Special relativity

Special relativity
Principle of relativity Theory of relativity Formulations
Foundations Einstein's postulates Inertial frame of reference Speed of light Maxwell's equations Lorentz transformation
Consequences Time dilation Length contraction Relativistic mass Mass–energy equivalence Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disk Bell's spaceship paradox Ehrenfest paradox
Spacetime Minkowski spacetime Spacetime diagram World line Light cone
Dynamics Proper time Proper mass Four-momentum
History Precursors Galilean relativity Galilean transformation Aether theories
People Einstein Sommerfeld Michelson Morley FitzGerald Herglotz Lorentz Poincaré Minkowski Fizeau Abraham Born Planck von Laue Ehrenfest Tolman Dirac
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In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, On the Electrodynamics of Moving Bodies, the theory is presented as being based on just two postulates:^{[p 1][1][2]}

1. The laws of physics are invariant (identical) in all inertial frames of reference (that is, frames of reference with no acceleration). This is known as the principle of relativity.
2. The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer. This is known as the principle of light constancy, or the principle of light speed invariance.

The first postulate was first formulated by Galileo Galilei (see Galilean invariance).

Special relativity was described by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".^{[p 1]} Maxwell's equations of electromagnetism appeared to be incompatible with Newtonian mechanics, and the Michelson–Morley experiment failed to detect the Earth's motion against the hypothesized luminiferous aether. These led to the development of the Lorentz transformations, by Hendrik Lorentz, which adjust distances and times for moving objects. Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as 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] These 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 ⁠${\displaystyle E=mc^{2}}$⁠, where ${\displaystyle c}$ is the speed of light in 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 several years later when Einstein developed general relativity, which introduced a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case".^{[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, with Hermann Minkowski's papers on spacetime.^[4]

The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, where the curvature of spacetime (a consequence of the energy–momentum tensor and representing gravity) is negligible.^{[8][note 2]} 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. But general relativity incorporates non-Euclidean geometry 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]

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:^{[p 1]}

• 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^[13] and the lack of evidence for the luminiferous ether.^[14] There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.^[15][16] 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.^[17] But 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.^[18]

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. 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 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.

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.^[19]: 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" does not strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.

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. But in the late 19th century the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance they called "aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). 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.^[20] 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.

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][21]}

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.^[22] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler^[11] and by Callahan.^[23] This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.

Define an event to have spacetime coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in a reference frame moving at a velocity v on the x-axis with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way: $${\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}}$$ where $${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$$ 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: $${\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}$$

This shows that the unprimed frame is moving with the velocity −v, as measured in the primed frame.^[24]

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 that is 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 (x′₁, t′₁), another event has coordinates (x₂, t₂) and (x′₂, t′₂), and the differences are defined as

• Eq. 1:    ${\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}$
• Eq. 2:    ${\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}$

we get

• Eq. 3:    ${\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }$ ${\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}$
• Eq. 4:    ${\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }$ ${\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}$

If we take differentials instead of taking differences, we get

• Eq. 5:    ${\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }$ ${\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}$
• Eq. 6:    ${\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }$ ${\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}$

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

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.^[21]

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig. 2-1.^{[21][25]: 155–199}

Fig. 3-1a. Draw the ${\displaystyle x}$ and ${\displaystyle t}$ axes of frame S. The ${\displaystyle x}$ axis is horizontal and the ${\displaystyle t}$ (actually ${\displaystyle ct}$) axis is vertical, which is the opposite of the usual convention in kinematics. The ${\displaystyle ct}$ axis is scaled by a factor of ${\displaystyle c}$ so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time ${\displaystyle t=0.}$ The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

Fig. 3-1b. Draw the ${\displaystyle x'}$ and ${\displaystyle ct'}$ axes of frame S'. The ${\displaystyle ct'}$ axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, ${\displaystyle v=c/2.}$ Both the ${\displaystyle ct'}$ and ${\displaystyle x'}$ axes are tilted from the unprimed axes by an angle ${\displaystyle \alpha =\tan ^{-1}(\beta ),}$ where ${\displaystyle \beta =v/c.}$ The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that ${\displaystyle t=0}$ when ${\displaystyle t'=0.}$

Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that ${\displaystyle (x',ct')}$ coordinates of ${\displaystyle (0,1)}$ in the primed coordinate system transform to ${\displaystyle (\beta \gamma ,\gamma )}$ in the unprimed coordinate system. Likewise, ${\displaystyle (x',ct')}$ coordinates of ${\displaystyle (1,0)}$ in the primed coordinate system transform to ${\displaystyle (\gamma ,\beta \gamma )}$ in the unprimed system. Draw gridlines parallel with the ${\displaystyle ct'}$ axis through points ${\displaystyle (k\gamma ,k\beta \gamma )}$ as measured in the unprimed frame, where ${\displaystyle k}$ is an integer. Likewise, draw gridlines parallel with the ${\displaystyle x'}$ axis through ${\displaystyle (k\beta \gamma ,k\gamma )}$ as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between ${\displaystyle ct'}$ units equals ${\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}}$ times the spacing between ${\displaystyle ct}$ units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as ${\displaystyle \beta \to 1.}$

Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time ${\displaystyle t'=0}$ still plot as 45° diagonal lines. The primed coordinates of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.

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

The consequences of special relativity can be derived from the Lorentz transformation equations.^[26] These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics 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.

In Galilean relativity, an object's length (⁠${\displaystyle \Delta r}$⁠)^{[note 3]} and the temporal separation between two events (⁠${\displaystyle \Delta t}$⁠) are independent invariants, the values of which do not change when observed from different frames of reference.^{[note 4][note 5]}

In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as ⁠${\displaystyle \Delta s^{2}}$⁠:^{[note 6]} $${\displaystyle \Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}$$

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

The form of ⁠${\displaystyle \Delta s^{2}}$⁠, 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.^{[30]: 33–34}

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed: $${\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}}$$

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:^[21] $${\displaystyle {\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}}$$

The value of ${\displaystyle \Delta s^{2}}$ is hence independent of the frame in which it is measured.

In considering the physical significance of ⁠${\displaystyle \Delta s^{2}}$⁠, there are three cases to note:^{[21][31]: 25–39}

• Δs² > 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 ⁠${\displaystyle \vert \Delta x/\Delta t\vert <c}$⁠, and given the Lorentz transformation ⁠${\displaystyle \Delta x'=\gamma \ (\Delta x-v\ \Delta t)}$⁠, it is evident that there exists a ${\displaystyle v}$ less than ${\displaystyle c}$ for which ${\displaystyle \Delta x'=0}$ (in particular, ⁠${\displaystyle v=\Delta x/\Delta t}$⁠). 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, ⁠${\displaystyle \Delta s/c}$⁠, is called the proper time.
• Δs² < 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 ⁠${\displaystyle \vert \Delta x/\Delta t\vert >c}$⁠, and given the Lorentz transformation ⁠${\displaystyle \Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2})}$⁠, there exists a ${\displaystyle v}$ less than ${\displaystyle c}$ for which ${\displaystyle \Delta t'=0}$ (in particular, ⁠${\displaystyle v=c^{2}\Delta t/\Delta x}$⁠). 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, ⁠${\displaystyle \textstyle {\sqrt {-\Delta s^{2}}}}$⁠, is called the proper distance, or proper length. For values of ${\displaystyle v}$ greater than and less than ⁠${\displaystyle c^{2}\Delta t/\Delta x}$⁠, the sign of ${\displaystyle \Delta t'}$ changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that ${\displaystyle v}$ could be greater than ${\displaystyle c^{2}\Delta t/\Delta x}$ would be if ⁠${\displaystyle v>c}$⁠.
• Δs² = 0: In this case, the two events are said to be lightlike separated. This implies that ⁠${\displaystyle \vert \Delta x/\Delta t\vert =c}$⁠, and this relationship is frame independent due to the invariance of ⁠${\displaystyle s^{2}}$⁠. From this, we observe that the speed of light is ${\displaystyle c}$ 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.

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) $${\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)}$$

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.^[32] Since relativity of simultaneity is a first order effect in ⁠${\displaystyle v}$⁠,^[21] instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity.^{[p 14]}

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

Suppose 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:

${\displaystyle \Delta t'=\gamma \,\Delta t}$ for events satisfying ${\displaystyle \Delta x=0\ .}$

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.^[33]

Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.^{[34]: 149–152}

The measurements that we would get if we actually looked at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the Doppler effect. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.^{[34]: 149–152}

Paul Langevin, an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass-energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909^[35]) which he used to perform a novel derivation of the Lorentz transformation.^[36]

A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.^{[37][38][39][40]} (Langevin himself made use of a light-clock oriented parallel to its line of motion.^[36])

Consider the scenario illustrated in Fig. 4-3A. Observer A holds a light-clock of length ${\displaystyle L}$ as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds ⁠${\displaystyle t_{\text{A}}=2L/c}$⁠. A time interval measured using a single clock which is motionless in a particular reference frame is called a proper time interval.^[41]

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of ⁠${\displaystyle v}$⁠. Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same ${\displaystyle c}$ that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as ${\textstyle \pm {\sqrt {c^{2}-v^{2}}},}$ so that the total round-trip time of the pulses is ${\textstyle t_{\text{B}}=2L{\big /}{\sqrt {c^{2}-v^{2}}}={}}$⁠${\displaystyle \textstyle t_{\text{A}}{\big /}{\sqrt {1-v^{2}/c^{2}}}}$⁠. Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore not a proper time interval because he did not measure it with a single resting clock.^[41]

In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of ${\displaystyle v}$ to the left, in which case observer A would perceive B's clock as running slower than her local clock.

There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with both A's and B's measurement of time.^[42]

The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called twin paradox, articulated in its present form by Langevin in 1911.^[43] Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older.

This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in order for the two observers to compare their proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.^[44]

Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the Twin paradox article. We will examine in the following one such solution to the paradox.

Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences. Fig 4-4 illustrates a scenario where the traveling twin flies at 0.6 c to and from a star 3 ly distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of ⁠${\displaystyle \textstyle f'=f{\sqrt {(1-\beta )/(1+\beta )}}}$⁠. Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of ⁠${\displaystyle \textstyle f''=f{\sqrt {(1+\beta )/(1-\beta )}}}$⁠. The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.^{[34]: 152–159}

Item	Measured by the stay-at-home	Fig 4-4	Measured by the traveler	Fig 4-4
Total time of trip	${\displaystyle T={\frac {2L}{v}}}$	10 yr	${\displaystyle T'={\frac {2L}{\gamma v}}}$	8 yr
Total number of pulses sent	${\displaystyle fT={\frac {2fL}{v}}}$	10	${\displaystyle fT'={\frac {2fL}{\gamma v}}}$	8
Time when traveler's turnaround is detected	${\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}}$	8 yr	${\displaystyle t_{1}'={\frac {L}{\gamma v}}}$	4 yr
Number of pulses received at initial ${\displaystyle f'}$ rate	${\displaystyle f't_{1}}$ ${\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}$	4	${\displaystyle f't_{1}'}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta )}$	2
Time for remainder of trip	${\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}}$	2 yr	${\displaystyle t_{2}'={\frac {L}{\gamma v}}}$	4 yr
Number of signals received at final ${\displaystyle f''}$ rate	${\displaystyle f''t_{2}}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}$	4	${\displaystyle f''t_{2}'}$ ${\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}$ ${\displaystyle ={\frac {fL}{v}}(1+\beta )}$	8
Total number of received pulses	${\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}}$ ${\displaystyle ={\frac {2fL}{\gamma v}}}$	8	${\displaystyle {\frac {2fL}{v}}}$	10
Twin's calculation as to how much the other twin should have aged	${\displaystyle T'={\frac {2L}{\gamma v}}}$	8 yr	${\displaystyle T={\frac {2L}{v}}}$	10 yr

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′:

${\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}}$  for events satisfying ${\displaystyle \Delta t'=0\ .}$

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.

Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector ⁠${\displaystyle \mathbf {u} }$⁠. What is its velocity ${\displaystyle \mathbf {u'} }$ in frame S′?

We can write

$${\displaystyle \mathbf {|u|} =u=dx/dt\,.}$$

(7)

$${\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.}$$

(8)

Substituting expressions for ${\displaystyle dx'}$ and ${\displaystyle dt'}$ from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed ${\displaystyle u}$ to ⁠${\displaystyle u'}$⁠:

$${\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.}$$

(9)

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing ${\displaystyle v}$ with ⁠${\displaystyle -v}$⁠.

$${\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.}$$

(10)

For ${\displaystyle \mathbf {u} }$ not aligned along the x-axis, we write:^{[12]: 47–49}

$${\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .}$$

(11)

$${\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .}$$

(12)

The forward and inverse transformations for this case are:

$${\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .}$$

(13)

$${\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .}$$

(14)

Equation 10 and Equation 14 can be interpreted as giving the resultant ${\displaystyle \mathbf {u} }$ of the two velocities ${\displaystyle \mathbf {v} }$ and ⁠${\displaystyle \mathbf {u'} }$⁠, and they replace the formula ${\displaystyle \mathbf {u=u'+v} }$ which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned with each other (although not necessarily in standard configuration).^{[12]: 47–49}

We note the following points:

• If an object (e.g., a photon) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±c), then it would also be moving at the speed of light in any other frame, moving at |v| < c.
• The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.
• If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g., |⁠u/c⁠| ≪ 1), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
• Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.

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.

Figure 4-5. Thomas–Wigner rotation

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

Thomas rotation results from the relativity of simultaneity. In Fig. 4-5a, a rod of length ${\displaystyle L}$ in its rest frame (i.e., having a proper length of ⁠${\displaystyle L}$⁠) rises vertically along the y-axis in the ground frame.

In Fig. 4-5b, the same rod is observed from the frame of a rocket moving at speed ${\displaystyle v}$ to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (not see) the clock at the right end of the rod as being advanced in time by ⁠${\displaystyle Lv/c^{2}}$⁠, and the rod is correspondingly observed as tilted.^{[31]: 98–99}

Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in 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.^{[31]: 169–174}

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".^{[p 15][31]: 98–99}

In Fig. 4-6, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; that is, there is a frame of reference in which events A and B occur at the 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. But no frames are accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result.

For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).^{[45][p 16]} A variety of causal paradoxes could then be constructed.

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

Consider the spacetime diagrams in Fig. 4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. 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. 4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the ${\displaystyle -x'}$ 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. 4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the ${\displaystyle +x}$ 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.^[46]

It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the ${\displaystyle x'}$ axis (and the signal from A to B slightly steeper than the ${\displaystyle x}$ axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the ${\displaystyle ct'}$ and ${\displaystyle x'}$ axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only slightly faster than the speed of light will result in causality violation.^[47]

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.^[48] For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).^[49][50]

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.^[51] The speed of light was measured in still water. What would be the speed of light in flowing water?

In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light.

According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its 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 ${\displaystyle u'=c/n}$ is the speed of light in still water, and ${\displaystyle v}$ is the speed of the water, and ${\displaystyle u_{\pm }}$ is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then $${\displaystyle u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .}$$

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 ${\displaystyle n}$ 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.^[52]

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.^[30]

${\displaystyle u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=}$ ${\displaystyle {\frac {c/n\pm v}{1\pm v/cn}}\approx }$ ${\displaystyle c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx }$ ${\displaystyle {\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)}$

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.^[53] (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.^[54]

The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light,^[55] and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.^[56] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,^[57] but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag.^[58]

Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include^{[30]: 57–60}

${\displaystyle \cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}}$   OR   ${\displaystyle \sin \theta '={\frac {\sin \theta }{\gamma [1+(v/c)\cos \theta ]}}}$   OR   ${\displaystyle \tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}}$

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.^[59][60]

Assume the receiver and the source are moving away from each other with a relative speed ${\displaystyle v}$ as measured by an observer on the receiver or the source (The sign convention adopted here is that ${\displaystyle v}$ is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then $${\displaystyle f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}}$$ where ${\displaystyle c_{s}}$ is the speed of sound.

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 $${\displaystyle f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}$$ where

• ${\displaystyle \beta =v/c}$  and
• ${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}$ is the Lorentz factor.

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.^[61][21]

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.^[21] In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of ⁠${\displaystyle \gamma }$⁠. In Fig. 5-3b, the light is redshifted by the same factor.

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. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

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][62][63]} A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although the surface of the sphere and the images on it will appear distorted.^[64][65]

Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig. 5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen.^[66] This illusion has come to be known 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.^[67][68][69] In Fig. 5-6, 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.^[70]

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.

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 mc². Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.

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

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0): it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c², 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c². As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c².

The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these do not talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.^{[p 1]} The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.^{[p 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.^[71] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.^[72]

Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.^{[p 22][note 10]}

In his fourth of his 1905 Annus mirabilis papers,^{[p 21]} Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time.

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

Figure 6-1. Einstein's 1905 derivation of E = mc²

Fig. 6-1 (top). Consider a system of plane waves of light having frequency ${\displaystyle f}$ traveling in direction ${\displaystyle \phi }$ relative to the x-axis of reference frame S. The frequency (and hence energy) of the waves as measured in frame S′ that is moving along the x-axis at velocity ${\displaystyle v}$ is given by the relativistic Doppler shift formula which Einstein had developed in his 1905 paper on special relativity:^{[p 1]}

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

Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame S. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle ${\displaystyle \phi }$ with respect to the x-axis. Each pulse has energy ⁠${\displaystyle L/2}$⁠. Because of conservation of momentum, the body remains stationary in S after emission of the two pulses. Let ${\displaystyle E_{0}}$ be the energy of the body before emission of the two pulses and ${\displaystyle E_{1}}$ after their emission.

Next, consider the same system observed from frame S′ that is moving along the x-axis at speed ${\displaystyle v}$ relative to frame S. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let ${\displaystyle H_{0}}$ be the energy of the body measured in reference frame S′ before emission of the two pulses and ${\displaystyle H_{1}}$ after their emission. We obtain the following relationships:^{[p 21]}

${\displaystyle {\begin{aligned}E_{0}&=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L\\[5mu]H_{0}&=H_{1}+{\tfrac {1}{2}}L{\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\frac {1+(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}=H_{1}+{\frac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}}$

From the above equations, we obtain the following:

${\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}$

(6-1)

The two differences of form ${\displaystyle H-E}$ seen in the above equation have a straightforward physical interpretation. Since ${\displaystyle H}$ and ${\displaystyle E}$ are the energies of the arbitrary body in the moving and stationary frames, ${\displaystyle H_{0}-E_{0}}$ and ${\displaystyle H_{1}-E_{1}}$ represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence,

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

(6-2)

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

${\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}}$

(6-3)

Comparing the above expression with the classical expression for kinetic energy, K.E. = ⁠1/2⁠mv², Einstein then noted: "If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²."

Rindler has observed that Einstein's heuristic argument suggested merely that energy contributes to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that all inertial mass represented a reserve of energy. "To equate all mass with energy required an act of aesthetic faith, very characteristic of Einstein."^{[12]: 81–84} Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.

For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.^[73]

Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems which exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant 1g, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by: $${\displaystyle v(t)={\frac {at}{\sqrt {1+a^{2}t^{2}/c^{2}}}}}$$ where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth.^{[p 23]} Therefore, after one year of accelerating at 9.81 m/s², the spaceship will be travelling at v = 0.712c and 0.946c after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth. During their journey, people on Earth will experience more time than they do - since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years.^[74] A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip 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.^[74] 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).^[75]

Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.^[76]

In Newtonian mechanics, analysis of collisions involves use of the conservation laws for mass, momentum and energy. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. (Inelastic collisions are discussed in Spacetime#Conservation laws. Radioactive decay may be considered a sort of time-reversed inelastic collision.^[76])

Elastic scattering of charged elementary particles deviates from ideality due to the production of Bremsstrahlung radiation.^[77][78]

Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls will subtend a right angle. (a) In the stationary frame, an incident sphere traveling at 2v strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±v. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±u. Energy conservation requires that |u| = |v|. (c) Reverting to the stationary frame, the rebound velocities are v ± u. The dot product (v + u) ⋅ (v − u) = v² − u² = 0, indicating that the vectors are orthogonal.^{[12]: 26–27}

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

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

${\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}}$

(6-4)

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

${\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}}$

(6-5)

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc²) equals the sum of the total energies of the emergent particles:

${\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}}$

(6-6)

Breaking down (6-5) into its components, replacing ${\displaystyle v}$ with the dimensionless ⁠${\displaystyle \beta }$⁠, and factoring out common terms from (6-5) and (6-6) yields the following:^{[p 24]}

${\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }}$

(6-7)

${\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }}$

(6-8)

${\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}}$

(6-9)

From these we obtain the following relationships:^{[p 24]}

${\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}$

(6-10)

${\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}$

(6-11)

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