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For a less technical and generally accessible introduction to the topic, see Introduction to special relativity.

Special relativity (SR) (aka the special theory of relativity) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies". It generalises Galileo's principle of relativity — that all uniform motion was relative, and that there is no absolute and well-defined state of rest (no privileged reference frames) — from mechanics to all the laws of physics, including electrodynamics.

To stress this point, Einstein not only widened the postulate of relativity, but added the second postulate that all observers will always measure the speed of light to be the same no matter what their state of uniform linear motion.^[1]

This theory has a variety of surprising consequences that seem to violate common sense, but all have been experimentally verified. Special relativity overthrows Newtonian notions of absolute space and time by stating that time and space are perceived differently in the sense that measurements of length and time intervals depend on the motion of the observer. It yields the equivalence of matter and energy, as expressed in the mass-energy equivalence formula E = mc², where c is the speed of light in a vacuum. 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 does not account for gravity, but it can deal with accelerations.

Although special relativity makes some quantities relative, such as time, that we would have imagined to be absolute based on everyday experience, it also makes absolute some others that were thought to be 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 accelerate to light speed.

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

Postulates

Main article: Postulates of special relativity

First postulate - Special principle of relativity - The laws of physics are the same in all inertial frames of reference. In other words, there are no privileged inertial frames of reference.
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 analysis of observations. An observer attempting to measure the speed of light's propagation will get exactly the same answer no matter how the observer or the system's components are moving.

Lack of an absolute reference frame

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 physicists to suggest that the universe was filled with a substance known as "aether", which would act as the medium through which these waves, or 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, any reference frame moving with uniform motion will observe the same laws of physics. 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

Main article: Consequences of special relativity

Einstein has said that all of the consequences of special relativity can be derived 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 traveling near the speed of light and returns to discover that his twin has aged much more).
Relativity 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 than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
Composition of velocities — velocities (and speeds) do not simply 'add', for example if a rocket is moving at ⅔ the speed of light relative to an observer, and the rocket fires a missile at ⅔ of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In 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 approaches the speed of light from an observer's point of view, its mass appears to increase thereby making it more and more difficult to accelerate it from within the observer's frame of reference.
Equivalence of mass and energy, E = mc² — The energy content of an object at rest with mass m equals $m c 2$ . Conservation of energy implies that in any reaction a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.

Reference frames, coordinates and the Lorentz transformation

Full article: Lorentz transformations

Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer.

In this animation, the vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower quarter of the diagram shows the events that are visible to the observer, and the upper quarter shows the light cone- those that will be able to see the observer. The small dots are arbitrary events in spacetime.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.

Relativity theory depends on "reference frames". A reference frame is an observational perspective 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 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 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 define 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 $v\,$ with respect to S along the $x\,$ -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 $(t, x, y, z)\,$ in system S and $(t', x', y', z')\,$ in S'. Then the Lorentz transformation specifies that these coordinates are related in the following way:

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

$x' = \gamma (x - v t)\,$

$y' = y\,$

$z' = z\,$

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

The $y\,$ and $z\,$ coordinates are unaffected, but the $x\,$ and $t\,$ 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

Main article: Relativity of simultaneity

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

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

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

Time dilation and length contraction

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

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

$\Delta x' = \gamma (\Delta x - v \Delta t)\,$

and

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

$\Delta x = \gamma (\Delta x' + v \Delta t')\,$

Suppose we have a clock at rest in the unprimed system S. Two consecutive ticks of this clock are then characterized by $Δ x = 0$ . 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:

$\Delta t' = \gamma \Delta t \qquad ( \,$ for events satisfying $\Delta x = 0 )\,$

This shows that the time $Δ t'$ between the two ticks as seen in the 'moving' frame S' is larger than the time $Δ t$ 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 $Δ x$ . 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 $x'$ to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by $Δ t' = 0$ , which we can combine with the fourth equation to find the relation between the lengths $Δ x$ and $Δ x'$ :

$\Delta x' = \frac{\Delta x}{\gamma} \qquad ( \,$ for events satisfying $\Delta t' = 0 )\,$

This shows that the length $Δ x'$ of the rod as measured in the 'moving' frame S' is shorter than the length $Δ x$ 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 events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spacial distance from each other when seen from another moving coordinate system.

Causality and prohibition of motion faster than light

See also: Causality

Diagram 2. Light cone

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 paradoxes of causality 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. Although this in itself won't give rise to a paradox, one can show ^[2] ^[3] that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.

Therefore, one of the consequences of special relativity is that (assuming causality is to be preserved), 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 $p = m γ v$ 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

Main article: Velocity-addition formula

If the observer in $S\!$ sees an object moving along the $x\!$ axis at velocity $w\!$ , then the observer in the $S'\!$ system, a frame of reference moving at velocity $v\!$ in the $x\!$ direction with respect to $S\!$ , will see the object moving with velocity $w'\!$ where

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

This equation can be derived from the space and time transformations above. Notice that if the object were moving at the speed of light in the $S\!$ system (i.e. $w=c\!$ ), then it would also be moving at the speed of light in the $S'\!$ 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: $w' \approx w-v\!.$

Mass, momentum, and energy

Main article: Mass in special relativity

Main article: Conservation of energy

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 m traveling at velocity v the energy and momentum are given (and even defined) by

$E = \gamma m c^2 \,\!$

$\vec p = \gamma m \vec v \,\!$

where γ (the Lorentz factor) is given by

$\gamma = \frac{1}{\sqrt{1 - \beta^2}}$

where $\beta = \frac{v}{c}$ is the ratio of the velocity and 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

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

which is referred to as the relativistic energy-momentum equation. It is interesting to observe that while the energy $E\,$ and the momentum $p\,$ are observer dependent (vary from frame to frame) the quantity $E^2 - (p c)^2 = (m c^2)^2 \,\!$ is observer independent.

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

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

$\vec p \approx m \vec v \,\!$

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:

$E_{rest} = m c^2 \,\!$

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.^[4]

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 that can be released by nuclear reactions, providing important information which was useful in the development of nuclear energy and, consequently, 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

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 invariant 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 m moving at some velocity relative to an observer's reference system. That observer defines the relativistic mass of that body as:

$M = \gamma m\!$

"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 applicable 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

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

$\vec f = d\vec p/dt$

and this is valid in relativity.

Many modern textbooks rewrite Newton's Second Law as

$\vec f = M \vec a$

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

This formula can be replaced in the relativistic case by

$\vec f = \gamma m \vec a + \gamma^3 m \frac{\vec v \cdot \vec a}{c^2} \vec v$

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

However the four-vector expression relating four-force $F^\mu\,$ to invariant mass m and four-acceleration $A^\mu\,$ restores the same equation form

$F^\mu = mA^\mu\,$

The geometry of space-time

Main article: Minkowski space

SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a