The special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies." Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest; a person on the deck of a ship may be at rest in his opinion, but someone observing from the shore would say that he was moving. Einstein's theory combines Galilean relativity with the postulate that all observers will always measure the speed of light to be the same no matter what their state of uniform linear motion is.

This theory has a variety of surprising consequences that seem to violate common sense, but which have been verified experimentally. Special relativity overthrows Newtonian notions of absolute space and time by stating that distance and time depend on the observer, and that time and space are perceived differently, depending on the observer. It yields the equivalence of matter and energy, as expressed in the famous equation E=mc², where c is the speed of light. Special relativity agrees with Newtonian mechanics in their common realm of applicability, in experiments in which all velocities are small compared to the speed of light.

The theory was called "special" because it applies the principle of relativity only to inertial frames. Einstein developed general relativity to apply the principle generally, that is, to any frame, and that theory includes the effects of gravity. Special relativity doesn't account for gravity, but it can deal with accelerations.

Although special relativity makes relative some quantities, such as time, that we would have imagined to be absolute based on everyday experience, it also makes absolute some others that we would have thought were relative. In particular, it states that the speed of light is the same for all observers, even if they are in motion relative to one another. Special relativity reveals that c is not just the velocity of a certain phenomenon -- light -- but rather a fundamental feature of the way space and time are tied together. In particular, special relativity states that it is impossible for any material object to travel as fast as light.

For history and motivation, see the article: history of special relativity

Postulates

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Main article: Postulates of special relativity

1. First postulate - Special principle of relativity - The laws of physics are the same in all inertial frames of reference.
2. Second postulate - Invariance of c - The speed of light in a vacuum is a universal constant (c) which is independent of the motion of the light source.

The power of Einstein's argument stems from the manner in which he derived startling and seemingly implausible results from two simple assumptions that were founded on observation. An observer attempting to measure the speed of light's propagation will get the same answer no matter how the observer or the system's components are moving.

Lack of an absolute reference frame

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The principle of relativity, which states that there is no stationary reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. However, in the late 19^th Century, the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance known as "aether," which would act as the medium through which the vibrations traveled. The aether was thought to constitute an absolute reference frame against which speeds could be measured. In other words, the aether was the only fixed or motionless thing in the universe. Aether supposedly had some wonderful properties: it was sufficiently elastic that it could support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson-Morley experiment, indicated that the Earth was always 'stationary' relative to the Aether — something that was difficult to explain, since the Earth is in orbit around the Sun. Einstein's elegant solution was to discard the notion of an aether and an absolute state of rest. Special Relativity is formulated so as to not assume that any particular frame of reference is special; rather, in relativity, a system appears to observe the same laws of physics independent of an observer's velocity with respect to it. In particular, the speed of light in a vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

Consequences

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Main article: consequences of special relativity

Einstein has said that all of the consequences of special relativity can be found from examination of the Lorentz transformations.

These transformations, and hence Special Relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive:

• 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 (e.g., the twin paradox which concerns a twin who flies off in a spaceship travelling near the speed of light and returns to discover that his twin has aged much more).
• Lack of simultaneity - two events happening in two different locations that occur simultaneously to one observer, may occur at different times to another observer (lack of absolute simultaneity).
• Lorentz contraction - the dimensions (e.g., length) of an object as measured by one observer may be smaller from the results of measurements of the same object made by another observer. (e.g., the ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage.)
• Composition of velocities - velocities (and speeds) do not simply 'add', for example if a rocket is moving at 2/3 the speed of light relative to an observer, and the rocket fires a missile at 2/3 of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In the this example, the observer would see the missile travel with a speed of 12/13 the speed of light.)
• Inertia and momentum - as an object's velocity gets close to the speed of light, it becomes more and more difficult to accelerate it.
• Equivalence of mass and energy, E=mc² - mass and energy can be converted to one another, and play equivalent roles (e.g., the gravitational force on a falling apple is partly due to the kinetic energies of the subatomic particle it is made of).

Reference frames, coordinates and The Lorentz transformation

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''Full article: Lorentz transformations

Relativity theory depends on "reference frames". A reference frame is a point in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has a clock moving with the reference frame, allowing the measurement of the time of events.

An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back 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 space-time coordinates: The time of occurrence and its 3-dimensional spatial location from a reference point. Let's call this reference frame S.

In relativity theory we often want to calculate the position of a point from a different reference point.

Suppose we have a second reference frame S', whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity $v\backslash ,$ with respect to S along the $x\backslash ,$axis.

Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always 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.

Let's define the event to have space-time coordinates $(t,\; x,\; y,\; z)\backslash ,$ in system S and $(t\text{'},\; x\text{'},\; y\text{'},\; z\text{'})\backslash ,$ in S'. Then the Lorentz transformation specifies that these coordinates are related in the following way:

$t\text{'}\; =\; \backslash gamma\; \backslash left(t\; -\; \backslash frac\{v\; x\}\{c^\{2\}\}\; \backslash right)$

$x\text{'}\; =\; \backslash gamma\; (x\; -\; v\; t)\backslash ,$

$y\text{'}\; =\; y\backslash ,$

$z\text{'}\; =\; z\backslash ,$

where $\backslash gamma\; \backslash equiv\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; v^2/c^2\}\}$ is called the Lorentz factor and $c$ is the speed of light in a vacuum.

The $y\backslash ,$ and $z\backslash ,$ coordinates are unaffected, but the $x\backslash ,$ and $t\backslash ,$ axes are mixed up by the transformation. In a way this transformation can be understood as a hyperbolic rotation.

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

Simultaneity

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From the first equation of the Lorentz transformation in terms of coordinate differences

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

it is clear that two events that are simultaneous in frame S (satisfying $\backslash Delta\; t\; =\; 0\backslash ,$), are not necessarily simultaneous in another inertial frame S' (satisfying $\backslash Delta\; t\text{'}\; =\; 0\backslash ,$). Only if these events are colocal in frame S (satisfying $\backslash Delta\; x\; =\; 0\backslash ,$), will they be simultaneous in another frame S'.

Time dilation and length contraction

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Writing the Lorentz Transformation and its inverse in terms of coordinate differences we get

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

$\backslash Delta\; x\text{'}\; =\; \backslash gamma\; (\backslash Delta\; x\; -\; v\; \backslash Delta\; t)\backslash ,$

and

$\backslash Delta\; t\; =\; \backslash gamma\; \backslash left(\backslash Delta\; t\text{'}\; +\; \backslash frac\{v\; \backslash Delta\; x\text{'}\}\{c^\{2\}\}\; \backslash right)$

$\backslash Delta\; x\; =\; \backslash gamma\; (\backslash Delta\; x\text{'}\; +\; v\; \backslash Delta\; t\text{'})\backslash ,$

Suppose we have a clock at rest in the unprimed system S. Two consecutive ticks of this clock are then characterized by $\backslash Delta\; x\; =\; 0$. If we want to know the relation between the times between these ticks as measured in both systems, we can use the first equation and find

$\backslash Delta\; t\text{'}\; =\; \backslash gamma\; \backslash Delta\; t\; \backslash ,$

This shows that the time $\backslash Delta\; t\text{'}$ between the two ticks as seen in the 'moving' frame S' is larger than the time $\backslash Delta\; t$ between these ticks as measured in the rest frame of the clock. This phenomenon is called Time Dilation.

Similarly, suppose we have a measuring rod at rest in the unprimed system. In this system, the length of this rod is written as $\backslash Delta\; x$. If we want to find the length of this rod as measured in the 'moving' system S', we must make sure to measure the distances $x\text{'}$ to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by $\backslash Delta\; t\text{'}\; =\; 0$, which we can combine with the fourth equation to find the relation between the lengths $\backslash Delta\; x$ and $\backslash Delta\; x\text{'}$:

$\backslash Delta\; x\text{'}\; =\; \backslash frac\{\backslash Delta\; x\}\{\backslash gamma\}$

This shows that the length $\backslash Delta\; x\text{'}$ of the rod as measured in the 'moving' frame S' is shorter than the length $\backslash Delta\; x$ in its own rest frame. This phenomenon is called Length Contraction or Lorentz Contraction.

These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between 'colocal' events and distances between simultaneous events.

See also the twin paradox.

Causality and prohibition of motion faster than light

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In diagram 2 the interval AB is 'time-like'; i.e., there is a frame of reference in which event A and event 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. It is hypothetically possible for matter (or information) to travel from A to 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'; i.e., there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. If it was possible for a cause-and-effect relationship to exist between events A and C, then logical paradoxes would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Another way of looking at it is that if there were a technology that allowed faster-than-light motion, it would also function as a time machine. Therefore, one of the consequences of special relativity is that (assuming causality is to be preserved as a logical principle), no information or material object can travel faster than light. On the other hand, the logical situation is not as clear in the case of general relativity, so it is an open question whether or not there is some fundamental principle that preserves causality (and therefore prevents motion faster than light) in general relativity.

Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F=dp/dt gives a momentum that grows without bound, but this is simply because $p=m\backslash gamma\; v$ approaches infinity as v approaches c. To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators.

Composition of velocities

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If the observer in $S$ sees an object moving along the $x$ axis at velocity $w$, then the observer in the $S\text{'}$ system will see the object moving with velocity $w\text{'}$ where

$w\text{'}=\backslash frac\{w-v\}\{1-wv/c^2\}$.

This equation can be derived from the space and time transformations above. Notice that if the object is moving at the speed of light in the $S$ system (i.e. $w=c$), then it will also be moving at the speed of light in the $S\text{'}$ system. Also, if both $w$ and $v$ are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: $w\text{'}=w-v$.

Mass, momentum, and energy

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In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.

There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.

Given an object of invariant mass m₀ traveling at velocity v the energy and momentum are given (and even defined) by

$E\; =\; \backslash gamma\; m\_0\; c^2\; \backslash ,\backslash !$

$\backslash vec\; p\; =\; \backslash gamma\; m\_0\; \backslash vec\; v\; \backslash ,\backslash !$

where γ (the Lorentz factor) is given by

$\backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; v^2/c^2\}\}\; \backslash ,\backslash !$

and c is the speed of light. The term γ occurs frequently in relativity, and comes from the Lorentz transformation equations.

Relativistic energy and momentum can be related through the formula

$E^2\; -\; (p\; c)^2\; =\; (m\_0\; c^2)^2\; \backslash ,\backslash !$

which is referred to as the relativistic energy-momentum equation.

For velocities much smaller than those of light, γ can be approximated using a Taylor series expansion and one finds that

$E\; \backslash approx\; m\_0\; c^2\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; m\_0\; v^2\; \backslash ,\backslash !$

$\backslash vec\; p\; \backslash approx\; m\_0\; \backslash vec\; v\; \backslash ,\backslash !$

Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.

Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and γ = 1) there is a non-zero energy remaining:

$E\; =\; m\_0\; c^2\; \backslash ,\backslash !$

This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful.

Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:

Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy. ^*

This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life have made it one of the most famous equations in all of science.

Relativistic mass

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Introductory physics courses and some older textbooks on special relativity sometimes define a relativistic mass which increases as the velocity of a body increases. According to the geometric interpretation of special relativity, this is often deprecated and the term 'mass' is reserved to mean 'rest mass' and is thus independent of the inertial frame, i.e., invariant.

Using the relativistic mass definition, the mass of an object may vary depending on the observer's inertial frame in the same way that other properties such as its length may do so. Defining such a quantity may sometimes be useful in that doing so simplifies a calculation by restricting it to a specific frame. For example, consider a body with an invariant mass m₀ moving at some velocity relative to an observer's reference system. That observer defines the relativistic mass of that body as:

$m\; =\; \backslash gamma\; m\_0\backslash !$

"Relativistic mass" should not be confused with the "longitudinal" and "transverse mass" definitions that were used around 1900 and that were based on an inconsistent application of the laws of Newton: those used F=ma for a variable mass, while relativistic mass corresponds to Newton's dynamic mass in which p=mv and F=dp/dt.

Note also that the body does not actually become more massive in its proper frame, since the relativistic mass is only different for an observer in a different frame. The only mass that is frame independent is the invariant mass. When using the relativistic mass, the used reference frame should be specified if it isn't already obvious or implied. It also goes almost without saying that the increase in relativistic mass does not come from an increased number of atoms in the object. Instead, the relativistic mass of each atom and subatomic particle has increased.

Physics textbooks sometimes use the relativistic mass as it allows the students to utilize their knowledge of Newtonian physics to gain some intuitive grasp of relativity in their frame of choice (usually their own!). "Relativistic mass" is also consistent with the concepts "time dilation" and "length contraction".

Force

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The classical definition of force F is given by Newton's Second Law in its original form:

$\backslash vec\; F\; =\; d\backslash vec\; p/dt$

and this is valid in relativity.

Many modern textbooks rewrite Newton's Second Law as

$\backslash vec\; F\; =\; m\; \backslash vec\; a$

This form is not valid in relativity or in other situations where the mass m is varying.

For constant m₀ this formula can be replaced in the relativistic case by

$\backslash vec\; F\; =\; \backslash gamma\; m\_0\; \backslash vec\; a\; +\; \backslash gamma^3\; m\_0\; \backslash frac\{\backslash vec\; v\; \backslash cdot\; \backslash vec\; a\}\{c^2\}\; \backslash vec\; v$

As seen from the equation, force and acceleration vectors are not necessarily parallel in relativity.

The geometry of space-time

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SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.

The differential of distance(ds) in cartesian 3D space is defined as:

$ds^2\; =\; dx\_1^2\; +\; dx\_2^2\; +\; dx\_3^2$

where $(dx\_1,dx\_2,dx\_3)$ are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of c, so that the equation for the differential of distance becomes:

$ds^2\; =\; dx\_1^2\; +\; dx\_2^2\; +\; dx\_3^2\; -\; c^2\; dt^2$

To make time coordinate symmetric to space coordinates, we must therefore treat time as imaginary: x₄ = ict . In this case the above equation becomes symmetric:

$ds^2\; =\; dx\_1^2\; +\; dx\_2^2\; +\; dx\_3^2\; +\; dx\_4^2$

This is, however, not just a simplifying mathematical technique, but has profound theoretical significance as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. The deeper development of the theory relies on the concept of the Minkowski metric as described below.

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space

$ds^2\; =\; dx\_1^2\; +\; dx\_2^2\; -\; c^2\; dt^2$

We see that the null geodesics lie along a dual-cone:

defined by the equation

$ds^2\; =\; 0\; =\; dx\_1^2\; +\; dx\_2^2\; -\; c^2\; dt^2$

or

$dx\_1^2\; +\; dx\_2^2\; =\; c^2\; dt^2$

Which is the equation of a circle with r=c*dt. If we extend this to three spatial dimensions, the null geodesics is 4-dimensional cone:

$ds^2\; =\; 0\; =\; dx\_1^2\; +\; dx\_2^2\; +\; dx\_3^2\; -\; c^2\; dt^2$

$dx\_1^2\; +\; dx\_2^2\; +\; dx\_3^2\; =\; c^2\; dt^2$

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event $d\; =\; \backslash sqrt\{x\_1^2+x\_2^2+x\_3^2\}$ meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

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

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

Physics in spacetime

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Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any reference frame) as

$\backslash eta^\{\backslash alpha\backslash beta\}\; =\backslash eta\_\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}

Then we recognise that co-ordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special case of motion along the x-axis, we have:

$\backslash Lambda^\backslash mu\{\}\_\backslash nu\; =\; \backslash begin\{pmatrix\}$

\gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}

where β and γ are defined as

$\backslash beta\; =\; \backslash frac\{v\}\{c\},\; \backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{1-\backslash beta^2\}\}$

This simplifies almost every formula ever encountered in special relativity. We understand that all proper physical quantities are given by tensors. So to transform from one frame to another, we use the well known tensor transformation law

$T^\{\backslash left*\}\_\{\backslash left*\}\; =$

\Lambda^{i_1'}{}_{i_1}\Lambda^{i_2'}{}_{i_2}...\Lambda^{i_p'}{}_{i_p} \Lambda_{j_1'}{}^{j_1}\Lambda_{j_2'}{}^{j_2}...\Lambda_{j_q'}{}^{j_q} T^{\left^*}_{\left^*}

To see how this is useful, we first recognise that position is a four vector, since in component form

$x\_\backslash nu=\backslash left(-ct,\; x,\; y,\; z\backslash right)$

So to transform it from an unprimed co-ordinate system S to a primed system S', we calculate

$$

\begin{pmatrix} ct'\\ x'\\ y'\\ z' \end{pmatrix} = x'^{\mu}=\Lambda^\mu{}_\nu x^\nu= \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct\\ x\\ y\\ z \end{pmatrix} = \begin{pmatrix} \gamma ct- \gamma\beta x\\ \gamma x - \beta \gamma ct \\ y\\ z \end{pmatrix}

which is a nice way of arriving at the cumbersome looking definition of the Lorentz transformation given above. But the true power becomes evident when you recognise that all tensors transform by the same rule. Firstly, note that the scalar (in fact the length of the position four-vector $x$) as constructed using

$\backslash mathbf\{x\}^2\; =\; x^\backslash mu\; x\_\backslash mu\; =\; \backslash eta\_\{\backslash mu\backslash nu\}x^\backslash mu\; x^\backslash nu\; =\; -(ct)^2+x^2+y^2+z^2\backslash ,$

is invariant - i.e. it takes the same value in all inertial frames, simply because it is a 0 rank tensor, and so no copies of the Lorentz appears in its transformation: x' = x. (Note that when $\backslash mathbf\{x\}^2$ is negative that $ix$ is a proper time, while when $\backslash mathbf\{x\}^2$ is positive, $x$ is a proper distance. Also note that proper time is represented below by $\backslash tau\; =\; ix$.)

Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector U is given by

$U^\backslash mu\; =\; \backslash frac\{dx^\backslash mu\}\{d\backslash tau\}\; =\; \backslash begin\{pmatrix\}\; \backslash gamma\; c\; \backslash \backslash \; \backslash gamma\; v\_x\; \backslash \backslash \; \backslash gamma\; v\_y\; \backslash \backslash \; \backslash gamma\; v\_z\; \backslash end\{pmatrix\}$

Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. U also has an invariant form:

$\{\backslash mathbf\; U\}^2\; =\; U\_\backslash mu\; U^\backslash mu\; =\; -c^2\backslash ,$

So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector is given by $A\_\backslash mu\; =\; d\{\backslash mathbf\; U\}/d\backslash tau$. Given this, differentiating the above equation by τ produces

$2\; A\_\backslash mu\; U^\backslash mu\; =\; 0\; \backslash ,$

So in relativity, the acceleration four-vector and the velocity 4-vector are orthogonal.

The energy-momentum four-vector is given by (and in component form)

$p\_\backslash mu\; =\; m\_0\; U\_\backslash mu\; =\; \backslash begin\{pmatrix\}$

E/c\\ p_x\\ p_y\\ p_z\end{pmatrix}

This gives us the invariant

$\backslash mathbf\{p\}^2=p^\backslash mu\; p\_\backslash mu\; =\; -E^2/c^2\; +\; p^2\backslash ,$

We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

$\backslash mathbf\{p\}^2=p^\backslash mu\; p\_\backslash mu\; =\; -E\_\{rest\}^2/c^2\; =\; -\; m\_0^2\; c^2\backslash ,$

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

The rest energy is related to the rest mass according to the celebrated equation discussed above:

$E\_\{rest\}=m\_0\; c^2\backslash ,$

Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.

Further examples of physical tensors are the electromagnetic field tensor, and the stress-energy tensor, and again their transformation is given by the simple tensor transformation law.

The electric field are now unified into the electromagnetic field tensor:

$$

F_{\mu\nu} \equiv \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}

The charge density $\backslash rho$ and current density $*$ are unified into the current-charge 4-vector:

$J^\backslash mu\; =\; \backslash begin\{pmatrix\}$

\rho c\\ J_x\\ J_y\\ J_z\end{pmatrix}

and Maxwell's equations are reduced to two in number. (see Maxwell's Equations#Formulation of Maxwell's equations in special relativity)

{| $\backslash partial^\backslash mu\; F\_\{\backslash mu\; \backslash nu\}\; =\; \backslash mu\_0\; J\_\{\backslash nu\}$    (Ampere-Gauss Law) $\backslash partial\_\backslash lambda\; F\_\{\backslash mu\backslash nu\}+\; \backslash partial\; \_\backslash mu\; F\_\{\backslash nu\; \backslash lambda\}+$

\partial_\nu F_{\lambda \mu} = 0||    (Faraday-Gauss Law)

Where $\backslash partial$ is the spacetime extension of gradient:

$\backslash partial\_\backslash nu\; =\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; x^\backslash nu\}$

Relativity and unifying electromagnetism

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The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.

Status

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Main article: Status of special relativity

Special relativity is accurate only when gravitational potential is much less than c²; in a strong gravitational field one must use general relativity (which becomes special relativity at the limit of weak field). At very small scales, such as at the Planck length and below quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10^-14) and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.

Because of the freedom one has to select how one defines units of length and time in physics, it is possible to make one of the two postulates of relativity a tautological consequence of the definitions, but one cannot do this for both postulates simultaneously, as when combined they have consequences which are independent of one's choice of definition of length and time.

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 Status of special relativity for a more detailed discussion.

A few key experiments can be mentioned that led to special relativity:

• The Trouton-Noble experiment showed that the torque on a capacitor is independent on position and inertial reference frame — such experiments led to the first postulate
• The famous Michelson-Morley experiment demonstrated the directional invariance of the two-way speed of light — "the speed of light" as defined in the second postulate.

A number of experiments have been conducted to test special relativity against rival theories. These include:

• Kaufman's experiment — electron deflection in exact accordance with Lorentz-Einstein prediction
• Hamar experiment — no "ether flow obstruction"
• Kennedy-Thorndike experiment — time dilation in accordance with Lorentz transformations
• Rossi-Hall experiment — relativistic effects on a fast-moving particle's half-life
• Experiments to test emitter theory demonstrated that the speed of light is independent of the speed of the emitter.

In addition, particle accelerators run almost every day somewhere in the world, and routinely accelerate and measure the properties of particles moving at near lightspeed. Many effects seen in particle accelerators are highly consistent with relativity theory and are deeply inconsistent with the earlier Newtonian mechanics.

See also

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People: Arthur Eddington | Albert Einstein | Hendrik Lorentz | Hermann Minkowski | Bernhard Riemann | Henri Poincaré | Alexander MacFarlane | Harry Bateman | Robert S. Shankland

Relativity: Theory of relativity | principle of relativity | general relativity | frame of reference | inertial frame of reference | Lorentz transformations | Einstein synchronisation

Physics: Newtonian Mechanics | spacetime | speed of light | simultaneity | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Moving magnet and conductor problem

Math: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number

Philosophy: actualism | convensionalism | formalism

External links

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Explanations of special relativity

• The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
• Relativity An introduction to special relativity at the undergraduate level, without calculus.
• Nothing but Relativity There are many ways to derive the Lorentz transformation without invoking Einstein's constancy of light postulate. The path preferred in this paper restates a simple, established approach.
• Reflections on Relativity A complete online book on relativity with an extensive bibliography.
• Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
• Special relativity theory made intuitive A new approach to explain the theoretical meaning of Special Relativity from an intuitive geometrical viewpoint
• Special Relativity Stanford University, Helen Quinn, 2003
• , by Albert Einstein
• Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
• Caltech Relativity Tutorial A basic introduction to concepts of Special and General Relativity, requiring only a knowledge of basic geometry.
• Greg Egan's Foundations.
• Understanding Special Relativity - The theory of special relativity in an easily understandable way.

Visualization

• Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
• Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program.
• Warp Special Relativity Simulator a computer program to show the effects of travelling close to the speed of light.

Other

• Test problems in mechanics and special relativity
• Brane World Mach Principles and the Michelson-Morley experiment
• "Einstein Was Right (Again): Experiments Confirm that E= mc²" A recent direct measurement of Einstein's famous equation accurate to "four-tenths of 1 part in 1 million."
• Relativity in its Historical Context The discovery of special relativity was inevitable, given the momentous discoveries that preceded it.

References

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Textbooks

• Einstein, Albert. "Relativity: The Special and the General Theory".
• Silberstein, Ludwik (1914) The Theory of Relativity.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman Company. ISBN 0716743450
• Schutz, Bernard F. A First Course in General Relativity, Cambridge University Press. ISBN 0521277035
• Taylor, Edwin, and Wheeler, John (1992). Spacetime Physics (2nd ed.). W.H. Freeman and Company. ISBN 0716723271
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1567311369
• Geroch, Robert (1981). General Relativity From A to B. University of Chicago Press. ISBN 0226288641
• Logunov, Anatoly A. (2005) Henri Poincaré and the Relativity Theory(transl. from Russian by G Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka, Moscow ^*.

Journal articles

• On the Electrodynamics of Moving Bodies, A. Einstein, Annalen der Physik, 17:891, June 30, 1905 (in English translation)
• Wolf, Peter and Gerard, Petit. "Satellite test of Special Relativity using the Global Positioning System," Physics Review A 56 (6), 4405-4409 (1997).
• Will, Clifford M. "Clock synchronization and isotropy of the one-way speed of light," Physics Review D 45, 403-411 (1992).
• Rizzi G. et al, "Synchronization Gauges and the Principles of Special Relativity", Found.Phys. 34 (2005) 1835-1887
• Alvager et al., "Test of the Second Postulate of Special Relativity in the GeV region," Physics Letters 12, 260 (1964).

Special relativity

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