Theory of interwoven space and time by Albert Einstein
Albert Einstein around 1905, the year his "annus mirabilis papers" were published. These includedZur Elektrodynamik bewegter Körper, the paper founding special relativity.
Thespeed of light invacuum 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.
Special relativity builds upon important physics ideas. The non-technical ideas include:
speed orvelocity, how the relative distance between an object and a reference point changes with time.[3]: 25
speed of light, the maximum speed of information, independent of the speed of the source and receiver,[4]: 39
event: something that happens at a definite place and time. For example, an explosion or a flash of light from an atom;[4]: 10 a generalization of a point in geometrical space,[3]: 43
clocks: relativity is all about time;[3] in relativity observers read clocks.[4]: 39
Two observers in relative motion receive information about two events via light signals traveling at constant speed, independent of either observer's speed. Their motion during the transit time causes them to get the information at different times on their local clock.
The more technical background ideas include:
invariance: when physical laws do not change when a specific circumstance changes, such as observations at different uniform velocities;[3]: 2
spacetime: a union of geometrical space and time.[4]: 18
spacetime interval between two events: a measure of separation that generalizes distance:[4]: 9
inertial reference frame: a limited region of spacetime within which all objects experience the same gravitational acceleration and, absent other forces, do not accelerate relative to each other,[4]: 44
relative velocity: the amount and direction of uniform, relative motion of objects in two reference frames,
The spacetime interval is an invariant between inertial frames, demonstrating the physical unity of spacetime.[4]: 15 Coordinate systems are not invariant between inertial frames and require transformations.[4]: 95
Unusual among modern topics in physics, the theory of special relativity needs only mathematics at high school level and yet it fundamentally alters our understanding, especially our understanding of the concept oftime.[3]: ix Built on just two postulates or assumptions, many interesting consequences follow.
The two postulates both concern observers moving at a constant speed relative to each other. The first postulate, the§ principle of relativity, says the laws of physics do not depend on objects being at absolute rest: for example, an observer on a train sees natural phenomena on that train that look the same whether the train is moving or not.[3]: 5 The second postulate, constant speed of light, says observers in a train station see light travel at the same speed whether they measure light from within the station or light from a moving train. A light signal from the station to the train has the same speed, no matter how fast a train goes.[3]: 25
In the theory of special relativity, the two postulates combine to change the definition of "relative speed". Rather than the simple concept of distance traveled divided by time spent, the new theory incorporates the speed of light as the maximum possible speed. In special relativity, covering ten times more distance on the ground in the same amount of time according to a moving watch does not result in a speed up as seen from the ground by a factor of ten.[3]: 28
Combined with other laws of physics, the two postulates of special relativity predict the equivalence ofmass andenergy, as expressed in themass–energy equivalence formula, where is thespeed of light in vacuum.[7][8]Special relativity replaced the conventional notion of an absolute, universal time with the notion of a time that is local to each observer.[9]: 33 Information about distant objects can arrive no faster than the speed of light so visual observations always report events that have happened in the past. This effect makes visual descriptions of the effects of special relativity especially prone to mistakes.[10]
Special relativity also has profound technical consequences.A defining feature of special relativity is the replacement ofEuclidean geometry withLorentzian geometry.[4]: 8 Distances in Euclidean geometry are calculated with thePythagorean theorem and only involved spatial coordinates. In Lorentzian geometry, 'distances' become 'intervals' and include a time coordinate with a minus sign. Unlike spatial distances, the interval between two events has the same value for all observers independent of their relative velocity. When comparing two sets of coordinates in relative motion isLorentz transformation replaceGalilean transformations of Newtonian mechanics.[4]: 98 Other effects include the relativistic corrects to theDoppler effect and theThomas precession.[1][2] It also explains how electricity and magnetism are related.[1][2]
The principle of relativity, forming one of the two postulates of special relativity, was described byGalileo Galilei in 1632 using a thought experiment involving observing natural phenomena on a moving ship.[11] His conclusions were summarized asGalilean relativity and used as the basis ofNewtonian mechanics.[3]: 1 This principle can be expressed as acoordinate transformation, between two coordinate systems.Isaac Newton noted that many transformations, such as those involving rotation or acceleration, will not preserve the observation of physical phenomena. Newton considered only those transformations involving motion with respect to an immovable absolute space, now called transformations between inertial frames.[12]: 17
In 1864James Clerk Maxwell presented a theory ofelectromagnetism which did not obey Galilean relativity. The theory specifically predicted a constant speed of light in vacuum, no matter the motion (velocity, acceleration, etc.) of the light emitter or receiver or its frequency, wavelength, direction, polarization, or phase. This, as yet untested theory, was thought at the time to be only valid in inertial frames fixed in anaether. Numerous experiments followed, attempting to measure the speed of light as Earth moved through the proposed fixed aether, culminating in the 1887Michelson–Morley experiment which only confirmed the constant speed of light.[12]: 18
Several fixes to the aether theory were proposed, with those ofGeorge Francis Fitzgerald,Hendrik Antoon Lorentz, andJules Henri Poincare all pointing in the direction of a result similar to the theory of special relativity. The final important step was taken by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".[p 1] Einstein applied theLorentz transformations known to be compatible withMaxwell's equations for electrodynamics to the classical laws of mechanics. This changed Newton's mechanics situations involving all motions, especially velocities close to that of light[12]: 18 (known asrelativistic velocities).
Another way to describe the advance made by the special theory is to say Einstein extended the Galilean principle so that it accounted for the constant speed of light,[4] a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all thelaws of physics, including both the laws of mechanics and ofelectrodynamics.[13]The theory became essentially complete in 1907, withHermann Minkowski's papers on spacetime.[14]
Special relativity has proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible.[15][14] Even so, the Newtonian model remains accurate at low velocities relative to the speed of light, for example, everyday motion on Earth.
When updating his 1911 book on relativity, to include general relativity in 1920,Robert Daniel Carmichael called the earlier work the "restricted theory" as a "special case" of the new general theory; he also used the phrase "special theory of relativity".[16] In comparing to the general theory in 1923 Einstein specifically called his earlier work "the special theory of relativity", saying he meant a restriction to frames uniform motion.[17]: 111 Just asGalilean relativity is accepted as an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weakgravitational fields, that is, at a sufficiently small scale (e.g., whentidal forces are negligible) and in conditions offree fall. But general relativity incorporatesnon-Euclidean geometry to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known asMinkowski space. As long as the universe can be modeled as apseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in thiscurved spacetime.
Traditional "two postulates" approach to special relativity
"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?"
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 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:[p 1]
Theprinciple 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.[p 1]
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).[p 1] That is, light in vacuum propagates with the speedc (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 constancy of the speed of light was motivated byMaxwell's theory of electromagnetism[18] and the lack of evidence for theluminiferous ether.[19] There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.[20][21] 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.
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 theisotropy andhomogeneity of space and the independence of measuring rods and clocks from their past history.[p 3]
Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocityv only along thex-axis, from the perspective of an observer stationary in the unprimed system. By theprinciple of relativity, an observer stationary in the primed system will view a likewise construction except that the velocity 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.
Reference frames 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 frame has the ability to determine measurements of the time of events using a "clock" (any reference device with uniform periodicity).
Anevent is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" inspacetime. 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.
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 frameS.
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 calledtransformation equations.
To gain insight into how the spacetime coordinates measured by observers in differentreference frames compare with each other, it is useful to work with a simplified setup with frames in astandard configuration.[22]: 107 With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, twoGalilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observerO, and frameS′ (pronounced "S prime" or "S dash") belongs to a second observerO′.
Thex,y,z axes of frame S are oriented parallel to the respective primed axes of frameS′.
FrameS′ moves, for simplicity, in a single direction: thex-direction of frame S with a constant velocityv as measured in frameS.
The origins of frames S and S′ are coincident when timet = 0 for frame S andt′ = 0 for frameS′.
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 becomoving. Therefore,S andS′ are notcomoving.
Theprinciple of relativity, which states that physical laws have the same form in eachinertial reference frame, dates back toGalileo, and was incorporated into Newtonian physics. But in the late 19th century the existence ofelectromagnetic waves led some physicists to suggest that the universe was filled with a substance they called "aether", 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 anabsolute reference frame 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 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.[23] 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 bec, even when measured by multiple systems that are moving at different (but constant) velocities.
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)it can be shown 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 vacuum.[p 4][24]: 511
Einstein combined the two postulates - of relativity - and of the invariance of the speed of light, into single postulate, the Lorentz transformation:
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 ...[p 2]
Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations,[25] but Einstein stuck to his approach throughout work.[p 5]
Henri Poincaré provided the mathematical framework for relativity theory by proving thatLorentz transformations are a subset of hisPoincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.
While the traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations,[26] other treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate ofMinkowski spacetime.[p 6][p 7] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler[4] and by Callahan.[27]
Define anevent to have spacetime coordinates(t,x,y,z) in systemS and(t′,x′,y′,z′) in a reference frameS′ moving at a velocityv along thex-axis. Then theLorentz transformation specifies that these coordinates are related in the following way:where is theLorentz factor andc is thespeed of light in vacuum, and the velocityv ofS′, relative toS, is parallel to thex-axis. For simplicity, they andz coordinates are unaffected; only thex andt coordinates are transformed. These Lorentz transformations form aone-parameter group oflinear mappings, that parameter being calledrapidity.
Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:
This shows that the unprimed frame is moving with the velocity −v, as measured in the primed frame.[28]
There is nothing special about thex-axis. The transformation can apply to they- orz-axis, or indeed in any direction parallel to the motion (which are warped by theγ factor) and perpendicular; see the articleLorentz transformation for details.
Writing the Lorentz transformation and its inverse in terms of coordinate differences, where 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 are defined as
Eq. 1:
Eq. 2:
we get
Eq. 3:
Eq. 4:
If we take differentials instead of taking differences, we get
Eq. 5:
Eq. 6:
Graphical representation of the Lorentz transformation
Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.
Spacetime diagrams (also calledMinkowski diagrams) 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.[24]: 536 To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig. 2-1.[29]: 155–199
Fig. 3-1a. Draw the and axes of frame S. The axis is horizontal and the (actually) axis is vertical, which is the opposite of the usual convention in kinematics. The axis is scaled by a factor of 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 theworldlines of two photons passing through the origin at time The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, and have been plotted on this graph so that their coordinates may be compared in the S and S' frames.
Fig. 3-1b. Draw the and axes of frame S'. The axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, Both the and axes are tilted from the unprimed axes by an angle where The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that when
Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that coordinates of in the primed coordinate system transform to in the unprimed coordinate system. Likewise, coordinates of in the primed coordinate system transform to in the unprimed system. Draw gridlines parallel with the axis through points as measured in the unprimed frame, where is an integer. Likewise, draw gridlines parallel with the axis through as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between units equals times the spacing between units, as measured in frame S. This ratio is always greater than 1, and approaches infinity as
Fig. 3-1d. Since the speed of light is an invariant, theworldlines of two photons passing through the origin at time still plot as 45° diagonal lines. The primed coordinates of and are related to the unprimed coordinates through the Lorentz transformations andcould 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 x-coordinates in the unprimed frame are now at the same position in space.
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 aCartesian plane, but the frames are actually equivalent.
Consequences derived from the Lorentz transformation
The consequences of special relativity can be derived from theLorentz transformation equations.[30] 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 most humans encounter that some of the effects predicted by relativity are initiallycounterintuitive.
In Galilean relativity, the spatial separation, (), and the temporal separation, (), between two events are independent invariants, the values of which do not change when observed from different frames of reference.In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of aninvariant interval, denoted as:In considering the physical significance of, there are three cases:[24]: 533 [4]: 25–39
Δs2 > 0: In this case, the two events are separated by more time than space, and they are hence said to betimelike separated. This implies that, and given the Lorentz transformation, it is evident that there exists a less than for which (in particular,). 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,, is called theproper time.
Δs2 < 0: In this case, the two events are separated by more space than time, and they are hence said to bespacelike separated. This implies that, and given the Lorentz transformation, there exists a less than for which (in particular,). 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,, is called theproper distance, orproper length. For values of greater than and less than, the sign of 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 could be greater than would be if.
Δs2 = 0: In this case, the two events are said to belightlike separated. This implies that, and this relationship is frame independent due to the invariance of. From this, we observe that the speed of light is 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 theory.
The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.
The form of, being thedifference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. The invariance of Δs2 under standard Lorentz transformation is analogous to the invariance of squared distances Δr2 under rotations in Euclidean space.[citation needed] Although space and time have an equalfooting 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,Minkowski space differs fromfour-dimensional Euclidean space. The invariance of this interval is a property of thegeneral Lorentz transform (also called thePoincaré transformation), making it anisometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is,Lorentz boosts, in the x-direction) with all othertranslations,reflections, androtations between any Cartesian inertial frame.[31]: 33–34
In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:
Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:[24]
The value of is hence independent of the frame in which it is measured.
Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observerO. In a reference frame moving atv = 0.3c, as measured byO, the events occur in the order C, B, A. In a reference frame moving atv = −0.5c with respect toO, the events occur in the order A, B, C. The white lines, thelines of simultaneity, 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 thelight cone with respect to the origin of all considered frames.
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 ofabsolute simultaneity).
FromEquation 3 (the forward Lorentz transformation in terms of coordinate differences)
It is clear that the two events that are simultaneous in frameS (satisfyingΔt = 0), are not necessarily simultaneous in another inertial frameS′ (satisfyingΔt′ = 0). Only if these events are additionally co-local in frameS (satisfyingΔx = 0), will they be simultaneous in another frameS′.
TheSagnac effect can be considered a manifestation of the relativity of simultaneity for local inertial frames comoving with a rotating Earth.[32] Instruments based on the Sagnac effect for their operation, such asring laser gyroscopes andfiber optic gyroscopes, are capable of extreme levels of sensitivity.[p 8]
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.
Suppose aclock is at rest in the unprimed systemS. The location of the clock on two different ticks is then characterized byΔx = 0. To find the relation between the times between these ticks as measured in both systems,Equation 3 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′), islonger than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speedmuons 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.[33]
Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame
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 the same point.[34]: 149–152
The measurements that we would get if we actuallylooked at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by theDoppler effect. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.[34]: 149–152
Figure 4–3. Thought experiment using a light-clock to explain time dilation
Paul Langevin, 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[35]), which he used to perform a novel derivation of the Lorentz transformation.[36]
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 mirrors.[37][38][39][40] (Langevin himself made use of a light-clock oriented parallel to its line of motion.[36])
Consider the scenario illustrated inFig. 4-3A. Observer A holds a light-clock of length 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 interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds. A time interval measured using a single clock that is motionless in a particular reference frame is called aproper time interval.[41]
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. 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 that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as so that the total round-trip time of the pulses is. 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 thereforenot a proper time interval because he did not measure it with a single resting clock.[41]
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 to the left, in which case observer A would perceive B's clock as running slower than her local clock.
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 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 withboth A's and B's measurement of time.[42]
The reciprocity of time dilation between two observers in separate inertial frames leads to the so-calledtwin paradox, articulated in its present form by Langevin in 1911.[43] 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.
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 order for the two observers to perform side-by-side comparisons of their elapsed 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.[44]
Figure 4-4. Doppler analysis of twin paradox
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 theTwin paradox article. We will examine in the following one such solution to the paradox.
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.Fig 4-4 illustrates a scenario where the traveling twin flies at0.6 c to and from a star3 ly 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. 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, she receives signals at the enhanced rate of. 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 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.[34]: 152–159
Item
Measured by the stay-at-home
Fig 4-4
Measured by the traveler
Fig 4-4
Total time of trip
10 yr
8 yr
Total number of pulses sent
10
8
Time when traveler's turnaround isdetected
8 yr
4 yr
Number of pulses received at initial rate
4
2
Time for remainder of trip
2 yr
4 yr
Number of signals received at final rate
4
8
Total number of received pulses
8
10
Twin's calculation as to how much theother twin should have aged
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).
Similarly, suppose ameasuring rod is at rest and aligned along thex-axis in the unprimed systemS. In this system, the length of this rod is written as Δx. To measure the length of this rod in the systemS′, in which the rod is moving, the distancesx′ to the end points of the rod must be measured simultaneously in that systemS′. In other words, the measurement is characterized byΔt′ = 0, which can be combined withEquation 4 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′), isshorter than its length (Δx) in its own rest frame (S).
Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuringtime intervals 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) aredifferent 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 not co-local, but are separated by distance (space), they willnot occur at the samespatial distance from each other when seen from another moving coordinate system.
Substituting expressions for and fromEquation 5 intoEquation 8, followed by straightforward mathematical manipulations and back-substitution fromEquation 7 yields the Lorentz transformation of the speed to:
Equation 10 andEquation 14 can be interpreted as giving theresultant of the two velocities and, and they replace the formula. which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as therelativistic velocity addition (or composition) formulas, valid for the three axes ofS andS′ being aligned with each other (although not necessarily in standard configuration).[13]: 47–49
We note the following points:
If an object (e.g., aphoton) were moving at the speed of light in one frame(i.e.,u = ±coru′ = ±c), then it would also be moving at the speed of light in any other frame, moving at|v| <c.
The resultant speed of two velocities with magnitude less thanc is always a velocity with magnitude less thanc.
If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g.,|u/c| ≪1), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.
There is nothing special about thex direction in the standard configuration. The aboveformalism 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. SeeVelocity-addition formula for details.
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.
Thomas rotation results from the relativity of simultaneity. In Fig. 4-5a, a rod of length in its rest frame (i.e., having aproper length of) rises vertically along the y-axis in the ground frame.
In Fig. 4-5b, the same rod is observed from the frame of a rocket moving at speed to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronizedin the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (notsee) the clock at the right end of the rod as being advanced in time by, and the rod is correspondingly observed as tilted.[4]: 98–99
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 thespin of moving particles, whereThomas precession is a relativistic correction that applies to thespin of an elementary particle or the rotation of a macroscopicgyroscope, relating theangular velocity of the spin of a particle following acurvilinear orbit to the angular velocity of the orbital motion.[4]: 169–174
Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".[p 9][4]: 98–99
Causality and prohibition of motion faster than light
In Fig. 4-6, the time interval between the events A (the "cause") and B (the "effect") is 'timelike'; that is, there is a frame of reference in which events A and B occur at thesame location in space, 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).
The interval AC in the diagram is 'spacelike'; 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 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.
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).[45][p 10] A variety of causal paradoxes could then be constructed.
Figure 4-7. Causality violation by the use of fictitious "instantaneous communicators"
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. Theworld lines of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ct′), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.
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 axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrivesearlier than it was sent.
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 axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, in the primed frames of C and D, B will receive the message before it was sent out, a violation of causality.[46]
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 axis (and the signal from A to B slightly steeper than the 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 and 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 onlyslightly faster than the speed of light will result in causality violation.[47]
Therefore,ifcausality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travelfaster than light in vacuum.
This is not to say thatall faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.[48] 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 (although this does not violate causality or any other relativistic phenomenon).[49][50]
Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.
In 1850,Hippolyte Fizeau andLéon Foucault independently established that light travels more slowly in water than in air, thus validating a prediction ofFresnel'swave theory of light and invalidating the corresponding prediction of Newton'scorpuscular theory.[51] The speed of light was measured in still water. What would be the speed of light in flowing water?
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 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.
According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speedthrough the medium plus the speedof 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 is the speed of light in still water, and is the speed of the water, and 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
Fizeau's results, although consistent with Fresnel's earlier hypothesis ofpartial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since depends on wavelength,the aether must be capable of sustaining different motions at the same time.[note 1] A variety of theoretical explanations were proposed to explainFresnel's dragging coefficient, 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.[52]
From the point of view of special relativity, Fizeau's result is nothing but an approximation toEquation 10, the relativistic formula for composition of velocities.[31]
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 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 theaberration of light. 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.[53] (2) If the source is in motion, the displacement would be the consequence oflight-time correction. 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.[54]
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,Arago used this expected phenomenon in a failed attempt to measure the speed of light,[55] and in 1870,George Airy 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.[56] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,[57] but was incompatible with the results of the Michelson–Morley experiment, which apparently demandedcomplete aether-drag.[58]
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[31]: 57–60
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 away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of atime dilation term, and that is the treatment described here.[59][60]
Assume the receiver and the source are movingaway from each other with a relative speed as measured by an observer on the receiver or the source (The sign convention adopted here is that isnegative if the receiver and the source are movingtowards each other). Assume that the source is stationary in the medium. Thenwhere is the speed of sound.
For light, and with the receiver moving at relativistic speeds, clocks on the receiver aretime dilated relative to clocks at the source. The receiver will measure the received frequency to bewhere
An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of thereceiver with a moving source.[61][24]: 540
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.
The transverseDoppler effect is one of the main novel predictions of the special theory of relativity.
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.
Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.[24]: 541 In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of. In Fig. 5-3b, the light is redshifted by the same factor.
Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.
Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact ofDoppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.
Scientists make a fundamental distinction betweenmeasurement orobservation on the one hand, versusvisual appearance, or what onesees. 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.
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.
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 beseen as length contracted. In 1959, James Terrell andRoger Penrose 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 wouldappear contracted, an approaching object wouldappear elongated, and a passing object would have a skew appearance that has been likened to a rotation.[p 13][p 14][62][63] A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, althoughthe surface of the sphere and the images on it will appear distorted.[64][65]
Figure 5–6. GalaxyM87 sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.
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 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.[66] This illusion has come to be known asTerrell rotation or theTerrell–Penrose effect.
Another example where visual appearance is at odds with measurement comes from the observation of apparentsuperluminal motion in variousradio galaxies,BL Lac objects,quasars, and other astronomical objects that ejectrelativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.[67][68][69] In Fig. 5-6, galaxyM87 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.[70]
Section§ Consequences derived from the Lorentz transformation dealt strictly withkinematics, 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.
Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form afour-vector 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(E/c, 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(E/c,Ev/c2, 0, 0). The momentum is equal to the energy multiplied by the velocity divided byc2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal toE/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 do not 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.[p 1] The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.[p 15] 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.[71] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.[72]
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.[p 16]
In his fourth of his 1905Annus mirabilis papers,[p 15] 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.
Fig. 6-1 (top). Consider a system of plane waves of light having frequency traveling in direction relative to the x-axis of reference frameS. The frequency (and hence energy) of the waves as measured in frameS′ that is moving along the x-axis at velocity is given by the relativistic Doppler shift formula that Einstein had developed in his 1905 paper on special relativity:[p 1]
Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frameS. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle with respect to the x-axis. Each pulse has energy. Because of conservation of momentum, the body remains stationary inS after emission of the two pulses. Let be the energy of the body before emission of the two pulses and after their emission.
Next, consider the same system observed from frameS′ that is moving along the x-axis at speed relative to frameS. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let be the energy of the body measured in reference frameS′ before emission of the two pulses and after their emission. We obtain the following relationships:[p 15]
From the above equations, we obtain the following:
The two differences of form seen in the above equation have a straightforward physical interpretation. Since and are the energies of the arbitrary body in the moving and stationary frames, and 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,
Comparing the above expression with the classical expression for kinetic energy,K.E. = 1/2mv2, Einstein then noted: "If a body gives off the energyL in the form of radiation, its mass diminishes byL/c2."
Rindler has observed that Einstein's heuristic argument suggested merely that energycontributes 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 thatall inertial mass represented a reserve of energy. "To equateall mass with energy required an act of aesthetic faith, very characteristic of Einstein."[13]: 81–84 Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.
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 shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.[73]
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 that exist within the limit of 100 light years from Earth. 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 constant1g, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by:wherev(t) is the velocity at a timet,a is the acceleration of the spaceship andt is the coordinate time as measured by people on Earth.[p 17] Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling atv = 0.712c and0.946c 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 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.[74] A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at1.1g will take148000 years and cover about140000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.[74] This same time dilation is why a muon travelling close toc is observed to travel much farther thanc times itshalf-life (when at rest).[75]
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 particles, requires special relativity.[76]
In Newtonian mechanics, analysis of collisions involves use of theconservation laws for mass,momentum andenergy. 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. (Inelastic collisions are discussed inSpacetime#Conservation laws. Radioactive decay may be considered a sort of time-reversed inelastic collision.[76])
Elastic scattering of charged elementary particles deviates from ideality due to the production ofBremsstrahlung radiation.[77][78]
Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle
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 stationary frame, an incident sphere traveling at 2v strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±v. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±u. Energy conservation requires that |u| = |v|. (c) Reverting to the stationary frame, the rebound velocities arev ±u. The dot product(v +u) ⋅ (v −u) =v2 −u2 = 0, indicating that the vectors are orthogonal.[13]: 26–27
Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle
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 smaller as the velocity of the incident particle approaches the speed of light:
The relativistic momentum and total relativistic energy of a particle are given by
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:
Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc2) equals the sum of the total energies of the emergent particles:
Breaking down (6-5) into its components, replacing with the dimensionless, and factoring out common terms from (6-5) and (6-6) yields the following:[p 18]
Figure 7-1a. A ray through theunit circlex2 +y2 = 1 in the point(cosa, sina), wherea is twice the area between the ray, the circle, and thex-axis.
Figure 7-1b. A ray through theunit hyperbolax2 −y2 = 1 in the point(cosha, sinha), wherea is twice the area between the ray, the hyperbola, and thex-axis.
Figure 7–2. Plot of the three basicHyperbolic functions: hyperbolic sine (sinh), hyperbolic cosine (cosh) and hyperbolic tangent (tanh). Sinh is red, cosh is blue and tanh is green.
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 formulas.
This nonlinearity is an artifact of our choice of parameters.[4]: 47–59 We have previously noted that in anx–ct 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.
The natural functions for expressing these relationships are thehyperbolic analogs of the trigonometric functions. Fig. 7-1a shows aunit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being thata is interpreted, not as the angle between the ray and thex-axis, but as twice the area of the sector swept out by the ray from thex-axis. Numerically, the angle and2 × area measures for the unit circle are identical. Fig. 7-1b shows aunit hyperbola with sinh(a) and cosh(a), wherea is likewise interpreted as twice the tinted area.[79] Fig. 7-2 presents plots of the sinh, cosh, and tanh functions.
For the unit circle, the slope of the ray is given by
In the Cartesian plane, rotation of point(x,y) into point(x',y') by angleθ is given by
In a spacetime diagram, the velocity parameter is the analog of slope. Therapidity,φ, is defined by[24]: 543
where
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;[24]: 544
or in other words,.
The Lorentz transformations take a simple form when expressed in terms of rapidity. Theγ factor can be written as
Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are calledboosts.
Substitutingγ andγβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in thex-direction may be written as
and the inverse Lorentz boost in thex-direction may be written as
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.[4]
Figure 10–1. Orthogonality and rotation of coordinate systems compared betweenleft:Euclidean space through circularangleφ,right: inMinkowski spacetime throughhyperbolic angleφ (red lines labelledc denote theworldlines of a light signal, a vector is orthogonal to itself if it lies on this line).[80]
The physical theory of special relativity was recast byHermann Minkowski in a 4-dimensional geometry now called Minkowski space. Minkowski spacetime appears to be very similar to the standard 3-dimensionalEuclidean space, but there is a crucial difference with respect to time.In 3D space, thedifferential of distance (line element)ds is defined bywheredx = (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinateX0 derived from time, such that the distance differential fulfillswheredX = (dX0,dX1,dX2,dX3) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply arotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1).[81] Just as Euclidean space uses aEuclidean metric, so spacetime uses aMinkowski metric.Basically, special relativity can be stated as theinvariance of any spacetime interval (that is the 4D distance between any two events) when viewed fromany inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (thePoincaré group) of Minkowski spacetime.
The actual form ofds above depends on the metric and on the choices for theX0 coordinate.To make the time coordinate look like the space coordinates, it can be treated asimaginary:X0 =ict (this is called aWick rotation).According toMisner, Thorne and Wheeler (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 takeX0 =ct, rather than a "disguised" Euclidean metric usingict as the time coordinate.
Some authors useX0 =t, with factors ofc elsewhere to compensate; for instance, spatial coordinates are divided byc or factors ofc±2 are included in the metric tensor.[82]These numerous conventions can be superseded by usingnatural units wherec = 1. Then space and time have equivalent units, and no factors ofc appear anywhere.
A four dimensional space has four-dimensional vectors, or "four-vectors". The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike componentct and spacelike componentx = (x,y,z), in acontravariantpositionfour-vector with components:where we defineX0 =ct so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.[83][84][85]
4‑vectors, and more generallytensors, simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that aremanifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance ofMaxwell's equations in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using thefield strength tensor formulation.[86]
A 4-tuple, is a "4-vector" if its componentAi transform between frames according to the Lorentz transformation.
If using coordinates,A is a4–vector if it transforms (in thex-direction) according to
which comes from simply replacingct withA0 andx withA1 in the earlier presentation of theLorentz transformation.
As usual, when we writex,t, etc. we generally mean Δx, Δt etc.
The last three components of a4–vector must be a standard vector in three-dimensional space. Therefore, a4–vector must transform like under Lorentz transformations as well as rotations.[87]: 36–59
Closure under linear combination: IfA andB are4-vectors, then is also a4-vector.
Inner-product invariance: IfA andB are4-vectors, 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 a3-vector. In the following, and are3-vectors:
In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in3-space.
Two vectors are said to beorthogonal if. Unlike the case with3-vectors, orthogonal4-vectors are not necessarily at right angles to each other. The rule is that two4-vectors 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 lightlike4-vector is orthogonal toitself.
Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a4-vector 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, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval and the invariant length of the relativistic momentum vector.[24]: 639 [87]: 36–59
Displacement 4-vector: Otherwise known as thespacetime separation, this is(Δt, Δx, Δy, Δz), or for infinitesimal separations,(dt,dx,dy,dz).
Velocity 4-vector: This results when the displacement4-vector is divided by, where is the proper time between the two events that yielddt,dx,dy, anddz.
Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
The4-velocity 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.
An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found that is momentarily comoving with the particle. This frame, themomentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles.
Since photons move on null lines, for a photon, and a4-velocity 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.
Energy–momentum 4-vector:
As indicated before, there are varying treatments for the energy–momentum4-vector so that one may also see it expressed as or. 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–momentum4-vector is a conserved quantity.
Acceleration 4-vector: This results from taking the derivative of the velocity4-vector with respect to.
Force 4-vector: This is the derivative of the momentum4-vector with respect to
As expected, the final components of the above4-vectors are all standard3-vectors corresponding to spatial3-momentum,3-force etc.[87]: 36–59
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 Einstein preferred to change the definition of momentum to one involving4-vectors rather than give up on conservation of momentum.
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 involving4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from4-vectors.[24]: 644 General relativity from the outset relies heavily on4‑vectors, and more generally tensors, representing physically relevant entities.
Special relativity does accommodateaccelerations as well asaccelerating frames of reference.[88]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.[89] It is only when gravitation is significant that general relativity is required.[90]
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, or rotating. To accommodate this difference, general relativity uses curved spacetime.[90]
In this section, we analyze several scenarios involving accelerated reference frames.
The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.[citation needed]
Figure 7–4. Dewan–Beran–Bell spaceship paradox
In Fig. 7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string that 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 direction along the line between them with the same constant proper acceleration. 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 measured. Will the string break?
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.
To observers in the rest frame, the spaceships start a distanceL apart and remain the same distance apart during acceleration. During acceleration,L is a length contracted distance of the distanceL' = γL in the frame of the accelerating spaceships. After a sufficiently long time,γ will increase to a sufficiently large factor that the string must break.
LetA andB 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.A says thatB has the same acceleration that he has, andB sees thatA matches her every move. So the spaceships stay the same distance apart, and the string does not break.
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 correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.
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 and after acceleration stops.
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 acceleration for proper time (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 turns out to be greater than the length along the line of simultaneity.
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. If and are the ships' positions in, the positions in frame are:[91]
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. As shown in Fig. 7-5, Bell's example asserts the moving lengths and measured in frame to be fixed, thereby forcing the rest frame length in frame to increase.
Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such asevent horizons. In the text accompanyingSection "Invariant hyperbola" of the article Spacetime, 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 justapproaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.
Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewedhere.
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 parameter approaches a limit of one as increases. Likewise, approaches infinity.
The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:
We remember that.
Since, we conclude that.
From the relativistic force law,.
Substituting from step 2 and the expression for from step 3 yields, which is a constant expression.[92]: 110–113
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 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 shenever receives any communications from Terence after 100 hours on his clock (dashed green lines).[92]: 110–113
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 toreceive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by anapparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.[92]: 110–113
Theoretical investigation inclassical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of theE andB fields required certain behaviors on charged particles. The general study of moving charges forms theLiénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of theelectric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called themagnetic field. Conversely, themagnetic field generated by a moving charge disappears and becomes a purelyelectrostatic field in a comoving frame of reference.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 ofelectromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in amanifestly covariant form, that is, in the language oftensor calculus.[86]
In 1928,Paul Dirac constructed an influentialrelativistic wave equation, now known as theDirac equation in his honour,[p 19] 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 calledspin, it also led to the prediction of theantiparticle of the electron (thepositron),[p 19][p 20] andfine structure could only be fully explained with special relativity. It was the first foundation ofrelativistic quantum mechanics.
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.[citation needed] Instead, a theory of particles interpreted as quantized fields, calledquantum field theory, becomes necessary; in which particles can becreated and destroyed throughout space and time.
Special relativity in itsMinkowski spacetime is accurate only when theabsolute value of thegravitational potential is much less thanc2 in the region of interest.[94] In a strong gravitational field, one must usegeneral relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at thePlanck length and below, quantum effects must be taken into consideration resulting inquantum gravity. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[95]and thus accepted by the physics community. Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.[96]
Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notablyquantum 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. SeeClassical 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,[97] 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.[21]
TheFizeau 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 travelling together at the same velocity at all times.
TheTrouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
TheExperiments of Rayleigh and Brace (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.
Particle accelerators 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 earlierNewtonian 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:
^The refractive index dependence of the presumed partial aether-drag was eventually confirmed byPieter Zeeman in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly toAmsterdam's main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).[p 11][p 12]
^Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920
^Yaakov Friedman (2004).Physical Applications of Homogeneous Balls. Progress in Mathematical Physics. Vol. 40. pp. 1–21.ISBN978-0-8176-3339-4.
^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 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.
^Das, A. (1993)The Special Theory of Relativity, A Mathematical Exposition, Springer,ISBN0-387-94042-1.
^Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited,ISBN0-582-31760-6.
^abcdefghijMermin, N. David (2009).It's about time: understanding Einstein's relativity (Sixth printing, and first paperback printing ed.). Princeton, NJ and Oxford: Princeton University Press.ISBN978-0-691-14127-5.
^Carmichael, R.D. (1920).The Theory of Relativity. United Kingdom: John Wiley & Sons, Incorporated.
^Einstein, Albert (1970). "The Foundation of the General Theory of Relativity". In Lorentz, Hendrik A.; Einstein, Albert (eds.).The principle of relativity: a collection of orig. memoirs on the special and general theory of relativity (Nachdr. d. Ausg. 1923 ed.). New York: Dover.ISBN978-0-486-60081-9.
^Michael Polanyi (1974)Personal Knowledge: Towards a Post-Critical Philosophy,ISBN0-226-67288-3, 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".[1]
^Collier, Peter (2017).A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity (3rd ed.). Incomprehensible Books.ISBN978-0-9573894-6-5.
^Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift",Einstein's generation. The origins of the relativity revolution, Chicago: University of Chicago Press,ISBN0-226-77057-5
^Debs, Talal A.; Redhead, Michael L.G. (1996). "The twin "paradox" and the conventionality of simultaneity".American Journal of Physics.64 (4):384–392.Bibcode:1996AmJPh..64..384D.doi:10.1119/1.18252.
^Kraus, Ute."The Ball is Round".Space Time Travel: Relativity visualized. Institut für Physik Universität Hildesheim. Archived fromthe original on 12 May 2017. Retrieved16 April 2017.
^Zensus, J. Anton; Pearson, Timothy J. (1987).Superluminal Radio Sources (1st ed.). Cambridge, New York: Cambridge University Press. p. 3.ISBN978-0-521-34560-6.
^Chase, Scott I."Apparent Superluminal Velocity of Galaxies".The Original Usenet Physics FAQ. Department of Mathematics, University of California, Riverside. Retrieved12 April 2017.
^Fernflores, Francisco (2018).Einstein's Mass–Energy Equation, Volume I: Early History and Philosophical Foundations. New York: Momentum Pres.ISBN978-1-60650-857-2.
^Halbert, M.L. (1972). "Review of Experiments on Nucleon-Nucleon Bremsstrahlung". In Austin, S.M.; Crawley, G.M. (eds.).The Two-Body Force in Nuclei. Boston, MA.: Springer.
^Thomas, George B.; Weir, Maurice D.; Hass, Joel; Giordano, Frank R. (2008).Thomas' Calculus: Early Transcendentals (Eleventh ed.). Boston: Pearson Education, Inc. p. 533.ISBN978-0-321-49575-4.
^J.A. Wheeler; C. Misner; K.S. Thorne (1973).Gravitation. W.H. Freeman & Co. p. 58.ISBN978-0-7167-0344-0.
^J.R. Forshaw; A.G. Smith (2009).Dynamics and Relativity. Wiley. p. 247.ISBN978-0-470-01460-8.
^abE. J. Post (1962).Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications Inc.ISBN978-0-486-65427-0.{{cite book}}:ISBN / Date incompatibility (help)
^abcSchutz, Bernard F. (1985).A first course in general relativity. Cambridge, UK: Cambridge University Press. p. 26.ISBN0-521-27703-5.
Logunov, Anatoly A. (2005).Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow.
Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press,ISBN0-19-927583-1;ISBN978-0-19-927583-0
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".Physics Letters.12 (3):260–262.Bibcode:1964PhL....12..260A.doi:10.1016/0031-9163(64)91095-9.
Einstein Light Anaward-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
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.
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).
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.
lightspeed An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.
Animation showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.
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