202: 16198: 16195: 16190: 16186: 16179: 16174: 16170: 16152: 16151: 16142: 16140: 16124: 16120: 16116: 16112: 16106: 16101: 16097: 16092: 16088: 16085: 16082: 16079: 16076: 16073: 16068: 16064: 16060: 16056: 16052: 16047: 16043: 16037: 16032: 16028: 16024: 16018: 16014: 16011: 16006: 16002: 15995: 15990: 15986: 15965: 15964: 15955: 15953: 15940: 15936: 15932: 15927: 15923: 15919: 15916: 15913: 15908: 15904: 15886: 15885: 15876: 15874: 15862: 15858: 15855: 15850: 15846: 15840: 15836: 15832: 15828: 15824: 15821: 15816: 15812: 15806: 15802: 15784: 15783: 15774: 15772: 15760: 15756: 15753: 15748: 15744: 15738: 15734: 15730: 15726: 15722: 15719: 15714: 15710: 15704: 15700: 15696: 15691: 15687: 15681: 15677: 15637: 15617: 15598: 15597: 15588: 15586: 15573: 15569: 15565: 15560: 15556: 15552: 15547: 15543: 15539: 15534: 15530: 15526: 15521: 15517: 15513: 15510: 15505: 15501: 15497: 15492: 15488: 15466: 15465: 15456: 15454: 15440: 15435: 15431: 15424: 15419: 15415: 15411: 15405: 15400: 15396: 15389: 15384: 15380: 15376: 15373: 15370: 15364: 15359: 15355: 15348: 15343: 15339: 15317: 15316: 15307: 15305: 15292: 15288: 15284: 15281: 15278: 15275: 15262: 15259: 15253: 15250: 15247: 15241: 15238: 15204: 15201: 15125: 15122: 15118:Bremsstrahlung 15086: 15083: 15008: 15004: 14999: 14993: 14989: 14983: 14979: 14975: 14972: 14967: 14964: 14958: 14955: 14952: 14949: 14946: 14920: 14917: 14856: 14855: 14846: 14844: 14831: 14827: 14819: 14815: 14811: 14804: 14801: 14796: 14791: 14787: 14783: 14778: 14774: 14753: 14752: 14743: 14741: 14729: 14725: 14722: 14714: 14710: 14705: 14699: 14695: 14691: 14688: 14684: 14678: 14674: 14671: 14666: 14662: 14658: 14653: 14649: 14619: 14615: 14611: 14606: 14602: 14579: 14575: 14571: 14566: 14562: 14541: 14521: 14501: 14498: 14495: 14482: 14481: 14472: 14470: 14458: 14454: 14451: 14443: 14439: 14434: 14428: 14424: 14420: 14417: 14413: 14407: 14403: 14400: 14397: 14392: 14388: 14384: 14379: 14375: 14371: 14368: 14365: 14360: 14356: 14352: 14347: 14343: 14339: 14321: 14320: 14300: 14296: 14291: 14285: 14281: 14277: 14274: 14270: 14265: 14260: 14256: 14252: 14244: 14240: 14235: 14229: 14225: 14221: 14218: 14212: 14208: 14205: 14202: 14199: 14195: 14191: 14188: 14185: 14182: 14176: 14170: 14167: 14161: 14153: 14149: 14144: 14138: 14134: 14130: 14127: 14121: 14117: 14114: 14111: 14108: 14104: 14100: 14097: 14094: 14091: 14085: 14079: 14076: 14070: 14065: 14061: 14057: 14054: 14052: 14048: 14044: 14040: 14039: 14036: 14033: 14028: 14024: 14020: 14017: 14011: 14008: 14002: 13999: 13993: 13990: 13984: 13979: 13975: 13971: 13968: 13966: 13962: 13958: 13954: 13953: 13928: 13924: 13894: 13890: 13865: 13833: 13829: 13806: 13802: 13777: 13773: 13769: 13749: 13734: 13733: 13717: 13713: 13708: 13702: 13698: 13694: 13691: 13685: 13681: 13678: 13675: 13672: 13668: 13664: 13661: 13658: 13655: 13649: 13644: 13640: 13637: 13611: 13580: 13560: 13536: 13529: 13528: 13527: 13518: 13517: 13516: 13515: 13514: 13500:, the laws of 13486: 13476: 13377:Main article: 13374: 13371: 13356: 13353: 13336:BL Lac objects 13332:radio galaxies 13264:, or what one 13234: 13231: 13218: 13199:Doppler effect 13186: 13183: 13175: 13174: 13171:Lorentz factor 13153: 13149: 13145: 13142: 13138: 13133: 13130: 13120: 13108: 13104: 13100: 13097: 13094: 13071: 13066: 13062: 13053: 13050: 13047: 13042: 13039: 13036: 13029: 13024: 13020: 13015: 13011: 13008: 13005: 13001: 12997: 12994: 12989: 12985: 12955: 12951: 12928: 12924: 12919: 12911: 12907: 12903: 12898: 12895: 12891: 12887: 12882: 12878: 12849: 12828: 12804: 12801: 12796:Main article: 12793: 12790: 12789: 12788: 12775: 12772: 12767: 12764: 12759: 12755: 12751: 12746: 12740: 12737: 12734: 12729: 12726: 12723: 12717: 12712: 12707: 12703: 12700: 12694: 12691: 12664: 12661: 12658: 12655: 12652: 12649: 12645: 12641: 12638: 12635: 12632: 12629: 12626: 12621: 12618: 12615: 12609: 12605: 12602: 12598: 12595: 12568: 12565: 12562: 12559: 12556: 12552: 12548: 12545: 12542: 12539: 12534: 12530: 12526: 12523: 12520: 12517: 12514: 12508: 12504: 12501: 12497: 12494: 12431: 12428: 12427: 12426: 12414: 12406: 12402: 12398: 12393: 12390: 12386: 12382: 12379: 12374: 12371: 12350: 12346: 12339: 12336: 12332: 12327: 12324: 12320: 12315: 12309: 12306: 12301: 12296: 12293: 12287: 12283: 12264: 12258: 12255: 12251: 12247: 12244: 12241: 12236: 12233: 12230: 12226: 12222: 12200: 12192: 12188: 12183: 12179: 12175: 12172: 12168: 12165: 12160: 12157: 12153: 12150: 12143: 12138: 12134: 12093: 12067: 12060: 12052: 12048: 12044: 12039: 12036: 12032: 12028: 12025: 12020: 12017: 12012: 12007: 12003: 11980: 11976: 11955: 11935: 11931: 11927: 11924: 11920: 11917: 11859:Main article: 11856: 11853: 11851: 11848: 11808: 11805: 11783: 11780: 11776: 11756: 11735: 11732: 11720: 11719: 11707: 11704: 11692: 11675: 11672: 11668: 11629: 11628: 11619: 11618: 11610: 11609: 11608: 11607: 11606: 11569: 11566: 11523: 11518: 11514: 11509: 11505: 11502: 11472: 11449: 11425: 11405: 11404: 11395: 11394: 11393: 11384: 11383: 11382: 11381: 11380: 11372: 11369: 11353: 11352: 11345: 11284:If both | 11282: 11271: 11194: 11191: 11187: 11184: 11180: 11177: 11156: 11151: 11148: 11125: 11103: 11075: 11074: 11065: 11063: 11052: 11042: 11036: 11032: 11027: 11023: 11019: 11015: 11011: 11007: 11004: 11000: 10996: 10991: 10987: 10983: 10977: 10972: 10968: 10963: 10953: 10947: 10943: 10938: 10934: 10930: 10926: 10922: 10918: 10915: 10911: 10907: 10902: 10898: 10894: 10888: 10883: 10879: 10874: 10863: 10859: 10854: 10850: 10846: 10842: 10838: 10834: 10831: 10826: 10823: 10819: 10815: 10811: 10804: 10799: 10795: 10779: 10778: 10769: 10767: 10756: 10746: 10740: 10736: 10731: 10727: 10722: 10718: 10714: 10711: 10707: 10703: 10697: 10693: 10687: 10683: 10679: 10675: 10670: 10660: 10654: 10650: 10645: 10641: 10636: 10632: 10628: 10625: 10621: 10617: 10611: 10607: 10601: 10597: 10593: 10589: 10584: 10573: 10569: 10564: 10560: 10555: 10551: 10547: 10544: 10539: 10536: 10531: 10527: 10520: 10516: 10512: 10508: 10489: 10488: 10479: 10477: 10466: 10460: 10456: 10453: 10449: 10445: 10440: 10437: 10433: 10427: 10423: 10420: 10416: 10412: 10407: 10404: 10400: 10394: 10390: 10387: 10383: 10379: 10374: 10371: 10367: 10364: 10361: 10358: 10354: 10350: 10346: 10339: 10335: 10331: 10327: 10320: 10316: 10312: 10308: 10304: 10301: 10296: 10293: 10276: 10275: 10266: 10264: 10253: 10247: 10244: 10241: 10237: 10233: 10230: 10224: 10221: 10218: 10214: 10210: 10207: 10201: 10198: 10195: 10191: 10187: 10184: 10181: 10178: 10175: 10170: 10166: 10159: 10154: 10150: 10143: 10138: 10134: 10130: 10127: 10123: 10096: 10082: 10081: 10072: 10070: 10059: 10051: 10047: 10042: 10038: 10034: 10031: 10027: 10024: 10019: 10016: 10012: 10009: 10002: 9999: 9975: 9969: 9966: 9946: 9933: 9932: 9923: 9921: 9910: 9899: 9895: 9890: 9887: 9880: 9877: 9872: 9869: 9866: 9860: 9850: 9847: 9842: 9839: 9826: 9822: 9818: 9812: 9809: 9804: 9801: 9794: 9791: 9786: 9783: 9773: 9766: 9757: 9753: 9748: 9745: 9741: 9734: 9731: 9728: 9724: 9720: 9715: 9712: 9709: 9705: 9702: 9699: 9696: 9693: 9690: 9684: 9677: 9674: 9670: 9664: 9661: 9657: 9651: 9647: 9644: 9617: 9614: 9593: 9554: 9551: 9547: 9526: 9523: 9519: 9506: 9505: 9496: 9494: 9483: 9478: 9475: 9471: 9467: 9462: 9459: 9455: 9452: 9448: 9445: 9441: 9436: 9431: 9428: 9423: 9406: 9405: 9396: 9394: 9383: 9379: 9376: 9372: 9368: 9365: 9362: 9359: 9356: 9351: 9347: 9343: 9304: 9301: 9279: 9275: 9241: 9238: 9220:time intervals 9187: 9186: 9175: 9169: 9166: 9162: 9159: 9155: 9124: 9120: 9117: 9111: 9107: 9104: 9100: 9018:ladder paradox 9007: 9004: 9001: 9000: 8995: 8982: 8978: 8975: 8969: 8966: 8956: 8951: 8937: 8934: 8929: 8926: 8920: 8916: 8913: 8902: 8892: 8891: 8888: 8875: 8871: 8868: 8865: 8852: 8849: 8835: 8832: 8827: 8824: 8821: 8815: 8794: 8790: 8786: 8782: 8776: 8772: 8768: 8765: 8762: 8757: 8753: 8750: 8747: 8734: 8730: 8729: 8726: 8715: 8712: 8709: 8706: 8703: 8698: 8694: 8691: 8685: 8664: 8660: 8656: 8651: 8645: 8642: 8639: 8634: 8631: 8628: 8622: 8615: 8611: 8607: 8603: 8597: 8593: 8589: 8586: 8583: 8578: 8574: 8571: 8565: 8545: 8541: 8537: 8532: 8529: 8518: 8515: 8502: 8498: 8494: 8490: 8484: 8480: 8476: 8473: 8470: 8465: 8461: 8458: 8452: 8431: 8427: 8423: 8418: 8412: 8409: 8406: 8401: 8398: 8395: 8389: 8384: 8381: 8378: 8375: 8372: 8367: 8363: 8360: 8354: 8333: 8329: 8324: 8321: 8310: 8297: 8294: 8281: 8280: 8275: 8261: 8258: 8254: 8249: 8245: 8241: 8237: 8226: 8221: 8208: 8205: 8200: 8195: 8192: 8187: 8182: 8178: 8167: 8163: 8162: 8159: 8148: 8145: 8142: 8139: 8136: 8131: 8127: 8124: 8118: 8096: 8092: 8088: 8083: 8077: 8074: 8071: 8066: 8063: 8060: 8054: 8047: 8043: 8039: 8035: 8029: 8025: 8021: 8018: 8015: 8010: 8006: 8003: 7997: 7977: 7973: 7969: 7964: 7961: 7950: 7947: 7934: 7930: 7926: 7922: 7916: 7912: 7908: 7905: 7902: 7897: 7893: 7890: 7884: 7862: 7858: 7854: 7849: 7843: 7840: 7837: 7832: 7829: 7826: 7820: 7815: 7812: 7809: 7806: 7803: 7798: 7794: 7791: 7785: 7764: 7760: 7755: 7752: 7741: 7728: 7725: 7712: 7711: 7706: 7692: 7689: 7685: 7680: 7676: 7672: 7668: 7657: 7652: 7639: 7636: 7631: 7626: 7623: 7618: 7613: 7609: 7598: 7591: 7590: 7587: 7573: 7570: 7565: 7562: 7559: 7553: 7549: 7546: 7542: 7532: 7529: 7516: 7512: 7509: 7506: 7500: 7497: 7494: 7484: 7480: 7479: 7474: 7460: 7457: 7452: 7449: 7443: 7439: 7436: 7425: 7420: 7407: 7403: 7400: 7394: 7391: 7381: 7377: 7376: 7373: 7368: 7365: 7360: 7341: 7336: 7333: 7330: 7327: 7324: 7320: 7316: 7313: 7310: 7307: 7304: 7299: 7296: 7292: 7289: 7267: 7262: 7259: 7256: 7253: 7250: 7246: 7242: 7239: 7236: 7233: 7230: 7225: 7222: 7218: 7215: 7156: 7153: 7132: 7119: 7116: 7098: 7091: 7087: 7082: 7076: 7072: 7068: 7065: 7058: 7051: 7047: 7025: 7018: 7014: 7010: 7005: 7001: 6993: 6988: 6985: 6982: 6977: 6973: 6952: 6945: 6941: 6937: 6932: 6928: 6922: 6902: 6881: 6878: 6849: 6846: 6842: 6838: 6835: 6832: 6823: 6801: 6765: 6762: 6758:Doppler effect 6704: 6703: 6692: 6686: 6683: 6680: 6677: 6651: 6648: 6644: 6641: 6637: 6634: 6630: 6584: 6581: 6560: 6481: 6473: 6469: 6464: 6461: 6457: 6451: 6448: 6445: 6441: 6437: 6434: 6430: 6427: 6423: 6350:Ladder paradox 6341: 6338: 6337: 6336: 6323: 6303: 6298: 6294: 6273: 6270: 6267: 6263: 6259: 6256: 6252: 6248: 6245: 6241: 6222: 6211: 6208: 6205: 6202: 6182: 6179: 6175: 6171: 6168: 6163: 6159: 6138: 6116: 6113: 6109: 6089: 6086: 6083: 6079: 6075: 6072: 6067: 6063: 6042: 6025:is called the 6014: 6007: 6003: 5999: 5996: 5974: 5971: 5967: 5963: 5960: 5955: 5951: 5947: 5944: 5924: 5921: 5917: 5914: 5910: 5890: 5870: 5850: 5847: 5842: 5838: 5833: 5829: 5826: 5823: 5820: 5817: 5814: 5811: 5805: 5802: 5798: 5795: 5791: 5771: 5768: 5765: 5761: 5757: 5754: 5750: 5746: 5743: 5739: 5720: 5715:is called the 5704: 5701: 5697: 5693: 5690: 5670: 5667: 5663: 5659: 5656: 5653: 5650: 5630: 5627: 5623: 5620: 5616: 5596: 5576: 5556: 5553: 5550: 5547: 5543: 5540: 5537: 5534: 5531: 5525: 5522: 5518: 5515: 5511: 5491: 5488: 5485: 5481: 5477: 5474: 5470: 5466: 5463: 5459: 5425: 5421: 5417: 5392: 5388: 5384: 5355: 5350: 5346: 5342: 5339: 5333: 5328: 5324: 5320: 5315: 5311: 5307: 5304: 5302: 5300: 5296: 5287: 5283: 5277: 5273: 5266: 5263: 5259: 5252: 5247: 5243: 5239: 5234: 5230: 5226: 5222: 5213: 5209: 5203: 5199: 5192: 5189: 5185: 5178: 5173: 5169: 5165: 5160: 5156: 5150: 5146: 5142: 5139: 5137: 5135: 5127: 5123: 5115: 5110: 5106: 5102: 5097: 5093: 5083: 5079: 5075: 5069: 5064: 5060: 5056: 5051: 5047: 5043: 5037: 5032: 5028: 5024: 5019: 5015: 5009: 5005: 5001: 4995: 4990: 4986: 4982: 4977: 4973: 4967: 4963: 4959: 4956: 4954: 4952: 4949: 4943: 4938: 4934: 4930: 4925: 4921: 4917: 4913: 4910: 4906: 4902: 4899: 4895: 4892: 4889: 4886: 4880: 4875: 4871: 4867: 4864: 4856: 4852: 4848: 4844: 4835: 4831: 4823: 4818: 4814: 4810: 4805: 4801: 4793: 4789: 4786: 4782: 4778: 4775: 4771: 4768: 4765: 4762: 4756: 4751: 4747: 4743: 4738: 4734: 4729: 4723: 4719: 4715: 4712: 4710: 4708: 4703: 4699: 4694: 4691: 4687: 4684: 4681: 4677: 4674: 4670: 4667: 4659: 4655: 4651: 4646: 4641: 4632: 4628: 4622: 4619: 4615: 4612: 4605: 4601: 4598: 4594: 4590: 4583: 4579: 4573: 4569: 4565: 4562: 4560: 4556: 4552: 4548: 4545: 4540: 4536: 4532: 4527: 4523: 4519: 4518: 4494: 4490: 4486: 4483: 4478: 4474: 4470: 4465: 4461: 4456: 4450: 4446: 4442: 4416:Lorentz boosts 4387: 4382: 4378: 4374: 4349: 4344: 4340: 4336: 4333: 4328: 4324: 4320: 4317: 4312: 4308: 4304: 4301: 4298: 4293: 4289: 4285: 4280: 4276: 4266: 4258: 4254: 4250: 4227: 4223: 4219: 4191: 4188: 4168: 4165: 4153: 4150: 4126: 4123: 4053: 4050: 4046: 4043: 4027:Fig. 3-1d 4013: 4010: 4007: 3987: 3984: 3962: 3957: 3953: 3949: 3946: 3943: 3939: 3935: 3930: 3926: 3922: 3919: 3916: 3893: 3890: 3886: 3866: 3863: 3860: 3857: 3854: 3851: 3848: 3845: 3824: 3821: 3800: 3780: 3777: 3774: 3771: 3768: 3765: 3762: 3759: 3738: 3735: 3731: 3711: 3708: 3705: 3702: 3699: 3696: 3676: 3673: 3670: 3667: 3664: 3644: 3640: 3637: 3633: 3630: 3626: 3623: 3619: 3599: 3596: 3593: 3590: 3587: 3584: 3564: 3561: 3558: 3555: 3552: 3532: 3528: 3525: 3521: 3518: 3514: 3511: 3507: 3496:Fig. 3-1c 3482: 3479: 3475: 3472: 3451: 3448: 3445: 3425: 3422: 3418: 3414: 3411: 3408: 3388: 3385: 3382: 3379: 3376: 3371: 3368: 3364: 3360: 3357: 3336: 3333: 3311: 3308: 3304: 3284: 3280: 3276: 3273: 3270: 3249: 3246: 3242: 3221: 3218: 3214: 3193: 3190: 3178:Fig. 3-1b 3163: 3116: 3113: 3110: 3086: 3066: 3063: 3043: 3040: 3020: 3000: 2980: 2960: 2949:Fig. 3-1a 2931: 2930: 2921: 2920: 2912: 2911: 2910: 2901: 2900: 2892: 2891: 2890: 2889: 2888: 2886: 2883: 2882: 2881: 2870: 2863: 2857: 2853: 2848: 2843: 2840: 2836: 2830: 2827: 2823: 2820: 2816: 2812: 2805: 2802: 2799: 2796: 2774: 2768: 2764: 2761: 2757: 2753: 2750: 2746: 2743: 2739: 2736: 2730: 2727: 2724: 2721: 2707: 2696: 2689: 2683: 2679: 2674: 2670: 2667: 2661: 2658: 2655: 2652: 2648: 2641: 2638: 2634: 2631: 2627: 2602: 2596: 2593: 2590: 2586: 2583: 2580: 2577: 2574: 2568: 2565: 2561: 2558: 2554: 2536: 2535: 2524: 2517: 2511: 2507: 2502: 2497: 2494: 2490: 2484: 2481: 2477: 2474: 2470: 2466: 2459: 2456: 2453: 2450: 2428: 2422: 2418: 2415: 2411: 2407: 2404: 2400: 2397: 2393: 2390: 2384: 2381: 2378: 2375: 2361: 2350: 2343: 2337: 2333: 2328: 2324: 2321: 2315: 2312: 2309: 2306: 2302: 2295: 2292: 2288: 2285: 2281: 2256: 2250: 2247: 2244: 2240: 2237: 2234: 2231: 2228: 2222: 2219: 2215: 2212: 2208: 2190: 2189: 2178: 2170: 2166: 2162: 2157: 2153: 2149: 2146: 2143: 2134: 2126: 2122: 2118: 2113: 2109: 2105: 2102: 2099: 2085: 2074: 2067: 2063: 2059: 2055: 2051: 2047: 2043: 2039: 2035: 2032: 2028: 2022: 2015: 2011: 2007: 2003: 1999: 1995: 1991: 1987: 1983: 1980: 1976: 1955: 1945: 1931: 1924: 1913: 1903: 1889: 1882: 1871:Lorentz scalar 1869:is known as a 1818: 1814: 1811: 1807: 1804: 1802: 1800: 1797: 1796: 1792: 1789: 1785: 1782: 1780: 1778: 1775: 1774: 1771: 1767: 1764: 1760: 1757: 1753: 1750: 1746: 1743: 1740: 1737: 1735: 1733: 1730: 1729: 1726: 1721: 1717: 1712: 1707: 1704: 1700: 1697: 1693: 1690: 1686: 1683: 1680: 1677: 1675: 1673: 1670: 1669: 1621:, relative to 1608:speed of light 1600:Lorentz factor 1582: 1578: 1573: 1567: 1563: 1559: 1556: 1552: 1547: 1544: 1520: 1517: 1514: 1511: 1509: 1506: 1503: 1499: 1498: 1495: 1492: 1489: 1487: 1484: 1481: 1477: 1476: 1473: 1470: 1467: 1464: 1461: 1458: 1452: 1449: 1446: 1444: 1441: 1438: 1434: 1433: 1430: 1425: 1421: 1416: 1412: 1409: 1406: 1403: 1400: 1394: 1391: 1388: 1386: 1383: 1380: 1376: 1375: 1284: 1281: 1248:Main article: 1245: 1242: 1237:Main article: 1234: 1231: 1221: 1218: 1184: 1181: 1157: 1156: 1125: 1103: 1046: 1043: 984: 981: 976:Main article: 973: 970: 963:Poincaré group 955:Henri Poincaré 895: 894: 883: 867:Main article: 851: 849: 846: 754:Henri Poincaré 724:speed of light 711: 689: 685: 681: 678: 675: 628:Doppler effect 569:Main article: 566: 563: 551: 550: 543:speed of light 539: 508:two postulates 496:space and time 478: 477: 475: 474: 467: 460: 452: 449: 448: 447: 446: 433: 416: 415: 411: 410: 405: 400: 395: 390: 385: 380: 375: 370: 365: 360: 355: 350: 345: 340: 335: 330: 324: 323: 317: 316: 315: 312: 311: 308: 307: 302: 297: 291: 287: 286: 283: 277: 276: 275: 274: 271: 270: 267: 266: 261: 256: 250: 242: 241: 240: 237: 236: 233: 232: 227: 222: 217: 211: 203: 202: 201: 198: 197: 194: 193: 188: 183: 178: 173: 168: 163: 158: 153: 148: 142: 136: 135: 134: 131: 130: 127: 126: 121: 116: 114:Speed of light 111: 106: 100: 94: 93: 92: 89: 88: 87: 86: 81: 76: 68: 67: 59: 58: 26: 9: 6: 4: 3: 2: 36180: 36169: 36166: 36164: 36161: 36160: 36158: 36148: 36143: 36138: 36136: 36126: 36124: 36114: 36112: 36107: 36102: 36100: 36090: 36088: 36083: 36078: 36077: 36074: 36061: 36058: 36056: 36053: 36051: 36048: 36046: 36043: 36041: 36038: 36036: 36033: 36031: 36028: 36026: 36023: 36021: 36018: 36016: 36013: 36011: 36008: 36006: 36003: 36001: 35998: 35997: 35995: 35993: 35989: 35979: 35976: 35974: 35971: 35969: 35966: 35964: 35961: 35959: 35956: 35954: 35951: 35949: 35946: 35944: 35941: 35939: 35936: 35935: 35933: 35929: 35923: 35920: 35918: 35915: 35913: 35910: 35908: 35905: 35903: 35900: 35898: 35897:Metric tensor 35895: 35893: 35890: 35888: 35885: 35884: 35882: 35878: 35875: 35871: 35865: 35862: 35860: 35857: 35855: 35852: 35850: 35847: 35845: 35842: 35840: 35837: 35835: 35832: 35830: 35827: 35825: 35822: 35820: 35817: 35815: 35812: 35810: 35809:Exterior form 35807: 35805: 35802: 35800: 35797: 35795: 35792: 35790: 35787: 35785: 35782: 35780: 35777: 35775: 35772: 35771: 35769: 35763: 35756: 35753: 35751: 35748: 35746: 35743: 35741: 35738: 35736: 35733: 35731: 35728: 35726: 35723: 35721: 35718: 35716: 35713: 35711: 35708: 35706: 35703: 35702: 35700: 35698: 35694: 35688: 35685: 35683: 35682:Tensor bundle 35680: 35678: 35675: 35673: 35670: 35668: 35665: 35663: 35660: 35658: 35655: 35653: 35650: 35648: 35645: 35643: 35640: 35639: 35637: 35631: 35625: 35622: 35620: 35617: 35615: 35612: 35610: 35607: 35605: 35602: 35600: 35597: 35595: 35592: 35590: 35587: 35585: 35582: 35581: 35579: 35575: 35565: 35562: 35560: 35557: 35555: 35552: 35550: 35547: 35545: 35542: 35541: 35539: 35534: 35531: 35529: 35526: 35525: 35522: 35516: 35513: 35511: 35508: 35506: 35503: 35501: 35498: 35496: 35493: 35491: 35488: 35486: 35483: 35481: 35478: 35477: 35475: 35473: 35469: 35466: 35462: 35458: 35457: 35451: 35447: 35440: 35435: 35433: 35428: 35426: 35421: 35420: 35417: 35405: 35402: 35400: 35397: 35395: 35392: 35391: 35389: 35385: 35379: 35376: 35374: 35371: 35370: 35368: 35364: 35358: 35355: 35353: 35350: 35348: 35345: 35344: 35342: 35340:Fizeau/Sagnac 35338: 35332: 35329: 35327: 35324: 35323: 35321: 35317: 35311: 35308: 35306: 35303: 35301: 35298: 35296: 35293: 35291: 35288: 35287: 35285: 35283: 35279: 35278:Time dilation 35275: 35269: 35266: 35264: 35261: 35259: 35256: 35254: 35251: 35249: 35246: 35244: 35241: 35239: 35236: 35234: 35231: 35230: 35228: 35224: 35218: 35215: 35213: 35210: 35208: 35205: 35203: 35200: 35198: 35195: 35193: 35190: 35188: 35185: 35184: 35182: 35178: 35174: 35167: 35162: 35160: 35155: 35153: 35148: 35147: 35144: 35132: 35129: 35127: 35124: 35122: 35119: 35118: 35116: 35114: 35110: 35104: 35101: 35099: 35096: 35094: 35091: 35089: 35086: 35085: 35083: 35081: 35077: 35071: 35068: 35066: 35063: 35061: 35058: 35056: 35053: 35051: 35048: 35046: 35043: 35041: 35038: 35036: 35033: 35031: 35028: 35026: 35025:Time dilation 35023: 35022: 35020: 35016: 35010: 35007: 35005: 35002: 35000: 34997: 34995: 34992: 34990: 34987: 34986: 34984: 34980: 34974: 34971: 34969: 34966: 34964: 34961: 34960: 34958: 34953: 34950: 34949: 34946: 34940: 34937: 34935: 34932: 34931: 34929: 34925: 34921: 34914: 34909: 34907: 34902: 34900: 34895: 34894: 34891: 34881: 34871: 34865: 34864: 34860: 34858: 34855: 34853: 34850: 34848: 34845: 34843: 34840: 34838: 34835: 34833: 34830: 34828: 34825: 34823: 34820: 34818: 34815: 34813: 34810: 34808: 34805: 34803: 34800: 34798: 34795: 34793: 34790: 34788: 34785: 34783: 34780: 34778: 34775: 34773: 34772:Chandrasekhar 34770: 34768: 34765: 34763: 34760: 34758: 34755: 34753: 34750: 34748: 34745: 34743: 34740: 34738: 34735: 34733: 34732:Schwarzschild 34730: 34728: 34725: 34723: 34720: 34718: 34715: 34713: 34710: 34709: 34707: 34703: 34693: 34689: 34688: 34685: 34682: 34680: 34677: 34675: 34671: 34670: 34667: 34664: 34662: 34659: 34657: 34654: 34652: 34649: 34646: 34642: 34638: 34637: 34634: 34631: 34628: 34625: 34623: 34619: 34618:Schwarzschild 34615: 34614: 34611: 34608: 34606: 34603: 34601: 34598: 34596: 34593: 34591: 34588: 34585: 34581: 34577: 34576: 34574: 34572: 34568: 34562: 34559: 34557: 34554: 34552: 34549: 34548: 34546: 34540: 34534: 34531: 34528: 34524: 34520: 34517: 34515: 34514:Shapiro delay 34512: 34510: 34507: 34504: 34500: 34496: 34493: 34490: 34486: 34483: 34482: 34479: 34476: 34473: 34470: 34468: 34465: 34463: 34460: 34458: 34457:collaboration 34454: 34450: 34447: 34445: 34441: 34438: 34437: 34434: 34431: 34429: 34426: 34424: 34423:Event horizon 34421: 34419: 34416: 34415: 34413: 34409: 34403: 34400: 34398: 34395: 34393: 34390: 34388: 34385: 34383: 34380: 34378: 34375: 34373: 34370: 34368: 34367:ADM formalism 34365: 34364: 34362: 34358: 34352: 34349: 34347: 34344: 34342: 34339: 34337: 34334: 34332: 34329: 34328: 34326: 34320: 34314: 34311: 34309: 34306: 34305: 34303: 34299: 34296: 34294: 34288: 34278: 34275: 34273: 34272:Biquaternions 34270: 34268: 34265: 34263: 34260: 34258: 34255: 34254: 34252: 34250: 34246: 34240: 34237: 34235: 34232: 34230: 34227: 34225: 34222: 34220: 34217: 34215: 34212: 34210: 34207: 34205: 34202: 34200: 34199:Time dilation 34197: 34196: 34194: 34190: 34184: 34181: 34180: 34178: 34174: 34168: 34165: 34163: 34160: 34158: 34155: 34153: 34152:Proper length 34150: 34148: 34145: 34143: 34140: 34138: 34135: 34133: 34130: 34128: 34125: 34124: 34122: 34116: 34110: 34107: 34105: 34102: 34099: 34096: 34094: 34090: 34087: 34086: 34084: 34080: 34077: 34075: 34069: 34065: 34058: 34053: 34051: 34046: 34044: 34039: 34038: 34035: 34023: 34022: 34013: 34012: 34009: 34003: 34000: 33998: 33995: 33993: 33990: 33988: 33985: 33983: 33980: 33978: 33975: 33973: 33970: 33968: 33965: 33963: 33960: 33958: 33955: 33953: 33950: 33948: 33945: 33943: 33940: 33938: 33935: 33933: 33930: 33928: 33925: 33924: 33922: 33918: 33911: 33908: 33905: 33902: 33899: 33896: 33893: 33890: 33887: 33884: 33881: 33878: 33875: 33874:Maja Einstein 33872: 33869: 33866: 33863: 33860: 33857: 33854: 33851: 33848: 33845: 33844:Elsa Einstein 33842: 33839: 33836: 33835: 33833: 33831: 33827: 33821: 33820: 33816: 33814: 33813: 33809: 33807: 33806: 33802: 33800: 33799: 33795: 33793: 33792: 33788: 33786: 33785: 33781: 33779: 33778: 33774: 33772: 33771: 33767: 33766: 33764: 33758: 33752: 33749: 33747: 33744: 33742: 33739: 33737: 33734: 33732: 33731:Kalinga Prize 33729: 33727: 33724: 33722: 33719: 33718: 33716: 33712: 33705: 33704: 33700: 33698:(2017 series) 33697: 33696: 33692: 33689: 33688: 33684: 33681: 33680: 33676: 33673: 33672: 33668: 33665: 33664: 33660: 33657: 33656: 33652: 33649: 33648: 33644: 33641: 33640: 33636: 33633: 33632: 33628: 33625: 33624: 33620: 33619: 33617: 33615: 33609: 33602: 33599: 33596: 33592: 33589: 33588: 33584: 33581: 33580: 33576: 33573: 33572: 33568: 33565: 33564: 33560: 33557: 33553: 33550: 33548: 33544: 33543: 33541: 33539: 33535: 33529: 33526: 33524: 33521: 33519: 33516: 33514: 33511: 33509: 33506: 33504: 33501: 33499: 33496: 33494: 33491: 33489: 33486: 33484: 33481: 33479: 33476: 33474: 33471: 33469: 33466: 33464: 33461: 33459: 33456: 33454: 33451: 33449: 33446: 33444: 33441: 33439: 33436: 33434: 33431: 33429: 33426: 33424: 33421: 33419: 33416: 33414: 33411: 33407: 33404: 33402: 33399: 33398: 33397: 33394: 33393: 33391: 33387: 33383: 33376: 33371: 33369: 33364: 33362: 33357: 33356: 33353: 33341: 33338: 33336: 33333: 33331: 33328: 33326: 33323: 33321: 33318: 33317: 33315: 33311: 33305: 33302: 33300: 33299:Ocean physics 33297: 33295: 33292: 33290: 33287: 33285: 33282: 33280: 33277: 33275: 33272: 33270: 33267: 33265: 33262: 33260: 33257: 33256: 33254: 33252: 33248: 33242: 33239: 33235: 33234:Modern optics 33232: 33230: 33227: 33225: 33222: 33221: 33220: 33217: 33215: 33212: 33210: 33207: 33205: 33202: 33198: 33195: 33193: 33190: 33189: 33188: 33185: 33184: 33182: 33180: 33176: 33168: 33165: 33163: 33160: 33159: 33158: 33155: 33151: 33148: 33146: 33143: 33142: 33141: 33138: 33136: 33133: 33131: 33128: 33124: 33121: 33119: 33116: 33114: 33111: 33109: 33106: 33105: 33104: 33101: 33100: 33098: 33096: 33092: 33084: 33083:Computational 33081: 33080: 33079: 33076: 33074: 33071: 33070: 33068: 33064: 33056: 33053: 33052: 33051: 33048: 33046: 33043: 33042: 33040: 33036: 33032: 33024: 33019: 33017: 33012: 33010: 33005: 33004: 33001: 32994: 32991: 32988: 32985: 32982: 32979: 32976: 32972: 32967: 32964: 32961: 32958: 32954: 32951: 32948: 32945: 32942: 32939: 32935: 32932: 32929: 32926: 32923: 32922: 32919:Visualization 32913: 32910: 32907: 32904: 32901: 32898: 32895: 32892: 32889: 32886: 32883: 32880: 32878: 32873: 32869: 32868: 32863: 32861: 32858: 32855: 32852: 32849: 32845: 32841: 32836: 32832: 32829: 32826: 32823: 32820: 32817: 32813: 32810: 32809: 32804: 32801: 32798: 32797: 32788: 32784: 32781: 32777: 32774: 32771: 32768: 32764: 32761: 32760: 32751: 32747: 32746: 32742: 32739: 32735: 32731: 32730: 32726: 32725: 32709: 32704: 32700: 32696: 32692: 32688: 32687: 32682: 32678: 32674: 32672: 32669: 32666: 32662: 32658: 32654: 32650: 32646: 32642: 32637: 32633: 32629: 32625: 32621: 32617: 32613: 32610:(4): 614–26. 32609: 32605: 32600: 32596: 32592: 32588: 32584: 32580: 32576: 32571: 32570: 32561: 32560:0-7167-4345-0 32557: 32553: 32549: 32546: 32545:0-7167-2327-1 32542: 32538: 32534: 32530: 32526: 32524:9783110515879 32520: 32516: 32515: 32509: 32505: 32499: 32495: 32490: 32486: 32480: 32476: 32475: 32469: 32466: 32462: 32458: 32452: 32448: 32443: 32439: 32433: 32429: 32425: 32421: 32418:. Singapore: 32417: 32416: 32411: 32410:Qadir, Asghar 32407: 32405: 32401: 32397: 32396:0-19-927583-1 32393: 32389: 32387: 32386:0-19-853952-5 32383: 32379: 32375: 32371: 32367: 32364: 32361: 32357: 32355: 32354:0-7167-0334-3 32351: 32347: 32343: 32339: 32335: 32332: 32331: 32322: 32321: 32316: 32314: 32313:1-56731-136-9 32310: 32306: 32302: 32299: 32298: 32293: 32292: 32275: 32269: 32265: 32264: 32256: 32250: 32249:0-521-31132-2 32246: 32242: 32238: 32233: 32227: 32226:0-7167-0344-0 32223: 32219: 32215: 32211: 32210:Kip S. Thorne 32207: 32202: 32196: 32195:0-07-032071-3 32192: 32188: 32182: 32174: 32168: 32164: 32163: 32155: 32147: 32141: 32137: 32130: 32122: 32116: 32112: 32105: 32097: 32093: 32089: 32085: 32081: 32077: 32074:(1): 45–105. 32073: 32069: 32065: 32058: 32042: 32038: 32031: 32025: 32018: 32014: 32010: 32006: 32002: 31998: 31993: 31988: 31984: 31980: 31971: 31965: 31963: 31956: 31950: 31946: 31945: 31937: 31929: 31923: 31919: 31915: 31914: 31906: 31898: 31892: 31887: 31886: 31877: 31875: 31873: 31864: 31860: 31856: 31852: 31848: 31844: 31839: 31834: 31830: 31826: 31819: 31803: 31799: 31795: 31788: 31786: 31777: 31771: 31767: 31760: 31758: 31756: 31747: 31741: 31737: 31730: 31728: 31719: 31713: 31709: 31702: 31694: 31687: 31679: 31675: 31671: 31667: 31663: 31659: 31652: 31636: 31632: 31625: 31623: 31615: 31611: 31608: 31603: 31588: 31581: 31579: 31570: 31564: 31560: 31553: 31545: 31539: 31535: 31534: 31531: 31527: 31519: 31511: 31505: 31501: 31500: 31495: 31489: 31473: 31469: 31465: 31458: 31442: 31438: 31434: 31427: 31411: 31407: 31400: 31392: 31390:9780521345606 31386: 31382: 31375: 31367: 31363: 31359: 31355: 31351: 31347: 31342: 31337: 31334:(6): 065025. 31333: 31329: 31322: 31314: 31310: 31306: 31302: 31298: 31294: 31287: 31271: 31267: 31263: 31256: 31237: 31233: 31226: 31219: 31203: 31200:Cook, Helen. 31196: 31188: 31184: 31180: 31178:9780201021165 31174: 31170: 31166: 31162: 31158: 31154: 31150: 31144: 31136: 31132: 31128: 31121: 31106: 31102: 31098: 31094: 31090: 31083: 31075: 31069: 31065: 31058: 31051: 31036: 31032: 31028: 31024: 31020: 31013: 30997: 30990: 30982: 30978: 30974: 30970: 30966: 30962: 30957: 30952: 30948: 30944: 30937: 30929: 30923: 30919: 30918: 30910: 30902: 30896: 30892: 30891: 30883: 30868: 30862: 30858: 30854: 30847: 30828: 30824: 30817: 30810: 30804: 30798: 30792: 30788: 30787: 30779: 30773: 30768: 30762: 30758: 30754: 30750: 30749: 30741: 30725: 30721: 30714: 30706: 30704:9781542323512 30700: 30696: 30689: 30673: 30666: 30658: 30657: 30649: 30641: 30637: 30633: 30629: 30625: 30621: 30614: 30598: 30594: 30590: 30583: 30575: 30569: 30565: 30561: 30560: 30552: 30544: 30542:0-471-81995-6 30538: 30534: 30527: 30525: 30509: 30503: 30499: 30498: 30490: 30482: 30476: 30472: 30468: 30467: 30459: 30451: 30445: 30441: 30437: 30436: 30428: 30420: 30414: 30410: 30406: 30405: 30397: 30381: 30377: 30370: 30363: 30361: 30345: 30341: 30337: 30333: 30329: 30325: 30321: 30314: 30306: 30304:0-393-09793-5 30300: 30296: 30289: 30287: 30285: 30276: 30270: 30266: 30262: 30261: 30253: 30245: 30241: 30236: 30231: 30226: 30221: 30217: 30213: 30209: 30205: 30201: 30194: 30185: 30184: 30175: 30173: 30171: 30169: 30160: 30154: 30150: 30143: 30141: 30139: 30123: 30119: 30115: 30111: 30104: 30097: 30089: 30083: 30079: 30072: 30064: 30058: 30054: 30047: 30045: 30036: 30034:9780471717249 30030: 30026: 30025: 30017: 30009: 30003: 29998: 29997: 29988: 29981: 29980:0-486-63282-2 29977: 29973: 29967: 29959: 29957:9781441931429 29953: 29949: 29942: 29934: 29930: 29926: 29922: 29918: 29914: 29909: 29904: 29900: 29896: 29889: 29882: 29881:1-139-46837-5 29878: 29874: 29868: 29866: 29864: 29862: 29860: 29858: 29856: 29854: 29852: 29850: 29848: 29846: 29844: 29842: 29840: 29838: 29831: 29830:0-226-77057-5 29827: 29823: 29817: 29809: 29807:9780957389465 29803: 29799: 29792: 29784: 29778: 29770: 29764: 29760: 29759: 29751: 29742: 29734: 29730: 29726: 29722: 29718: 29714: 29709: 29704: 29700: 29696: 29689: 29687: 29680: 29677: 29676:0-226-67288-3 29673: 29669: 29665: 29660: 29646: 29642: 29636: 29621: 29620:Physics World 29617: 29611: 29603: 29597: 29593: 29592: 29584: 29582: 29580: 29578: 29576: 29567: 29565:0-7167-2327-1 29561: 29557: 29556: 29548: 29546: 29544: 29542: 29540: 29533: 29528: 29522: 29518: 29517: 29509: 29503: 29498: 29492: 29488: 29487: 29479: 29473: 29467: 29461: 29456: 29448: 29442: 29438: 29437: 29429: 29415: 29411: 29404: 29396: 29390: 29386: 29382: 29376: 29374: 29365: 29363:0-201-02918-9 29359: 29355: 29354: 29349: 29343: 29335: 29333:0-471-30932-X 29329: 29325: 29324: 29319: 29313: 29311: 29309: 29300: 29294: 29290: 29289: 29284: 29278: 29276: 29274: 29269: 29255: 29250: 29246: 29242: 29238: 29234: 29230: 29226: 29225:C.D. Anderson 29220: 29212: 29208: 29203: 29198: 29194: 29190: 29186: 29182: 29178: 29174: 29168: 29166: 29157: 29153: 29148: 29143: 29139: 29135: 29131: 29127: 29123: 29116: 29114: 29112: 29092: 29085: 29078: 29077: 29071: 29064: 29060: 29057:A. Einstein, 29056: 29051: 29049: 29047: 29038: 29034: 29030: 29026: 29022: 29018: 29014: 29010: 29009: 29001: 28993: 28989: 28985: 28981: 28977: 28973: 28972: 28964: 28956: 28952: 28948: 28944: 28940: 28933: 28925: 28921: 28917: 28913: 28909: 28902: 28894: 28890: 28886: 28882: 28878: 28874: 28870: 28866: 28859: 28851: 28847: 28843: 28839: 28835: 28831: 28827: 28823: 28816: 28808: 28804: 28800: 28796: 28792: 28788: 28784: 28780: 28776: 28772: 28765: 28757: 28756: 28751: 28744: 28729: 28725: 28721: 28717: 28710: 28703: 28701: 28685: 28681: 28677: 28673: 28666: 28659: 28652: 28651:0-582-31760-6 28648: 28642: 28635: 28634:0-387-94042-1 28631: 28627: 28621: 28613: 28607: 28603: 28596: 28586: 28577: 28568: 28566: 28556: 28547: 28540: 28534: 28527: 28526:Megh Nad Saha 28523: 28519: 28515: 28511: 28507: 28506: 28501: 28496: 28494: 28492: 28490: 28488: 28486: 28484: 28479: 28464: 28456: 28445: 28441: 28435: 28425: 28418: 28412: 28405: 28401: 28400:Pieter Zeeman 28395: 28388: 28385:differs from 28384: 28379: 28375: 28371: 28365: 28359:of spacetime. 28358: 28354: 28350: 28349: 28344: 28340: 28336: 28332: 28328: 28322: 28315: 28311: 28300: 28296: 28292: 28285: 28278: 28271: 28267: 28257: 28253: 28249: 28245: 28239: 28232: 28226: 28205: 28188: 28178: 28174: 28165: 28162: 28160: 28157: 28155: 28152: 28150: 28147: 28146: 28143: 28140: 28139: 28136: 28133: 28131: 28128: 28127: 28124: 28121: 28120: 28117: 28114: 28112: 28111:Lorentz group 28109: 28108: 28105: 28102: 28101: 28098: 28095: 28093: 28092:Born rigidity 28090: 28088: 28085: 28083: 28080: 28078: 28075: 28073: 28070: 28068: 28065: 28063: 28060: 28058: 28055: 28053: 28050: 28049: 28046: 28043: 28042: 28039: 28036: 28034: 28031: 28029: 28026: 28024: 28021: 28019: 28016: 28014: 28011: 28009: 28006: 28005: 28002: 27999: 27998: 27995: 27992: 27990: 27987: 27985: 27982: 27980: 27977: 27975: 27972: 27970: 27967: 27966: 27963: 27960: 27959: 27953: 27951: 27947: 27943: 27938: 27936: 27932: 27928: 27924: 27920: 27916: 27911: 27896: 27892: 27888: 27885: 27879: 27876: 27869: 27865: 27861: 27855: 27850: 27846: 27837: 27833: 27829: 27824: 27822: 27817: 27813: 27800: 27795: 27791: 27787: 27784: 27775: 27765: 27762: 27749: 27744: 27736: 27733: 27727: 27724: 27719: 27714: 27709: 27700: 27694: 27689: 27686: 27681: 27664: 27651: 27646: 27642: 27638: 27633: 27628: 27623: 27620: 27615: 27610: 27607: 27602: 27598: 27592: 27588: 27582: 27579: 27575: 27571: 27566: 27551: 27547: 27537: 27523: 27520: 27515: 27511: 27505: 27501: 27495: 27492: 27488: 27484: 27476: 27472: 27456: 27450: 27446: 27442: 27439: 27434: 27430: 27424: 27420: 27414: 27411: 27407: 27403: 27398: 27383: 27378: 27376: 27369: 27366: 27359:is positive, 27358: 27355: 27352:, while when 27351: 27344: 27341: 27332: 27329: 27326: 27308: 27300: 27297: 27291: 27286: 27278: 27275: 27269: 27264: 27256: 27253: 27247: 27242: 27234: 27231: 27227: 27221: 27218: 27213: 27209: 27205: 27199: 27195: 27191: 27185: 27182: 27178: 27174: 27169: 27165: 27161: 27155: 27151: 27147: 27144: 27139: 27129: 27121: 27103: 27099: 27095: 27090: 27080: 27077: 27068: 27062: 27059: 27054: 27050: 27036: 27021: 27013: 27008: 26999: 26995: 26989: 26980: 26976: 26970: 26961: 26957: 26953: 26948: 26939: 26935: 26929: 26920: 26916: 26912: 26907: 26898: 26894: 26884: 26867: 26863: 26857: 26853: 26847: 26824: 26815: 26797: 26792: 26788: 26782: 26779: 26775: 26769: 26765: 26761: 26756: 26752: 26746: 26743: 26739: 26733: 26729: 26725: 26720: 26716: 26710: 26706: 26697: 26693: 26689: 26684: 26668: 26658: 26655: 26644: 26634: 26631: 26619: 26616: 26611: 26608: 26603: 26599: 26594: 26591: 26587: 26578: 26573: 26571: 26553: 26550: 26546: 26523: 26517: 26512: 26507: 26502: 26495: 26490: 26485: 26480: 26473: 26468: 26463: 26458: 26451: 26446: 26441: 26436: 26433: 26427: 26422: 26417: 26414: 26410: 26397: 26394: 26390: 26389:inner product 26386: 26385:metric tensor 26376: 26374: 26370: 26366: 26362: 26358: 26354: 26349: 26347: 26343: 26339: 26335: 26330: 26314: 26304: 26301: 26273: 26263: 26260: 26232: 26229: 26226: 26223: 26218: 26215: 26212: 26209: 26205: 26199: 26189: 26186: 26177: 26172: 26162: 26159: 26148: 26138: 26135: 26124: 26114: 26111: 26102: 26097: 26087: 26084: 26073: 26063: 26060: 26051: 26045: 26042: 26038: 26034: 26031: 26026: 26023: 26016: 26013: 26009: 26005: 26002: 25997: 25994: 25989: 25981: 25977: 25972: 25969: 25953: 25943: 25940: 25912: 25902: 25899: 25873: 25868: 25864: 25858: 25848: 25845: 25836: 25830: 25827: 25822: 25813: 25809: 25804: 25802: 25786: 25778: 25774: 25761: 25756: 25744: 25739: 25726: 25716: 25713: 25704: 25698: 25692: 25689: 25660: 25655: 25649: 25644: 25639: 25634: 25627: 25622: 25617: 25612: 25605: 25600: 25595: 25590: 25587: 25584: 25577: 25572: 25567: 25564: 25561: 25556: 25550: 25543: 25533: 25525: 25509: 25501: 25485: 25477: 25461: 25453: 25440: 25437: 25428: 25423: 25418: 25407: 25404: 25395: 25378: 25375: 25366: 25349: 25346: 25337: 25320: 25317: 25308: 25295: 25292: 25283: 25273: 25269: 25268:four-gradient 25264: 25262: 25258: 25254: 25249: 25236: 25230: 25227: 25220: 25216: 25212: 25206: 25201: 25197: 25188: 25183: 25181: 25177: 25161: 25157: 25148: 25143: 25140: 25134: 25130: 25126: 25120: 25116: 25112: 25107: 25103: 25099: 25094: 25090: 25086: 25081: 25078: 25072: 25068: 25060: 25056: 25052: 25047: 25043: 25039: 25034: 25030: 25026: 25023: 25014: 25008: 25005: 25002: 24997: 24993: 24989: 24986: 24981: 24977: 24968: 24967:four-momentum 24965: 24964:contravariant 24944: 24938: 24932: 24929: 24917: 24899: 24895: 24891: 24885: 24879: 24876: 24873: 24866: 24861: 24848: 24843: 24838: 24834: 24830: 24825: 24820: 24816: 24812: 24807: 24802: 24798: 24794: 24789: 24785: 24775: 24771: 24766: 24760: 24756: 24752: 24749: 24745: 24740: 24734: 24728: 24708: 24697: 24694: 24685: 24679: 24676: 24668: 24664: 24660: 24655: 24651: 24647: 24642: 24638: 24634: 24631: 24622: 24616: 24613: 24607: 24604: 24597: 24593: 24589: 24583: 24578: 24574: 24565: 24549: 24544: 24540: 24532: 24531:four-velocity 24527: 24512: 24508: 24502: 24492: 24489: 24480: 24474: 24471: 24466: 24457: 24439: 24435: 24427: 24422: 24420: 24402: 24392: 24389: 24363: 24355: 24337: 24333: 24327: 24317: 24314: 24305: 24299: 24296: 24291: 24280: 24276: 24259: 24248: 24245: 24242: 24236: 24230: 24227: 24224: 24221: 24218: 24215: 24212: 24209: 24203: 24195: 24191: 24187: 24182: 24178: 24174: 24169: 24165: 24161: 24156: 24152: 24145: 24140: 24136: 24127: 24124: 24121: 24120:contravariant 24115: 24111: 24107: 24103: 24098: 24093: 24090: 24086: 24082: 24078: 24074: 24070: 24064: 24051: 24046: 24040: 24033: 24026: 24023: 24020: 24017: 24014: 24011: 24008: 24001: 23998: 23995: 23992: 23989: 23986: 23983: 23977: 23972: 23967: 23961: 23954: 23947: 23940: 23937: 23931: 23924: 23918: 23913: 23908: 23903: 23896: 23891: 23886: 23881: 23874: 23869: 23864: 23859: 23856: 23853: 23846: 23841: 23836: 23833: 23830: 23825: 23819: 23814: 23809: 23802: 23799: 23790: 23787: 23778: 23775: 23766: 23763: 23759: 23753: 23743: 23741: 23736: 23722: 23720: 23716: 23711: 23709: 23705: 23700: 23697: 23677: 23672: 23668: 23664: 23659: 23654: 23650: 23646: 23641: 23636: 23632: 23626: 23623: 23615: 23610: 23608: 23599: 23595: 23582: 23577: 23573: 23569: 23564: 23560: 23556: 23551: 23546: 23542: 23538: 23535: 23530: 23525: 23521: 23517: 23514: 23509: 23504: 23500: 23496: 23474: 23470: 23466: 23461: 23457: 23453: 23448: 23443: 23439: 23435: 23432: 23427: 23422: 23418: 23414: 23411: 23406: 23401: 23397: 23393: 23390: 23387: 23384: 23379: 23375: 23371: 23357: 23355: 23339: 23334: 23330: 23326: 23321: 23317: 23313: 23308: 23303: 23299: 23295: 23292: 23287: 23282: 23278: 23274: 23252: 23248: 23244: 23239: 23235: 23231: 23226: 23221: 23217: 23213: 23210: 23205: 23200: 23196: 23192: 23189: 23186: 23183: 23178: 23174: 23170: 23162: 23159: 23143: 23138: 23134: 23130: 23125: 23121: 23117: 23112: 23107: 23103: 23099: 23096: 23091: 23086: 23082: 23078: 23075: 23070: 23066: 23062: 23049: 23040: 23038: 23032: 23027: 23026:natural units 23023: 23019: 23015: 23010: 23006: 23000: 22998: 22993: 22989: 22984: 22980: 22979:Wick rotation 22975: 22968: 22963: 22959: 22955: 22950: 22948: 22944: 22940: 22936: 22932: 22928: 22919: 22912: 22905: 22898: 22894: 22891: 22874: 22869: 22864: 22860: 22856: 22853: 22848: 22843: 22839: 22835: 22832: 22827: 22822: 22818: 22814: 22811: 22806: 22801: 22797: 22793: 22790: 22787: 22782: 22778: 22774: 22766: 22757: 22750: 22743: 22739: 22736: 22719: 22714: 22709: 22705: 22701: 22698: 22693: 22688: 22684: 22680: 22677: 22672: 22667: 22663: 22659: 22656: 22648: 22645: 22637: 22634: 22629: 22625: 22621: 22613: 22609: 22604: 22602: 22598: 22590: 22586: 22582: 22579: 22575: 22571: 22567: 22564: 22560: 22557: 22552: 22547: 22531: 22518: 22514: 22511: 22507: 22504: 22501: 22498: 22495: 22492: 22489: 22486: 22483: 22480: 22477: 22474: 22471: 22470: 22469: 22467: 22463: 22456: 22452: 22449: 22445: 22441: 22438: 22434: 22433: 22432: 22428: 22426: 22420: 22418: 22417:string theory 22414: 22409: 22407: 22403: 22402:Planck length 22399: 22395: 22391: 22387: 22383: 22377: 22373: 22363: 22361: 22357: 22356: 22349: 22347: 22346: 22341: 22337: 22333: 22329: 22328: 22322: 22318: 22314: 22309: 22308:of the time. 22307: 22303: 22299: 22295: 22290: 22288: 22284: 22280: 22276: 22274: 22269: 22265: 22261: 22257: 22248: 22246: 22242: 22238: 22234: 22232: 22228: 22224: 22223:electrostatic 22220: 22216: 22212: 22207: 22205: 22201: 22197: 22193: 22187: 22183: 22173: 22171: 22167: 22162: 22160: 22139: 22136: 22132: 22126: 22122: 22118: 22115: 22112: 22092: 22069: 22066: 22060: 22053:Substituting 22052: 22038: 22032: 22029: 22023: 22019: 22010: 22006: 22002: 21999: 21996: 21990: 21987: 21981: 21978: 21972: 21968: 21964: 21961: 21958: 21939: 21936: 21933: 21929: 21925: 21922: 21919: 21916: 21908: 21894: 21890: 21882: 21878: 21874: 21869: 21865: 21859: 21855: 21833: 21826: 21822: 21818: 21815: 21809: 21805: 21802: 21799: 21792: 21778: 21771: 21767: 21763: 21758: 21754: 21748: 21744: 21737: 21733: 21730: 21727: 21721: 21718: 21712: 21692: 21687: 21683: 21679: 21674: 21670: 21666: 21661: 21657: 21651: 21647: 21638: 21624: 21621: 21617: 21613: 21610: 21607: 21604: 21596: 21595: 21594: 21591: 21577: 21557: 21554: 21534: 21521: 21520: 21514: 21510: 21508: 21504: 21500: 21494: 21490: 21480: 21478: 21477:to increase. 21463: 21460: 21438: 21435: 21430: 21427: 21406: 21385: 21382: 21377: 21374: 21353: 21350: 21330: 21302: 21299: 21296: 21292: 21288: 21284: 21280: 21276: 21272: 21268: 21264: 21262: 21256: 21253: 21244: 21240: 21237: 21234: 21231: 21228: 21223: 21219: 21214: 21210: 21207: 21205: 21199: 21195: 21191: 21182: 21178: 21175: 21172: 21167: 21163: 21158: 21154: 21151: 21149: 21143: 21139: 21135: 21123: 21122: 21121: 21106: 21103: 21082: 21079: 21059: 21056: 21051: 21047: 21043: 21038: 21034: 21011: 21007: 20986: 20982: 20979: 20968: 20954: 20951: 20930: 20927: 20922: 20919: 20898: 20878: 20864: 20860: 20853: 20849: 20845: 20841: 20837: 20833: 20829: 20826: 20821: 20812: 20808: 20804: 20803: 20802: 20798: 20790: 20786: 20784: 20778: 20768: 20766: 20763: 20759: 20754: 20744: 20732: 20719: 20683: 20673: 20667: 20661: 20658: 20653: 20650: 20643: 20639: 20620: 20616: 20609: 20606: 20595: 20589: 20583: 20577: 20574: 20569: 20566: 20559: 20555: 20536: 20530: 20527: 20522: 20519: 20513: 20510: 20503: 20502: 20489: 20486: 20474: 20471: 20454: 20447: 20444: 20430: 20424: 20418: 20412: 20406: 20403: 20398: 20395: 20388: 20384: 20365: 20353: 20347: 20344: 20341: 20332: 20329: 20325: 20304: 20298: 20295: 20290: 20287: 20281: 20278: 20271: 20270: 20257: 20254: 20242: 20239: 20238: 20218: 20212: 20203: 20197: 20194: 20162: 20156: 20153: 20138: 20137: 20115: 20111: 20107: 20102: 20098: 20094: 20089: 20085: 20081: 20078: 20074: 20070: 20064: 20052: 20046: 20043: 20039: 20035: 20029: 20026: 20019: 20018: 20017: 20014: 20013: 19993: 19990: 19987: 19984: 19976: 19973: 19969: 19962: 19961: 19951: 19939: 19905: 19899: 19896: 19893: 19871: 19867: 19860: 19857: 19852: 19849: 19843: 19837: 19834: 19829: 19826: 19820: 19814: 19811: 19806: 19803: 19797: 19794: 19790: 19786: 19767: 19761: 19757: 19753: 19750: 19742: 19739: 19736: 19733: 19730: 19727: 19724: 19721: 19718: 19715: 19712: 19703: 19697: 19694: 19689: 19686: 19680: 19677: 19670: 19669: 19667: 19663: 19647: 19644: 19624: 19621: 19609: 19606: 19587: 19584: 19581: 19578: 19575: 19572: 19569: 19566: 19563: 19560: 19557: 19551: 19548: 19545: 19538: 19537: 19533: 19525: 19519: 19515: 19512: 19511: 19491: 19486: 19482: 19476: 19472: 19468: 19463: 19459: 19436: 19432: 19428: 19423: 19419: 19413: 19409: 19388: 19385: 19382: 19379: 19376: 19373: 19360: 19357: 19356: 19352: 19320: 19317: 19314: 19311: 19308: 19300: 19296: 19289: 19288: 19265: 19259: 19250: 19244: 19239: 19235: 19229: 19225: 19205: 19200: 19196: 19190: 19186: 19182: 19177: 19173: 19167: 19163: 19159: 19154: 19150: 19144: 19140: 19136: 19131: 19127: 19121: 19117: 19097: 19094: 19091: 19088: 19081: 19080: 19056: 19027: 19008: 19004: 19000: 18997: 18976: 18973: 18970: 18967: 18964: 18961: 18958: 18944: 18940: 18936: 18933: 18932: 18926: 18907: 18901: 18898: 18892: 18889: 18883: 18880: 18874: 18852: 18850: 18846: 18842: 18838: 18833: 18832: 18831: 18823: 18819: 18812: 18808: 18783: 18779: 18775: 18773: 18767: 18763: 18759: 18749: 18745: 18741: 18739: 18733: 18729: 18725: 18716: 18710: 18706: 18699: 18695: 18691: 18685: 18680: 18676: 18671: 18667: 18664: 18662: 18656: 18652: 18648: 18639: 18633: 18629: 18622: 18618: 18614: 18608: 18603: 18599: 18594: 18590: 18587: 18585: 18579: 18575: 18571: 18559: 18558: 18557: 18553: 18544: 18541:coordinates, 18523: 18520: 18517: 18514: 18511: 18508: 18505: 18502: 18488: 18486: 18464: 18458: 18454: 18450: 18445: 18441: 18437: 18432: 18428: 18424: 18419: 18415: 18410: 18406: 18403: 18387: 18385: 18381: 18368: 18367:formulation. 18366: 18362: 18358: 18354: 18340: 18330: 18327: 18325: 18306: 18301: 18294: 18291: 18282: 18279: 18275: 18269: 18262: 18256: 18253: 18250: 18245: 18242: 18239: 18232: 18229: 18226: 18221: 18218: 18215: 18209: 18204: 18199: 18193: 18186: 18183: 18177: 18168: 18167: 18166: 18162: 18142: 18137: 18131: 18124: 18121: 18115: 18108: 18102: 18099: 18096: 18091: 18088: 18085: 18082: 18075: 18072: 18069: 18066: 18061: 18058: 18055: 18049: 18044: 18039: 18032: 18029: 18020: 18017: 18013: 18007: 17998: 17997: 17996: 17992: 17987: 17983: 17980:Substituting 17978: 17976: 17957: 17954: 17951: 17948: 17945: 17923: 17920: 17915: 17911: 17907: 17904: 17899: 17896: 17893: 17887: 17879: 17875: 17871: 17868: 17864: 17859: 17856: 17853: 17846: 17832: 17829: 17826: 17823: 17820: 17798: 17795: 17790: 17786: 17782: 17779: 17775: 17770: 17762: 17758: 17754: 17751: 17747: 17742: 17739: 17732: 17731: 17730: 17728: 17723: 17710: 17705: 17701: 17697: 17692: 17688: 17684: 17681: 17658: 17650: 17646: 17642: 17637: 17633: 17626: 17623: 17604: 17596: 17592: 17588: 17585: 17580: 17576: 17572: 17569: 17566: 17563: 17556: 17552: 17548: 17545: 17542: 17537: 17533: 17529: 17526: 17504: 17496: 17492: 17486: 17482: 17478: 17475: 17468: 17464: 17460: 17455: 17451: 17444: 17441: 17434: 17433: 17432: 17414: 17406: 17403: 17399: 17395: 17390: 17386: 17378: 17375: 17371: 17367: 17362: 17358: 17351: 17345: 17342: 17339: 17334: 17331: 17328: 17322: 17319: 17316: 17313: 17306: 17305: 17304: 17287: 17282: 17279: 17274: 17271: 17268: 17265: 17262: 17259: 17252: 17251: 17250: 17248: 17244: 17228: 17205: 17200: 17194: 17187: 17181: 17174: 17168: 17165: 17162: 17157: 17154: 17151: 17144: 17141: 17138: 17135: 17130: 17127: 17124: 17118: 17113: 17108: 17101: 17098: 17089: 17086: 17079: 17070: 17069: 17068: 17066: 17057: 17050: 17042: 17038: 17017: 17011: 17008: 17005: 17000: 16997: 16994: 16988: 16985: 16982: 16979: 16976: 16964: 16963: 16962: 16959: 16957: 16953: 16949: 16945: 16935: 16928: 16923: 16919: 16915: 16911: 16907: 16902: 16894: 16887: 16883: 16879: 16875: 16870: 16860: 16856: 16850: 16846: 16841:in the point 16838: 16834: 16830: 16823: 16814: 16810: 16804: 16800: 16795:in the point 16792: 16788: 16784: 16777: 16767: 16748: 16741: 16739: 16720: 16716: 16712: 16704: 16701: 16696: 16691: 16687: 16683: 16680: 16675: 16671: 16666: 16662: 16653: 16649: 16643: 16639: 16635: 16632: 16623: 16622: 16619: 16617: 16616: 16599: 16594: 16590: 16586: 16581: 16577: 16556: 16553: 16550: 16537: 16530: 16528: 16509: 16505: 16501: 16493: 16490: 16485: 16481: 16477: 16474: 16471: 16468: 16463: 16459: 16453: 16445: 16442: 16437: 16433: 16420: 16416: 16413: 16409: 16405: 16402: 16396: 16393: 16388: 16384: 16374: 16367: 16364: 16361: 16354: 16351: 16342: 16341: 16334: 16327: 16325: 16306: 16302: 16298: 16288: 16283: 16279: 16274: 16267: 16264: 16261: 16255: 16250: 16246: 16242: 16238: 16234: 16229: 16225: 16219: 16214: 16210: 16200: 16196: 16193: 16188: 16184: 16177: 16172: 16168: 16158: 16157: 16150: 16143: 16141: 16122: 16118: 16114: 16104: 16099: 16095: 16090: 16083: 16080: 16077: 16071: 16066: 16062: 16058: 16054: 16050: 16045: 16041: 16035: 16030: 16026: 16016: 16012: 16009: 16004: 16000: 15993: 15988: 15984: 15974: 15973: 15970: 15963: 15956: 15954: 15938: 15934: 15930: 15925: 15921: 15917: 15914: 15911: 15906: 15902: 15892: 15891: 15884: 15877: 15875: 15860: 15856: 15853: 15848: 15844: 15838: 15834: 15830: 15826: 15822: 15819: 15814: 15810: 15804: 15800: 15790: 15789: 15782: 15775: 15773: 15758: 15754: 15751: 15746: 15742: 15736: 15732: 15728: 15724: 15720: 15717: 15712: 15708: 15702: 15698: 15694: 15689: 15685: 15679: 15675: 15665: 15664: 15661: 15659: 15658: 15653: 15652: 15635: 15615: 15607: 15606: 15596: 15589: 15587: 15571: 15567: 15563: 15558: 15554: 15550: 15545: 15541: 15537: 15532: 15528: 15524: 15519: 15515: 15511: 15508: 15503: 15499: 15495: 15490: 15486: 15476: 15475: 15472: 15464: 15457: 15455: 15433: 15429: 15422: 15417: 15413: 15409: 15398: 15394: 15387: 15382: 15378: 15374: 15371: 15368: 15357: 15353: 15346: 15341: 15337: 15327: 15326: 15323: 15315: 15308: 15306: 15290: 15286: 15282: 15279: 15276: 15273: 15257: 15251: 15248: 15245: 15236: 15224: 15223: 15220: 15217: 15209: 15200: 15196: 15192: 15188: 15184: 15180: 15176: 15169: 15165: 15159: 15153: 15148: 15144: 15140: 15130: 15121: 15119: 15114: 15112: 15108: 15104: 15100: 15096: 15091: 15082: 15080: 15076: 15072: 15068: 15064: 15060: 15055: 15051: 15047: 15043: 15039: 15035: 15031: 15027: 15006: 15002: 14997: 14991: 14987: 14981: 14977: 14973: 14970: 14965: 14962: 14956: 14950: 14944: 14936: 14935: 14926: 14916: 14912: 14910: 14906: 14901: 14896: 14894: 14890: 14886: 14882: 14864: = 14863: 14854: 14847: 14845: 14829: 14825: 14817: 14813: 14809: 14802: 14799: 14794: 14789: 14785: 14781: 14776: 14772: 14762: 14761: 14758: 14751: 14744: 14742: 14727: 14723: 14720: 14712: 14708: 14703: 14697: 14693: 14689: 14686: 14682: 14676: 14672: 14669: 14664: 14660: 14656: 14651: 14647: 14637: 14636: 14633: 14617: 14613: 14609: 14604: 14600: 14577: 14573: 14569: 14564: 14560: 14539: 14519: 14499: 14496: 14493: 14480: 14473: 14471: 14456: 14452: 14449: 14441: 14437: 14432: 14426: 14422: 14418: 14415: 14411: 14405: 14401: 14398: 14390: 14386: 14382: 14377: 14373: 14366: 14358: 14354: 14350: 14345: 14341: 14328: 14327: 14324: 14298: 14294: 14289: 14283: 14279: 14275: 14272: 14268: 14263: 14258: 14254: 14250: 14242: 14238: 14233: 14227: 14223: 14219: 14216: 14210: 14206: 14203: 14197: 14193: 14189: 14183: 14180: 14174: 14168: 14165: 14159: 14151: 14147: 14142: 14136: 14132: 14128: 14125: 14119: 14115: 14112: 14106: 14102: 14098: 14092: 14089: 14083: 14077: 14074: 14068: 14063: 14059: 14055: 14053: 14046: 14042: 14034: 14031: 14026: 14022: 14018: 14015: 14009: 14006: 14000: 13997: 13991: 13988: 13982: 13977: 13973: 13969: 13967: 13960: 13956: 13944: 13943: 13942: 13926: 13922: 13910: 13892: 13888: 13879: 13863: 13852: 13847: 13831: 13827: 13804: 13800: 13791: 13775: 13771: 13767: 13747: 13739: 13715: 13711: 13706: 13700: 13696: 13692: 13689: 13683: 13679: 13676: 13670: 13666: 13662: 13656: 13653: 13647: 13642: 13638: 13635: 13625: 13624: 13623: 13609: 13598: 13594: 13578: 13558: 13544: 13540: 13533: 13522: 13513: 13511: 13507: 13503: 13499: 13494: 13492: 13485: 13481: 13475: 13471: 13467: 13465: 13461: 13457: 13451: 13447: 13443: 13439: 13431: 13427: 13421: 13416: 13413: 13409: 13404: 13399: 13397: 13393: 13388: 13386: 13380: 13370: 13368: 13364: 13363: 13352: 13349: 13345: 13341: 13337: 13333: 13329: 13324: 13322: 13318: 13317: 13307: 13302: 13298: 13295: 13291: 13286: 13285:Roger Penrose 13282: 13273: 13269: 13267: 13263: 13259: 13255: 13250: 13248: 13247:Doppler shift 13239: 13230: 13216: 13206: 13202: 13200: 13191: 13182: 13180: 13172: 13151: 13147: 13143: 13140: 13136: 13131: 13128: 13121: 13106: 13102: 13098: 13095: 13092: 13085: 13084: 13083: 13069: 13064: 13060: 13051: 13048: 13045: 13040: 13037: 13034: 13027: 13022: 13018: 13013: 13009: 13006: 13003: 12999: 12995: 12992: 12987: 12983: 12974: 12969: 12953: 12949: 12926: 12922: 12917: 12909: 12905: 12901: 12896: 12893: 12889: 12885: 12880: 12876: 12867: 12863: 12847: 12826: 12818: 12813: 12811: 12810:time dilation 12799: 12773: 12770: 12765: 12762: 12757: 12753: 12749: 12744: 12738: 12735: 12732: 12727: 12724: 12721: 12715: 12710: 12705: 12701: 12698: 12692: 12689: 12682: 12681: 12659: 12656: 12653: 12647: 12643: 12639: 12633: 12630: 12624: 12619: 12616: 12613: 12607: 12603: 12600: 12596: 12593: 12586: 12585: 12566: 12563: 12560: 12554: 12550: 12546: 12540: 12537: 12532: 12528: 12524: 12521: 12518: 12515: 12512: 12506: 12502: 12499: 12495: 12492: 12485: 12484: 12483: 12479: 12478:aether-drag. 12477: 12472: 12468: 12462: 12460: 12456: 12446: 12441: 12437: 12412: 12404: 12400: 12396: 12391: 12388: 12384: 12380: 12377: 12372: 12369: 12348: 12344: 12337: 12334: 12330: 12325: 12322: 12318: 12313: 12307: 12304: 12299: 12294: 12291: 12285: 12281: 12262: 12256: 12253: 12249: 12245: 12242: 12239: 12234: 12231: 12228: 12224: 12220: 12198: 12190: 12186: 12181: 12177: 12173: 12170: 12166: 12163: 12158: 12155: 12151: 12148: 12141: 12136: 12132: 12124: 12123: 12122: 12120: 12119: 12113: 12111: 12107: 12091: 12083: 12078: 12065: 12058: 12050: 12046: 12042: 12037: 12034: 12030: 12026: 12023: 12018: 12015: 12010: 12005: 12001: 11978: 11974: 11953: 11933: 11929: 11925: 11922: 11918: 11915: 11906: 11902: 11897: 11893: 11891: 11887: 11884: 11880: 11879:Léon Foucault 11876: 11867: 11862: 11847: 11844: 11839: 11837: 11833: 11830: 11825: 11823: 11806: 11803: 11781: 11778: 11774: 11754: 11733: 11730: 11705: 11702: 11693: 11690: 11673: 11670: 11666: 11657: 11656: 11655: 11653: 11646: 11642: 11623: 11614: 11605: 11602: 11598: 11596: 11589: 11584: 11579: 11575: 11565: 11562: 11560: 11556: 11552: 11548: 11544: 11540: 11535: 11521: 11516: 11512: 11507: 11503: 11500: 11492: 11491: 11486: 11470: 11461: 11447: 11439: 11438:proper length 11423: 11414: 11399: 11388: 11378: 11368: 11367:for details. 11366: 11362: 11358: 11350: 11346: 11332: 11324: 11309: 11300: 11294: 11288: 11283: 11280: 11276: 11272: 11268: 11263: 11254: 11247: 11241: 11237: 11231: 11227: 11226: 11225: 11222: 11217: 11213: 11209: 11185: 11154: 11149: 11093: 11089: 11088: 11083: 11082: 11073: 11066: 11064: 11050: 11040: 11034: 11030: 11025: 11021: 11017: 11013: 11009: 11005: 11002: 10998: 10994: 10989: 10985: 10981: 10975: 10970: 10966: 10961: 10951: 10945: 10941: 10936: 10932: 10928: 10924: 10920: 10916: 10913: 10909: 10905: 10900: 10896: 10892: 10886: 10881: 10877: 10872: 10861: 10857: 10852: 10848: 10844: 10840: 10836: 10832: 10829: 10824: 10821: 10817: 10813: 10809: 10802: 10797: 10793: 10785: 10784: 10777: 10770: 10768: 10754: 10744: 10738: 10734: 10729: 10725: 10720: 10716: 10712: 10709: 10705: 10701: 10695: 10691: 10685: 10681: 10677: 10673: 10668: 10658: 10652: 10648: 10643: 10639: 10634: 10630: 10626: 10623: 10619: 10615: 10609: 10605: 10599: 10595: 10591: 10587: 10582: 10571: 10567: 10562: 10558: 10553: 10549: 10545: 10542: 10537: 10534: 10529: 10525: 10518: 10514: 10510: 10506: 10498: 10497: 10494: 10487: 10480: 10478: 10464: 10454: 10451: 10447: 10443: 10438: 10435: 10431: 10425: 10421: 10418: 10414: 10410: 10405: 10402: 10398: 10392: 10388: 10385: 10381: 10377: 10372: 10369: 10365: 10359: 10352: 10348: 10344: 10337: 10333: 10329: 10325: 10318: 10314: 10310: 10306: 10299: 10294: 10282: 10281: 10274: 10267: 10265: 10251: 10242: 10239: 10235: 10231: 10228: 10222: 10219: 10216: 10212: 10208: 10205: 10199: 10196: 10193: 10189: 10185: 10182: 10176: 10168: 10164: 10157: 10152: 10148: 10141: 10136: 10132: 10125: 10113: 10112: 10109: 10080: 10073: 10071: 10057: 10049: 10045: 10040: 10036: 10032: 10029: 10025: 10022: 10017: 10014: 10010: 10007: 10000: 9997: 9990: 9989: 9986: 9973: 9967: 9964: 9944: 9931: 9924: 9922: 9908: 9897: 9893: 9888: 9885: 9878: 9875: 9870: 9867: 9864: 9858: 9848: 9845: 9840: 9837: 9824: 9820: 9816: 9810: 9807: 9802: 9799: 9792: 9789: 9784: 9781: 9771: 9764: 9755: 9751: 9746: 9743: 9739: 9732: 9729: 9726: 9722: 9718: 9710: 9707: 9703: 9700: 9697: 9694: 9688: 9682: 9675: 9672: 9668: 9662: 9659: 9655: 9649: 9645: 9642: 9634: 9633: 9630: 9615: 9612: 9591: 9583: 9582: 9577: 9576: 9571: 9570: 9552: 9549: 9545: 9524: 9521: 9517: 9504: 9497: 9495: 9481: 9476: 9473: 9469: 9465: 9460: 9457: 9453: 9450: 9446: 9443: 9439: 9429: 9412: 9411: 9404: 9397: 9395: 9381: 9377: 9374: 9370: 9366: 9363: 9360: 9357: 9354: 9332: 9331: 9328: 9327:We can write 9325: 9320: 9302: 9277: 9264: 9257: 9253: 9247: 9237: 9235: 9231: 9226: 9221: 9216: 9214: 9210: 9206: 9199: 9192: 9173: 9167: 9164: 9160: 9157: 9122: 9118: 9109: 9105: 9102: 9091: 9090: 9089: 9084: 9080: 9076: 9075: 9065: 9056: 9052: 9042: 9038: 9034: 9030: 9026: 9025:measuring rod 9021: 9019: 9013: 8996: 8980: 8976: 8973: 8967: 8964: 8957: 8952: 8935: 8932: 8927: 8924: 8918: 8914: 8911: 8903: 8900: 8899: 8894: 8893: 8889: 8873: 8869: 8866: 8863: 8853: 8850: 8833: 8830: 8825: 8822: 8819: 8813: 8792: 8788: 8784: 8774: 8770: 8766: 8763: 8755: 8751: 8748: 8745: 8735: 8732: 8731: 8727: 8710: 8707: 8704: 8696: 8692: 8689: 8683: 8662: 8658: 8654: 8649: 8643: 8640: 8637: 8632: 8629: 8626: 8620: 8613: 8609: 8605: 8595: 8591: 8587: 8584: 8576: 8572: 8569: 8563: 8543: 8539: 8535: 8530: 8527: 8519: 8516: 8500: 8496: 8492: 8482: 8478: 8474: 8471: 8463: 8459: 8456: 8450: 8429: 8425: 8421: 8416: 8410: 8407: 8404: 8399: 8396: 8393: 8387: 8379: 8376: 8373: 8365: 8361: 8358: 8352: 8331: 8327: 8322: 8319: 8311: 8295: 8292: 8283: 8282: 8276: 8259: 8256: 8252: 8247: 8243: 8239: 8235: 8227: 8222: 8206: 8203: 8198: 8193: 8190: 8185: 8180: 8176: 8168: 8165: 8164: 8160: 8143: 8140: 8137: 8129: 8125: 8122: 8116: 8094: 8090: 8086: 8081: 8075: 8072: 8069: 8064: 8061: 8058: 8052: 8045: 8041: 8037: 8027: 8023: 8019: 8016: 8008: 8004: 8001: 7995: 7975: 7971: 7967: 7962: 7959: 7951: 7948: 7932: 7928: 7924: 7914: 7910: 7906: 7903: 7895: 7891: 7888: 7882: 7860: 7856: 7852: 7847: 7841: 7838: 7835: 7830: 7827: 7824: 7818: 7810: 7807: 7804: 7796: 7792: 7789: 7783: 7762: 7758: 7753: 7750: 7742: 7726: 7723: 7714: 7713: 7707: 7690: 7687: 7683: 7678: 7674: 7670: 7666: 7658: 7653: 7637: 7634: 7629: 7624: 7621: 7616: 7611: 7607: 7599: 7597: 7593: 7592: 7588: 7571: 7568: 7563: 7560: 7557: 7551: 7547: 7544: 7540: 7533: 7530: 7514: 7510: 7507: 7504: 7498: 7495: 7492: 7485: 7482: 7481: 7475: 7458: 7455: 7450: 7447: 7441: 7437: 7434: 7426: 7421: 7405: 7401: 7398: 7392: 7389: 7382: 7379: 7378: 7374: 7369: 7366: 7361: 7358: 7357: 7354: 7339: 7331: 7328: 7325: 7318: 7311: 7308: 7305: 7297: 7294: 7290: 7287: 7265: 7257: 7254: 7251: 7244: 7237: 7234: 7231: 7223: 7220: 7216: 7213: 7190: 7188: 7178: 7174: 7170: 7168: 7162: 7152: 7150: 7144: 7130: 7115: 7113: 7096: 7089: 7085: 7080: 7074: 7070: 7066: 7063: 7049: 7045: 7023: 7016: 7012: 7008: 7003: 6999: 6986: 6983: 6980: 6975: 6971: 6950: 6943: 6939: 6935: 6930: 6926: 6920: 6900: 6879: 6876: 6867: 6865: 6864: 6847: 6844: 6840: 6836: 6833: 6830: 6821: 6799: 6786: 6782: 6780: 6779:Paul Langevin 6772: 6768: 6761: 6759: 6755: 6750: 6742: 6738: 6735: 6731: 6727: 6723: 6716: 6709: 6690: 6684: 6681: 6678: 6649: 6642: 6639: 6635: 6632: 6621: 6620: 6619: 6617: 6616: 6609: 6603: 6599: 6594: 6590: 6589:Time dilation 6583:Time dilation 6580: 6578: 6574: 6558: 6550: 6549:Sagnac effect 6545: 6540: 6534: 6528: 6519: 6510: 6504: 6498: 6493: 6479: 6471: 6467: 6462: 6455: 6449: 6446: 6439: 6435: 6432: 6428: 6425: 6413: 6412: 6406: 6404: 6395: 6391: 6387: 6382: 6378: 6373: 6369: 6365: 6361: 6356: 6351: 6347: 6321: 6301: 6296: 6292: 6271: 6268: 6265: 6257: 6250: 6246: 6230: 6226: 6223: 6209: 6206: 6203: 6200: 6180: 6173: 6169: 6161: 6157: 6136: 6114: 6111: 6087: 6084: 6077: 6073: 6065: 6061: 6040: 6032: 6031:proper length 6028: 6012: 6005: 6001: 5994: 5972: 5965: 5961: 5953: 5949: 5945: 5942: 5922: 5919: 5915: 5912: 5888: 5868: 5848: 5840: 5836: 5831: 5827: 5821: 5818: 5815: 5803: 5800: 5796: 5793: 5769: 5766: 5763: 5755: 5748: 5744: 5728: 5724: 5721: 5718: 5702: 5699: 5695: 5691: 5668: 5661: 5657: 5651: 5648: 5628: 5625: 5621: 5618: 5594: 5574: 5554: 5548: 5541: 5538: 5535: 5523: 5520: 5516: 5513: 5489: 5486: 5483: 5475: 5468: 5464: 5448: 5444: 5441: 5440: 5439: 5423: 5419: 5406: 5390: 5386: 5375:The value of 5373: 5353: 5348: 5344: 5337: 5331: 5326: 5322: 5313: 5309: 5305: 5303: 5294: 5285: 5281: 5275: 5271: 5264: 5261: 5257: 5250: 5245: 5241: 5232: 5228: 5224: 5220: 5211: 5207: 5201: 5197: 5190: 5187: 5183: 5176: 5171: 5167: 5158: 5154: 5148: 5144: 5140: 5138: 5125: 5121: 5113: 5108: 5104: 5095: 5091: 5081: 5077: 5073: 5067: 5062: 5058: 5049: 5045: 5041: 5035: 5030: 5026: 5017: 5013: 5007: 5003: 4999: 4993: 4988: 4984: 4975: 4971: 4965: 4961: 4957: 4955: 4941: 4936: 4932: 4923: 4919: 4915: 4911: 4908: 4900: 4897: 4890: 4887: 4884: 4878: 4873: 4869: 4854: 4850: 4846: 4842: 4833: 4829: 4821: 4816: 4812: 4803: 4799: 4791: 4787: 4784: 4776: 4773: 4766: 4763: 4760: 4754: 4749: 4745: 4736: 4732: 4727: 4721: 4717: 4713: 4711: 4701: 4692: 4689: 4682: 4679: 4675: 4672: 4657: 4653: 4649: 4644: 4639: 4630: 4626: 4620: 4617: 4610: 4603: 4599: 4596: 4588: 4581: 4577: 4571: 4567: 4563: 4561: 4554: 4550: 4543: 4538: 4534: 4525: 4521: 4507: 4492: 4488: 4481: 4476: 4472: 4463: 4459: 4454: 4448: 4444: 4431: 4429: 4425: 4421: 4417: 4413: 4409: 4405: 4401: 4385: 4380: 4376: 4363: 4360: 4342: 4338: 4331: 4326: 4322: 4315: 4310: 4306: 4296: 4291: 4287: 4278: 4274: 4264: 4256: 4252: 4225: 4221: 4209:, denoted as 4208: 4203: 4189: 4166: 4149: 4147: 4142: 4136: 4132: 4122: 4120: 4114: 4111: 4051: 4048: 4044: 4041: 4032: 4028: 4024: 4011: 4005: 3985: 3982: 3955: 3951: 3947: 3944: 3937: 3928: 3924: 3920: 3917: 3905:units equals 3891: 3888: 3884: 3861: 3858: 3855: 3852: 3849: 3846: 3836:axis through 3822: 3819: 3798: 3775: 3772: 3769: 3766: 3763: 3760: 3736: 3733: 3729: 3706: 3703: 3700: 3697: 3671: 3668: 3665: 3638: 3635: 3631: 3628: 3624: 3621: 3594: 3591: 3588: 3585: 3559: 3556: 3553: 3526: 3523: 3519: 3516: 3512: 3509: 3497: 3493: 3480: 3477: 3473: 3470: 3449: 3446: 3443: 3423: 3420: 3416: 3412: 3409: 3406: 3386: 3380: 3374: 3369: 3366: 3362: 3358: 3355: 3334: 3331: 3309: 3306: 3302: 3282: 3278: 3274: 3271: 3268: 3247: 3244: 3240: 3219: 3216: 3212: 3191: 3188: 3179: 3175: 3161: 3114: 3111: 3108: 3100: 3084: 3064: 3061: 3041: 3038: 3018: 2998: 2978: 2958: 2950: 2946: 2943: 2941: 2925: 2916: 2905: 2896: 2868: 2861: 2855: 2851: 2846: 2841: 2838: 2834: 2828: 2825: 2821: 2818: 2814: 2810: 2803: 2800: 2797: 2794: 2772: 2762: 2759: 2755: 2751: 2748: 2744: 2741: 2737: 2728: 2725: 2722: 2719: 2712: 2708: 2694: 2687: 2681: 2677: 2672: 2668: 2665: 2659: 2656: 2653: 2650: 2646: 2639: 2636: 2632: 2629: 2625: 2600: 2591: 2588: 2584: 2581: 2578: 2575: 2566: 2563: 2559: 2556: 2552: 2545: 2541: 2540: 2539: 2522: 2515: 2509: 2505: 2500: 2495: 2492: 2482: 2479: 2475: 2472: 2464: 2457: 2454: 2451: 2426: 2416: 2413: 2405: 2402: 2398: 2395: 2382: 2379: 2376: 2366: 2362: 2348: 2341: 2335: 2331: 2326: 2322: 2313: 2310: 2307: 2300: 2293: 2290: 2286: 2283: 2254: 2245: 2238: 2235: 2232: 2220: 2217: 2213: 2210: 2199: 2195: 2194: 2193: 2176: 2168: 2164: 2160: 2155: 2151: 2147: 2144: 2132: 2124: 2120: 2116: 2111: 2107: 2103: 2100: 2090: 2086: 2072: 2065: 2061: 2057: 2053: 2049: 2045: 2041: 2037: 2033: 2030: 2020: 2013: 2009: 2005: 2001: 1997: 1993: 1989: 1985: 1981: 1978: 1967: 1963: 1962: 1961: 1951: 1941: 1930: 1923: 1909: 1899: 1888: 1881: 1874: 1872: 1868: 1863: 1862:for details. 1861: 1857: 1853: 1849: 1845: 1840: 1838: 1833: 1816: 1812: 1809: 1805: 1803: 1798: 1790: 1787: 1783: 1781: 1776: 1765: 1762: 1758: 1755: 1751: 1748: 1741: 1738: 1736: 1731: 1719: 1715: 1710: 1705: 1702: 1698: 1695: 1691: 1688: 1681: 1678: 1676: 1671: 1658: 1656: 1652: 1648: 1644: 1640: 1636: 1632: 1628: 1624: 1617: 1613: 1609: 1605: 1601: 1580: 1576: 1571: 1565: 1561: 1557: 1554: 1550: 1545: 1542: 1518: 1515: 1512: 1510: 1504: 1501: 1493: 1490: 1488: 1482: 1479: 1468: 1465: 1462: 1459: 1450: 1447: 1445: 1439: 1436: 1423: 1419: 1414: 1410: 1407: 1404: 1401: 1392: 1389: 1387: 1381: 1378: 1365: 1358: 1354: 1350: 1341: 1334: 1327: 1320: 1314: 1308: 1304: 1300: 1296: 1290: 1280: 1278: 1274: 1268: 1266: 1260: 1255: 1251: 1240: 1230: 1227: 1217: 1215: 1210: 1206: 1202: 1198: 1194: 1190: 1180: 1178: 1171: 1167: 1164:. Therefore, 1163: 1151: 1142: 1135: 1126: 1123: 1119: 1115: 1108: 1104: 1098: 1094: 1090: 1086: 1082: 1081: 1080: 1075: 1068: 1064: 1060: 1056: 1052: 1042: 1040: 1035: 1033: 1027: 1025: 1021: 1016: 1013: 1006: 1002: 998: 994: 989: 979: 969: 966: 964: 960: 956: 951: 949: 942: 938: 934: 930: 925: 921: 919: 915: 911: 906: 904: 900: 892: 888: 884: 881: 877: 876: 875: 870: 864: 862: 855: 845: 843: 839: 835: 831: 827: 823: 821: 817: 813: 809: 805: 801: 797: 793: 788: 786: 782: 781:accelerations 778: 774: 770: 766: 761: 759: 755: 751: 747: 742: 740: 736: 732: 727: 725: 709: 687: 683: 679: 676: 673: 665: 661: 657: 653: 649: 645: 641: 637: 633: 629: 625: 624:time dilation 621: 617: 612: 610: 602: 598: 594: 590: 586: 582: 578: 572: 562: 560: 556: 548: 544: 540: 537: 533: 529: 525: 521: 517: 513: 512: 511: 509: 505: 501: 497: 493: 489: 485: 473: 468: 466: 461: 459: 454: 453: 451: 450: 444: 434: 431: 426: 420: 419: 418: 417: 409: 406: 404: 401: 399: 396: 394: 391: 389: 386: 384: 381: 379: 376: 374: 371: 369: 366: 364: 361: 359: 356: 354: 351: 349: 346: 344: 341: 339: 336: 334: 331: 329: 326: 325: 314: 313: 306: 303: 301: 298: 296: 293: 292: 284: 282: 279: 278: 273: 272: 265: 264:Four-momentum 262: 260: 257: 255: 252: 251: 246: 239: 238: 231: 228: 226: 223: 221: 218: 216: 213: 212: 207: 200: 199: 192: 189: 187: 184: 182: 179: 177: 174: 172: 169: 167: 164: 162: 159: 157: 154: 152: 149: 147: 146:Time dilation 144: 143: 133: 132: 125: 122: 120: 117: 115: 112: 110: 107: 105: 102: 101: 91: 90: 85: 82: 80: 77: 75: 72: 71: 70: 69: 65: 61: 60: 57: 54: 53: 47: 43: 41: 36: 32: 19: 36060:Hermann Weyl 35864:Vector space 35849:Pseudotensor 35814:Fiber bundle 35767:abstractions 35662:Mixed tensor 35647:Tensor field 35454: 35366:Alternatives 35018:Consequences 34919: 34862: 34556:Kaluza–Klein 34308:Introduction 34234:Twin paradox 34103: 34071: 34019: 33987:Einsteinhaus 33862:Pauline Koch 33840:(first wife) 33838:Mileva Marić 33817: 33810: 33803: 33796: 33789: 33782: 33775: 33768: 33701: 33693: 33685: 33677: 33669: 33661: 33653: 33645: 33637: 33629: 33621: 33585: 33577: 33569: 33561: 33546: 33400: 33259:Astrophysics 33191: 33073:Experimental 32865: 32843: 32839: 32807: 32806:Greg Egan's 32749: 32744: 32728: 32690: 32686:Scholarpedia 32684: 32671:Scholarpedia 32644: 32640: 32607: 32603: 32578: 32574: 32551: 32536: 32513: 32493: 32473: 32464: 32446: 32414: 32369: 32359: 32345: 32318: 32304: 32295: 32262: 32255: 32240: 32232: 32217: 32201: 32186: 32181: 32161: 32154: 32135: 32129: 32110: 32104: 32071: 32067: 32057: 32045:. Retrieved 32040: 32030: 31982: 31978: 31970: 31961: 31943: 31936: 31912: 31905: 31884: 31828: 31824: 31818: 31806:. Retrieved 31797: 31765: 31735: 31707: 31701: 31692: 31686: 31661: 31657: 31651: 31639:. Retrieved 31634: 31602: 31590:. Retrieved 31558: 31552: 31533: 31529: 31525: 31518: 31498: 31488: 31476:. Retrieved 31472:the original 31467: 31462:Keel, Bill. 31457: 31445:. Retrieved 31441:the original 31436: 31426: 31414:. Retrieved 31409: 31399: 31380: 31374: 31331: 31327: 31321: 31296: 31292: 31286: 31274:. Retrieved 31270:the original 31265: 31260:Kraus, Ute. 31255: 31243:. Retrieved 31236:the original 31231: 31218: 31206:. Retrieved 31195: 31164: 31143: 31126: 31120: 31108:. Retrieved 31096: 31092: 31082: 31066:. Springer. 31063: 31050: 31038:. Retrieved 31026: 31022: 31012: 31000:. Retrieved 30989: 30946: 30942: 30936: 30916: 30909: 30889: 30882: 30870:. Retrieved 30856: 30846: 30834:. Retrieved 30827:the original 30822: 30809: 30785: 30778: 30747: 30740: 30728:. Retrieved 30723: 30713: 30694: 30688: 30676:. Retrieved 30665: 30655: 30648: 30623: 30619: 30613: 30601:. Retrieved 30596: 30592: 30582: 30558: 30551: 30532: 30511:. Retrieved 30496: 30489: 30465: 30458: 30434: 30427: 30403: 30396: 30384:. Retrieved 30379: 30375: 30347:. Retrieved 30327: 30323: 30313: 30294: 30259: 30252: 30207: 30203: 30193: 30182: 30148: 30125:. Retrieved 30113: 30109: 30096: 30077: 30071: 30052: 30023: 30016: 29995: 29987: 29971: 29966: 29947: 29941: 29898: 29894: 29888: 29872: 29821: 29816: 29797: 29791: 29757: 29750: 29741: 29698: 29694: 29667: 29659: 29648:. Retrieved 29644: 29635: 29624:. Retrieved 29622:. 2006-12-01 29619: 29610: 29590: 29554: 29515: 29508: 29485: 29478: 29466: 29455: 29435: 29428: 29417:. Retrieved 29413: 29403: 29384: 29352: 29342: 29322: 29287: 29236: 29232: 29219: 29184: 29180: 29173:P.A.M. Dirac 29129: 29125: 29098:. Retrieved 29084: 29075: 29070: 29062: 29058: 29012: 29006: 29000: 28975: 28969: 28963: 28946: 28942: 28932: 28915: 28911: 28901: 28868: 28864: 28858: 28825: 28821: 28815: 28774: 28770: 28764: 28754: 28743: 28731:. Retrieved 28719: 28715: 28687:. Retrieved 28675: 28671: 28658: 28641: 28628:, Springer, 28625: 28620: 28601: 28595: 28585: 28576: 28555: 28546: 28538: 28533: 28509: 28504: 28463: 28458:resemblance. 28434: 28424: 28416: 28411: 28394: 28377: 28373: 28369: 28364: 28352: 28347: 28342: 28338: 28330: 28321: 28313: 28309: 28298: 28294: 28290: 28283: 28277: 28255: 28238: 28230: 28225: 28187: 28177: 27994:Mileva Marić 27979:Max von Laue 27949: 27945: 27941: 27939: 27934: 27930: 27926: 27922: 27918: 27912: 27827: 27825: 27818: 27814: 27766: 27763: 27665: 27543: 27474: 27470: 27381: 27379: 27367: 27364: 27356: 27353: 27342: 27339: 27330: 27327: 27325:line element 27119: 27042: 26885: 26822: 26816: 26695: 26691: 26685: 26574: 26395: 26382: 26357:tensor field 26350: 26331: 25973: 25970: 25811: 25807: 25805: 25272:scalar field 25265: 25260: 25256: 25252: 25250: 25184: 25175: 24862: 24528: 24423: 24278: 24274: 24113: 24109: 24105: 24101: 24096: 24094: 24069:four-vectors 24065: 23744: 23739: 23737: 23733: 23712: 23707: 23703: 23701: 23695: 23611: 23606: 23604: 23363: 23360:4D spacetime 23353: 23054: 23043:3D spacetime 23036: 23030: 23021: 23017: 23013: 23008: 23004: 23001: 22996: 22991: 22987: 22973: 22966: 22957: 22953: 22951: 22942: 22938: 22917: 22910: 22903: 22896: 22892: 22889: 22764: 22755: 22748: 22741: 22737: 22734: 22611: 22608:differential 22605: 22594: 22584: 22580: 22569: 22565: 22555: 22546:line element 22460: 22429: 22421: 22410: 22393: 22379: 22353: 22350: 22343: 22332:antiparticle 22325: 22310: 22302:alkali metal 22291: 22272: 22255: 22254: 22235: 22230: 22222: 22218: 22208: 22199: 22195: 22189: 22169: 22165: 22163: 22158: 22155: 21592: 21525: 21518: 21506: 21496: 21479: 21321: 20969: 20870: 20857: 20851: 20847: 20843: 20839: 20835: 20831: 20824: 20815: 20810: 20806: 20799: 20795: 20780: 20767: 20764: 20760: 20756: 20747:Acceleration 20733: 20725: 20701: 20472: 20240: 20015: 19971: 19665: 19661: 19607: 19531: 19523: 19517: 19513: 19358: 19350: 19298: 19006: 19002: 18998: 18942: 18938: 18934: 18853: 18848: 18844: 18840: 18836: 18834: 18829: 18821: 18817: 18810: 18806: 18804: 18551: 18542: 18489: 18481: 18393: 18383: 18379: 18369: 18356: 18342: 18328: 18321: 18160: 18157: 17990: 17985: 17981: 17979: 17974: 17972: 17726: 17724: 17673: 17429: 17302: 17246: 17242: 17220: 17067:is given by 17064: 17055: 17048: 17040: 17036: 17032: 16960: 16955: 16951: 16947: 16933: 16926: 16921: 16917: 16913: 16903: 16895: 16891: 16858: 16854: 16848: 16844: 16836: 16832: 16812: 16808: 16802: 16798: 16790: 16786: 16742: 16613: 16542: 16531: 16328: 16144: 15968: 15957: 15878: 15776: 15655: 15649: 15603: 15601: 15590: 15469: 15458: 15320: 15309: 15218: 15214: 15194: 15190: 15186: 15182: 15178: 15174: 15167: 15163: 15157: 15151: 15146: 15142: 15138: 15135: 15115: 15106: 15092: 15088: 15074: 15070: 15066: 15062: 15058: 15053: 15049: 15045: 15041: 15037: 15033: 15029: 15025: 14933: 14928: 14913: 14908: 14904: 14899: 14897: 14892: 14888: 14884: 14880: 14861: 14859: 14848: 14756: 14745: 14485: 14474: 14322: 13908: 13877: 13850: 13848: 13789: 13737: 13735: 13596: 13592: 13550: 13542: 13538: 13495: 13488: 13483: 13479: 13472: 13468: 13463: 13459: 13455: 13449: 13445: 13441: 13437: 13429: 13425: 13417: 13411: 13407: 13400: 13395: 13391: 13389: 13382: 13361: 13358: 13325: 13320: 13314: 13311: 13293: 13289: 13280: 13278: 13265: 13261: 13257: 13253: 13251: 13244: 13207: 13203: 13196: 13178: 13176: 12973:time dilated 12970: 12865: 12861: 12816: 12814: 12806: 12679: 12678: 12583: 12582: 12480: 12475: 12463: 12451: 12116: 12114: 12105: 12079: 11904: 11900: 11898: 11894: 11872: 11842: 11840: 11828: 11826: 11821: 11721: 11688: 11648: 11644: 11637: 11603: 11599: 11594: 11592: 11586:Figure 4–6. 11563: 11536: 11489: 11484: 11462: 11415: 11411: 11356: 11354: 11348: 11330: 11322: 11307: 11298: 11292: 11286: 11278: 11274: 11266: 11265:| < 11261: 11252: 11245: 11239: 11235: 11223: 11215: 11211: 11207: 11091: 11085: 11079: 11078: 11067: 10771: 10492: 10481: 10268: 10085: 10074: 9936: 9925: 9579: 9573: 9567: 9509: 9498: 9398: 9326: 9318: 9262: 9255: 9251: 9249: 9233: 9229: 9224: 9219: 9217: 9212: 9208: 9204: 9197: 9190: 9188: 9082: 9078: 9072: 9063: 9054: 9047: 9040: 9036: 9032: 9028: 9022: 9015: 8897: 8896: 7595: 7372:the traveler 7364:stay-at-home 7191: 7187:Twin paradox 7183: 7171: 7167:twin paradox 7164: 7161:Twin paradox 7155:Twin paradox 7148: 7145: 7121: 7111: 6868: 6861: 6787: 6783: 6777: 6767: 6753: 6751: 6747: 6729: 6725: 6721: 6714: 6707: 6705: 6613: 6607: 6601: 6595: 6592: 6546: 6538: 6532: 6529:(satisfying 6526: 6517: 6514:(satisfying 6508: 6502: 6499:(satisfying 6496: 6494: 6409: 6407: 6399: 6389: 6385: 6380: 6376: 6371: 6367: 6363: 6359: 6228: 6224: 6193:would be if 6100:the sign of 6030: 6026: 5726: 5722: 5716: 5446: 5442: 5407: 5374: 4508: 4432: 4420:translations 4403: 4399: 4365:The form of 4364: 4361: 4206: 4204: 4155: 4138: 4131:Twin paradox 4115: 4109: 4030: 4026: 4025: 3495: 3494: 3177: 3176: 3098: 2948: 2947: 2944: 2937: 2537: 2191: 1949: 1939: 1928: 1921: 1907: 1897: 1886: 1879: 1875: 1864: 1855: 1851: 1847: 1843: 1841: 1836: 1834: 1659: 1642: 1638: 1634: 1630: 1626: 1622: 1615: 1611: 1603: 1356: 1352: 1348: 1339: 1332: 1325: 1318: 1312: 1306: 1302: 1298: 1294: 1286: 1269: 1262: 1257: 1253: 1223: 1213: 1186: 1176: 1169: 1165: 1161: 1158: 1149: 1140: 1133: 1121: 1117: 1113: 1106: 1096: 1092: 1088: 1084: 1073: 1066: 1065:, and frame 1062: 1054: 1048: 1038: 1036: 1031: 1028: 1017: 1010: 1004: 996: 992: 967: 953: 947: 940: 936: 932: 928: 927: 922: 907: 896: 890: 886: 872: 860: 857: 853: 824: 800:tidal forces 789: 765:special case 762: 743: 728: 613: 604: 574: 552: 532:acceleration 491: 487: 481: 138:Consequences 84:Formulations 55: 45: 39: 36123:Outer space 36000:Élie Cartan 35948:Spin tensor 35922:Weyl tensor 35880:Mathematics 35844:Multivector 35635:definitions 35533:Engineering 35472:Mathematics 35126:Proper mass 35121:Proper time 34982:Foundations 34645:Kerr–Newman 34616:Spherical: 34485:Other tests 34428:Singularity 34360:Formulation 34322:Fundamental 34176:Formulation 34157:Proper time 34118:Fundamental 33992:Einsteinium 33760:Books about 33706:(2023 film) 33703:Oppenheimer 33682:(2003 play) 33674:(1994 film) 33666:(1993 play) 33658:(1988 film) 33650:(1985 film) 33493:EPR paradox 33162:Statistical 33078:Theoretical 33055:Engineering 32808:Foundations 32693:(2): 8520. 32346:Gravitation 32218:Gravitation 32111:Gravitation 31099:: 105–111. 31040:20 November 31029:: 103–107. 31002:20 November 30724:Physics FAQ 30127:15 November 29645:www.aps.org 28949:: 398–408. 28918:: 445–451. 28733:15 November 28722:: 809–831. 28689:15 November 28678:: 669–678. 28444:Lie algebra 28442:inside the 28314:True states 28103:Mathematics 28077:Shape waves 27836:four-vector 27350:proper time 24564:proper time 24426:four-vector 24126:four-vector 22587:denote the 19337:orthogonal 18394:A 4-tuple, 18339:Four-vector 17045:into point 16950:) and cosh( 16910:unit circle 16783:unit circle 15120:radiation. 14900:contributes 13420:four-vector 13330:in various 13258:observation 13254:measurement 12471:George Airy 11838:in vacuum. 11827:Therefore, 11641:world lines 11559:curvilinear 7370:Measured by 5717:proper time 4424:reflections 3180:. Draw the 2951:. Draw the 1362:. Then the 918:homogeneity 783:as well as 259:Proper mass 254:Proper time 96:Foundations 36157:Categories 35829:Linear map 35697:Operations 35131:4-momentum 35103:Light cone 35093:World line 34797:Zel'dovich 34705:Scientists 34684:Alcubierre 34491:of Mercury 34489:precession 34418:Black hole 34301:Background 34293:relativity 34262:World line 34257:Light cone 34082:Background 34074:relativity 34064:Relativity 33997:Max Talmey 33894:(grandson) 33852:(daughter) 33612:In popular 33279:Geophysics 33269:Biophysics 33113:Analytical 33066:Approaches 32987:lightspeed 32338:Kip Thorne 31775:0521277035 31494:Max Jammer 31299:(5): 283. 31187:2010938208 31110:11 October 30730:31 October 30678:31 October 29650:2024-03-22 29626:2024-03-22 29419:2008-09-17 29265:References 29100:22 January 28122:Philosophy 28000:Relativity 27969:Max Planck 27832:four-force 27477:produces: 24079:under the 23267:or simply 22589:worldlines 22544:See also: 22313:Paul Dirac 22292:The early 22275:relativity 22247:calculus. 21527:parameter 21507:approaches 20850:sees that 20842:says that 20712:3-momentum 20008:4-velocity 19965:4-velocity 19662:dt, dx, dy 19520:, this is 19335:3-vectors, 19299:orthogonal 18991:is also a 18554:-direction 18357:manifestly 18163:-direction 17993:-direction 16946:with sinh( 16916:) and cos( 16893:formulas. 15077:times its 14923:See also: 13432:, 0, 0, 0) 13367:kinematics 13119: and 11588:Light cone 11572:See also: 11375:See also: 9244:See also: 9010:See also: 7159:See also: 6790:Fig. 4-3A. 6596:Suppose a 6587:See also: 6394:light cone 6344:See also: 5901:for which 5881:less than 5723:Δs < 0: 5607:for which 5587:less than 5443:Δs > 0: 4400:difference 4398:being the 4129:See also: 4031:worldlines 3099:worldlines 3031:(actually 1311:in system 1287:Define an 1148:for frame 587:, and the 526:(that is, 348:FitzGerald 333:Sommerfeld 285:Precursors 230:Light cone 225:World line 36099:Astronomy 35968:EM tensor 35804:Dimension 35755:Transpose 35080:Spacetime 34927:Overviews 34767:Robertson 34752:Friedmann 34747:Eddington 34737:de Sitter 34571:Solutions 34449:detectors 34444:astronomy 34411:Phenomena 34346:Geodesics 34249:Spacetime 34192:Phenomena 33229:Molecular 33130:Acoustics 33123:Continuum 33118:Celestial 33108:Newtonian 33095:Classical 33038:Divisions 32993:Animation 32328:Textbooks 32220:, pg 51, 32024:this page 31838:0906.1919 31641:2 January 31592:30 August 31366:118498333 31341:1410.4583 30981:119528074 30564:CRC Press 30382:: 113–142 30349:22 August 29908:0907.0902 29777:cite book 29733:119220040 29708:0908.1545 29233:Phys. Rev 29156:123018629 29037:123023118 28893:121124132 28850:119855914 28828:(1): 72. 28468:measured. 28404:Amsterdam 28141:Paradoxes 28130:actualism 27897:ν 27880:τ 27870:ν 27851:ν 27728:− 27690:− 27611:− 27603:ν 27593:μ 27583:ν 27580:μ 27576:η 27516:ν 27506:μ 27496:ν 27493:μ 27489:η 27443:− 27435:μ 27425:ν 27415:μ 27412:ν 27408:η 27222:− 27214:ν 27200:μ 27186:ν 27183:μ 27179:η 27170:μ 27156:μ 27104:ν 27091:ν 27078:μ 27073:Λ 27060:μ 27009:α 27000:γ 26990:γ 26981:β 26971:β 26962:α 26949:α 26940:β 26930:β 26921:α 26908:α 26899:α 26868:α 26858:α 26793:β 26783:β 26780:α 26776:η 26770:α 26757:β 26747:β 26744:α 26740:η 26734:α 26721:α 26711:α 26669:β 26656:ν 26651:Λ 26645:α 26632:μ 26627:Λ 26617:ν 26609:μ 26604:η 26595:β 26592:α 26588:η 26554:β 26551:α 26547:η 26434:− 26418:β 26415:α 26411:η 26315:ψ 26302:χ 26297:Λ 26274:ψ 26261:χ 26256:Λ 26233:ρ 26230:⋯ 26227:ν 26224:μ 26219:ϕ 26216:⋯ 26213:υ 26210:σ 26200:ϕ 26187:κ 26182:Λ 26178:⋯ 26173:υ 26160:ι 26155:Λ 26149:σ 26136:θ 26131:Λ 26125:ρ 26112:ζ 26107:Λ 26103:⋯ 26098:ν 26085:β 26080:Λ 26074:μ 26061:α 26056:Λ 26043:ζ 26039:⋯ 26032:β 26024:α 26014:κ 26010:⋯ 26003:ι 25995:θ 25954:ν 25941:μ 25936:Λ 25913:ν 25900:μ 25895:Λ 25869:ν 25859:ν 25846:μ 25841:Λ 25828:μ 25779:μ 25771:∂ 25767:∂ 25762:≡ 25757:μ 25753:∂ 25745:ϕ 25740:ν 25736:∂ 25727:ν 25714:μ 25709:Λ 25699:ϕ 25690:μ 25685:∂ 25596:γ 25591:γ 25588:β 25568:γ 25565:β 25557:γ 25531:∂ 25526:ϕ 25523:∂ 25507:∂ 25502:ϕ 25499:∂ 25483:∂ 25478:ϕ 25475:∂ 25459:∂ 25454:ϕ 25451:∂ 25401:∂ 25396:ϕ 25393:∂ 25372:∂ 25367:ϕ 25364:∂ 25343:∂ 25338:ϕ 25335:∂ 25314:∂ 25309:ϕ 25306:∂ 25231:τ 25221:μ 25202:μ 25009:γ 24998:μ 24982:μ 24933:γ 24880:γ 24753:− 24729:γ 24680:γ 24617:γ 24608:τ 24598:μ 24579:μ 24545:μ 24513:ν 24503:ν 24490:μ 24485:Λ 24472:μ 24440:ν 24403:ν 24390:μ 24385:Λ 24364:ν 24338:ν 24328:ν 24315:μ 24310:Λ 24297:μ 24141:ν 24021:γ 24018:β 24015:− 24009:γ 23999:β 23996:γ 23993:− 23984:γ 23865:γ 23860:γ 23857:β 23854:− 23837:γ 23834:β 23831:− 23826:γ 23607:c dt 23454:− 23354:c dt 23232:− 23161:geodesics 23118:− 22962:imaginary 22791:− 22646:⋅ 22597:spacetime 22311:In 1928, 22093:γ 22061:β 22000:γ 21997:β 21875:− 21823:β 21819:− 21800:γ 21764:− 21713:β 21667:− 21605:β 21578:γ 21535:β 21452:in frame 21300:γ 21281:− 21235:− 21211:γ 21173:− 21155:γ 20899:σ 20820:= γL 20741:4-vectors 20737:4-vectors 20729:4-vectors 20708:3-vectors 20704:4-vectors 20677:→ 20640:γ 20599:→ 20556:γ 20531:τ 20514:≡ 20487:τ 20434:→ 20425:γ 20399:γ 20385:γ 20357:→ 20348:γ 20342:γ 20333:τ 20299:τ 20282:≡ 20255:τ 20207:→ 20166:→ 20056:→ 20030:≡ 19988:τ 19909:→ 19900:γ 19894:γ 19787:γ 19762:γ 19698:τ 19681:≡ 19648:τ 19625:τ 19552:≡ 19469:− 19429:− 19377:⋅ 19343:4-vectors 19339:4-vectors 19312:⋅ 19269:→ 19260:⋅ 19254:→ 19245:− 19206:≡ 19183:− 19160:− 19137:− 19098:≡ 19092:⋅ 19077:3-vectors 19060:→ 19031:→ 19011:4-vectors 18947:4-vectors 18905:Δ 18896:Δ 18887:Δ 18878:Δ 18686:− 18668:γ 18609:− 18591:γ 18490:If using 18376:4‑vectors 18372:4‑vectors 18349:4‑vectors 18333:4‑vectors 18257:ϕ 18254:⁡ 18246:ϕ 18243:⁡ 18233:ϕ 18230:⁡ 18222:ϕ 18219:⁡ 18103:ϕ 18100:⁡ 18092:ϕ 18089:⁡ 18083:− 18076:ϕ 18073:⁡ 18067:− 18062:ϕ 18059:⁡ 17955:ϕ 17952:⁡ 17924:ϕ 17921:⁡ 17908:− 17900:ϕ 17897:⁡ 17876:β 17872:− 17865:β 17857:β 17854:γ 17830:ϕ 17827:⁡ 17799:ϕ 17796:⁡ 17783:− 17759:β 17755:− 17740:γ 17702:ϕ 17689:ϕ 17682:ϕ 17647:ϕ 17634:ϕ 17627:⁡ 17593:ϕ 17589:⁡ 17577:ϕ 17573:⁡ 17553:ϕ 17549:⁡ 17534:ϕ 17530:⁡ 17493:β 17483:β 17465:β 17452:β 17442:β 17407:ϕ 17404:− 17391:ϕ 17379:ϕ 17376:− 17368:− 17363:ϕ 17346:ϕ 17343:⁡ 17335:ϕ 17332:⁡ 17320:ϕ 17317:⁡ 17275:≡ 17272:ϕ 17269:⁡ 17263:≡ 17260:β 17229:β 17169:θ 17166:⁡ 17158:θ 17155:⁡ 17145:θ 17142:⁡ 17136:− 17131:θ 17128:⁡ 17063:by angle 17009:⁡ 16998:⁡ 16983:⁡ 16954:), where 16912:with sin( 16702:− 16688:β 16672:γ 16650:β 16640:θ 16636:⁡ 16591:β 16578:β 16557:θ 16551:ϕ 16494:θ 16491:⁡ 16472:θ 16469:⁡ 16434:γ 16421:θ 16417:⁡ 16410:θ 16406:⁡ 16394:− 16385:γ 16368:θ 16362:ϕ 16355:⁡ 16280:γ 16268:θ 16262:ϕ 16256:⁡ 16239:θ 16235:⁡ 16211:β 16201:θ 16197:⁡ 16185:β 16169:β 16096:γ 16084:θ 16078:ϕ 16072:⁡ 16055:ϕ 16051:⁡ 16027:β 16017:ϕ 16013:⁡ 16001:β 15985:β 15935:γ 15922:γ 15903:γ 15861:ϕ 15857:⁡ 15845:γ 15835:β 15827:θ 15823:⁡ 15811:γ 15801:β 15759:ϕ 15755:⁡ 15743:γ 15733:β 15725:θ 15721:⁡ 15709:γ 15699:β 15686:γ 15676:β 15636:β 15555:γ 15529:γ 15487:γ 15439:→ 15414:γ 15404:→ 15379:γ 15363:→ 15338:γ 15280:γ 15261:→ 15249:γ 15240:→ 15107:Inelastic 15079:half-life 15052:and 0.946 14782:− 14721:− 14690:− 14657:− 14610:− 14570:− 14497:− 14450:− 14419:− 14383:− 14367:− 14351:− 14276:− 14220:− 14211:ϕ 14207:⁡ 14129:− 14120:ϕ 14116:⁡ 14093:− 13748:ϕ 13693:− 13684:ϕ 13680:⁡ 13657:− 13579:ϕ 13403:heuristic 13217:γ 13148:β 13144:− 13129:γ 13093:β 13052:β 13041:β 13038:− 13010:β 13007:− 12996:γ 12897:− 12771:θ 12766:⁡ 12725:− 12699:θ 12693:⁡ 12660:θ 12657:⁡ 12625:γ 12620:θ 12617:⁡ 12601:θ 12597:⁡ 12567:θ 12564:⁡ 12519:θ 12516:⁡ 12500:θ 12496:⁡ 12392:− 12378:± 12349:≈ 12326:∓ 12300:± 12263:≈ 12243:± 12232:± 12167:± 12156:± 12137:± 12038:− 12024:± 12006:± 11979:± 11883:Fresnel's 11873:In 1850, 11832:causality 11667:− 11551:gyroscope 11361:formalism 11338:| ≪ 11092:resultant 10995:γ 10906:γ 10713:− 10702:γ 10627:− 10616:γ 10546:− 10535:− 9965:− 9879:− 9868:− 9811:− 9800:− 9733:− 9719:γ 9701:− 9689:γ 9317:in frame 9225:different 9154:Δ 9123:γ 9116:Δ 9099:Δ 8933:γ 8831:γ 8771:β 8767:− 8711:β 8644:β 8641:− 8633:β 8592:β 8588:− 8479:β 8475:− 8411:β 8408:− 8400:β 8380:β 8377:− 8257:γ 8199:− 8144:β 8141:− 8076:β 8065:β 8062:− 8024:β 8020:− 7911:β 7907:− 7842:β 7831:β 7828:− 7811:β 7688:γ 7569:γ 7456:γ 7332:β 7329:− 7312:β 7258:β 7238:β 7235:− 7067:− 7009:− 6936:− 6921:± 6676:Δ 6647:Δ 6643:γ 6629:Δ 6460:Δ 6450:− 6444:Δ 6436:γ 6422:Δ 6255:Δ 6244:Δ 6229:lightlike 6178:Δ 6167:Δ 6108:Δ 6082:Δ 6071:Δ 5998:Δ 5995:− 5970:Δ 5959:Δ 5909:Δ 5825:Δ 5819:− 5813:Δ 5804:γ 5790:Δ 5753:Δ 5742:Δ 5727:spacelike 5689:Δ 5666:Δ 5655:Δ 5615:Δ 5546:Δ 5539:− 5533:Δ 5524:γ 5510:Δ 5473:Δ 5462:Δ 5416:Δ 5383:Δ 5341:Δ 5338:− 5319:Δ 5265:− 5238:Δ 5229:γ 5225:− 5191:− 5164:Δ 5145:γ 5101:Δ 5078:γ 5055:Δ 5046:γ 5042:− 5023:Δ 5004:γ 5000:− 4981:Δ 4962:γ 4929:Δ 4905:Δ 4894:Δ 4866:Δ 4851:γ 4847:− 4809:Δ 4781:Δ 4770:Δ 4742:Δ 4718:γ 4686:Δ 4669:Δ 4654:γ 4650:− 4614:Δ 4593:Δ 4578:γ 4547:Δ 4544:− 4531:Δ 4485:Δ 4482:− 4469:Δ 4441:Δ 4428:rotations 4373:Δ 4335:Δ 4319:Δ 4303:Δ 4297:− 4284:Δ 4249:Δ 4218:Δ 4187:Δ 4164:Δ 4009:→ 4006:β 3952:β 3948:− 3925:β 3862:γ 3853:γ 3850:β 3776:γ 3773:β 3764:γ 3707:γ 3704:β 3698:γ 3595:γ 3589:γ 3586:β 3407:β 3381:β 3375:⁡ 3367:− 3356:α 3295:Both the 2804:γ 2729:γ 2657:− 2640:γ 2582:− 2567:γ 2489:Δ 2469:Δ 2458:γ 2449:Δ 2410:Δ 2392:Δ 2383:γ 2374:Δ 2320:Δ 2311:− 2305:Δ 2294:γ 2280:Δ 2243:Δ 2236:− 2230:Δ 2221:γ 2207:Δ 2161:− 2142:Δ 2117:− 2098:Δ 2054:− 2027:Δ 2002:− 1975:Δ 1742:γ 1682:γ 1558:− 1543:γ 1463:− 1451:γ 1405:− 1393:γ 1273:Spacetime 1024:spacetime 804:free fall 520:invariant 398:Ehrenfest 368:Minkowski 338:Michelson 206:Spacetime 35834:Manifold 35819:Geodesic 35577:Notation 35113:Dynamics 34880:Category 34757:Lemaître 34722:Einstein 34712:Poincaré 34672:Others: 34656:Taub–NUT 34622:interior 34544:theories 34542:Advanced 34509:redshift 34324:concepts 34142:Rapidity 34120:concepts 34021:Category 33942:Memorial 33888:(cousin) 33876:(sister) 33870:(father) 33864:(mother) 33762:Einstein 33597:" (1949) 33558:" (1905) 32953:Archived 32934:Archived 32831:Archived 32812:Archived 32776:Archived 32679:(2011). 32632:26997100 32624:16011297 32535:(1992). 32412:(1989). 32368:(1991). 32243:, pg 4, 32189:, pg 5, 32096:17459755 32017:17286330 31863:18059490 31802:Archived 31610:Archived 31496:(1997). 31478:29 April 31447:20 April 31416:12 April 31276:16 April 31245:12 April 31208:12 April 31135:5947329M 30872:17 April 30593:Scientia 30344:20022495 30244:28163638 30210:(1): 1. 30116:: 1–22. 29933:20444859 29227:(1933). 29175:(1930). 28799:20208844 28502:(1905) " 28331:negative 28327:distance 28310:apparent 28268:and the 27989:Max Born 27956:See also 27081:′ 27063:′ 26659:′ 26635:′ 26620:′ 26612:′ 26402:matrix: 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27440:= 27431:U 27421:U 27404:= 27399:2 27394:U 27382:U 27368:X 27365:d 27357:X 27354:d 27343:X 27340:d 27338:− 27331:X 27328:d 27309:2 27305:) 27301:z 27298:d 27295:( 27292:+ 27287:2 27283:) 27279:y 27276:d 27273:( 27270:+ 27265:2 27261:) 27257:x 27254:d 27251:( 27248:+ 27243:2 27239:) 27235:t 27232:d 27228:c 27225:( 27219:= 27210:X 27206:d 27196:X 27192:d 27175:= 27166:X 27162:d 27152:X 27148:d 27145:= 27140:2 27135:X 27130:d 27100:X 27096:d 27069:= 27055:X 27051:d 27022:, 27014:= 26996:T 26977:T 26958:T 26954:, 26936:T 26917:T 26913:, 26895:T 26864:T 26854:T 26848:= 26844:| 26839:T 26834:| 26823:T 26819:Λ 26798:= 26789:S 26766:T 26762:= 26753:S 26730:T 26726:= 26717:S 26707:T 26696:S 26692:T 26600:= 26524:) 26518:1 26513:0 26508:0 26503:0 26496:0 26491:1 26486:0 26481:0 26474:0 26469:0 26464:1 26459:0 26452:0 26447:0 26442:0 26437:1 26428:( 26423:= 26396:η 26206:T 26052:= 25990:T 25874:, 25865:T 25837:= 25823:T 25787:. 25775:x 25748:) 25732:( 25705:= 25702:) 25681:( 25661:. 25656:) 25650:1 25645:0 25640:0 25635:0 25628:0 25623:1 25618:0 25613:0 25606:0 25601:0 25585:+ 25578:0 25573:0 25562:+ 25551:( 25544:) 25534:z 25510:y 25486:x 25462:t 25441:c 25438:1 25429:( 25424:= 25419:) 25405:z 25376:y 25347:x 25318:t 25296:c 25293:1 25284:( 25237:. 25228:d 25217:U 25213:d 25207:= 25198:A 25176:m 25162:. 25158:) 25153:p 25149:, 25144:c 25141:E 25135:( 25131:= 25127:) 25121:z 25117:p 25113:, 25108:y 25104:p 25100:, 25095:x 25091:p 25087:, 25082:c 25079:E 25073:( 25069:= 25066:) 25061:z 25057:v 25053:, 25048:y 25044:v 25040:, 25035:x 25031:v 25027:, 25024:c 25021:( 25018:) 25015:v 25012:( 25006:m 25003:= 24994:U 24990:m 24987:= 24978:P 24949:v 24945:m 24942:) 24939:v 24936:( 24930:= 24926:p 24900:2 24896:c 24892:m 24889:) 24886:v 24883:( 24877:= 24874:E 24849:. 24844:2 24839:z 24835:v 24831:+ 24826:2 24821:y 24817:v 24813:+ 24808:2 24803:x 24799:v 24795:= 24790:2 24786:v 24776:2 24772:c 24767:/ 24761:2 24757:v 24750:1 24746:1 24741:= 24738:) 24735:v 24732:( 24709:. 24706:) 24702:v 24698:, 24695:c 24692:( 24689:) 24686:v 24683:( 24677:= 24674:) 24669:z 24665:v 24661:, 24656:y 24652:v 24648:, 24643:x 24639:v 24635:, 24632:c 24629:( 24626:) 24623:v 24620:( 24614:= 24605:d 24594:X 24590:d 24584:= 24575:U 24550:, 24541:U 24509:T 24481:= 24467:T 24436:T 24334:X 24306:= 24292:X 24275:X 24260:. 24257:) 24253:x 24249:, 24246:t 24243:c 24240:( 24237:= 24234:) 24231:z 24228:, 24225:y 24222:, 24219:x 24216:, 24213:t 24210:c 24207:( 24204:= 24201:) 24196:3 24192:X 24188:, 24183:2 24179:X 24175:, 24170:1 24166:X 24162:, 24157:0 24153:X 24149:( 24146:= 24137:X 24116:) 24114:z 24110:y 24106:x 24102:x 24052:. 24047:) 24041:z 24034:y 24027:t 24024:c 24012:x 24002:x 23990:t 23987:c 23978:( 23973:= 23968:) 23962:z 23955:y 23948:x 23941:t 23938:c 23932:( 23925:) 23919:1 23914:0 23909:0 23904:0 23897:0 23892:1 23887:0 23882:0 23875:0 23870:0 23847:0 23842:0 23820:( 23815:= 23810:) 23800:z 23788:y 23776:x 23764:t 23760:c 23754:( 23740:x 23708:t 23704:t 23678:2 23673:3 23669:x 23665:+ 23660:2 23655:2 23651:x 23647:+ 23642:2 23637:1 23633:x 23627:= 23624:d 23583:. 23578:2 23574:t 23570:d 23565:2 23561:c 23557:= 23552:2 23547:3 23543:x 23539:d 23536:+ 23531:2 23526:2 23522:x 23518:d 23515:+ 23510:2 23505:1 23501:x 23497:d 23475:2 23471:t 23467:d 23462:2 23458:c 23449:2 23444:3 23440:x 23436:d 23433:+ 23428:2 23423:2 23419:x 23415:d 23412:+ 23407:2 23402:1 23398:x 23394:d 23391:= 23388:0 23385:= 23380:2 23376:s 23372:d 23340:, 23335:2 23331:t 23327:d 23322:2 23318:c 23314:= 23309:2 23304:2 23300:x 23296:d 23293:+ 23288:2 23283:1 23279:x 23275:d 23253:2 23249:t 23245:d 23240:2 23236:c 23227:2 23222:2 23218:x 23214:d 23211:+ 23206:2 23201:1 23197:x 23193:d 23190:= 23187:0 23184:= 23179:2 23175:s 23171:d 23144:, 23139:2 23135:t 23131:d 23126:2 23122:c 23113:2 23108:2 23104:x 23100:d 23097:+ 23092:2 23087:1 23083:x 23079:d 23076:= 23071:2 23067:s 23063:d 23037:c 23031:c 23022:c 23018:c 23014:c 23009:t 23005:X 22988:X 22970:0 22967:X 22958:X 22923:) 22921:3 22914:2 22907:1 22900:0 22893:X 22890:d 22875:, 22870:2 22865:3 22861:X 22857:d 22854:+ 22849:2 22844:2 22840:X 22836:d 22833:+ 22828:2 22823:1 22819:X 22815:d 22812:+ 22807:2 22802:0 22798:X 22794:d 22788:= 22783:2 22779:s 22775:d 22765:X 22761:) 22759:3 22752:2 22745:1 22738:x 22735:d 22720:, 22715:2 22710:3 22706:x 22702:d 22699:+ 22694:2 22689:2 22685:x 22681:d 22678:+ 22673:2 22668:1 22664:x 22660:d 22657:= 22653:x 22649:d 22642:x 22638:d 22635:= 22630:2 22626:s 22622:d 22585:c 22581:φ 22566:φ 22394:c 22200:B 22196:E 22140:, 22137:s 22133:/ 22127:2 22123:c 22119:m 22116:= 22113:F 22073:) 22070:t 22067:c 22064:( 22039:. 22036:) 22033:t 22030:c 22027:( 22024:d 22020:/ 22016:) 22011:2 22007:c 22003:m 21994:( 21991:d 21988:= 21985:) 21982:t 21979:c 21976:( 21973:d 21969:/ 21965:c 21962:p 21959:d 21940:= 21937:t 21934:d 21930:/ 21926:p 21923:d 21920:= 21917:F 21895:s 21891:/ 21883:2 21879:s 21870:2 21866:t 21860:2 21856:c 21834:= 21827:2 21816:1 21810:/ 21806:1 21803:= 21779:. 21772:2 21768:s 21759:2 21755:t 21749:2 21745:c 21738:/ 21734:t 21731:c 21728:= 21725:) 21722:t 21719:c 21716:( 21693:, 21688:2 21684:s 21680:= 21675:2 21671:x 21662:2 21658:t 21652:2 21648:c 21625:. 21622:x 21618:/ 21614:t 21611:c 21608:= 21558:t 21555:c 21461:S 21436:B 21428:A 21407:S 21383:B 21375:A 21354:B 21351:A 21331:S 21303:L 21297:= 21289:A 21285:x 21273:B 21269:x 21265:= 21254:L 21245:) 21241:t 21238:v 21232:L 21229:+ 21224:A 21220:x 21215:( 21208:= 21196:B 21192:x 21183:) 21179:t 21176:v 21168:A 21164:x 21159:( 21152:= 21140:A 21136:x 21104:S 21083:, 21080:S 21060:L 21057:+ 21052:A 21048:x 21044:= 21039:B 21035:x 21012:A 21008:x 20987:. 20980:S 20955:B 20952:A 20928:B 20920:A 20879:k 20852:A 20848:B 20844:B 20840:A 20836:B 20832:A 20816:L 20811:L 20807:L 20684:) 20674:f 20668:, 20662:t 20659:d 20654:E 20651:d 20644:( 20621:= 20617:) 20610:t 20607:d 20596:p 20590:d 20584:, 20578:t 20575:d 20570:E 20567:d 20560:( 20537:= 20528:d 20523:P 20520:d 20511:F 20490:. 20455:) 20448:t 20445:d 20440:) 20431:v 20422:( 20419:d 20413:, 20407:t 20404:d 20396:d 20389:( 20366:= 20363:) 20354:v 20345:, 20339:( 20330:d 20326:d 20305:= 20296:d 20291:V 20288:d 20279:A 20258:. 20219:. 20216:) 20213:c 20204:p 20198:, 20195:E 20192:( 20172:) 20163:p 20157:, 20154:E 20151:( 20121:) 20116:z 20112:p 20108:, 20103:y 20099:p 20095:, 20090:x 20086:p 20082:, 20079:c 20075:/ 20071:E 20068:( 20065:= 20062:) 20053:p 20047:, 20044:c 20040:/ 20036:E 20033:( 20027:P 19994:0 19991:= 19985:d 19915:) 19906:v 19897:, 19891:( 19872:= 19868:) 19861:t 19858:d 19853:z 19850:d 19844:, 19838:t 19835:d 19830:y 19827:d 19821:, 19815:t 19812:d 19807:x 19804:d 19798:, 19795:1 19791:( 19768:= 19758:/ 19754:t 19751:d 19746:) 19743:z 19740:d 19737:, 19734:y 19731:d 19728:, 19725:x 19722:d 19719:, 19716:t 19713:d 19710:( 19704:= 19695:d 19690:S 19687:d 19678:V 19645:d 19622:d 19591:) 19588:z 19585:d 19582:, 19579:y 19576:d 19573:, 19570:x 19567:d 19564:, 19561:t 19558:d 19555:( 19549:S 19546:d 19534:) 19530:( 19522:( 19492:. 19487:2 19483:c 19477:2 19473:p 19464:2 19460:E 19437:2 19433:x 19424:2 19420:t 19414:2 19410:c 19389:, 19386:0 19383:= 19380:A 19374:A 19353:. 19318:= 19315:B 19309:A 19294:. 19266:B 19251:A 19240:0 19236:B 19230:0 19226:A 19201:3 19197:B 19191:3 19187:A 19178:2 19174:B 19168:2 19164:A 19155:1 19151:B 19145:1 19141:A 19132:0 19128:B 19122:0 19118:A 19095:B 19089:A 19057:B 19028:A 19007:B 19003:A 18995:. 18977:B 18974:a 18971:+ 18968:A 18965:a 18962:= 18959:C 18943:B 18939:A 18911:) 18908:z 18902:, 18899:y 18893:, 18890:x 18884:, 18881:t 18875:c 18872:( 18841:t 18837:x 18825:1 18822:A 18818:x 18814:0 18811:A 18784:3 18780:A 18776:= 18764:3 18760:A 18750:2 18746:A 18742:= 18730:2 18726:A 18717:) 18711:0 18707:A 18703:) 18700:c 18696:/ 18692:v 18689:( 18681:1 18677:A 18672:( 18665:= 18653:1 18649:A 18640:) 18634:1 18630:A 18626:) 18623:c 18619:/ 18615:v 18612:( 18604:0 18600:A 18595:( 18588:= 18576:0 18572:A 18552:x 18543:A 18527:) 18524:z 18521:, 18518:y 18515:, 18512:x 18509:, 18506:t 18503:c 18500:( 18484:i 18482:A 18465:) 18459:3 18455:A 18451:, 18446:2 18442:A 18438:, 18433:1 18429:A 18425:, 18420:0 18416:A 18411:( 18407:= 18404:A 18307:. 18302:) 18292:x 18280:t 18276:c 18270:( 18263:) 18210:( 18205:= 18200:) 18194:x 18187:t 18184:c 18178:( 18161:x 18143:, 18138:) 18132:x 18125:t 18122:c 18116:( 18109:) 18050:( 18045:= 18040:) 18030:x 18018:t 18014:c 18008:( 17991:x 17958:. 17946:= 17916:2 17905:1 17888:= 17880:2 17869:1 17860:= 17833:, 17821:= 17791:2 17780:1 17776:1 17771:= 17763:2 17752:1 17748:1 17743:= 17711:. 17706:2 17698:+ 17693:1 17685:= 17659:, 17656:) 17651:2 17643:+ 17638:1 17630:( 17605:= 17597:2 17581:1 17567:+ 17564:1 17557:2 17543:+ 17538:1 17505:= 17497:2 17487:1 17479:+ 17476:1 17469:2 17461:+ 17456:1 17445:= 17415:. 17400:e 17396:+ 17387:e 17372:e 17359:e 17352:= 17323:= 17288:, 17283:c 17280:v 17206:. 17201:) 17195:y 17188:x 17182:( 17175:) 17119:( 17114:= 17109:) 17099:y 17087:x 17080:( 17061:) 17056:y 17049:x 17047:( 17043:) 17041:y 17037:x 17035:( 17018:. 17012:a 17001:a 16989:= 16986:a 16977:= 16956:a 16952:a 16948:a 16934:x 16927:x 16922:a 16918:a 16914:a 16859:x 16855:a 16851:) 16849:a 16845:a 16837:y 16833:x 16813:x 16809:a 16805:) 16803:a 16799:a 16791:y 16787:x 16747:) 16743:( 16721:2 16717:/ 16713:1 16709:} 16705:2 16697:2 16692:1 16684:3 16681:+ 16676:1 16667:/ 16663:2 16660:{ 16654:1 16644:= 16612:( 16600:, 16595:3 16587:= 16582:2 16554:= 16536:) 16532:( 16510:2 16506:/ 16502:1 16498:} 16486:2 16478:4 16475:+ 16464:2 16454:2 16450:) 16446:1 16443:+ 16438:1 16430:( 16427:{ 16400:) 16397:1 16389:1 16381:( 16375:= 16371:) 16365:+ 16359:( 16333:) 16329:( 16307:2 16303:/ 16299:1 16295:} 16289:2 16284:1 16275:/ 16271:) 16265:+ 16259:( 16251:2 16243:+ 16230:2 16220:2 16215:1 16207:{ 16189:1 16178:= 16173:3 16149:) 16145:( 16123:2 16119:/ 16115:1 16111:} 16105:2 16100:1 16091:/ 16087:) 16081:+ 16075:( 16067:2 16059:+ 16046:2 16036:2 16031:1 16023:{ 16005:1 15994:= 15989:2 15962:) 15958:( 15939:3 15931:+ 15926:2 15918:= 15915:1 15912:+ 15907:1 15883:) 15879:( 15849:3 15839:3 15831:= 15815:2 15805:2 15781:) 15777:( 15747:3 15737:3 15729:+ 15713:2 15703:2 15695:= 15690:1 15680:1 15616:v 15595:) 15591:( 15572:2 15568:c 15564:m 15559:3 15551:+ 15546:2 15542:c 15538:m 15533:2 15525:= 15520:2 15516:c 15512:m 15509:+ 15504:2 15500:c 15496:m 15491:1 15463:) 15459:( 15434:3 15430:v 15423:m 15418:3 15410:+ 15399:2 15395:v 15388:m 15383:2 15375:= 15372:0 15369:+ 15358:1 15354:v 15347:m 15342:1 15314:) 15310:( 15291:2 15287:c 15283:m 15277:= 15274:E 15258:v 15252:m 15246:= 15237:p 15195:u 15191:v 15187:u 15183:v 15179:u 15175:v 15173:( 15168:u 15164:v 15158:v 15152:u 15147:u 15143:v 15139:v 15075:c 15071:c 15067:g 15063:g 15059:g 15054:c 15050:c 15046:v 15042:t 15038:a 15034:t 15030:t 15028:( 15026:v 15007:2 15003:c 14998:/ 14992:2 14988:t 14982:2 14978:a 14974:+ 14971:1 14966:t 14963:a 14957:= 14954:) 14951:t 14948:( 14945:v 14934:g 14932:1 14893:c 14891:/ 14889:L 14885:L 14875:2 14872:/ 14869:1 14853:) 14849:( 14830:2 14826:v 14818:2 14814:c 14810:L 14803:2 14800:1 14795:= 14790:1 14786:K 14777:0 14773:K 14750:) 14746:( 14728:) 14724:1 14713:2 14709:c 14704:/ 14698:2 14694:v 14687:1 14683:1 14677:( 14673:L 14670:= 14665:1 14661:K 14652:0 14648:K 14618:1 14614:E 14605:1 14601:H 14578:0 14574:E 14565:0 14561:H 14540:E 14520:H 14500:E 14494:H 14479:) 14475:( 14457:) 14453:1 14442:2 14438:c 14433:/ 14427:2 14423:v 14416:1 14412:1 14406:( 14402:L 14399:= 14396:) 14391:1 14387:E 14378:1 14374:H 14370:( 14364:) 14359:0 14355:E 14346:0 14342:H 14338:( 14299:2 14295:c 14290:/ 14284:2 14280:v 14273:1 14269:L 14264:+ 14259:1 14255:H 14251:= 14243:2 14239:c 14234:/ 14228:2 14224:v 14217:1 14201:) 14198:c 14194:/ 14190:v 14187:( 14184:+ 14181:1 14175:L 14169:2 14166:1 14160:+ 14152:2 14148:c 14143:/ 14137:2 14133:v 14126:1 14110:) 14107:c 14103:/ 14099:v 14096:( 14090:1 14084:L 14078:2 14075:1 14069:+ 14064:1 14060:H 14056:= 14047:0 14043:H 14035:L 14032:+ 14027:1 14023:E 14019:= 14016:L 14010:2 14007:1 14001:+ 13998:L 13992:2 13989:1 13983:+ 13978:1 13974:E 13970:= 13961:0 13957:E 13927:1 13923:H 13912:′ 13909:S 13893:0 13889:H 13878:S 13864:v 13854:′ 13851:S 13832:1 13828:E 13805:0 13801:E 13790:S 13776:2 13772:/ 13768:L 13738:S 13716:2 13712:c 13707:/ 13701:2 13697:v 13690:1 13674:) 13671:c 13667:/ 13663:v 13660:( 13654:1 13648:= 13643:f 13636:f 13610:v 13600:′ 13597:S 13593:S 13559:f 13539:E 13480:E 13464:c 13462:/ 13460:E 13456:c 13450:c 13448:/ 13442:c 13440:/ 13438:E 13436:( 13430:c 13428:/ 13426:E 13424:( 13408:E 13392:m 13173:. 13152:2 13141:1 13137:1 13132:= 13107:c 13103:/ 13099:v 13096:= 13070:. 13065:s 13061:f 13049:+ 13046:1 13035:1 13028:= 13023:s 13019:f 13014:) 13004:1 13000:( 12993:= 12988:r 12984:f 12954:s 12950:c 12927:s 12923:f 12918:) 12910:s 12906:c 12902:v 12894:1 12890:( 12886:= 12881:r 12877:f 12848:v 12827:v 12774:2 12758:2 12754:/ 12750:1 12745:) 12739:v 12736:+ 12733:c 12728:v 12722:c 12716:( 12711:= 12706:2 12663:] 12651:) 12648:c 12644:/ 12640:v 12637:( 12634:+ 12631:1 12628:[ 12608:= 12558:) 12555:c 12551:/ 12547:v 12544:( 12541:+ 12538:1 12533:c 12529:/ 12525:v 12522:+ 12507:= 12413:) 12405:2 12401:n 12397:1 12389:1 12385:( 12381:v 12373:n 12370:c 12345:) 12338:n 12335:c 12331:v 12323:1 12319:( 12314:) 12308:c 12305:v 12295:n 12292:1 12286:( 12282:c 12257:n 12254:c 12250:/ 12246:v 12240:1 12235:v 12229:n 12225:/ 12221:c 12199:= 12191:2 12187:c 12182:/ 12178:v 12171:u 12164:1 12159:v 12149:u 12142:= 12133:u 12092:n 12066:. 12059:) 12051:2 12047:n 12043:1 12035:1 12031:( 12027:v 12019:n 12016:c 12011:= 12002:u 11975:u 11954:v 11934:n 11930:/ 11926:c 11923:= 11916:u 11804:x 11779:t 11775:c 11755:x 11731:x 11706:x 11703:+ 11671:x 11651:′ 11522:, 11517:2 11513:c 11508:/ 11504:v 11501:L 11471:v 11448:L 11424:L 11357:x 11342:) 11340:1 11331:c 11327:/ 11323:u 11311:′ 11308:v 11302:′ 11299:u 11293:v 11287:u 11281:. 11279:c 11275:c 11270:. 11267:c 11262:v 11255:) 11253:c 11249:′ 11246:u 11240:c 11236:u 11219:′ 11216:S 11212:S 11193:v 11190:+ 11183:u 11179:= 11176:u 11155:, 11147:u 11124:v 11102:u 11072:) 11068:( 11051:. 11041:) 11035:2 11031:c 11026:/ 11022:v 11014:1 11010:u 11006:+ 11003:1 10999:( 10986:3 10982:u 10976:= 10971:3 10967:u 10962:, 10952:) 10946:2 10942:c 10937:/ 10933:v 10925:1 10921:u 10917:+ 10914:1 10910:( 10897:2 10893:u 10887:= 10882:2 10878:u 10873:, 10862:2 10858:c 10853:/ 10849:v 10841:1 10837:u 10833:+ 10830:1 10825:v 10822:+ 10814:1 10810:u 10803:= 10798:1 10794:u 10776:) 10772:( 10755:. 10745:) 10739:2 10735:c 10730:/ 10726:v 10721:1 10717:u 10710:1 10706:( 10696:3 10692:u 10686:= 10678:3 10674:u 10669:, 10659:) 10653:2 10649:c 10644:/ 10640:v 10635:1 10631:u 10624:1 10620:( 10610:2 10606:u 10600:= 10592:2 10588:u 10583:, 10572:2 10568:c 10563:/ 10559:v 10554:1 10550:u 10543:1 10538:v 10530:1 10526:u 10519:= 10511:1 10507:u 10486:) 10482:( 10465:. 10459:) 10452:t 10448:d 10444:/ 10436:z 10432:d 10426:, 10419:t 10415:d 10411:/ 10403:y 10399:d 10393:, 10386:t 10382:d 10378:/ 10370:x 10366:d 10363:( 10360:= 10357:) 10349:3 10345:u 10338:, 10330:2 10326:u 10319:, 10311:1 10307:u 10303:( 10300:= 10292:u 10273:) 10269:( 10252:. 10246:) 10243:t 10240:d 10236:/ 10232:z 10229:d 10223:, 10220:t 10217:d 10213:/ 10209:y 10206:d 10200:, 10197:t 10194:d 10190:/ 10186:x 10183:d 10180:( 10177:= 10174:) 10169:3 10165:u 10158:, 10153:2 10149:u 10142:, 10137:1 10133:u 10129:( 10126:= 10122:u 10095:u 10079:) 10075:( 10058:. 10050:2 10046:c 10041:/ 10037:v 10030:u 10026:+ 10023:1 10018:v 10015:+ 10008:u 10001:= 9998:u 9974:. 9968:v 9945:v 9930:) 9928:9 9926:( 9909:. 9898:2 9894:c 9889:v 9886:u 9876:1 9871:v 9865:u 9859:= 9849:t 9846:d 9841:x 9838:d 9825:2 9821:c 9817:v 9808:1 9803:v 9793:t 9790:d 9785:x 9782:d 9772:= 9765:) 9756:2 9752:c 9747:x 9744:d 9740:v 9730:t 9727:d 9723:( 9714:) 9711:t 9708:d 9704:v 9698:x 9695:d 9692:( 9683:= 9673:t 9669:d 9660:x 9656:d 9650:= 9643:u 9613:u 9592:u 9550:t 9546:d 9522:x 9518:d 9503:) 9501:8 9499:( 9482:. 9474:t 9470:d 9466:/ 9458:x 9454:d 9451:= 9444:u 9440:= 9435:| 9427:u 9422:| 9403:) 9401:7 9399:( 9382:. 9378:t 9375:d 9371:/ 9367:x 9364:d 9361:= 9358:u 9355:= 9350:| 9346:u 9342:| 9322:′ 9319:S 9300:u 9278:. 9274:u 9263:S 9259:′ 9256:S 9252:S 9213:S 9209:x 9201:′ 9198:S 9194:′ 9191:x 9174:. 9168:0 9165:= 9158:t 9119:x 9110:= 9103:x 9086:′ 9083:x 9079:x 9067:′ 9064:t 9062:Δ 9058:′ 9055:S 9050:′ 9048:x 9044:′ 9041:S 9037:x 9033:S 9029:x 8981:v 8977:L 8974:2 8968:= 8965:T 8936:v 8928:L 8925:2 8919:= 8912:T 8874:v 8870:L 8867:f 8864:2 8834:v 8826:L 8823:f 8820:2 8814:= 8793:2 8789:/ 8785:1 8781:) 8775:2 8764:1 8761:( 8756:v 8752:L 8749:f 8746:2 8714:) 8708:+ 8705:1 8702:( 8697:v 8693:L 8690:f 8684:= 8663:2 8659:/ 8655:1 8650:) 8638:1 8630:+ 8627:1 8621:( 8614:2 8610:/ 8606:1 8602:) 8596:2 8585:1 8582:( 8577:v 8573:L 8570:f 8564:= 8540:2 8536:t 8528:f 8501:2 8497:/ 8493:1 8489:) 8483:2 8472:1 8469:( 8464:v 8460:L 8457:f 8451:= 8430:2 8426:/ 8422:1 8417:) 8405:1 8397:+ 8394:1 8388:( 8383:) 8374:1 8371:( 8366:v 8362:L 8359:f 8353:= 8332:2 8328:t 8320:f 8293:f 8260:v 8253:L 8248:= 8240:2 8236:t 8207:c 8204:L 8194:v 8191:L 8186:= 8181:2 8177:t 8147:) 8138:1 8135:( 8130:v 8126:L 8123:f 8117:= 8095:2 8091:/ 8087:1 8082:) 8073:+ 8070:1 8059:1 8053:( 8046:2 8042:/ 8038:1 8034:) 8028:2 8017:1 8014:( 8009:v 8005:L 8002:f 7996:= 7972:1 7968:t 7960:f 7933:2 7929:/ 7925:1 7921:) 7915:2 7904:1 7901:( 7896:v 7892:L 7889:f 7883:= 7861:2 7857:/ 7853:1 7848:) 7839:+ 7836:1 7825:1 7819:( 7814:) 7808:+ 7805:1 7802:( 7797:v 7793:L 7790:f 7784:= 7763:1 7759:t 7751:f 7724:f 7691:v 7684:L 7679:= 7671:1 7667:t 7638:c 7635:L 7630:+ 7625:v 7622:L 7617:= 7612:1 7608:t 7572:v 7564:L 7561:f 7558:2 7552:= 7545:T 7541:f 7515:v 7511:L 7508:f 7505:2 7499:= 7496:T 7493:f 7459:v 7451:L 7448:2 7442:= 7435:T 7406:v 7402:L 7399:2 7393:= 7390:T 7340:. 7335:) 7326:1 7323:( 7319:/ 7315:) 7309:+ 7306:1 7303:( 7298:f 7295:= 7288:f 7266:. 7261:) 7255:+ 7252:1 7249:( 7245:/ 7241:) 7232:1 7229:( 7224:f 7221:= 7214:f 7131:v 7097:. 7090:2 7086:c 7081:/ 7075:2 7071:v 7064:1 7057:/ 7050:A 7046:t 7024:= 7017:2 7013:v 7004:2 7000:c 6992:/ 6987:L 6984:2 6981:= 6976:B 6972:t 6951:, 6944:2 6940:v 6931:2 6927:c 6901:c 6880:. 6877:v 6848:. 6845:c 6841:/ 6837:L 6834:2 6831:= 6826:A 6822:t 6800:L 6730:S 6726:t 6718:′ 6715:S 6711:′ 6708:t 6691:. 6685:0 6682:= 6679:x 6650:t 6640:= 6633:t 6608:x 6606:Δ 6602:S 6559:v 6542:′ 6539:S 6533:x 6531:Δ 6527:S 6521:′ 6518:t 6516:Δ 6512:′ 6509:S 6503:t 6501:Δ 6497:S 6480:) 6472:2 6468:c 6463:x 6456:v 6447:t 6440:( 6433:= 6426:t 6386:O 6381:c 6377:v 6372:O 6368:c 6364:v 6360:O 6322:c 6302:. 6297:2 6293:s 6272:, 6269:c 6266:= 6262:| 6258:t 6251:/ 6247:x 6240:| 6210:. 6207:c 6201:v 6181:x 6174:/ 6170:t 6162:2 6158:c 6137:v 6112:t 6088:, 6085:x 6078:/ 6074:t 6066:2 6062:c 6041:v 6013:, 6006:2 6002:s 5973:x 5966:/ 5962:t 5954:2 5950:c 5946:= 5943:v 5923:0 5920:= 5913:t 5889:c 5869:v 5849:, 5846:) 5841:2 5837:c 5832:/ 5828:x 5822:v 5816:t 5810:( 5801:= 5794:t 5770:, 5767:c 5760:| 5756:t 5749:/ 5745:x 5738:| 5719:. 5703:, 5700:c 5696:/ 5692:s 5669:t 5662:/ 5658:x 5652:= 5649:v 5629:0 5626:= 5619:x 5595:c 5575:v 5555:, 5552:) 5549:t 5542:v 5536:x 5530:( 5521:= 5514:x 5490:, 5487:c 5480:| 5476:t 5469:/ 5465:x 5458:| 5424:2 5420:s 5391:2 5387:s 5354:2 5345:x 5332:2 5323:t 5314:2 5310:c 5306:= 5295:) 5286:2 5282:c 5276:2 5272:v 5262:1 5258:( 5251:2 5242:x 5233:2 5221:) 5212:2 5208:c 5202:2 5198:v 5188:1 5184:( 5177:2 5168:t 5159:2 5155:c 5149:2 5141:= 5126:2 5122:c 5114:2 5105:x 5096:2 5092:v 5082:2 5074:+ 5068:2 5059:x 5050:2 5036:2 5027:t 5018:2 5014:v 5008:2 4994:2 4985:t 4976:2 4972:c 4966:2 4958:= 4948:) 4942:2 4933:t 4924:2 4920:v 4916:+ 4909:t 4898:x 4891:v 4888:2 4885:+ 4879:2 4870:x 4863:( 4855:2 4843:) 4834:2 4830:c 4822:2 4813:x 4804:2 4800:v 4792:+ 4785:t 4774:x 4767:v 4764:2 4761:+ 4755:2 4746:t 4737:2 4733:c 4728:( 4722:2 4714:= 4702:2 4698:) 4690:t 4683:v 4680:+ 4673:x 4666:( 4658:2 4645:2 4640:) 4631:2 4627:c 4618:x 4611:v 4604:+ 4597:t 4589:( 4582:2 4572:2 4568:c 4564:= 4555:2 4551:x 4539:2 4535:t 4526:2 4522:c 4493:2 4489:x 4477:2 4473:t 4464:2 4460:c 4455:= 4449:2 4445:s 4386:, 4381:2 4377:s 4348:) 4343:2 4339:z 4332:+ 4327:2 4323:y 4316:+ 4311:2 4307:x 4300:( 4292:2 4288:t 4279:2 4275:c 4265:= 4257:2 4253:s 4240:: 4226:2 4222:s 4190:t 4167:r 4095:B 4073:A 4052:0 4049:= 4042:t 3986:t 3983:c 3961:) 3956:2 3945:1 3942:( 3938:/ 3934:) 3929:2 3921:+ 3918:1 3915:( 3889:t 3885:c 3865:) 3859:k 3856:, 3847:k 3844:( 3820:x 3799:k 3779:) 3770:k 3767:, 3761:k 3758:( 3734:t 3730:c 3710:) 3701:, 3695:( 3675:) 3672:0 3669:, 3666:1 3663:( 3643:) 3636:t 3632:c 3629:, 3622:x 3618:( 3598:) 3592:, 3583:( 3563:) 3560:1 3557:, 3554:0 3551:( 3531:) 3524:t 3520:c 3517:, 3510:x 3506:( 3478:= 3471:t 3450:0 3447:= 3444:t 3424:. 3421:c 3417:/ 3413:v 3410:= 3387:, 3384:) 3378:( 3370:1 3359:= 3332:x 3307:t 3303:c 3279:/ 3275:c 3272:= 3269:v 3245:t 3241:c 3217:t 3213:c 3189:x 3162:, 3158:B 3136:A 3112:= 3109:t 3085:c 3065:t 3062:c 3042:t 3039:c 3019:t 2999:x 2979:t 2959:x 2869:. 2862:) 2856:2 2852:c 2847:/ 2839:x 2835:d 2829:v 2826:+ 2819:t 2815:d 2811:( 2801:= 2798:t 2795:d 2773:, 2767:) 2760:t 2756:d 2752:v 2749:+ 2742:x 2738:d 2735:( 2726:= 2723:x 2720:d 2695:. 2688:) 2682:2 2678:c 2673:/ 2669:x 2666:d 2660:v 2654:t 2651:d 2647:( 2637:= 2630:t 2626:d 2601:, 2595:) 2592:t 2589:d 2585:v 2579:x 2576:d 2573:( 2564:= 2557:x 2553:d 2523:. 2516:) 2510:2 2506:c 2501:/ 2493:x 2483:v 2480:+ 2473:t 2465:( 2455:= 2452:t 2427:, 2421:) 2414:t 2406:v 2403:+ 2396:x 2389:( 2380:= 2377:x 2349:. 2342:) 2336:2 2332:c 2327:/ 2323:x 2314:v 2308:t 2301:( 2291:= 2284:t 2255:, 2249:) 2246:t 2239:v 2233:x 2227:( 2218:= 2211:x 2177:. 2169:1 2165:t 2156:2 2152:t 2148:= 2145:t 2133:, 2125:1 2121:x 2112:2 2108:x 2104:= 2101:x 2073:. 2062:1 2058:t 2046:2 2042:t 2038:= 2031:t 2021:, 2010:1 2006:x 1994:2 1990:x 1986:= 1979:x 1958:) 1956:2 1953:′ 1950:t 1946:2 1943:′ 1940:x 1938:( 1934:) 1932:2 1929:t 1925:2 1922:x 1920:( 1916:) 1914:1 1911:′ 1908:t 1904:1 1901:′ 1898:x 1896:( 1892:) 1890:1 1887:t 1883:1 1880:x 1878:( 1856:γ 1852:z 1848:y 1844:x 1837:v 1817:. 1810:z 1806:= 1799:z 1788:y 1784:= 1777:y 1770:) 1763:t 1759:v 1756:+ 1749:x 1745:( 1739:= 1732:x 1725:) 1720:2 1716:c 1711:/ 1703:x 1699:v 1696:+ 1689:t 1685:( 1679:= 1672:t 1643:t 1639:x 1635:z 1631:y 1627:x 1623:S 1619:′ 1616:S 1612:v 1604:c 1581:2 1577:c 1572:/ 1566:2 1562:v 1555:1 1551:1 1546:= 1519:, 1516:z 1513:= 1502:z 1494:y 1491:= 1480:y 1472:) 1469:t 1466:v 1460:x 1457:( 1448:= 1437:x 1429:) 1424:2 1420:c 1415:/ 1411:x 1408:v 1402:t 1399:( 1390:= 1379:t 1360:′ 1357:S 1353:x 1349:v 1345:) 1343:′ 1340:z 1336:′ 1333:y 1329:′ 1326:x 1322:′ 1319:t 1317:( 1313:S 1309:) 1307:z 1303:y 1299:x 1295:t 1293:( 1214:c 1173:′ 1170:S 1166:S 1155:. 1153:′ 1150:S 1144:′ 1141:t 1134:t 1129:′ 1124:. 1122:S 1118:v 1114:x 1110:′ 1107:S 1102:. 1100:′ 1097:S 1093:z 1089:y 1085:x 1077:′ 1074:O 1070:′ 1067:S 1063:O 1032:S 1005:v 997:x 993:v 950:. 948:K 944:′ 941:K 933:K 891:c 887:c 710:c 688:2 684:c 680:m 677:= 674:E 538:. 471:e 464:t 457:v 20:)

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