special relativity


  • 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).

  • A more mathematical statement of the principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is: Special principle of
    relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively
    to K.[18] Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations.

  • Relativity without the second postulate[edit] 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.

  • He wrote: The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new
    type (“Lorentz transformation”) are postulated for the conversion of coordinates and times of events …

  • 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.

  • 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

  • Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula
    , where is the speed of light in a vacuum.

  • Invariant interval[edit] In Galilean relativity, an object’s length ()[note 3] 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 his initial presentation of special relativity in 1905 he expressed these postulates as:[p 1] • The principle of relativity – the laws by which the states of physical systems
    undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.

  • The equations that relate measurements made in different frames are called transformation equations.

  • Let’s call this reference frame S. In relativity theory, we often want to calculate the coordinates of an event from differing reference frames.

  • 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).

  • The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including
    the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.

  • 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.

  • [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).

  • This is a restricting principle for natural laws …[p 5] Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance,
    or, equivalently, on the single postulate of Minkowski spacetime.

  • 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.

  • 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.

  • [1][2] It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position.

  • 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.

  • Lack of an absolute reference frame[edit] The principle of relativity, which states that physical laws have the same form in each inertial reference frame, dates back to Galileo,
    and was incorporated into Newtonian physics.

  • [9][10] Just as Galilean relativity is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation
    of general relativity that is valid for weak gravitational fields, that is, at a sufficiently small scale (e.g., when tidal forces are negligible) and in conditions of free fall.

  • 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.

  • 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.

  • By the principle of relativity, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −v.

  • [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:[21] The value of is hence independent of the frame in which it is measured.

  • Standard configuration[edit] To gain insight into how the spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to
    work with a simplified setup with frames in a standard configuration.

  • [21] To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S’, in standard configuration, as shown in Fig.

  • The transformation can apply to the y- or z-axis, or indeed in any direction parallel to the motion (which are warped by the γ factor) and perpendicular; see the article Lorentz
    transformation for details.

  • 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.

  • 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).

  • [12] 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.

  • Lorentz transformation and its inverse[edit] Define an event to have spacetime coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in a reference frame moving at a velocity
    v on the x axis with respect to that frame, S′.

  • [p 1] That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the “stationary system”),
    regardless of the state of motion of the light source.

  • The aether was thought to be an absolute 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.

  • [27] 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.

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

  • Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic

  • Einstein extended this principle so that it accounted for the constant speed of light,[11] a phenomenon that had been observed in the Michelson–Morley experiment.

  • In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

  • As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small
    neighborhood of each point in this curved spacetime.

  • Then the Lorentz transformation specifies that these coordinates are related in the following way: where is the Lorentz factor and c is the speed of light in vacuum, and the
    velocity v of S′, relative to S, is parallel to the x-axis.

  • 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 ultimately it approaches infinity as Fig.

  • 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.

  • The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.

  • The primed coordinates of and are related to the unprimed coordinates through the Lorentz transformations and could 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.

  • [p 6] Following Einstein’s original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.

  • Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible.

  • [p 8][21] Lorentz invariance as the essential core of special relativity Alternative approaches to special relativity[edit] Main article: Derivations of the Lorentz transformations
    Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance.

  • But in the late 19th century the existence of electromagnetic 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).

  • Albert Einstein: Autobiographical Notes[p 5] 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.

  • [p 7] Principle of relativity Reference frames and relative motion[edit] Figure 2–1.

  • This implies that and given the Lorentz transformation there exists a less than for which (in particular, ).

  • [17] But the most common set of postulates remains those employed by Einstein in his original paper.

  • [35]: 152–159  Length contraction[edit] See also: Lorentz contraction The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results
    of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).

  • [12]: 47–49  We note the following points: • If an object (e.g., a photon) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±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 than c is always a velocity with magnitude less than c. • 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.

  • In 1959, James Terrell and Roger 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.

  • 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.

  • 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.

  • 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′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S).

  • 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.

  • 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.

  • These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the
    first, unless the events, in addition to being co-local, are also simultaneous.

  • For the interval between these two events, observer A finds A time interval measured using a single clock which is motionless in a particular reference frame is called a proper
    time interval.

  • 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.

  • Time dilation is explicitly related to our way of measuring time intervals between events that occur at the same place in a given coordinate system (called “co-local” events).

  • Illustration of stellar aberration 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.

  • 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.

  • In other words, the measurement is characterized by Δt′ = 0, which can be combined with Equation 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′), is shorter than its length (Δx) in its own rest frame (S).

  • Relativity of simultaneity[edit] See also: Relativity of simultaneity and Ladder paradox Consider two events happening in two different locations that occur simultaneously
    in the reference frame of one inertial observer.

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

  • [48] Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light
    in vacuum.

  • [p 14] Time dilation[edit] See also: Time dilation The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds
    of the observers’ reference frames.

  • [note 9] Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects,
    quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer.

  • 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 of partial aether dragging, were extremely disconcerting to physicists of the time.

  • [35]: 149–152  The measurements that we would get if we actually looked at a moving clock would, in general, not at all be the same thing, because the time that would see
    would be delayed by the finite speed of light, i.e the times that we see would be distorted by the Doppler effect.

  • 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 a time dilation term, and that is the treatment described here.

  • [32]: 98–99  Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities.

  • 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.

  • The world 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.

  • Measurements of these effects are not an artifact of Doppler 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.

  • The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.

  • If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate

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

  • If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the
    rocket frame to observe (not see) the clock at the right end of the rod as being advanced in time by and the rod is correspondingly observed as tilted.

  • In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light,[56] 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.

  • Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons 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.

  • 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.

  • Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice.

  • 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.

  • 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.

  • From Equation 3 (the forward Lorentz transformation in terms of coordinate differences) It is clear that the two events that are simultaneous in frame S (satisfying Δt = 0),
    are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0).

  • 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).

  • 4-6, the time interval between the events A (the “cause”) and B (the “effect”) is ‘time-like’; that is, there is a frame of reference in which events A and B occur at the
    same location in space, separated only by occurring at different times.

  • There is nothing special about the x direction in the standard configuration.

  • An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.

  • 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.

  • 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.

  • If C’s reference frame coincides with neither A’s frame nor B’s frame, then C’s measurement of time will disagree with both A’s and B’s measurement of 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.

  • Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0.

  • We can write Substituting expressions for and from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7
    yields the Lorentz transformation of the speed to : The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing with For not aligned along the x-axis, we write:[12]: 47–49  The forward and inverse transformations
    for this case are: Equation 10 and Equation 14 can be interpreted as giving the resultant of the two velocities and and they replace the formula which is valid in Galilean relativity.

  • Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees.

  • With this modified setup, it can be demonstrated that even signals only slightly faster than the speed of light will result in causality violation.

  • In reality, there is no paradox at all, because in order for the two observers to compare their proper times, the symmetry of the situation must be broken: At least one of
    the two observers must change their state of motion to match that of the other.

  • 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.

  • They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).

  • An apparent optical illusion results giving the appearance of faster than light travel.

  • 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.

  • [52] The speed of light was measured in still water.

  • To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system

  • [33] Since relativity of simultaneity is a first order effect in ,[21] instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber
    optic gyroscopes, are capable of extreme levels of sensitivity.

  • Lorentz transformation of velocities[edit] See also: Velocity-addition formula Consider two frames S and S′ in standard configuration.

  • Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned
    with each other (although not necessarily in standard configuration).

  • 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.

  • 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.

  • 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.

  • [53] From the point of view of special relativity, Fizeau’s result is nothing but an approximation to Equation 10, the relativistic formula for composition of velocities.

  • [49] 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).

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

  • This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),[p
    25][p 26] and fine structure could only be fully explained with special relativity.

  • Relativistic dynamics and invariance[edit] The invariant magnitude of the momentum 4-vector generates the energy–momentum relation: We can work out what this invariant is
    by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

  • 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 23]
    From these we obtain the following relationships:[p 23] For the symmetrical case in which and (6-12) takes on the simpler form:[p 23] How far can you travel from the Earth?

  • The Lorentz transformation of the electric field of a moving charge into a non-moving observer’s reference frame results in the appearance of a mathematical term commonly
    called the magnetic field.

  • 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.

  • An example of a four-dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum,
    and the other three are related to the boost of the center of mass of the system.

  • 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.

  • At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity.

  • Equivalence of mass and energy[edit] Main article: Mass–energy equivalence As an object’s speed approaches the speed of light from an observer’s point of view, its relativistic
    mass increases thereby making it more and more difficult to accelerate it from within the observer’s frame of reference.

  • A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.

  • Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general
    relativity (in the limiting case of negligible gravitational fields).

  • Physics in spacetime[edit] Transformations of physical quantities between reference frames[edit] Above, the Lorentz transformation for the time coordinate and three space
    coordinates illustrates that they are intertwined.

  • The rest energy is related to the mass according to the celebrated equation discussed above: The mass of systems measured in their center of momentum frame (where total momentum
    is zero) is given by the total energy of the system in this frame.

  • Consider a system of plane waves of light having frequency traveling in direction relative to the x-axis of reference frame S. The frequency (and hence energy) of the waves
    as measured in frame S′ that is moving along the x-axis at velocity is given by the relativistic Doppler shift formula which Einstein had developed in his 1905 paper on special relativity:[p 1] Fig.

  • The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components
    (the momentum) in a non-trivial way.

  • Metric[edit] The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector.

  • 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.

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

  • According to Misner, 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 take , rather than a “disguised” Euclidean metric using ict as the time coordinate.

  • 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).

  • 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.

  • The covariant version of the four-force is: In the rest frame of the object, the time component of the four-force is zero unless the “invariant mass” of the object is changing
    (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times

  • 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.

  • The 4-velocity Uμ has an invariant form: which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being
    at coordinate rest in relativity: at the least, you are always moving forward through time.

  • Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is: Invariant means that it takes the same value in all inertial
    frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation.

  • 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 earlier Newtonian mechanics.

  • Status Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c2 in the region of interest.

  • 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 four-acceleration is the proper time derivative of 4-velocity: The transformation rules for three-dimensional velocities and accelerations are very awkward; even above
    in standard configuration the velocity equations are quite complicated owing to their non-linearity.

  • The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.

  • We obtain the following relationships:[p 21] From the above equations, we obtain the following: The two differences of form seen in the above equation have a straightforward
    physical interpretation.

  • 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.

  • Maxwell’s equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe.

  • This is described by: where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth.

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

  • The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics
    of the time.

  • The magnitude of the 4-vector T is the positive square root of the inner product with itself: One can extend this idea to tensors of higher order, for a second order tensor
    we can form the invariants: similarly for higher order tensors.

  • Then space and time have equivalent units, and no factors of c appear anywhere.

  • Some examples: • Tests of relativistic energy and momentum – testing the limiting speed of particles • Ives–Stilwell experiment – testing relativistic Doppler effect and time
    dilation • Experimental testing of time dilation – relativistic effects on a fast-moving particle’s half-life • Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations • Hughes–Drever experiment – testing isotropy
    of space and mass • Modern searches for Lorentz violation – various modern tests • Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.

  • [82] In a strong gravitational field, one must use general relativity.

  • Maxwell’s equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant
    form, that is, in the language of tensor calculus.

  • Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout
    space and time.

  • So to transform from one frame to another, we use the well-known tensor transformation law[91] where is the reciprocal matrix of .

  • Einstein took as starting assumptions his recently discovered formula for relativistic Doppler shift, the laws of conservation of energy and conservation of momentum, and
    the relationship between the frequency of light and its energy as implied by Maxwell’s equations.

  • But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy [83] and thus
    accepted by the physics community.

  • As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields.

  • Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles.

  • Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.

  • Dynamics Section Consequences derived from the Lorentz transformation dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without
    considering the forces that caused the motion.

  • [81] In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,[p 25] that is fully compatible both with special
    relativity and with the final version of quantum theory existing after 1926.

  • Let be the energy of the body measured in reference frame S′ before emission of the two pulses and after their emission.

  • In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.

  • [74] Elastic collisions[edit] 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.

  • If a particle is not traveling at c, one can transform the 3D force from the particle’s co-moving reference frame into the observer’s reference frame.

  • The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought experiments in special relativity.

  • [p 1] The first of Einstein’s papers on this subject was “Does the Inertia of a Body Depend upon its Energy Content?”

  • Consider an arbitrary body that is stationary in reference frame S. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle with respect to
    the x-axis.

  • More generally, all contravariant components of a four-vector transform from one frame to another frame by a Lorentz transformation: Examples of other 4-vectors include the
    four-velocity defined as the derivative of the position 4-vector with respect to proper time: where the Lorentz factor is: The relativistic energy and relativistic momentum of an object are respectively the timelike and spacelike components
    of a contravariant four-momentum vector: where m is the invariant mass.

  • If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space we see that the null geodesics lie along a dual-cone (see Fig.

  • How general relativity and quantum mechanics can be unified is one of the unsolved problems in physics; quantum gravity and a “theory of everything”, which require a unification
    including general relativity too, are active and ongoing areas in theoretical research.

  • To use Newton’s third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate.

  • Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs.

  • 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.

  • 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

  • In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least
    four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc2.

  • Next, consider the same system observed from frame S′ that is moving along the x-axis at speed relative to frame S. In this frame, light from the forwards and reverse pulses
    will be relativistically Doppler-shifted.

  • 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

  • The Lorentz transformation in standard configuration above, that is, for a boost in the x-direction, can be recast into matrix form as follows: In Newtonian mechanics, quantities
    that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time.

  • • The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable.


Works Cited

[‘1. Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes “The word ‘special’ is meant to intimate that the principle is restricted to the case …”. See p. 111 of The Principle of Relativity, A. Einstein,
H. A. Lorentz, H. Weyl, H. Minkowski, Dover reprint of 1923 translation by Methuen and Company.]
2. ^ Wald, General Relativity, p. 60: “… the special theory of relativity asserts that spacetime is the manifold with a flat metric of Lorentz signature
defined on it. Conversely, the entire content of special relativity … is contained in this statement …”
3. ^ In a spacetime setting, the length of a moving rigid object is the spatial distance between the ends of the object measured at the same
time. In the rest frame of the object the simultaneity is not required.
4. ^ The results of the Michelson–Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz
believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.[28]: 62–68
Lorentz expected that length contraction would result
in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation[p 11] and induction of double refraction,[p 12] and the induction of torques on
charged condensers moving at an angle with respect to the aether.[p 12] Lorentz was perplexed by experiments such as the Trouton–Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.[28]
5. ^
For mathematical consistency, Lorentz proposed a new time variable, the “local time”, called that because it depended on the position of a moving body, following the relation .[p 13] Lorentz considered local time not to be “real”; rather, it represented
an ad hoc change of variable.[29]: 51, 80
Impressed by Lorentz’s “most ingenious idea”, Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer’s clocks.
On the other hand, Poincaré did not consider this measured time to be the “true time” that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation
described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.[30]
6. ^ This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may
assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of is also
an invariant, generated by a variant of the metric signature of spacetime.
7. ^ The invariance of Δs2 under standard Lorentz transformation in analogous to the invariance of squared distances Δr2 under rotations in Euclidean space. Although space
and time have an equal footing 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 from four-dimensional Euclidean space.
8. ^ The refractive index dependence of the presumed partial aether-drag was eventually confirmed by Pieter 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 to Amsterdam’s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).[p
17][p 18]
9. ^ Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein’s Cosmos (W. W. Norton &
Company, 2004. p. 65): “… imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length.”
10. ^
In a letter to Carl Seelig in 1955, Einstein wrote “I had already previously found that Maxwell’s theory did not account for the micro-structure of radiation and could therefore have no general validity.”, Einstein letter to Carl Seelig, 1955.
1. ^ Jump up to:a b c d e f g Albert Einstein (1905) “Zur Elektrodynamik bewegter Körper”, Annalen der Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by George Barker Jeffery and Wilfrid Perrett (1923);
Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920).
2. ^ “Science and Common Sense”, P. W. Bridgman, The Scientific Monthly, Vol. 79, No. 1 (Jul. 1954), pp. 32–39.
3. ^ The Electromagnetic Mass and Momentum
of a Spinning Electron, G. Breit, Proceedings of the National Academy of Sciences, Vol. 12, p.451, 1926
4. ^ Kinematics of an electron with an axis. Phil. Mag. 3:1-22. L. H. Thomas.]
5. ^ Jump up to:a b Einstein, Autobiographical Notes, 1949.
6. ^
Einstein, “Fundamental Ideas and Methods of the Theory of Relativity”, 1920
7. ^ 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.
8. ^ Yaakov Friedman (2004). Physical Applications of Homogeneous Balls. Progress in Mathematical Physics. Vol. 40. pp. 1–21.
ISBN 978-0-8176-3339-4.
9. ^ Das, A. (1993) The Special Theory of Relativity, A Mathematical Exposition, Springer, ISBN 0-387-94042-1.
10. ^ Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, ISBN 0-582-31760-6.
11. ^
Lorentz, H.A. (1902). “The rotation of the plane of polarization in moving media” (PDF). Huygens Institute – Royal Netherlands Academy of Arts and Sciences (KNAW). 4: 669–678. Bibcode:1901KNAB….4..669L. Retrieved 15 November 2018.
12. ^ Jump up
to:a b Lorentz, H. A. (1904). “Electromagnetic phenomena in a system moving with any velocity smaller than that of light” (PDF). Huygens Institute – Royal Netherlands Academy of Arts and Sciences (KNAW). 6: 809–831. Bibcode:1903KNAB….6..809L. Retrieved
15 November 2018.
13. ^ Lorentz, Hendrik (1895). “Investigation of oscillations excited by oscillating ions”. Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen
in bewegten Körpern). Leiden: E. J. Brill. (subsection § 31).
14. ^ Lin, Shih-Chun; Giallorenzi, Thomas G. (1979). “Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer”. Applied Optics. 18 (6): 915–931. Bibcode:1979ApOpt..18..915L.
doi:10.1364/AO.18.000915. PMID 20208844. S2CID 5343180.
15. ^ Shaw, R. (1962). “Length Contraction Paradox”. American Journal of Physics. 30 (1): 72. Bibcode:1962AmJPh..30…72S. doi:10.1119/1.1941907. S2CID 119855914.
16. ^ G. A. Benford; D.
L. Book & W. A. Newcomb (1970). “The Tachyonic Antitelephone”. Physical Review D. 2 (2): 263–265. Bibcode:1970PhRvD…2..263B. doi:10.1103/PhysRevD.2.263. S2CID 121124132.
17. ^ Zeeman, Pieter (1914). “Fresnel’s coefficient for light of different
colours. (First part)”. Proc. Kon. Acad. Van Weten. 17: 445–451. Bibcode:1914KNAB…17..445Z.
18. ^ Zeeman, Pieter (1915). “Fresnel’s coefficient for light of different colours. (Second part)”. Proc. Kon. Acad. Van Weten. 18: 398–408. Bibcode:1915KNAB…18..398Z.
19. ^
Terrell, James (15 November 1959). “Invisibility of the Lorentz Contraction”. Physical Review. 116 (4): 1041–1045. Bibcode:1959PhRv..116.1041T. doi:10.1103/PhysRev.116.1041.
20. ^ Penrose, Roger (24 October 2008). “The Apparent Shape of a Relativistically
Moving Sphere”. Mathematical Proceedings of the Cambridge Philosophical Society. 55 (1): 137–139. Bibcode:1959PCPS…55..137P. doi:10.1017/S0305004100033776. S2CID 123023118.
21. ^ Jump up to:a b c Does the inertia of a body depend upon its energy
content? A. Einstein, Annalen der Physik. 18:639, 1905 (English translation by W. Perrett and G.B. Jeffery)
22. ^ On the Inertia of Energy Required by the Relativity Principle, A. Einstein, Annalen der Physik 23 (1907): 371–384
23. ^ Jump up to:a
b c Champion, Frank Clive (1932). “On some close collisions of fast β-particles with electrons, photographed by the expansion method”. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character.
136 (830). The Royal Society Publishing: 630–637. Bibcode:1932RSPSA.136..630C. doi:10.1098/rspa.1932.0108. S2CID 123018629.
24. ^ Baglio, Julien (26 May 2007). “Acceleration in special relativity: What is the meaning of “uniformly accelerated movement”
?” (PDF). Physics Department, ENS Cachan. Retrieved 22 January 2016.
25. ^ Jump up to:a b P.A.M. Dirac (1930). “A Theory of Electrons and Protons”. Proceedings of the Royal Society. A126 (801): 360–365. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013.
JSTOR 95359.
26. ^ C.D. Anderson (1933). “The Positive Electron”. Phys. Rev. 43 (6): 491–494. Bibcode:1933PhRv…43..491A. doi:10.1103/PhysRev.43.491.
27. Griffiths, David J. (2013). “Electrodynamics and Relativity”. Introduction
to Electrodynamics (4th ed.). Pearson. Chapter 12. ISBN 978-0-321-85656-2.
28. ^ Jump up to:a b c Jackson, John D. (1999). “Special Theory of Relativity”. Classical Electrodynamics (3rd ed.). John Wiley & Sons, Inc. Chapter 11. ISBN 0-471-30932-X.
29. ^
Goldstein, Herbert (1980). “Chapter 7: Special Relativity in Classical Mechanics”. Classical Mechanics (2nd ed.). Addison-Wesley Publishing Company. ISBN 0-201-02918-9.
30. ^ Jump up to:a b Lanczos, Cornelius (1970). “Chapter IX: Relativistic Mechanics”.
The Variational Principles of Mechanics (4th ed.). Dover Publications. ISBN 978-0-486-65067-8.
31. ^ Tom Roberts & Siegmar Schleif (October 2007). “What is the experimental basis of Special Relativity?”. Usenet Physics FAQ. Retrieved 2008-09-17.
32. ^
Albert Einstein (2001). Relativity: The Special and the General Theory (Reprint of 1920 translation by Robert W. Lawson ed.). Routledge. p. 48. ISBN 978-0-415-25384-0.
33. ^ The Feynman Lectures on Physics Vol. I Ch. 15-9: Equivalence of mass and
34. ^ Sean Carroll, Lecture Notes on General Relativity, ch. 1, “Special relativity and flat spacetime,” http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html
35. ^ Koks, Don (2006). Explorations in Mathematical Physics: The
Concepts Behind an Elegant Language (illustrated ed.). Springer Science & Business Media. p. 234. ISBN 978-0-387-32793-8. Extract of page 234
36. ^ Steane, Andrew M. (2012). Relativity Made Relatively Easy (illustrated ed.). OUP Oxford. p. 226.
ISBN 978-0-19-966286-9. Extract of page 226
37. ^ Edwin F. Taylor & John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. W. H. Freeman. ISBN 978-0-7167-2327-1.
38. ^ Jump up to:a b c d e Rindler, Wolfgang (1977).
Essential Relativity: Special, General, and Cosmological (illustrated ed.). Springer Science & Business Media. p. §1,11 p. 7. ISBN 978-3-540-07970-5.
39. ^ “James Clerk Maxwell: a force for physics”. Physics World. 2006-12-01. Retrieved 2024-03-22.
40. ^
“November 1887: Michelson and Morley report their failure to detect the luminiferous ether”. www.aps.org. Retrieved 2024-03-22.
41. ^ Michael Polanyi (1974) Personal Knowledge: Towards a Post-Critical Philosophy, ISBN 0-226-67288-3, 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]
42. ^
Jump up to:a b Jeroen van Dongen (2009). “On the role of the Michelson–Morley experiment: Einstein in Chicago”. Archive for History of Exact Sciences. 63 (6): 655–663. arXiv:0908.1545. Bibcode:2009arXiv0908.1545V. doi:10.1007/s00407-009-0050-5. S2CID
43. ^ For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990
44. ^ Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. (1952). The Principle of Relativity: a collection of original memoirs
on the special and general theory of relativity. Courier Dover Publications. p. 111. ISBN 978-0-486-60081-9.
45. ^ Collier, Peter (2017). A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity (3rd
ed.). Incomprehensible Books. ISBN 9780957389465.
46. ^ 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,
ISBN 0-226-77057-5
47. ^ Jump up to:a b c d e f g h David Morin (2007) Introduction to Classical Mechanics, Cambridge University Press, Cambridge, chapter 11, Appendix I, ISBN 1-139-46837-5.
48. ^ Miller, D. J. (2010). “A constructive approach
to the special theory of relativity”. American Journal of Physics. 78 (6): 633–638. arXiv:0907.0902. Bibcode:2010AmJPh..78..633M. doi:10.1119/1.3298908. S2CID 20444859.
49. ^ Taylor, Edwin; Wheeler, John Archibald (1992). Spacetime Physics (2nd
ed.). W.H. Freeman & Co. ISBN 978-0-7167-2327-1.
50. ^ Callahan, James J. (2011). The Geometry of Spacetime: An Introduction to Special and General Relativity. New York: Springer. ISBN 9781441931429.
51. ^ P. G. Bergmann (1976) Introduction to
the Theory of Relativity, Dover edition, Chapter IV, page 36 ISBN 0-486-63282-2.
52. ^ Mermin, N. David (1968). Space and Time in Special Relativity. McGraw-Hill. ISBN 978-0881334203.
53. ^ Robert Resnick (1968). Introduction to special relativity.
Wiley. pp. 62–63. ISBN 9780471717249.
54. ^ Jump up to:a b Miller, Arthur I. (1998). Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911). Mew York: Springer-Verlag. ISBN 978-0-387-94870-6.
55. ^
Bernstein, Jeremy (2006). Secrets of the Old One: Einstein, 1905. Copernicus Books (imprint of Springer Science + Business Media). ISBN 978-0387-26005-1.
56. ^ Darrigol, Olivier (2005). “The Genesis of the Theory of Relativity” (PDF). Séminaire
Poincaré. 1: 1–22. Bibcode:2006eins.book….1D. Retrieved 15 November 2018.
57. ^ Jump up to:a b c Rindler, Wolfgang (1977). Essential Relativity (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-10090-6.
58. ^ Jump up to:a b c d Taylor, Edwin
F.; Wheeler, John Archibald (1966). Spacetime Physics (1st ed.). San Francisco: W. H. Freeman and Company.
59. ^ Ashby, Neil (2003). “Relativity in the Global Positioning System”. Living Reviews in Relativity. 6 (1): 1. Bibcode:2003LRR…..6….1A.
doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
60. ^ Daniel Kleppner & David Kolenkow (1973). An Introduction to Mechanics. McGraw-Hill. pp. 468–70. ISBN 978-0-07-035048-9.
61. ^ Jump up to:a b c French, A. P. (1968). Special Relativity.
New York: W. W. Norton & Company. ISBN 0-393-09793-5.
62. ^ Lewis, Gilbert Newton; Tolman, Richard Chase (1909). “The Principle of Relativity, and Non-Newtonian Mechanics”. Proceedings of the American Academy of Arts and Sciences. 44 (25): 709–726.
doi:10.2307/20022495. JSTOR 20022495. Retrieved 22 August 2023.
63. ^ Jump up to:a b Cuvaj, Camillo (1971). “Paul Langeyin and the Theory of Relativity” (PDF). Japanese Studies in the History of Science. 10: 113–142. Retrieved 12 June 2023.
64. ^
Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002). Understanding Physics. Springer-Verlag. p. 422. ISBN 978-0-387-98756-9.
65. ^ Cutner, Mark Leslie (2003). Astronomy, A Physical Perspective. Cambridge University Press. p.
128. ISBN 978-0-521-82196-4.
66. ^ Ellis, George F. R.; Williams, Ruth M. (2000). Flat and Curved Space-times (2n ed.). Oxford University Press. pp. 28–29. ISBN 978-0-19-850657-7.
67. ^ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew
(2011). The feynman lectures on physics; vol I: The new millennium edition. Basic Books. p. 15-5. ISBN 978-0-465-02414-8. Retrieved 12 June 2023.
68. ^ Jump up to:a b Halliday, David; Resnick, Robert (1988). Fundamental Physics: Extended Third Edition.
New York: John Wiley & sons. pp. 958–959. ISBN 0-471-81995-6.
69. ^ Adams, Steve (1997). Relativity: An introduction to space-time physics. CRC Press. p. 54. ISBN 978-0-7484-0621-0.
70. ^ Langevin, Paul (1911). “L’Évolution de l’espace et du
temps”. Scientia. 10: 31–54. Retrieved 20 June 2023.
71. ^ 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.
72. ^
Tolman, Richard C. (1917). The Theory of the Relativity of Motion. Berkeley: University of California Press. p. 54.
73. ^ Takeuchi, Tatsu. “Special Relativity Lecture Notes – Section 10”. Virginia Tech. Retrieved 31 October 2018.
74. ^ Morin,
David (2017). Special Relativity for the Enthusiastic Beginner. CreateSpace Independent Publishing Platform. pp. 90–92. ISBN 9781542323512.
75. ^ Gibbs, Philip. “Is Faster-Than-Light Travel or Communication Possible?”. Physics FAQ. Department of
Mathematics, University of California, Riverside. Retrieved 31 October 2018.
76. ^ Ginsburg, David (1989). Applications of Electrodynamics in Theoretical Physics and Astrophysics (illustrated ed.). CRC Press. p. 206. Bibcode:1989aetp.book…..G.
ISBN 978-2-88124-719-4. Extract of page 206
77. ^ Wesley C. Salmon (2006). Four Decades of Scientific Explanation. University of Pittsburgh. p. 107. ISBN 978-0-8229-5926-7., Section 3.7 page 107
78. ^ Lauginie, P. (2004). “Measuring Speed of Light:
Why? Speed of what?” (PDF). Proceedings of the Fifth International Conference for History of Science in Science Education. Archived from the original (PDF) on 4 July 2015. Retrieved 3 July 2015.
79. ^ Stachel, J. (2005). “Fresnel’s (dragging) coefficient
as a challenge to 19th century optics of moving bodies”. In Kox, A.J.; Eisenstaedt, J (eds.). The universe of general relativity. Boston: Birkhäuser. pp. 1–13. ISBN 978-0-8176-4380-5. Retrieved 17 April 2012.
80. ^ Richard A. Mould (2001). Basic
Relativity (2nd ed.). Springer. p. 8. ISBN 978-0-387-95210-9.
81. ^ Seidelmann, P. Kenneth, ed. (1992). Explanatory Supplement to the Astronomical Almanac. ill Valley, Calif.: University Science Books. p. 393. ISBN 978-0-935702-68-2.
82. ^ Ferraro,
Rafael; Sforza, Daniel M. (2005). “European Physical Society logo Arago (1810): the first experimental result against the ether”. European Journal of Physics. 26 (1): 195–204. arXiv:physics/0412055. Bibcode:2005EJPh…26..195F. doi:10.1088/0143-0807/26/1/020.
S2CID 119528074.
83. ^ Dolan, Graham. “Airy’s Water Telescope (1870)”. The Royal Observatory Greenwich. Retrieved 20 November 2018.
84. ^ Hollis, H. P. (1937). “Airy’s water telescope”. The Observatory. 60: 103–107. Bibcode:1937Obs….60..103H.
Retrieved 20 November 2018.
85. ^ Janssen, Michel; Stachel, John (2004). “The Optics and Electrodynamics of Moving Bodies” (PDF). In Stachel, John (ed.). Going Critical. Springer. ISBN 978-1-4020-1308-9.
86. ^ Sher, D. (1968). “The Relativistic
Doppler Effect”. Journal of the Royal Astronomical Society of Canada. 62: 105–111. Bibcode:1968JRASC..62..105S. Retrieved 11 October 2018.
87. ^ Gill, T. P. (1965). The Doppler Effect. London: Logos Press Limited. pp. 6–9. OL 5947329M.
88. ^ Feynman,
Richard P.; Leighton, Robert B.; Sands, Matthew (February 1977). “Relativistic Effects in Radiation”. The Feynman Lectures on Physics: Volume 1. Reading, Massachusetts: Addison-Wesley. pp. 34–7 f. ISBN 9780201021165. LCCN 2010938208.
89. ^ Cook,
Helen. “Relativistic Distortion”. Mathematics Department, University of British Columbia. Retrieved 12 April 2017.
90. ^ Signell, Peter. “Appearances at Relativistic Speeds” (PDF). Project PHYSNET. Michigan State University, East Lansing, MI. Archived
from the original (PDF) on 13 April 2017. Retrieved 12 April 2017.
91. ^ Kraus, Ute. “The Ball is Round”. Space Time Travel: Relativity visualized. Institut für Physik Universität Hildesheim. Archived from the original on 12 May 2017. Retrieved
16 April 2017.
92. ^ Boas, Mary L. (1961). “Apparent Shape of Large Objects at Relativistic Speeds”. American Journal of Physics. 29 (5): 283. Bibcode:1961AmJPh..29..283B. doi:10.1119/1.1937751.
93. ^ Müller, Thomas; Boblest, Sebastian (2014).
“Visual appearance of wireframe objects in special relativity”. European Journal of Physics. 35 (6): 065025. arXiv:1410.4583. Bibcode:2014EJPh…35f5025M. doi:10.1088/0143-0807/35/6/065025. S2CID 118498333.
94. ^ Zensus, J. Anton; Pearson, Timothy
J. (1987). Superluminal Radio Sources (1st ed.). Cambridge, New York: Cambridge University Press. p. 3. ISBN 9780521345606.
95. ^ Chase, Scott I. “Apparent Superluminal Velocity of Galaxies”. The Original Usenet Physics FAQ. Department of Mathematics,
University of California, Riverside. Retrieved 12 April 2017.
96. ^ Richmond, Michael. “”Superluminal” motions in astronomical sources”. Physics 200 Lecture Notes. School of Physics and Astronomy, Rochester Institute of Technology. Archived from
the original on 16 February 2017. Retrieved 20 April 2017.
97. ^ Keel, Bill. “Jets, Superluminal Motion, and Gamma-Ray Bursts”. Galaxies and the Universe – WWW Course Notes. Department of Physics and Astronomy, University of Alabama. Archived from
the original on 1 March 2017. Retrieved 29 April 2017.
98. ^ Max Jammer (1997). Concepts of Mass in Classical and Modern Physics. Courier Dover Publications. pp. 177–178. ISBN 978-0-486-29998-3.
99. ^ John J. Stachel (2002). Einstein from B to
Z. Springer. p. 221. ISBN 978-0-8176-4143-6.
100. ^ Fernflores, Francisco (2018). Einstein’s Mass-Energy Equation, Volume I: Early History and Philosophical Foundations. New York: Momentum Pres. ISBN 978-1-60650-857-2.
101. ^ Jump up to:a b Idema,
Timon (17 April 2019). “Mechanics and Relativity. Chapter 14: Relativistic Collisions”. LibreTexts Physics. California State University Affordable Learning Solutions Program. Retrieved 2 January 2023.
102. ^ Nakel, Werner (1994). “The elementary
process of bremsstrahlung”. Physics Reports. 243 (6): 317–353. Bibcode:1994PhR…243..317N. doi:10.1016/0370-1573(94)00068-9.
103. ^ 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.
104. ^ Jump up to:a b Philip Gibbs & Don Koks. “The Relativistic Rocket”. Retrieved 30 August 2012.
105. ^ The special theory of relativity shows that time and space are affected by
motion Archived 2012-10-21 at the Wayback Machine. Library.thinkquest.org. Retrieved on 2013-04-24.
106. ^ E. J. Post (1962). Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications Inc. ISBN 978-0-486-65427-0.
107. ^
R. Resnick; R. Eisberg (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. pp. 114–116. ISBN 978-0-471-87373-0.
108. ^ Øyvind Grøn & Sigbjørn Hervik (2007). Einstein’s general theory of relativity:
with modern applications in cosmology. Springer. p. 195. ISBN 978-0-387-69199-2. Extract of page 195 (with units where c=1)
109. ^ The number of works is vast, see as example:
Sidney Coleman; Sheldon L. Glashow (1997). “Cosmic Ray and Neutrino
Tests of Special Relativity”. Physics Letters B. 405 (3–4): 249–252. arXiv:hep-ph/9703240. Bibcode:1997PhLB..405..249C. doi:10.1016/S0370-2693(97)00638-2. S2CID 17286330.
An overview can be found on this page
110. ^ John D. Norton, John D. (2004).
“Einstein’s Investigations of Galilean Covariant Electrodynamics prior to 1905”. Archive for History of Exact Sciences. 59 (1): 45–105. Bibcode:2004AHES…59…45N. doi:10.1007/s00407-004-0085-6. S2CID 17459755.
111. ^ J.A. Wheeler; C. Misner;
K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 978-0-7167-0344-0.
112. ^ J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. p. 247. ISBN 978-0-470-01460-8.
113. ^ R. Penrose (2007). The Road to Reality. Vintage books.
ISBN 978-0-679-77631-4.
114. ^ Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, ISBN 0-07-032071-3
115. ^ Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, pg 51, ISBN 0-7167-0344-0
116. ^ George Sterman, An Introduction
to Quantum Field Theory, pg 4, ISBN 0-521-31132-2
117. ^ Sean M. Carroll (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. p. 22. ISBN 978-0-8053-8732-2.
Photo credit: https://www.flickr.com/photos/evilerin/3474066174/’]