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Special relativity From Wikipedia, the free encyclopedia For history and motivation, see History of special relativity. For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. General relativity

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Fundamental concepts[hide] Special relativity Equivalence principle World line • Riemannian geometry

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USSR postage stamp dedicated to Albert Einstein Special relativity (SR, also known as the special theory of relativity or STR) is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré[1] and others) in the paper "On the Electrodynamics of Moving Bodies".[2] It generalizes Galileo's principle of relativity—that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)—from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be.[3] Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.[4] This theory has a wide range of consequences which have been experimentally verified,[5] including counter-intuitive ones such as length contraction, time dilationand relativity of simultaneity, contradicting the classical notion that the duration of the time interval between two events is equal for all observers. (On the other hand, it introduces the space-time interval, which is invariant.) Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = mc2, where c is the speed of light in a vacuum.[6][7] The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared with the speed of light. Special relativity reveals that c is not just the velocity of a certain phenomenon—namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. One of the consequences of the theory is that it is impossible for any particle that has rest mass to be accelerated to the speed of light. The theory was originally termed "special" because it applied the principle of relativity only to the special case of inertial reference frames, i.e. frames of reference in uniform relative motion with respect to each other.[8] Einstein developed general relativity to apply the principle in the more general case, that is, to any frame so as to handle general coordinate transformations, and that theory includes the effects of gravity. The term is currently used more generally to refer to any case in which gravitation is not significant. General relativity is the generalization of special relativity to include gravitation. In general relativity, gravity is described using noneuclidean geometry, so that gravitational effects are represented by curvature of spacetime; special relativity is restricted to flat spacetime. Just as the curvature of the earth's surface is not noticeable in everyday life, the curvature of spacetime can be neglected on small scales, so that locally, special relativity is a valid approximation to general relativity.[9] The presence of gravity becomes undetectable in a sufficiently small, free-falling laboratory. Contents [hide] •	1 Postulates •	2 Mass–energy equivalence •	3 Lack of an absolute reference frame •	4 Consequences •	5 Reference frames, coordinates and the Lorentz transformation •	6 Simultaneity, time dilation and length contraction o	6.1 How far can one travel from the Earth? •	7 Causality and prohibition of motion faster than light •	8 Composition of velocities •	9 Geometry of space-time o	9.1 Physics in spacetime •	10 Relativity in electromagnetism •	11 Status •	12 Relativistic quantum mechanics •	13 See also •	14 References o	14.1 Textbooks o	14.2 Journal articles •	15 External links o	15.1 Original works o	15.2 Special relativity for a general audience (no math knowledge required) o	15.3 Special relativity explained (using simple or more advanced math) o	15.4 Visualization

[edit]Postulates “	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? ” —Albert Einstein: Autobiographical Notes[10]

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 and the independence of physical laws (especially 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:[2] 	The Principle of Relativity – 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.[2] 	The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body." (from the preface).[2] That is, light in vacuum propagates with the speed c (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. The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.[11] Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.[12] However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is: Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.[13] Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms. Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.[14] 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: 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 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...[10] Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.[15][16] 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) one can show that the space-time 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 vacuum.[17][18] The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson-Morley experiment.[19][20] 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. [edit]Mass–energy equivalence See also: Mass–energy equivalence 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 heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc2. Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a nontrivial way. For an object at rest, the energy-momentum four-vector is (E, 0, 0, 0): 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 approximately (E, Ev/c2, 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c2. 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 don't talk about matter or radiation, they only talk about space and time. The derivation 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.[2] The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.[21] 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.[22] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.[23] 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 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.[24][25] [edit]Lack of an absolute reference frame The principle of relativity, which states that there is no preferred inertial reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. However, in the late 19th century, the existence ofelectromagnetic waves led physicists to suggest that the universe was filled with a substance known as "aether", which would act as the medium through which these waves, or vibrations traveled. The aether was thought to constitute an absolute reference frame against which speeds could be measured, and could be considered fixed and motionless. Aether supposedly had some wonderful properties: it was sufficiently elastic that it could support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson–Morley experiment, indicated that the Earth was always 'stationary' relative to the aether – something that was difficult to explain, since the Earth is in orbit around the Sun. Einstein's solution was to discard the notion of an aether and an absolute state of rest. Special relativity is formulated so as to not assume that any particular frame of reference is special; rather, in relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in a vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities. [edit]Consequences The consequences of special relativity can be derived from the Lorentz transformation equations.[26] These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counterintuitive: 	Time dilation – 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 (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more). 	Relativity of simultaneity – two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer, may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity). 	Lorentz contraction – 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., theladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage). 	Composition of velocities – velocities (and speeds) do not simply 'add', for example if a rocket is moving at 2⁄3 the speed of light relative to an observer, and the rocket fires a missile at 2⁄3 of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In this example, the observer would see the missile travel with a speed of 12⁄13 the speed of light.) 	Thomas rotation - the orientation of an object (i.e. the alignment of its axes with the observer's axes) may be different for different observers. Unlike other relativistic effects, this effect becomes quite significant at fairly low velocities as can be seen in the spin of moving particles. 	Inertia and momentum – as an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference. 	Equivalence of mass and energy, E = mc2 – The energy content of an object at rest with mass m equals mc2. 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. [edit]Reference frames, coordinates and the Lorentz transformation Main article: Lorentz transformation The primed system is in motion relative to the unprimed system with constant speed v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the principle of relativity, an observer stationary in the primed system will view a likewise construction except that the speed they record will be -v. 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. Changing views of spacetime along theworld line of a rapidly accelerating observer. Relativity theory depends on "reference frames". The term reference frame as used here is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity). An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S. In relativity theory we often want to calculate the position of a point from a different reference point. Suppose we have a second reference frame S′, whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity v with respect to S along the x-axis. Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always 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 comoving. Therefore S and S′ are not comoving. In space-time, events are defined by the coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in S′. Then the Lorentz transformation is a linear transformation of coordinates in the following way: where is the Lorentz factor and c is the speed of light in a vacuum. The y and z coordinates are unaffected; only the x and t axes transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity. A quantity invariant under Lorentz transformations is known as a Lorentz scalar. For further generalizations, see the main article Lorentz transformations. The case of an accelerated observer, rather than one traveling at uniform velocity, is shown in the animation on the right. The vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events that are visible to the observer, and the upper quarter shows the light cone- those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. [edit]Simultaneity, time dilation and length contraction Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame. Main article: Relativity of simultaneity See also: Twin paradox Writing the Lorentz transformation and its inverse in terms of coordinate differences, where for instance one event has coordinates (x1, t1) and (x′1, t′1), another event has coordinates (x2, t2) and (x′2, t′2), and the differences given by we get and these equations imply that two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′(satisfying Δt′ = 0). Only if these events are colocal in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′. Similarly, Suppose a clock is at rest in the unprimed system S. Two different ticks of this clock are then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, the first equation can be used to find: for events satisfying This shows that the time Δt′ between the two ticks as seen in the frame in which the clock is moving S′, is longer than the time Δt between these ticks as measured in the rest frame of the clock S. This phenomenon is called time dilation. Time dilation explains a number of physical phenomena; for example, the decay rate of muons produced by cosmic rays impinging on the Earth's atmosphere.[27] Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S', in which the clock is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S'. In other words, the measurement is characterized by Δt′ = 0, which can be combined with the fourth equation to find the relation between the lengths Δx and Δx′: for events satisfying This shows that the length Δx′ of the rod as measured in the frame in which it is moving S′, is shorter than its length Δx in its own rest frame S. This phenomenon is called length contraction or Lorentz contraction. These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. However, the space-time interval will be the same for all observers. The underlying reality remains the same. Only our perspective changes. [edit]How far can one travel from the Earth? Since one can not travel faster than light, one might conclude that a human can never travel further from Earth than 40 light years, if the traveler is active between the age of 20 and 60. One would easily think that a traveler would never be able to reach more than the very few solar systems which exist within the limit of 20-40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1g, it will after a little less than a year be traveling at almost the speed of light as seen from Earth. Time dilation will increase his life span as seen from the reference system of the Earth, but his lifespan measured by a clock traveling with him will not thereby change. During his journey, people on Earth will experience more time than he does. A 5 year round trip for him will take 6½ Earth years and cover a distance of over 6 light-years. A 20 year round trip for him will land him back on Earth having traveled for 336 Earth years and a distance of 314 light years. A full 40 year trip at 1 g 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 g will take 148,000 Earth years and cover about 140,000 light years. This same time dilation is why a muon traveling close to c is observed to travel much further than c times itshalf-life (when at rest).[28] [edit]Causality and prohibition of motion faster than light See also: Causality and Tachyonic antitelephone Diagram 2. Light cone In diagram 2 the interval AB is 'time-like'; i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect). The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which events A and C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show[29][30] that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously. Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in a vacuum. However, some "things" can still 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 turned rapidly.[31] Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F = dp/dt gives a momentum that grows without bound, but this is simply because approaches infinity as v approaches c. To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators, where each charged particle is accelerated by the electromagnetic force. Theoretical and experimental tunneling studies carried out by Günter Nimtz and Petrissa Eckle wrongfully claimed that under special conditions signals may travel faster than light.[32][33][34][35] It was measured that fiber digital signals were traveling up to 5 times c and a zero-time tunneling electron carried the information that the atom is ionized, with photons, phonons and electrons spending zero time in the tunneling barrier. According to Nimtz and Eckle, in this superluminal process only the Einstein causality and the Special Relativity but not the primitive causality are violated: Superluminal propagation does not result in any kind of time travel.[36][37] Several scientists have, however, stated not only that Nimtz' interpretations were erroneous, but that the experiment actually provided a trivial experimental confirmation of the Special relativity theory.[38][39][40] [edit]Composition of velocities Main articles: Velocity-addition formula and Relativistic mechanics If the observer in S measures an object moving along the x axis at velocity u, then the observer in the S′ system, a frame of reference moving at velocity v in the x direction with respect to S, will observe the object moving with velocity u′, given by differentials of the Lorentz transformations above: that is Notice that if the object were moving at the speed of light in the S system (i.e. u = c), then it would also be moving at the speed of light in the S′ system. Also, if both u and v are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: u' ≈ u − v. The usual example given is that of a train (call it system K) traveling due east with a velocity v with respect to the tracks (system K′). A child inside the train throws a baseball due east with a velocity u with respect to the train. In classical physics, an observer at rest on the tracks will measure the velocity of the baseball as v + u. In special relativity, this is no longer true. Instead, an observer on the tracks will measure the velocity of the baseball as (v + u)/[1 + (vu/c2)]. If u and v are small compared to c, then the above expression approaches the classical sum v + u. More generally, the baseball need not travel in the same direction as the train. To obtain the general formula for Einstein velocity addition, suppose an observer at rest in system K measures the velocity of an object as u. Let K′ be an inertial system such that the relative velocity of K to K′ is v, where u and v are now 3d vectors. An observer at rest in K′ will then measure the velocity of the object as[17] where  and   are the components of u parallel and perpendicular to v respectively, and Einstein's addition of colinear velocities is consistent with the Fizeau experiment which determined the speed of light in a fluid moving parallel to the light, but no experiment has ever tested the formula for the general case of non-parallel velocities. [edit]Geometry of space-time Main article: Minkowski space SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space. The differential of distance (ds) in cartesian 3D space is defined as: where  are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes: . If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict (this is called a Wick rotation). In this case the above equation becomes symmetric: . This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space.[citation needed] Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate. Three dimensional dual-cone. Null spherical space. If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space , we see that the null geodesics lie along a dual-cone: defined by the equation or simply , which is the equation of a circle of radius c dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone: . This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars 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  away and a time d/c 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 picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".) The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'. The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought-experiments in special relativity. [edit]Physics in spacetime See also: tensor index notation, covariance and contravariance of vectors, and raising and lowering indices [edit]The metric and line element Main articles: 4-vector, line element, and metric tensor The equations of special relativity are written in a manifestly covariant form, using 4-vectors. There are several such vectors but the most fundamental one is the position of an event in spacetime: given by acontravariant 4-position whose components are: where (x1, x2, x3) = (x, y, z) are the usual spatial coordinates. The zeroth component is the time coordinate: x0 = ct, the factor of c arises by dimensional analysis (i.e. x0 has the same dimensions of distance as the other spatial coordinates), but more fundamentally the magnitude of the 4-position is the spacetime interval: where τ is the proper time. This places space and time on an equal footing, more so if natural units are used (c = 1).[41][42][43] In special relativity, Minkowski space-time uses the Minkowski metric η, in matrix form it is symmetric: and is equal to its inverse matrix. The metric is valid in any inertial reference frame. The squared length of the differential of the position four-vector  is constructed using the metric Notice that when the line element dx2 is negative that dτ = √(-dx2)/c is the differential of proper time, while when dx2 is positive, √(dx2) is differential of the proper distance. [edit]Lorentz transformations of 4-vectors and tensors Main article: Lorentz transformations The coordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special (and simplest) case of motion along the x-axis, we have: where β = v/c, which is simply the matrix of a boost (like a rotation) between the x and ct coordinates. The μ' index indicates the row and ν indicates the column. The Lorentz transformations above for the position and time of an event from an unprimed coordinate system S to a primed system S' can be written as: More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy: where there is an implied summation (Einstein summation convention) of  and   from 0 to 3 on the right-hand side. The Poincaré group is the most general group of transformations which preserves theMinkowski metric and this is the physical symmetry underlying special relativity. Most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law[44] where  is the inverse matrix of   All tensors transform by this rule. The magnitude (length) of a 4-vector is invariant, meaning that the magnitude has the same value in all inertial frames because it is a scalar (0 rank tensor). The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also 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. [edit]Relativity in electromagnetism Main articles: Covariant formulation of classical electromagnetism and Classical electromagnetism and special relativity Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the electric andmagnetic fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity. In the frame of a stationary electric charge (or charge distribution), an observer measures the electric field due to the charge to be electrostatic, however an observer in an inertial frame moving relative to the charge will notice an electric current, because the charge moves relative to the observer, and hence a magnetic field. Mathematically, the Lorentz transformations from one frame to another result in coupled equations in the electric and magnetic field components, just like space and time coordinates are coupled. Maxwell's equations 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 electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame. [edit]Status Main articles: Tests of special relativity and Criticism of relativity theory Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c2 in the region of interest.[45] In a strong gravitational field, one must usegeneral relativity. General relativity becomes special relativity at the limit of weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[46] 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. Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields). 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 Classical mechanics for a more detailed discussion. 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,[47] and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.[20] 	The Fizeau experiment (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. 	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 source and observer's velocity, as both source and observer were traveling together at the same velocity at all times. 	The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame. 	The Experiments of Rayleigh and Brace‎ (1902, 1904) showed that length contraction doesn't lead to birefringence for a co-moving observer, in accordance with the relativity principle. Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. 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: 	Tests of relativistic energy and momentum – testing the limiting speed of particles 	Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation 	Time dilation of moving particles – relativistic effects on a fast-moving particle's half-life 	Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations 	Hughes–Drever experiment – testing isotropy of space and mass 	Modern searches for Lorentz violation – various modern tests 	Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter. 	Experiments to test Aether drag hypothesis – no "aether flow obstruction" [edit]Relativistic quantum mechanics Special relativity and quantum mechanics can be combined into a fundamental theoretical framework - quantum field theory[48], and has been crucial for developing the standard model of particle physicsbecause of very small scales and high energies which occur in particle interactions.[49] The early Bohr-Sommerfeld atomic model explained the fine structure of alkaline atoms by using both special relativity and the preliminary knowledge on quantum mechanics of the time. Paul Dirac developed a relativistic wave equation; the Dirac equation[50], fully compatible both with special relativity and with the final version of quantum theory existing after 1926. Fine structure could not be fully explained without special relativity. On the other hand, the existence of antiparticles makes obvious that one is not dealing with a naive unification of special relativity and quantum mechanics. Instead a theory is necessary where one is dealing with quantized fields, and where particles can be created and destroyed as in quantum electrodynamics or quantum chromodynamics (which cannot happen according to the non-relativistic Schrödinger equation). Later after Dirac, more relativistic wave equations followed. In contrast, it is an unsolved question if and how general relativity can be unified with quantum mechanics, to formulate quantum gravitation. [edit]See also Physics portal

People: Hendrik Lorentz | Henri Poincaré | Albert Einstein | Max Planck | Hermann Minkowski | Max von Laue | Arnold Sommerfeld | Max Born | Gustav Herglotz | Richard C. Tolman Relativity: Theory of relativity | History of special relativity | Principle of relativity | General relativity | Fundamental Speed | Frame of reference | Inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijk-Putnam Argument | Special relativity (alternative formulations) | Criticism of relativity theory | Relativity priority dispute | Relativistic mechanics Physics: Newtonian Mechanics | spacetime | speed of light | simultaneity | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem | Shape waves| Relativistic heat conduction Mathematics: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number | Relativity in the APS formalism Philosophy: actualism | conventionalism | formalism Paradoxes: Twin paradox | Ehrenfest paradox | Ladder paradox | Bell's spaceship paradox | Velocity composition paradox [hide] 	V 	T 	E Tests of special relativity

Speed/Isotropy	Michelson–Morley experiment • Kennedy–Thorndike experiment  • Moessbauer rotor experiments  • Resonator experiments  • de Sitter double star experiment  • Hammar experiment  • Measurements of neutrino speed

Lorentz invariance	Modern searches for Lorentz violation • Hughes–Drever experiment  • Trouton–Noble experiment  • Experiments of Rayleigh and Brace  • Trouton–Rankine experiment  • Antimatter tests of Lorentz violation  •Lorentz-violating neutrino oscillations

Time dilation Length contraction Ives–Stilwell experiment • Moessbauer rotor experiments  • Time dilation of moving particles  • Hafele–Keating experiment  • Length contraction confirmations

Relativistic energy	Tests of relativistic energy and momentum • Kaufmann–Bucherer–Neumann experiments

Fizeau/Sagnac	Fizeau experiment • Sagnac experiment  • Michelson–Gale–Pearson experiment

Alternatives	Refutations of aether theory • Refutations of emission theory

General	One-way speed of light • Test theories of special relativity  • Standard-Model Extension

[edit]References 1.	^ Isaacson, Walter. Einstein: His Life and Universe. Simon and Schuster, 2008. 2.	^ a b c d e Albert Einstein (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920). 3.	^ Wolfgang Rindler (1977). Essential Relativity. Birkhäuser. p. §1,11 p. 7. ISBN 3-540-07970-X. 4.	^ Edwin F. Taylor and John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. W. H. Freeman. ISBN 0-7167-2327-1. 5.	^ Tom Roberts and Siegmar Schleif (October 2007). "What is the experimental basis of Special Relativity?". Usenet Physics FAQ. Retrieved 2008-09-17. 6.	^ Albert Einstein (2001). Relativity: The Special and the General Theory (Reprint of 1920 translation by Robert W. Lawson ed.). Routledge. p. 48. ISBN 0-415-25384-5. 7.	^ Richard Phillips Feynman (1998). Six Not-so-easy Pieces: Einstein's relativity, symmetry, and space-time (Reprint of 1995 ed.). Basic Books. p. 68. ISBN 0-201-32842-9. 8.	^ Albert Einstein, Relativity — The Special and General Theory, chapter 18 9.	^ Charles W. Misner, Kip S. Thorne & John A. Wheeler,Gravitation, pg 172, 6.6 The local coordinate system of an accelerated observer, ISBN 0-7167-0344-0 10.	^ a b Einstein, Autobiographical Notes, 1949. 11.	^ Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920 12.	^ For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990 13.	^ Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. (1952). The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity. Courier Dover Publications. p. 111.ISBN 0-486-60081-5. 14.	^ 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 Special and General Theory, 1916; The Principle 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. 15.	^ Das, A., The Special Theory of Relativity, A Mathematical Exposition, Springer, 1993. 16.	^ Schutz, J., Independent Axioms for Minkowski Spacetime, 1997. 17.	^ a b Yaakov Friedman, Physical Applications of Homogeneous Balls, Progress in Mathematical Physics 40 Birkhäuser, Boston, 2004, pages 1-21. 18.	^ David Morin, Introduction to Classical Mechanics, Cambridge University Press, Cambridge, 2007, chapter 11, Appendix I 19. ^ Michael Polanyi, Personal Knowledge: Towards a Post-Critical Philosophy, 1974, ISBN 0-226-67288-3, footnote page 10-11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morely experiment had no role in the foundation of the theory." and "..the theory of relativity was not founded to explain its outcome at all."[1] 20.	^ a b Dongen, Jeroen van (2009). "On the role of the Michelson–Morley experiment: Einstein in Chicago". Eprint arXiv:0908.1545 0908: 1545. arXiv:0908.1545. Bibcode2009arXiv0908.1545V. 21.	^ Does the inertia of a body depend upon its energy content? A. Einstein, Annalen der Physik.18:639, 1905 (English translation by W. Perrett and G.B. Jeffery) 22.	^ Max Jammer (1997). Concepts of Mass in Classical and Modern Physics. Courier Dover Publications. pp. 177–178. ISBN 0-486-29998-8. 23.	^ John J. Stachel (2002). Einstein from B to Z. Springer. p. 221. ISBN 0-8176-4143-2. 24.	^ On the Inertia of Energy Required by the Relativity Principle, A. Einstein, Annalen der Physik 23 (1907): 371-384 25.	^ 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, 1955. 26.	^ Resnick, Robert (1968). Introduction to special relativity. Wiley. pp. 62–63. 27.	^ Kleppner, Daniel; Kolenkow, David (1973). An Introduction to Mechanics. pp. 468–70. 28.	^ http://library.thinkquest.org/C0116043/specialtheorytext.htm Thinkquest org 29.	^ Tolman, R. C. (1917). The theory of the Relativity of Motion. Berkeley. p. 54. 30.	^ Benford, G. A.; Book, D. L.; Newcomb, W. A. (1970). "The Tachyonic Antitelephone". Phys. Rev. D 2(2): 263–265. DOI:10.1103/PhysRevD.2.263. 31.	^ Salmon, Wesley C. (2006). Four Decades of Scientific Explanation. University of Pittsburgh. p. 107.ISBN 0-8229-5926-7., Section 3.7 page 107 32.	^ F. Low and P. Mende, A Note on the Tunneling Time Problem, Ann. Phys. NY, 210, 380-387 (1991) 33.	^ A. Enders and G. Nimtz, On superluminal barrier traversal, J. Phys. I, France 2, 1693-1698 (1992) 34.	^ Longhi, S.; et al. (2002). "Measurement of superluminal optical tunneling times in double-barrier photonic band gaps". Phys.Rev. E 65: 06610 1-6. DOI:10.1103/PhysRevE.65.046610. 35.	^ Eckle, P.; et al. (2008). "Attosecond Ionization and Tunneling Delay Time Measurements in Helium".Science 322 (5907): 1525–1529. DOI:10.1126/science.1163439. 36.	^ Nimtz, G. (2006). "Do Evanescent Modes Violate Relativistic Causality?". Lect.Notes Phys. 702: 506–531. DOI:10.1007/3-540-34523-X_19. 37.	^ Nimtz, G. (2010). "Tunneling Violates Special Relativity". Preprint. arXiv:1003.3944. 38.	^ Herbert Winful (2007-09-18). "Comment on "Macroscopic violation of special relativity" by Nimtz and Stahlhofen". arXiv:0709.2736 [quant-ph]. 39.	^ Chris Lee (2007-08-16). "Latest "faster than the speed of light" claims wrong (again)". 40.	^ Winful, Herbert G. (December 2006). "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox". Physics Reports 436 (1-2): 1–69. Bibcode2006PhR...436....1W. DOI:10.1016/j.physrep.2006.09.002. 41.	^ Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, ISBN 0-07-032071-3 42.	^ Charles W. Misner, Kip S. Thorne & John A. Wheeler,Gravitation, pg 51, ISBN 0-7167-0344-0 43.	^ George Sterman, An Introduction to Quantum Field Theory, pg 4, ISBN 0-521-31132-2 44.	^ M. Carroll, Sean (2004). Spacetime and Geometry: An Introduction to General Relativity(illustrated ed.). Addison Wesley. p. 22. ISBN 0-8053-8732-3. 45.	^ Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's general theory of relativity: with modern applications in cosmology. Springer. p. 195. ISBN 0-387-69199-5., Extract of page 195 (with units where c=1) 46.	^ The number of works is vast, see as example: Sidney Coleman, Sheldon L. Glashow, Cosmic Ray and Neutrino Tests of Special Relativity, Phys. Lett. B405 (1997) 249-252, online An overview can be found on this page 47.	^ Norton, John D., John D. (2004), "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905", Archive for History of Exact Sciences 59: 45–105, Bibcode 2004AHES...59...45N,DOI:10.1007/s00407-004-0085-6 48.	^ This "standard relativistic quantum field theory" is still - in some respects - at its infancy, since one of its particles, "the Higgs" has not yet been found. 49.	^ Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008,ISBN 978-0-470-03294-7 50.	^ Dirac, P.A.M. (1930). "A Theory of Electrons and Protons". Proc. R. Soc. A126: 360. Bibcode1930RSPSA.126..360D. DOI:10.1098/rspa.1930.0013. JSTOR 95359. [edit]Textbooks 	Einstein, Albert (1920). Relativity: The Special and General Theory. 	Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9 	Freund, Jürgen (2008) Special Relativity for Beginners - A Textbook for Undergraduates World Scientific. ISBN 981-277-160-3 	Logunov, Anatoly A. (2005) Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka, Moscow. 	Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. ISBN 0-7167-0334-3 	Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications. 	Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. ISBN 978-0-19-853952-0; ISBN 0-19-853952-5 	Harvey R. Brown (2005). Physical relativity: space-time structure from a dynamical perspective, Oxford University Press, ISBN 0-19-927583-1; ISBN 978-0-19-927583-0 	Qadir, Asghar (1989). Relativity: An Introduction to the Special Theory. Singapore: World Scientific Publications. pp. 128. ISBN 9971-5-0612-2. 	Silberstein, Ludwik (1914) The Theory of Relativity. 	Lawrence Sklar (1977). Space, Time and Spacetime. University of California Press. ISBN 0-520-03174-1. 	Lawrence Sklar (1992). Philosophy of Physics. Westview Press. ISBN 0-8133-0625-6. 	Taylor, Edwin, and John Archibald Wheeler (1992) Spacetime Physics (2nd ed.). W.H. Freeman & Co. ISBN 0-7167-2327-1 	Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. ISBN 0-7167-4345-0 [edit]Journal articles 	Alvager, et al.; Farley, F. J. M.; Kjellman, J.; Wallin, L. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters 12 (3): 260. Bibcode 1964PhL....12..260A.DOI:10.1016/0031-9163(64)91095-9. 	Darrigol, Olivier (2004). "The Mystery of the Poincaré-Einstein Connection". Isis 95 (4): 614–26. DOI:10.1086/430652. PMID 16011297. 	Feigenbaum, Mitchell (2008). "The Theory of Relativity - Galileo's Child". Eprint arXiv:0806.1234 0806: 1234. arXiv:0806.1234. Bibcode 2008arXiv0806.1234F. 	Gulevich, D. R. et al.; Kusmartsev, F. V.; Savel'Ev, Sergey; Yampol'Skii, V. A.; Nori, Franco (2008). "Shape waves in 2D Josephson junctions: Exact solutions and time dilation". Phys. Rev. Lett. 101 (12): 127002. Bibcode 2008PhRvL.101l7002G. DOI:10.1103/PhysRevLett.101.127002. PMID 18851404. 	Rizzi, G. et al. (2005). "Synchronization Gauges and the Principles of Special Relativity". Found. Phys 34: 1835–87. arXiv:gr-qc/0409105. Bibcode 2004FoPh...34.1835R. DOI:10.1007/s10701-004-1624-3. 	Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System". Physical Review A 56 (6): 4405–09. Bibcode 1997PhRvA..56.4405W.DOI:10.1103/PhysRevA.56.4405. [edit]External links Wikisource has original works on the topic: Relativity

Wikibooks has a book on the topic of Special Relativity

Wikiversity has learning materials about Special Relativity

Look up special relativity in Wiktionary, the free dictionary. [edit]Original works 	Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Bern 1905 	On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity. [edit]Special relativity for a general audience (no math knowledge required) 	Wikibooks: Special Relativity 	Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics. 	Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics. 	Audio: Cain/Gay (2006) - Astronomy Cast. Einstein's Theory of Special Relativity [edit]Special relativity explained (using simple or more advanced math) 	Greg Egan's Foundations. 	The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus. 	Motion Mountain, Volume II - A modern introduction to relativity, including its visual effects. 	MathPages - Reflections on Relativity A complete online book on relativity with an extensive bibliography. 	Relativity An introduction to special relativity at the undergraduate level, without calculus. 	Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein 	Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University. 	Understanding Special Relativity The theory of special relativity in an easily understandable way. 	An Introduction to the Special Theory of Relativity (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) 	Lecture Notes on Special Relativity by J D Cresser Department of Physics Macquarie University [edit]Visualization 	Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity. 	Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program. 	Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes. 	Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images. 	Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light. 	Animation clip visualizing the Lorentz transformation. 	Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc. 	Relativistic Optics at the ANU [show] 	V 	T 	E Branches of physics

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