Astronomy

Lorentz transformation for stars

Lorentz transformation for stars

If a star that has regular pulses(standard candle) is flying away from us at relativistic velocity(very distant), would that pulses look like very sluggish compared to a non-relativistic velocity(close to earth)… something like a clock ticking in a relativistic spaceship and… ofcourse red-shifted.Thanks.


They would if we could see them far enough away, but cepheid variables are only visible up to about 20 million light years distant. That may sound like a long way, but in astronomical terms it is nearby.


(in the case of a moving body)

Let us consider two frames S and S_1 in relative motion to each other, which coincide when the times t and t_1 are equal to 0 (i.e. the instants of time in which the frames are coincident constitute the same initial time instant for both) and with the x axis and x_1 axis always superimposed.

In particular, reference is always made to th e well-known case in which the origin O_1 of the frame S_1 moves away from the origin O of the frame S. (but with a velocity v which, even if it has only a horizontal component along the abscissa axis x, is not always constant)

Now let’s imagine that in the origin of the frame S_1 a body is positioned which, starting from the origin of the frame S (at the zero initial time t = 0 and at zero initial speed), arrives at a point of the x axis of positive abscissa d with respect to the frame S.

For the frame S, the frame S_1 at time t is in motion at speed v(t) while, for the frame S_1, the frame S at time t_1 is in motion at speed v_1(t_1).
We thus obtain that v(t) = v_1(t_1) (in modulo and for t_1 which corresponds to the travel time measured in S_1 when for the frame S the measured travel time is t) and the two Lorentz factors γ(t) and γ(t_1) coincide.

From the Lorentz transformation x_1 = γ*(x -v*t), for x_1=0 as the body is positioned in the origin of the frame S_1, we obtain x -v*t=0, from which x=v*t. Turning to the differentials we obtain the relation dx = v(t)*dt.
From the other Lorentz transformation x = γ*(x_1+v*t_1), again for x_1=0, we verify that x = γ*v*t_1 and passing to the differentials we obtain the following relation dx = γ(t_1)*v _1(t_1)*dt_1.

The last equation is correct, in fact in the meantime dx_1 = v_1(t_1)*dt_1 for the frame S_1 the infinitesimal distance dx is in motion at speed v_1(t_1) and, in the frame S_1, it is less than dx and measures dx/γ (t_1). From dx_1=v_1(t_1)*dt_1 and from dx_1 = dx/γ(t_1) we get dx = γ(t_1)*v_1 (t_1)*dt_1.
From dx = γ (t_1)*v_1(t_1)*dt_1 we then pass to

v(t)*dt = γ(t_1)*v_1(t_1)*dt_1 finally it is possible to obtain the known result dt_1 = γ^(-1)(t)*dt, and this is true since v(t) = v_1 (t_1)and γ (t)=γ (t_1).

Since dt_1 < dt, proceeding to integrate we also obtain t_1 < t. (the body in motion is “younger” than all the other stationary bodies in the frame S)

dx_1 is less than dx, also x_1(t_1) < x(t) and this is what causes the symmetry to break.

The infinitesimal time dt is the time of the body to travel the infinitesimal distance dx as measured in the frame S, the infinitesimal time dt_1 instead represents the time for the distance dx_1 (dx in motion) “overtakes” the origin of the frame S_1. In the frame S_1, dt_1 therefore corresponds to the infinitesimal travel time of the body which, even if stationary in its frame, waits for the infinitesimal distance dx_1 to meet it with the opposite speed to its own.

For accuracy, if the abscissa d is positive, in the two Lorentz transformations examined the two speeds v(t) and v_1(t_1) are also to be considered as positive values, even the exact expression of the differential dx_1 is

dx_1 = -v_1(t_1)*dt_1. In the demonstration shown dx_1 an infinitesimal distance (and therefore positive) has been considered, also pay attention to the signs of the various quantities in case the body goes back.

It should also be considered that if the body crosses areas where gravitational fields are particularly intense, it will be even “younger” than all the bodies that remain stationary in the frame S, but ideally it is always possible to consider a point-like body of mass negligible in motion (regardless of the gravitational forces acting on massive bodies).

As regards the distance between the two origins of the frames S and S_1 measured at the instant of time t_1 by an observer in the frame S_1, it, indicated with d_1m (t_1), at the instant of time t_1 is equal to x(t)*γ^(-1)(t_1) or d_1m (t_1) = x (t)*γ^(-1) (t).

The last relation is important as it also means that, in the case where the speed of the body is equal to zero, d_1m(t_1)= x(t). When the body is stationary in the frame S all distances have the same measure for all observers of the frame itself. The now stationary body will therefore seem to have traveled a distance equal to d, although this does not correspond to the actual distance traveled from the origin of the frame S with respect to the origin of the frame S_1.

In summary, the useful relationships are:

dt_1 = γ^(-1)(t)*dt

dx = γ(t_1)*v_1 (t_1)*dt_1

dx_1 = dx/γ(t_1)= dx/γ(t) = v_1(t_1)*dt_1

d_1m(t_1) = x(t)*γ^(-1) (t) (the only one in which there are no differentials).

This discussion also applies in the event that the body goes back along a further distance d to return to the origin of the frame S with zero final speed. (situation that represents the so-called “twin paradox”)

Although the relation dx = v(t)*dt is trivial at first sight, it is important because it states that there is a body in motion within the frame S. (for example the motion of a body in the Earth’s frame or better in the frame of the fixed stars). If a body is moving within a frame S and travels a distance d, in the body’s frame S_1 it is not only the origin O that moves.(but it is the whole frame S that “travels” a distance less than d)

Remember that the frame S in which the movement takes place is not a privileged frame, it is the frame in which the body at the end of the journey is “younger” than the other bodies that have always remained stationary in the frame S.

And, as has been shown in this article, the Lorentz transformations are sufficient to prove this simply and elegantly.


Lorentz transformation of blackbody radiation

In a paper entitled "Lorentz transformation of blackbody radiation" (Phys. Rev. E 88, 044101, 2013), G. W. Ford andRobert O'Connell have solved a problem which numerous authors have worked on over a time span exceeding a hundred years. This is the question of how temperature behaves under a Lorentz transformation. Both Einstein in 1907 and Planck in 1908 published results which disagreed with each other and both of which proved to be incorrect. The reason why the problem remained unsolved for so many years was the fact that no experimental evidence existed to provide a check on the huge variety of theoretical methods employed. By constrast, Ford and O'Connell employed the experimental results for the spectrum of the universal cosmic blackbody radiation measured by Earth observers who are in different reference frames because of their motion through the 2.7K radiation. Despite the fact that kT behaves like an energy, it turns out that T does not change in a Lorentz transformation.


Lorentz transformations: 1+1 spacetime only

Yes, that's what we have been arguing. Or, to avoid the double negatives:

The product of two Lorentz boosts produces a constant velocity relationship (Lorentz boost plus fixed rotation).

I'm coming into this discussion pretty late, but for the record, it's not the case that a combination of two Lorentz transformations is another Lorentz transformation. There are 10 independent linear transformations such that any two inertial coordinate systems are related by some combination of the 10:

  1. A translation in the x-direction (that is, the transformation ##x' = x + delta x##.
  2. A translation in the y-direction
  3. A translation in the z-direction
  4. A translation in the t-direction (##t' = t + delta t##)
  5. A rotation about the x-direction (the transformation ##y' = y cos( heta) + z sin( heta), z' = z cos( heta) - y sin( heta)##)
  6. A rotation about the y-direction
  7. A rotation about the z-direction
  8. A boost in the x-direction (##x' = gamma (x - v t), t' = gamma (t - fracx)##
  9. A boost in the y-direction
  10. A boost in the z-direction

Lorentz transformations themselves don't form a group (in more than one spatial dimension), but only the combination of Lorentz transformations + rotations.

Of course the Lorentz group is a group and thus the composition of two Lorentz transformations is again a Lorentz transformation. It's all linear transformations which leave the Minkowski product invariant. For four-vector components it reads
$V^ = >_ < u>V^< u>,$
and the matrix must fulfill the pseudo-orthogonality relation
$eta_ >_ < ho>>_=eta_< ho sigma>$
with ##(eta_)=mathrm(1,-1,-1,-1)##.
These matrices build the group ##mathrm(1,3)##. Physically relevant for the space-time description is a priori only the subgroup connected continuously to the identity, and that's the proper orthchronous Lorentz group ##mathrm(1,3)^##, i.e., all Lorentz-trafo matrices with determinent 1 and ##_0 geq 1##.

While the rotations form a subgroup the rotation-free boosts are not a subgroup only those in one fixed direction form an Abelian subgroup. The composition of two rotation-free boosts in different directions is of course again a Lorentz transformation but not a rotation-free boost, but a rotation-free boost followed by a rotation (the Wigner rotation).

Minkowski space is the affine pseudo-Euclidean space with a fundamental form of signature (1,3), also called an affine Lorentzian space, and as such the full symmetry group is the Poincare group and is the corresponding semidirect product generated by Lorentz transformations and spatio-temporal translations. Again a priori physically relevant is the proper orthochronous Poincare group, and indeed Nature is described well with a space-time model obeying this symmetry group. The larger group built from the proper orthochronous Poincare group by including time, space, and space-time reflections is not a symmetry group of Nature. The weak interaction violates both time reversal as well as space reflections. Within relativistic local QFT you have, however, necessarily an additional symmetry, which is charge conjugation, i.e., where all particles in a reaction are exchanged by their antiparticles. The weak interaction also breaks the C-symmetry. Today, it has been independently observed that the weak interaction breaks all these discrete symmetries, i.e., T, P, and CP. Relativistic local QFT predicts however that necessarily the "grand reflection" CPT must be a symmetry, and so far this symmetry indeed has passed all tests.

Together with all the other tests of Lorentz symmetry, it is pretty safe to say that Poincare symmetry is obeyed by all phenomena with an amazing accuracy, as is the extension to GR and the corresponding gauge symmetry leading to it from SR and its global Poincare symmetry.


Contents

Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies". [p 1] The incompatibility of Newtonian mechanics with Maxwell's equations of electromagnetism and, experimentally, the Michelson-Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized luminiferous aether did not exist. This led to Einstein's development of special relativity, which corrects mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities ). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible. [3] [4] Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.

Special relativity has a wide range of consequences that have been experimentally verified. [5] They include the relativity of simultaneity, length contraction, time dilation, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession. [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. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = m c 2 > , where c is the speed of light in a vacuum. [6] [7] It also explains how the phenomena of electricity and magnetism are related. [1] [2]

A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into a single continuum known as "spacetime". Events that occur at the same time for one observer can occur at different times for another.

Until Einstein developed general relativity, introducing a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity" "special" really means "special case". [p 2] [p 3] [p 4] [note 1] Some of the work of Albert Einstein in special relativity is built on the earlier work by Hendrik Lorentz and Henri Poincaré. The theory became essentially complete in 1907. [4]

The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, the curvature of spacetime, described by the energy–momentum tensor and causing gravity, is negligible. [8] [note 2] In order to correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate accelerations as well as accelerating frames of reference. [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. General relativity, however, incorporates non-Euclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as Minkowski space. 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.

Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. 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. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics. [12]

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. These propositions were the constancy of the speed of light in a 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]

  • 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. [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 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 constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment. [13] [14] 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 the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history. [p 6]

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations. [15] However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively to K. [16]

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.

Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles. [p 7]

Reference frames and relative motion Edit

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).

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

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

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

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. [17] : 107 With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two Galilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′.

  • The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
  • Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.
  • The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.

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

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. However, 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). 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. 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. [18] Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

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. 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 8] [19]

Alternative approaches to special relativity Edit

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

The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events . The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws . [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. [p 9] [p 10]

Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations. [20] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler [21] and by Callahan. [22] This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.

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 with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:

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. For simplicity, the y and z coordinates are unaffected only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.

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

Enforcing this inverse Lorentz transformation to coincide with the Lorentz transformation from the primed to the unprimed system, shows the unprimed frame as moving with the velocity v′ = −v, as measured in the primed frame.

There is nothing special about the x-axis. 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.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x1, t1) and (x1, t1) , another event has coordinates (x2, t2) and (x2, t2) , and the differences are defined as

If we take differentials instead of taking differences, we get

Graphical representation of the Lorentz transformation Edit

Spacetime diagrams (Minkowski 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. [19]

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1. [19] [23] : 155–199

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

The consequences of special relativity can be derived from the Lorentz transformation equations. [24] 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 initially counterintuitive.

Invariant interval Edit

In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as Δ s 2 > :

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

The form of Δ s 2 , ,> being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. [note 7] The invariance of this interval is a property of the general Lorentz transform (also called the Poincaré transformation), making it an isometry 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 other translations, reflections, and rotations between any Cartesian inertial frame. [28] : 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: [19]

In considering the physical significance of Δ s 2 > , there are three cases to note: [19] [29] : 25–39

  • Δs 2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that | Δ x / Δ t | < c , and given the Lorentz transformation Δ x ′ = γ ( Δ x − v Δ t ) , it is evident that there exists a v less than c for which Δ x ′ = 0 (in particular, v = Δ x / Δ t ). 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, Δ s / c , is called the proper time.
  • Δs 2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that | Δ x / Δ t | > c , and given the Lorentz transformation Δ t ′ = γ ( Δ t − v Δ x / c 2 ) , ),> there exists a v less than c for which Δ t ′ = 0 (in particular, v = c 2 Δ t / Δ x Delta t/Delta x> ). 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, − Δ s 2 , >>,> is called the proper distance, or proper length. For values of v greater than and less than c 2 Δ t / Δ x , Delta t/Delta x,> the sign of Δ t ′ changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. The temporal order of timelike-separated events, however, is absolute, since the only way that v could be greater than c 2 Δ t / Δ x Delta t/Delta x> would be if v > c .
  • Δs 2 = 0: In this case, the two events are said to be lightlike separated. This implies that | Δ x / Δ t | = c , and this relationship is frame independent due to the invariance of s 2 . .> From this, we observe that the speed of light is c 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.

Relativity of simultaneity Edit

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

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 ). Only if these events are additionally co-local in frame S (satisfying Δx = 0 ), will they be simultaneous in another frame S′.

The Sagnac effect can be considered a manifestation of the relativity of simultaneity. [30] Since relativity of simultaneity is a first order effect in v , [19] 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. [p 14]

Time dilation Edit

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels).

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 . To find the relation between the times between these ticks as measured in both systems, Equation 3 can be used to find:

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). 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. [31]

Length contraction Edit

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).

Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0 , which can be combined with Equation 4 to find the relation between the lengths Δx and Δx′:

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).

Time dilation and length contraction are not merely appearances. 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). 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. 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 will not occur at the same spatial distance from each other when seen from another moving coordinate system.

Lorentz transformation of velocities Edit

Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector u . .> What is its velocity u ′ > in frame S′ ?

Substituting expressions for d x ′ and d t ′ from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed u to u ′ :

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing v with − v .

The forward and inverse transformations for this case are:

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.

There is nothing special about the x direction in the standard configuration. The above formalism applies to any direction and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.

Thomas rotation Edit

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.

Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion. [29] : 169–174

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox". [p 15] [29] : 98–99

Causality and prohibition of motion faster than light Edit

In Fig. 4-3, 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. 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 'space-like' that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. However, there are no frames 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, then 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). [32] [p 16] A variety of causal paradoxes could then be constructed.

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

  1. Fig. 4-4a. 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 − x ′ axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.
  2. Fig. 4-4b. 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 + x axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality. [33]

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.

This is not to say that all faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light. [35] 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). [36] [37]

Dragging effects Edit

In 1850, Hippolyte Fizeau and Léon Foucault independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory. [38] 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 speed through the medium plus the speed of 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 u ′ = c / n is the speed of light in still water, and v is the speed of the water, and u ± > is the water-bourne 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. Among other things, the presence of an index of refraction term meant that, since n depends on wavelength, the aether must be capable of sustaining different motions at the same time. [note 8] A variety of theoretical explanations were proposed to explain Fresnel'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. [39]

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. [28]

Relativistic aberration of light Edit

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 the aberration 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. [40] (2) If the source is in motion, the displacement would be the consequence of light-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. [41]

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, [42] 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. [43] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag, [44] but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag. [45]

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 [28] : 57–60

Relativistic Doppler effect Edit

Relativistic longitudinal Doppler effect Edit

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 a time dilation term, and that is the treatment described here. [46] [47]

For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source. [48] [19]

Transverse Doppler effect Edit

The transverse Doppler 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. [19] In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of γ . In Fig. 5-3b, the light is redshifted by the same factor.

Measurement versus visual appearance Edit

Time dilation and length contraction are not optical illusions, but genuine effects. 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.

Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. 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 visual appearance of an object, however, is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

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. 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. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation. [p 19] [p 20] [49] [50] A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted. [51]

Fig. 5-4 illustrates a cube 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. This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect. [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. An apparent optical illusion results giving the appearance of faster than light travel. [52] [53] [54] In Fig. 5-5, galaxy M87 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. [55]

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. 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.

Equivalence of mass and energy Edit

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.

The energy content of an object at rest with mass m equals mc 2 . Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.

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 = mc 2 .

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a 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/c 2 , 0, 0) . The momentum is equal to the energy multiplied by the velocity divided by c 2 . As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c 2 .

The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations. [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 21] Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong. [56] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions. [57]

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 22] [note 10]

How far can one travel from the Earth? Edit

Since one can not travel faster than light, one might conclude that a human can never travel farther from Earth than 40 light years if the traveler is active between the ages of 20 and 60. One would easily think that a traveler would never be able to reach more than the very few solar systems which exist within the limit of 20–40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1g, it will, after a little less than a year, be travelling at almost the speed of light as seen from Earth. This is described by:

where v(t) is the velocity at a time t, a is the acceleration of 1g and t is the time as measured by people on Earth. [p 23] Therefore, after one year of accelerating at 9.81 m/s 2 , the spaceship will be travelling at v = 0.77c relative to Earth. Time dilation will increase the travellers life span as seen from the reference frame of the Earth to 2.7 years, but his lifespan measured by a clock travelling with him will not change. During his journey, people on Earth will experience more time than he does. A 5-year round trip for him will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for him (5 years accelerating, 5 decelerating, twice each) will land him back on Earth having travelled for 335 Earth years and a distance of 331 light years. [58] 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 at 1.1g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy. [58] This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its half-life (when at rest). [59]

Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.

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. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic 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 of electromagnetic 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 a manifestly covariant form, that is, in the language of tensor calculus. [60]

Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. 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.

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. [61]

In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour, [p 24] that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only describe the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron), [p 24] [p 25] and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic 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. 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.

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c 2 in the region of interest. [62] In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10 −20 ) [63] and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.

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

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

Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905, [64] 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. [14]

  • The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
  • The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
  • The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
  • The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.

Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:

    – testing the limiting speed of particles – testing relativistic Doppler effect and time dilation – relativistic effects on a fast-moving particle's half-life – time dilation in accordance with Lorentz transformations – testing isotropy of space and mass – various modern tests
  • Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
  • Experiments to test the aether drag hypothesis – no "aether flow obstruction".

Geometry of spacetime Edit

Comparison between flat Euclidean space and Minkowski space Edit

Special relativity uses a 'flat' 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.

In 3D space, the differential of distance (line element) ds is defined by

where dx = (dx1, dx2, dx3) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X 0 derived from time, such that the distance differential fulfills

where dX = (dX0, dX1, dX2, dX3) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1). [66] Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.

The actual form of ds above depends on the metric and on the choices for the X 0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X0 = ict (this is called a Wick rotation). 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 X 0 = ct , rather than a "disguised" Euclidean metric using ict as the time coordinate.

Some authors use X 0 = t , with factors of c elsewhere to compensate for instance, spatial coordinates are divided by c or factors of c ±2 are included in the metric tensor. [67] These numerous conventions can be superseded by using natural units where c = 1 . Then space and time have equivalent units, and no factors of c appear anywhere.

3D spacetime Edit

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. 10-2) defined by the equation

which is the equation of a circle of radius c dt.

4D spacetime Edit

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

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

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance d = x 1 2 + x 2 2 + x 3 2 ^<2>+x_<2>^<2>+x_<3>^<2>>>> away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the Fig. 10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

The cone in the −t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

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

Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see Center of mass (relativistic).

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. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.

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. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "four vectors", in Minkowski spacetime. The components of vectors are written using tensor index notation, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.

The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z) , in a contravariant position four vector with components:

where we define X 0 = 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. [68] [69] [70] Now the transformation of the contravariant components of the position 4-vector can be compactly written as:

where the Lorentz factor is:

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. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.

The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:

which is the transpose of:

only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.

More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:

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

More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law [71]

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

The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.

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. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates) which can be arranged in a 4 × 4 matrix:

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

and this is the physical symmetry underlying special relativity.

The metric can be used for raising and lowering indices on vectors and tensors. 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. 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. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.

Relativistic kinematics and invariance Edit

The coordinate differentials transform also contravariantly:

so the squared length of the differential of the position four-vector dX μ constructed using

is an invariant. Notice that when the line element dX 2 is negative that √ −dX 2 is the differential of proper time, while when dX 2 is positive, √ dX 2 is differential of the proper distance.

The 4-velocity U μ 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. Differentiating the above equation by τ produces:

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

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 doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

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

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. It may not be equal to the sum of individual system masses measured in other frames.

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. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.

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. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. 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 c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, dp/dt while the four force is defined by the rate of change of momentum with respect to proper time, that is, dp/dτ.

In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.


Lorentz transformation for stars - Astronomy

Interpretive Assignments
500 words

1. Lorentz Transformation of Special Relativity

In Ch.XI of Einstein's popular text, Relativity: The Special and the General Theory, Einstein shows that the light postulate is respected by the Lorentz transformation. A light signal moves according to x=ct. He transforms it to a new inertial frame of reference in which it turns out also be described by x'=ct'. (See the chapter for notation.)

(a) How does this vindicate the light postulate?
(b) Redo Einstein's calculation for the case of x = - ct. (Note the "minus.")
(c) Show that a corresponding calculation for the Galilean transformation (specified on p. 40.) fails.

At the bottom on one of the pages of Section 3 of Einstein's special relativity paper of Einstein remarks on a simpler derivation of the Lorentz transformation

Reconstruct Einstein's derivation.

To make life a little easier, consider only the transformation that takes (t, x) to (τ, ξ). Note also that Einstein's condition is only strong enough to recover the Lorentz transformation with the arbitrary factor φ(v).

I. Assume the transformation is linear and write it in the general form
τ = φ(v)(t + bx) ξ = ψ(x-vt)

II. Substitute these equations into ξ 2 = c 2 τ 2

III. For this equation to reduce to x 2 = c 2 t 2 , the "xt" terms must vanish, which puts a significant constraint on the constants in the transformation equations.

IV. Keep working the algebra until you have arrived at x 2 = c 2 t 2 . You should find that the condition needed to achieve this amounts to the Lorentz transformation for the simpler case of (t, x) to (τ, ξ).

2. Principle and Constructive Theories OR E = mc 2

Einstein's pathway to his special theory of relativity included him making use of a distinction between what he called "principle theories" and "constructive theories." The distinction is described in a 1919 article written for The Times (London), "What is the Theory of Relativity?" He also recounts the role the distinction played in his discovery of special relativity in a paragraph in his Autobiographical Notes.

(a) What is the distinction Einstein draws?
(b) How did it play a role in his thinking concerning the discovery of special relativity?
(c) Aside from Einstein's examples, can you find an example of a principle theory and a constructive theory? Which are they? Why are they properly categorized so?
(d) Einstein claimed certain advantages for each type of theory in his 1919 article. Do these benefits arise in your examples? Explain.

Since this assignment requires a simple text answer, please do not handwrite it. Please type it and hand in a printed version.

(a) Reconstruct the derivation.
(b) Einstein says his derivation uses the expression for the aberration of starlight from astronomy. How is it possible to use this expression when there are no stars talked of in the derivation?
(c) There is a typo in the derivation associated with the effect of aberration. What is it?
(d) How does this derivation differ from the original derivation of 1905?
(e) Is it improved? How?

(mathematical, harder, but I think it is easy once you see how to set it up)

In Section 1 of A. Einstein, "The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy," (1906) Doc. 35 in Einstein Papers, Vol 2, Einstein presents a "special case" thought experiment in which he uses the conservation of the motion of the center of gravity to arrive at the inertia of energy. The thought experiment analyses a cyclic process involving very small quantities of mass and very small dislocations in space. The computation requires the selective neglecting of certain small quantities.

The basic idea of the thought experiment can be shown without using very small quantities if they are replaced by differential operators. To do this, consider a mass m that emits radiation at a steady rate s in a fixed direction. Compute the center of mass of the total system, assuming that the radiant energy emitted has inertial mass. The position of this center of mass must remain fixed. That is, its time derivative must vanish.

Use the condition of the vanishing of this time derivative at the moment of first emission of the radiation to recover the inertia of energy.

(mathematical, harder, but satisfying if you can crack it!)

In Section 2 and later of A. Einstein, "The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy," (1906) Doc. 35 in Einstein Papers, Vol 2, Einstein provides another demonstration of the inertial of energy this time using a result pertaining to the center of gravity. The proof is given in terms of field components. That is "X, Y, Z" instead of the electric field vector E.

Convert Einstein's analysis into the now standard vector notation using "div, grad, curl" etc. What precisely is the result shown?

NB.The editorial notes to the German text in Einstein Papers, Vol.2, contains corrections to typographical errors in the text. And, of course, Einstein's "V" is now our "c."

3. Orthogonal Transformations

In the opening pages of Einstein's text on relativity theory, Meaning of Relativity, Einstein introduces what are now called "orthogonal transformations." They are the analog in ordinary three dimensional Euclidean space of the Lorentz transformation in Minkowski spacetime. They are defined as the transformation of the three Cartesian coordinates (x, y, z) = (x1, x2, x3) that preserves the Euclidean line element

That is, we transform to new coordinates (x', y', z') = (x'1, x'2, x'3) such that

Einstein shows that the transformations that preserve this Euclidean line element are given by his equation (3) on p. 7:

where bνα are constants that satisfy the orthogonality condition

(a) Show that a translation

where aν = (a1, a2, a3) are three constants, is an orthogonal transformation that satisfies (4).

(b) A rotation by an angle theta around the x3 = z axis is given by bνα's that satisfy

Show that this transformation is an orthogonal transformation that satisfies (4).

(c) Use (3) and (4) to prove that an orthogonal transformation leaves the Euclidean interval invariant that is, orthogonal transformations satisfy (*).

(d) (Optional) On p.7, Einstein uses a power series expansion to establish that an orthogonal transformation from xν to x'ν must be linear. Supply the missing steps in his argument. (NB. This might be hard. I haven't been able to see how to answer this one.)

Hint: If you know some matrix algebra, the three coordinates (x1, x2, x3) can be represented by a column vector x = (x1, x2, x3) T . The transformation (3) is then just

x' = a + b x

Einstein's orthogonality condition (4) can be represented as

b T b = I or, equivalently b T = b -1

Using this matrix representation simplifies the presentation of the answers.


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In: Chinese Journal of Physics , Vol. 35, No. 4, 01.12.1997, p. 407-417.

Research output : Contribution to journal › Article › peer-review

T1 - Generalized lorentz transformations for linearly accelerated frames with limiting four-dimensional symmetry

N2 - Based on the principle of limiting four-dimensional symmetry, we discuss coordinate transformations for constant-linear-acceleration (CLA) frames. We derive a new "Wu transformation" which reduces to the Lorentz transformation in the limit of zero acceleration. The time for an accelerated frame can be realized by "computerized clocks". A 'CLA coordinate' (w, x, y, z) is preferred for the accelerated transformation and has as much physical meaning for an accelerated frame F as (wI, xI, yI/, zI) for an inertial frame FI. Furthermore, constant-linear-acceleration αa must be constant increase of a particle's "energy" per unit length, in consistent with what has been realized in high energy linear accelerators. Some experimental implications are discussed. PACS. 04.20.-q - Classical general relativity. PACS. 11.30.Cp - Lorentz and Poincare invariance. PACS. 03.30,+p - Special relativity.

AB - Based on the principle of limiting four-dimensional symmetry, we discuss coordinate transformations for constant-linear-acceleration (CLA) frames. We derive a new "Wu transformation" which reduces to the Lorentz transformation in the limit of zero acceleration. The time for an accelerated frame can be realized by "computerized clocks". A 'CLA coordinate' (w, x, y, z) is preferred for the accelerated transformation and has as much physical meaning for an accelerated frame F as (wI, xI, yI/, zI) for an inertial frame FI. Furthermore, constant-linear-acceleration αa must be constant increase of a particle's "energy" per unit length, in consistent with what has been realized in high energy linear accelerators. Some experimental implications are discussed. PACS. 04.20.-q - Classical general relativity. PACS. 11.30.Cp - Lorentz and Poincare invariance. PACS. 03.30,+p - Special relativity.


Einstein's core idea about gravity just passed an extreme, whirling test in deep space

Once again, physicists have confirmed one of Albert Einstein's core ideas about gravity — this time with the help of a neutron star flashing across space.

The new work makes an old idea even more certain: that heavy and light objects fall at the same rate. Einstein wasn't the first person to realize this there are contested accounts of Galileo Galilei demonstrating the principle by dropping weights off the Tower of Pisa in the 16th century. And suggestions of the idea appear in the work of the 12th-century philosopher Abu'l-Barakāt al-Baghdādī. This concept eventually made its way into Isaac Newton's model of physics, and then Einstein's theory of general relativity as the gravitational "strong equivalence principle" (SEP). This new experiment demonstrates the truth of the SEP, using a falling neutron star, with more precision than ever.

The SEP has appeared to be true for a long time. You might have seen this video of Apollo astronauts dropping a feather and a hammer in the vacuum of the moon, showing that they fall at the same rate in lunar gravity.

But small tests in the relatively weak gravitational fields of Earth, the moon or the sun don't really put the SEP through its paces, according to Sharon Morsink, an astrophysicist at the University of Alberta in Canada, who wasn't involved in the new study.

"At some level, the majority of physicists believe that Einstein's theory of gravity, called general relativity, is correct. However, that belief is mainly based on observations of phenomena taking place in regions of space with weak gravity, while Einstein's theory of gravity is meant to explain phenomena taking place near really strong gravitational fields," Morsink told Live Science. "Neutron stars and black holes are the objects that have the strongest known gravitational fields, so any test of gravity that involves these objects really test the heart of Einstein's gravity theory."

Neutron stars are the collapsed cores of dead stars. Super dense, but not dense enough to form black holes, they can pack masses greater than that of our sun into whirling spheres just a few miles wide.

The researchers focused on a type of neutron star called a pulsar, which from Earth's perspective seems to flash as it spins. That flashing is a result of a bright spot on the star's surface whirling in and out of view, 366 times per second. This spinning is regular enough to keep time by.

This pulsar, known as J0337+1715, is special even among pulsars: It's locked in a tight binary orbit with a white dwarf star. The two stars orbit each other as they circle a third star, also a white dwarf, just like Earth and the moon do as they circle the sun.

(Researchers have already shown that the SEP is true for orbits like this in our solar system: Earth and the moon are affected to exactly the same degree by the sun's gravity, measurements suggest.)

The precise timekeeping of J0337+1715, combined with its relationship to those two gravity fields created by the two white dwarf stars, offers astronomers a unique opportunity to test the principle.

The pulsar is much heavier than the other two stars in the system. But the pulsar still falls toward each of them a little bit as they fall toward the pulsar's larger mass. (The same thing happens with you and Earth. When you jump, you fall back toward the planet very quickly. But the planet falls toward you as well — very slowly, due to your own low gravity, but at the exact same rate as a feather or a hammer would if you ignore air resistance.) And because J0337+1715 is such a precise timekeeper, astronomers on Earth can track how the gravitational fields of the two stars affect the pulsar's period.

To do so, the astronomers carefully timed the arrival of light from J0337+1715 using large radio telescopes, in particular the Nançay Radio Observatory in France. As the star moved around each of its neighbors — one in a quick little orbit and one in a longer, slower orbit — the pulsar got closer and farther from Earth. As the neutron star moved farther away from Earth, the light from its pulses had to travel longer distances to reach the telescope. So, to a tiny degree, the gaps between the pulses seemed to get longer.

As the pulsar swung back toward Earth, the gaps between the pulses got shorter. That allowed physicists to build a robust model of the neutron star's movement through space, explaining precisely how it interacted with the gravity fields of its neighbors. Their work built on a technique used in an earlier paper, published in the journal Nature in 2018, to study the same system.

The new paper, published online June 10 in the journal Astronomy and Astrophysics, showed that the objects in this system behaved as Einstein's theory predicts — or at least didn't differ from Einstein's predictions by more than 1.8 parts per million. That's the absolute limit of the precision of their telescope data analysis. They reported 95% confidence in their findings.

Morsink, who uses X-ray data to study the mass, widths, and surface patterns of neutron stars, said that this confirmation isn't surprising, but it is important for her research.

"In that work, we have to assume that Einstein's theory of gravity is correct, since the data analysis is already very complex," Morsink told Live Science in an in an email. "So tests of Einstein's gravity using neutron stars really make me feel better about our assumption that Einstein's theory describes the gravity of a neutron star correctly!"

Without understanding the SEP, Einstein would never have been able to develop his ideas of relativity. In an insight he described as "the most fortunate thought in my life," he recognized that objects in free fall don't feel the gravitational fields tugging on them.

(This is why astronauts in orbit around the Earth float. In constant free fall, they don't experience the gravitational field that holds them in orbit. Without windows, they wouldn't know Earth was there at all.)

Most of Einstein's key insights about the universe begin with the universality of free fall. So, in this way, the cornerstone of general relativity has been made that much stronger.

For a limited time, you can take out a digital subscription to any of our best-selling science magazines for just $2.38 per month, or 45% off the standard price for the first three months.View Deal

It seems that many physicists still don't know that Einstein's relativity is logically wrong and can be easily disproved through Lorentz Transformation of clock time i.e. our physical time between two inertial reference frames:

Let's look at the twin paradox which is designed to demonstrate that relative speed would generate time dilation as predicted by special relativity which claims that when the speed of a clock relative to an observer was close to the speed of light, the observer would see the clock slow down close to stop. But, it is pretty ironic as shown on Wikipedia, the final conclusion of the twin paradox becomes that, after a high speed space travel, it is the acceleration of the traveling twin (not his speed relative to his twin brother) that made him younger than his twin brother staying on the earth because both twins had experienced exactly the same speed relative to each other during the entire trip. Is it funny that the original argument that relative speed generates time dilation is completely lost, although relativists still think that the paradox has been solved? In fact, this paradox has simply confirmed that relative speed can never generate time dilation and special relativity is wrong.

Actually Einstein's relativity has already been disproved both theoretically and experimentally for more than four years. The fatal mistake of Einstein's relativity is that it uses Lorentz Transformation to redefine time and space, and the newly defined time is no longer the physical time we measure with physical clocks. The claim of the constant speed of light is very similar to the claim that everybody had the same height if the height is measured with a new ruler - an elastic band a ruler. Obviously, such claims do not make any sense.

In a physics theory, the physical time shown on a physical clock is T = tf/k where t is the theoretical time, f is the frequency of the clock and k is a reference frame independent calibration constant.

In Newton's mechanics, the absolute Galilean time makes frequency f a reference frame independent constant. Therefore, we can set k = f to make the clock show the theoretical time i.e. the absolute Galilean time t: T = tf/k = tf/f = t.

But in special relativity, the relative relativistic time makes frequency f a reference frame dependent variable and can't be eliminated by setting k = f in the clock formula. Thus, clock time can't be simply calculated by the formula: T = tf/k != t in special relativity. Therefore, we need to verify whether clock time T and relativistic time t have the same property in Lorentz Transformation.

When a clock is observed in another inertial reference frame, we have t' = rt and f' = f/r and T' = t'f'/k = rt(f/r)/k = tf/k = T, where r = 1/sqrt(1 - v^2/c^2), which means that the physical time T won't change with the change of the inertial reference frame, and is Lorentz invariant and absolute, completely different from relativistic time. That is, a clock still measures the absolute time in special relativity.

Some people may argue that relativistic time has to be shown on two clocks. OK, here is it.
If you have a clock (clock 1) with you and watch my clock (clock 2) in motion and both clocks are set to be synchronized to show the same physical time T relative to your inertial reference frame, you will see your clock time: T1 = tf1/k1 = T and my clock time: T2 = tf2/k2 = T, where t is relativistic time of your frame, f1 and f2 are the frequencies of clock 1 and clock 2 respectively observed in your inertial reference frame, k1 and k2 are calibration constants of the clocks. The two events:

(Clock1, T1=T, x1=0, y1=0, z1=0, t1=t)
and
(Clock2, T2=T, x2=vt, y2=0, z2=0, t2=t)

are simultaneous measured with both relativistic time t and clock time T in your reference frame. When these two clocks are observed by me in the moving inertial reference frame, according to special relativity, we can use Lorentz Transformation to get the events in my frame (x', y', z', t'):

(clock1, T1', x1'=-vt1', y1'=0, z1'=0, t1')
and
(clock2, T2', x2'=0, y2'=0, z2'=0, t2')

t1' = r(t1-vx1/c^2) = r(t-0) = rt
t2' = r(t2-vx2/c^2) = r(t-tv^2/c^2) = rt/r^2 = t/r
T1' = t1'f1'/k1 = (rt)(f1/r)/k1 = tf1/k1 = T1 = T
T2' = t2'f2'/k2 = (t/r)(rf2)/k2 = tf2/k2 = T2 = T

That is, no matter observed from which inertial reference frame, the two events are still simultaneous measured with physical time T i.e. the two clocks are always synchronized measured with clock time T i.e. clock time T is absolute, but not synchronized measured with relativistic time t'. In real observations, we can only see clock time T but not relativistic time. Therefore, clock time is our physical time and absolute, totally different from relativistic time in Lorentz Transformation and thus relativistic time is a fake time without physical meaning. The change of the reference frame only makes changes of the relativistic time from t to t' and the relativistic frequency from f to f', which cancel each other in the formula: T= tf/k to make the physical time T unchanged. This proves that even in special relativity our physical time is still absolute. Therefore, special relativity based on the fake relativistic time is wrong.

That the physical time (i.e. clock time) is absolute has been clearly confirmed by the physical fact that all the atomic clocks on the GPS satellites are synchronized not only relative to the ground clocks but also relative to each other to show the same absolute physical time, which directly denies the claim of special relativity that clocks can never be synchronized relative to more than one inertial reference frame no matter how you correct them because "time is relative".

You will find the mathematical proofs that in special relativity, the real speed of light still follows Newton's velocity addition law, and both time dilation and length contraction are simply illusions in my peer-reviewed journal paper and conference paper which are available free of charge at: https://www.researchgate.net/publication/297527784_Challenge_to_the_Special_Theory_of_Relativity and https://www.researchgate.net/publication/297528348_Clock_Time_Is_Absolute_and_Universal

The puzzle of a constant velocity has always been a mystery. This is because everyone has been taught that EM waves are alternating and continuous, like media waves are. This is probably because of what we measure when we detect them.

But media waves have different and relative velocities. Why not light? How does one solve this mystery with an omnipresent time.

EM radiation is the instant emission, of discreet volumes(or lengths or durations) with a 50% duty cycle, is does not have frequency until detected. It has duty cycle. Alternation is not required. Single pole particles emit.

If you understand RF, one precision rectified half-wave sine signal, fed into a dipole will show you this at the absorber/receiver.

It seems that many physicists still don't know that Einstein's relativity is logically wrong and can be easily disproved through Lorentz Transformation of clock time i.e. our physical time between two inertial reference frames:

Let's look at the twin paradox which is designed to demonstrate that relative speed would generate time dilation as predicted by special relativity which claims that when the speed of a clock relative to an observer was close to the speed of light, the observer would see the clock slow down close to stop. But, it is pretty ironic as shown on Wikipedia, the final conclusion of the twin paradox becomes that, after a high speed space travel, it is the acceleration of the traveling twin (not his speed relative to his twin brother) that made him younger than his twin brother staying on the earth because both twins had experienced exactly the same speed relative to each other during the entire trip. Is it funny that the original argument that relative speed generates time dilation is completely lost, although relativists still think that the paradox has been solved? In fact, this paradox has simply confirmed that relative speed can never generate time dilation and special relativity is wrong.

Actually Einstein's relativity has already been disproved both theoretically and experimentally for more than four years. The fatal mistake of Einstein's relativity is that it uses Lorentz Transformation to redefine time and space, and the newly defined time is no longer the physical time we measure with physical clocks. The claim of the constant speed of light is very similar to the claim that everybody had the same height if the height is measured with a new ruler - an elastic band a ruler. Obviously, such claims do not make any sense.

In a physics theory, the physical time shown on a physical clock is T = tf/k where t is the theoretical time, f is the frequency of the clock and k is a reference frame independent calibration constant.

In Newton's mechanics, the absolute Galilean time makes frequency f a reference frame independent constant. Therefore, we can set k = f to make the clock show the theoretical time i.e. the absolute Galilean time t: T = tf/k = tf/f = t.

But in special relativity, the relative relativistic time makes frequency f a reference frame dependent variable and can't be eliminated by setting k = f in the clock formula. Thus, clock time can't be simply calculated by the formula: T = tf/k != t in special relativity. Therefore, we need to verify whether clock time T and relativistic time t have the same property in Lorentz Transformation.

When a clock is observed in another inertial reference frame, we have t' = rt and f' = f/r and T' = t'f'/k = rt(f/r)/k = tf/k = T, where r = 1/sqrt(1 - v^2/c^2), which means that the physical time T won't change with the change of the inertial reference frame, and is Lorentz invariant and absolute, completely different from relativistic time. That is, a clock still measures the absolute time in special relativity.

Some people may argue that relativistic time has to be shown on two clocks. OK, here is it.
If you have a clock (clock 1) with you and watch my clock (clock 2) in motion and both clocks are set to be synchronized to show the same physical time T relative to your inertial reference frame, you will see your clock time: T1 = tf1/k1 = T and my clock time: T2 = tf2/k2 = T, where t is relativistic time of your frame, f1 and f2 are the frequencies of clock 1 and clock 2 respectively observed in your inertial reference frame, k1 and k2 are calibration constants of the clocks. The two events:

(Clock1, T1=T, x1=0, y1=0, z1=0, t1=t)
and
(Clock2, T2=T, x2=vt, y2=0, z2=0, t2=t)

are simultaneous measured with both relativistic time t and clock time T in your reference frame. When these two clocks are observed by me in the moving inertial reference frame, according to special relativity, we can use Lorentz Transformation to get the events in my frame (x', y', z', t'):

(clock1, T1', x1'=-vt1', y1'=0, z1'=0, t1')
and
(clock2, T2', x2'=0, y2'=0, z2'=0, t2')

t1' = r(t1-vx1/c^2) = r(t-0) = rt
t2' = r(t2-vx2/c^2) = r(t-tv^2/c^2) = rt/r^2 = t/r
T1' = t1'f1'/k1 = (rt)(f1/r)/k1 = tf1/k1 = T1 = T
T2' = t2'f2'/k2 = (t/r)(rf2)/k2 = tf2/k2 = T2 = T

That is, no matter observed from which inertial reference frame, the two events are still simultaneous measured with physical time T i.e. the two clocks are always synchronized measured with clock time T i.e. clock time T is absolute, but not synchronized measured with relativistic time t'. In real observations, we can only see clock time T but not relativistic time. Therefore, clock time is our physical time and absolute, totally different from relativistic time in Lorentz Transformation and thus relativistic time is a fake time without physical meaning. The change of the reference frame only makes changes of the relativistic time from t to t' and the relativistic frequency from f to f', which cancel each other in the formula: T= tf/k to make the physical time T unchanged. This proves that even in special relativity our physical time is still absolute. Therefore, special relativity based on the fake relativistic time is wrong.

That the physical time (i.e. clock time) is absolute has been clearly confirmed by the physical fact that all the atomic clocks on the GPS satellites are synchronized not only relative to the ground clocks but also relative to each other to show the same absolute physical time, which directly denies the claim of special relativity that clocks can never be synchronized relative to more than one inertial reference frame no matter how you correct them because "time is relative".

You will find the mathematical proofs that in special relativity, the real speed of light still follows Newton's velocity addition law, and both time dilation and length contraction are simply illusions in my peer-reviewed journal paper and conference paper which are available free of charge at: https://www.researchgate.net/publication/297527784_Challenge_to_the_Special_Theory_of_Relativity and https://www.researchgate.net/publication/297528348_Clock_Time_Is_Absolute_and_Universal

The puzzle of a constant velocity has always been a mystery. This is because everyone has been taught that EM waves are alternating and continuous, like media waves are. This is probably because of what we measure when we detect them.

But media waves have different and relative velocities. Why not light? How does one solve this mystery with an omnipresent time.

EM radiation is the instant emission, of discreet volumes(or lengths or durations) with a 50% duty cycle, is does not have frequency until detected. It has duty cycle. Alternation is not required. Single pole particles emit.

If you understand RF, one precision rectified half-wave sine signal, fed into a dipole will show you this at the absorber/receiver.


The Lorentz Transformation: Relativity: Chapter 11.


If we drop these hypotheses, then the dilemma of Section VII disappears, because the theorem of the addition of velocities derived in Section VI becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of Section VI in order to remove the apparent disagreement between these two fundamental results of experience? This question leads to a general one. In the discussion of Section VI we have to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity? In other words: Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another.

Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in Section II we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework. Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as “co-ordinate planes” (“co-ordinate system”). A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K' to the train. An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate planes, and with regard to time by a time-value t. Relative to K', the same event would be fixed in respect of space and time by corresponding values x', y', z', t', which of course are not identical with x, y, z, t. It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements.

Obviously our problem can be exactly formulated in the following manner. What are the values x', y', z', t' of an event with respect to K', when the magnitudes x, y, z, t, of the same event with respect to K are given? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every ray) with respect to K and K'. For the relative orientation in space of the co-ordinate systems indicated in the diagram (Fig. 2), this problem is solved by means of the equations:

This system of equations is known as the “Lorentz transformation.”


Lorentz transformation for stars - Astronomy

What Einstein 's special theory of relativity says is that to understand why the speed of light is constant, we have to modify the way in which we translate the observation in one inertial frame to that of another. The Galilei transformation

is wrong. The correct relation is

This is called the Lorentz transformation .

You can see that if the relative velocity v between the two frames are much smaller than the speed of light c , then the ratio v/c can be neglected in this relation and we recover the Galilei transformation . So the reason why we did not have any problems with the Galilei transformation up to now is that v was small enough for it to be a good approximation of the Lorentz transformation .

Let's check that this relation does indeed show that the speed of light is the same in both frames (x,t) and (x',t') . Let's say that a beam of light started out from the origin x'=x=0 at time t'=t=0 . Since the speed of light is c , at time t=T , the beam of light would have traveled to the point x=cT in the (x,t) frame. In the other frame, this point is observed as

so the speed of light in the (x',t') frame would also be:

The following figure shows a graphical representation of the Lorentz transformation :

Spacetime diagram explaining why the speed of light is the same in both frames: