Special relativity

thumb|[[Albert Einstein around 1905, the year his paper on special relativity was published]]

In physics, the special theory of relativity, or simply special relativity, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, [[Annus Mirabilis papers#Special relativity|

"On the Electrodynamics of Moving Bodies"]], the theory is presented as being based on just two postulates:

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.
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).

Overview

Relativity is a theory that accurately describes objects moving at speeds far beyond normal experience. Relativity replaces the idea that time flows equally everywhere in the universe with a new concept that time flows differently for every independent object. The flow of time can be expressed by counting ticks on a clock. Moving clocks run slower. Two events measured at the same time on a stationary clock occur at different times if measured on moving clocks. At speeds encountered in normal experience, the slow down cannot be observed. Near the speed of light the slow down is significant. At these relativistic speeds, many other physical effects can only be understood by including the effects of special relativity. The conceptual effects include:

The relativity of simultaneity events that appear simultaneous to one observer may not be simultaneous to an observer in motion

Special relativity replaced the conventional notion of an absolute, universal time with the notion of a time that is local to each observer. Information about distant objects can arrive no faster than the speed of light so visual observations always report events that have happened in the past. This effect makes visual descriptions of the effects of special relativity especially prone to mistakes.

Special relativity also has profound technical consequences.

A defining feature of special relativity is the replacement of Euclidean geometry with Lorentzian geometry. His conclusions were summarized as Galilean relativity and used as the basis of Newtonian mechanics.

In 1864 James Clerk Maxwell presented a theory of electromagnetism which did not obey Galilean relativity. The theory specifically predicted a constant speed of light in vacuum, no matter the motion (velocity, acceleration, etc.) of the light emitter or receiver or its frequency, wavelength, direction, polarization, or phase. This, as yet untested theory, was thought at the time to be only valid in inertial frames fixed in an aether. Numerous experiments followed, attempting to measure the speed of light as Earth moved through the proposed fixed aether, culminating in the 1887 Michelson–Morley experiment which only confirmed the constant speed of light. Einstein applied the Lorentz transformations known to be compatible with Maxwell's equations for electrodynamics to the classical laws of mechanics. This changed Newton's mechanics situations involving all motions, especially velocities close to that of light

The theory became essentially complete in 1907, with Hermann Minkowski's papers on spacetime. Even so, the Newtonian model remains accurate at low velocities relative to the speed of light, for example, everyday motion on Earth.

In comparing to the general theory of relativity, Einstein specifically called his earlier work "special theory of relativity" (German: Spezielle Relativitätstheorie) in two short papers published in November 1915 and in a long review article published in 1916, saying he meant a restriction to frames in uniform motion, and was featured in the title of Einstein's popular book Relativity: The Special and the General Theory first published in 1916. Just as Galilean relativity is accepted as an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for 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.

Terminology

Special relativity builds upon important physics ideas. Among the most basic of these are the following:

speed or velocity, how fast an object moves relative to a reference point.
speed of light, the maximum speed of information, independent of the speed of the source and receiver,

<math display="block">(\text{interval})^2 = \left[ \text{event separation in time} \right]^2 - \left[ \text{event separation in space} \right]^2 </math>

coordinate system or reference frame: a way to locate events in spacetime. Events have coordinates x, y, z for space and t for time. The coordinates of the event are different in a different reference frame.
inertial reference frame: a region of a reference frame where objects at rest with respect to the frame stay as rest, or if in uniform motion, stay in motion; also called a free-float frame.

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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:

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. 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.

Principle of relativity

Reference frames and relative motion

thumb|Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the [[principle of relativity, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −v. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.]]

Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a "clock" (any reference device with uniform periodicity).

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

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. 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 (pronounced "S prime" or "S dash") belongs to a second observer .

The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame .
Frame 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 are coincident when time for frame S and for frame .

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 are not comoving.

Lack of an absolute reference frame

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. 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

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.

Lorentz transformation

Two- vs one- postulate approaches

Einstein combined the two postulates – of relativity – and of the invariance of the speed of light, into a single postulate, the Lorentz transformation:

</math> is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of , 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:

<math display="block">\begin{align}

t &= \gamma ( t' + v x'/c^2) \\

x &= \gamma ( x' + v t') \\

y &= y' \\

z &= z'.

\end{align}</math>

This shows that the unprimed frame is moving with the velocity −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 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 and , another event has coordinates and , and the differences are defined as

<math>\Delta x' = x'_2-x'_1 \ , \ \Delta t' = t'_2-t'_1 \ .</math>
<math>\Delta x = x_2-x_1 \ , \ \ \Delta t = t_2-t_1 \ .</math>

we get

<math>\Delta x' = \gamma \ (\Delta x - v \,\Delta t) \ ,\ \ </math> <math>\Delta t' = \gamma \ \left(\Delta t - v \ \Delta x / c^{2} \right) \ . </math>
<math>\Delta x = \gamma \ (\Delta x' + v \,\Delta t') \ , \ </math> <math>\Delta t = \gamma \ \left(\Delta t' + v \ \Delta x' / c^{2} \right) \ . </math>

If we take differentials instead of taking differences, we get

<math>dx' = \gamma \ (dx - v\,dt) \ ,\ \ </math> <math>dt' = \gamma \ \left( dt - v \ dx / c^{2} \right) \ . </math>
<math>dx = \gamma \ (dx' + v\,dt') \ , \ </math> <math>dt = \gamma \ \left(dt' + v \ dx' / c^{2} \right) \ . </math>

Graphical representation of the Lorentz transformation

Spacetime diagrams (also called 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.

Fig. 3-1a. Draw the <math>x</math> and <math>t</math> axes of frame S. The <math>x</math> axis is horizontal and the <math>ct</math> (time written in units of space) axis is vertical, which is the opposite of the usual convention in kinematics. The <math>ct</math> axis is scaled by a factor of <math>c</math> 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 <math>t = 0.</math> The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, <math>\text{A}</math> and <math>\text{B},</math> have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

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

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

Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time <math>t' = 0</math> still plot as 45° diagonal lines. The primed coordinates of <math>\text{A}</math> and <math>\text{B}</math> 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. The frames are equivalent.

Consequences derived from the Lorentz transformation

The consequences of special relativity can be derived from the Lorentz transformation equations. 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

In Galilean relativity, the spatial separation, (), and the temporal separation, (), between two events are independent invariants, the values of which do not change when observed from different frames of reference. In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as :

<math display="block"> \Delta s^2 \; \overset\text{def}{=} \; c^2 \Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) </math>

In considering the physical significance of , there are three cases:

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:

<math display="block">\Delta s^2 \, = \, c^2 \Delta t^2 - \Delta x^2</math>

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration: 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.

Time dilation

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

Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by . To find the relation between the times between these ticks as measured in both systems, can be used to find:

: <math>\Delta t' = \gamma\, \Delta t </math>for events satisfying<math>\Delta x = 0 \ .</math>

This shows that the time (Δ) between the two ticks as seen in the frame in which the clock is moving (), 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.

thumb|Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame

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

Langevin's light-clock

thumb|Figure 4–3. Thought experiment using a light-clock to explain time dilation

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

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. (Langevin himself made use of a light-clock oriented parallel to its line of motion.

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of . Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same <math>c</math> that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as <math display=inline>\pm \sqrt{c^2 - v^2},</math> so that the total round-trip time of the pulses is <math display=inline>t_\text{B} = 2L \big/ \sqrt{ c^2 - v^2 } = {}</math>. 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.

Twin paradox

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. Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older.

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

thumb|Figure 4-4. Doppler analysis of twin paradox

Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in 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. illustrates a scenario where the traveling twin flies at to and from a star distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of . Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of . The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox. A variety of causal paradoxes could then be constructed.

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 ('), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.

Fig. 4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the <math>-x'</math> 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.
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 <math>+x</math> axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, in the primed frames of C and D, B will receive the message before it was sent out, a violation of causality.

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 <math>x'</math> axis (and the signal from A to B slightly steeper than the <math>x</math> 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 <math>ct'</math> and <math>x'</math> 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.

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.

Only matter and energy are limited by the speed of light. Various trivial situations can be described where some imaginary points move faster than light. For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly. The light beam is not solid and it does not instantly follow the motion of the search light and thus does not violate causality or any other relativistic phenomenon.

Optical effects

Dragging effects

thumb|Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.

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. 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 <math>u' = c/n</math> is the speed of light in still water, and <math>v</math> is the speed of the water, and <math> u_{\pm} </math> 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

<math display="block">u_{\pm} =\frac{c}{n} \pm v\left(1-\frac{1}{n^2}\right) \ . </math>

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 <math>n</math> depends on wavelength, the aether must be capable of sustaining different motions at the same time. 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.

From the point of view of special relativity, Fizeau's result is nothing but an approximation to , the relativistic formula for composition of velocities. (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.

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

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

Assume the receiver and the source are moving away from each other with a relative speed <math>v</math> as measured by an observer on the receiver or the source (The sign convention adopted here is that <math>v</math> is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then

<math display="block">f_{r} = \left(1 - \frac v {c_s} \right) f_s</math>

where <math>c_s</math> 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

<math display="block">f_r = \gamma\left(1 - \beta\right) f_s = \sqrt{\frac{1 - \beta}{1 + \beta\,f_s.</math>

where

<math>\beta = v/c </math>  and
<math>\gamma = \frac{1}{\sqrt{1 - \beta^2</math> 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. 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.

thumb|Figure 5–6. Galaxy [[Messier 87|M87 sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.]]

Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is 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. This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect.

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. 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.

Dynamics

Section ' 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

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 : 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 . 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. 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. Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.

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.

Einstein's 1905 demonstration of E = mc2

In his fourth of his 1905 Annus mirabilis papers,

How far can you travel from the Earth?

Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~ 100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems that exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant 1g, it will, after one 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 the spaceship and t is the coordinate time as measured by people on Earth. Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at and 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. 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 will take years and cover about 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.

Elastic collisions

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.

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.

Newtonian analysis

thumb|Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle

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

{ \{ \beta_1^2 \sin^2{\phi} + \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} } </math> |6-10

{ \{ \beta_1^2 \sin^2{\theta} + \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} } </math> |6-11

For the symmetrical case in which <math> \phi = \theta</math> and , () takes on the simpler form: Fig. 7-2 presents plots of the sinh, cosh, and tanh functions.

For the unit circle, the slope of the ray is given by

: <math>\text{slope} = \tan a = \frac{\sin a }{\cos a }.</math>

In the Cartesian plane, rotation of point into point by angle θ is given by

: <math>

\begin{pmatrix}

x' \\

y' \\

\end{pmatrix} = \begin{pmatrix}

\cos \theta & -\sin \theta \\

\sin \theta & \cos \theta \\

\end{pmatrix}\begin{pmatrix}

x \\

y \\

\end{pmatrix}.</math>

In a spacetime diagram, the velocity parameter <math>\beta \equiv \frac{v}{c}</math> is the analog of slope. The rapidity, φ, is defined by

Minkowski spacetime

thumb|Figure 10–1. Orthogonality and rotation of coordinate systems compared between left: [[Euclidean space through circular angle φ, right: in Minkowski spacetime through hyperbolic angle φ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).]]

The physical theory of special relativity was recast by Hermann Minkowski in a 4-dimensional geometry now called Minkowski space. 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

<math display="block"> ds^2 = d\mathbf{x} \cdot d\mathbf{x} = dx_1^2 + dx_2^2 + dx_3^2, </math>

where are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X0 derived from time, such that the distance differential fulfills

where 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). 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 form of ds above depends on the metric and on the choices for the X0 coordinate.

To make the time coordinate look like the space coordinates, it can be treated as imaginary: (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 , rather than a "disguised" Euclidean metric using ict as the time coordinate.

Some authors use , 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.

These numerous conventions can be superseded by using natural units where . Then space and time have equivalent units, and no factors of c appear anywhere.

A four dimensional space has four-dimensional vectors, or "four-vectors". The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component , in a contravariant position four-vector with components:

<math display="block">X^\nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z) = (ct, \mathbf{x} ).</math>

where we define so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.

4‑vectors

, and more generally tensors, simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the field strength tensor formulation.

Definition of 4-vectors

A 4-tuple, is a "4-vector" if its component Ai transform between frames according to the Lorentz transformation.

If using coordinates, A is a if it transforms (in the ) according to

: <math>\begin{align}

A_0' &= \gamma \left( A_0 - (v/c) A_1 \right) \\

A_1' &= \gamma \left( A_1 - (v/c) A_0 \right)\\

A_2' &= A_2 \\

A_3' &= A_3

\end{align} ,</math>

which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation.

As usual, when we write x, t, etc. we generally mean Δx, Δt etc.

The last three components of a must be a standard vector in three-dimensional space. Therefore, a must transform like under Lorentz transformations as well as rotations.

Properties of 4-vectors

Closure under linear combination: If A and B are , then is also a .
Inner-product invariance: If A and B are , then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a . In the following, <math>\vec{A}</math> and <math>\vec{B}</math> are :
: <math>A \cdot B \equiv </math> <math>A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 \equiv </math> <math>A_0 B_0 - \vec{A} \cdot \vec{B}</math>

: In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in .

: Two vectors are said to be orthogonal if . Unlike the case with , orthogonal are not necessarily at right angles to each other. The rule is that two are orthogonal if they are offset by equal and opposite angles from the 45° line, which is the world line of a light ray. This implies that a lightlike is orthogonal to itself.

Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which , while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval <math>c^2 t^2 - x^2</math> and the invariant length of the relativistic momentum vector .

It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. It is only when gravitation is significant that general relativity is required.

Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.

thumb|Figure 7–4. Dewan–Beran–Bell spaceship paradox

In Fig. 7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string that is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration. In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Will the string break?

When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.

To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.
Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.

The problem with the first argument is that there is no "frame of the spaceships". There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.

thumb|Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.

A spacetime diagram (Fig. 7-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude <math>k</math> acceleration for proper time <math>\sigma</math> (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity <math>A'B</math> turns out to be greater than the length along the line of simultaneity .

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 7-5, the acceleration is finished, the ships will remain at a constant offset in some frame . If <math>x_{A}</math> and <math>x_{B}=x_{A}+L</math> are the ships' positions in , the positions in frame <math>S'</math> are:

: <math>\begin{align}

x'_{A}& = \gamma\left(x_{A}-vt\right)\\

x'_{B}& = \gamma\left(x_{A}+L-vt\right)\\

L'& = x'_{B}-x'_{A} =\gamma L

\end{align}</math>

The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame . As shown in Fig. 7-5, Bell's example asserts the moving lengths <math>AB</math> and <math>A'B'</math> measured in frame <math>S</math> to be fixed, thereby forcing the rest frame length <math>A'B</math> in frame <math>S'</math> to increase.

Accelerated observer with horizon

Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Section "Invariant hyperbola" of the article Spacetime, the magenta hyperbolae represented paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

thumb|Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed [[:File:ConstantAcceleration02.jpg|here. ]]

Fig. 7-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter <math>\beta</math> approaches a limit of one as <math>ct</math> increases. Likewise, <math>\gamma</math> approaches infinity.

The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:

We remember that .
Since , we conclude that .
<math>\gamma = 1/\sqrt{1 - \beta ^2} = </math> <math>\sqrt{c^2 t^2 - s^2}/s</math>
From the relativistic force law, <math>F = dp/dt = </math>.
Substituting <math>\beta(ct)</math> from step 2 and the expression for <math>\gamma</math> from step 3 yields , which is a constant expression.

Fig. 7-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after 100 hours on his clock (dashed green lines).

In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour, that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron), 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.

Status

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c2 in the region of interest. 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. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)

and thus accepted by the physics community. Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.

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, 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.

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

Tests of relativistic energy and momentum – testing the limiting speed of particles
Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation
Experimental testing of time dilation – relativistic effects on a fast-moving particle's half-life
Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations
Hughes–Drever experiment – testing isotropy of space and mass
Modern searches for Lorentz violation – various modern tests
Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
Experiments to test the aether drag hypothesis – no "aether flow obstruction".

Notes

Primary sources

References

External links

Original works

Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Bern 1905
On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity.