For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. In Physics, special relativity is a fundamental Theory about Space and Time, developed by Albert Einstein in 1905 as a modification

Special relativity (SR) (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after considerable contributions of Hendrik Lorentz and Henri Poincaré) in the paper "On the Electrodynamics of Moving Bodies". Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Hendrik Antoon Lorentz ( July 18, 1853 &ndash February 4, 1928) was a Dutch Physicist who shared the 1902 Nobel Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician The Annus Mirabilis Papers (from Latin, Annus mirabilis, for 'extraordinary year' are the papers of Albert Einstein published in the " [1] It generalizes Galileo's principle of relativity – that all uniform motion is 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 both the laws of mechanics and of electrodynamics, whatever they may be. Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest In Theoretical physics, a preferred or privileged frame is usually a special Hypothetical Frame of reference in which the Laws of physics Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently In addition, special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source. The term observer in Special relativity refers most commonly to an Inertial reference frame. [2]

This theory has a wide range of consequences which have been experimentally verified. Special relativity overthrows Newtonian notions of absolute space and time by stating that time and space are perceived differently by observers in different states of motion. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another 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. Matter is commonly defined as being anything that has mass and that takes up space. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared to the speed of light.

The theory is termed "special" because it applies the principle of relativity only to inertial frames. A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Einstein developed general relativity to apply the principle generally, that is, to any frame, and that theory includes the effects of gravity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another Strictly, special relativity cannot be applied in accelerating frames or in gravitational fields.

Special relativity reveals that c is not just the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS A consequence of this is that it is impossible for any particle that has mass to be accelerated to the speed of light.

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

## Postulates

In his autobiographical notes published in November 1949 Einstein described how he had arrived at the two fundamental postulates on which he based the special theory of relativity. The History of special relativity consists of many theoretical and empirical results of physicists like Hendrik Lorentz and Henri Poincaré, which culminated in the After describing in detail the state of both mechanics and electrodynamics at the beginning of the 20th century, he wrote

"Reflections of this type made it clear to me as long ago as shortly after 1900, i. e. , shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results… How, then, could such a universal principle be found?"[3]

He discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of either the (then) known laws of mechanics or electrodynamics. These propositions were (1) the constancy of the velocity of light, and (2) the independence of physical laws (especially the constancy of the velocity of light) from the choice of inertial system. In his initial presentation of special relativity in 1905[4] 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 of inertial coordinates in uniform translatory motion.
• The Principle of Invariant Light Speed - Light in vacuum propagates with the speed c (a fixed constant) in terms of any system of inertial coordinates, regardless of the state of motion of the light source.

It should be noted that the derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (which are made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history. Isotropy is uniformity in all directions Precise definitions depend on the subject area [5].

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations[6]. However, the most common set of postulates remains those employed by Einstein in his original paper. These postulates refer to the axiomatic basis of the Lorentz transformation, which is the essential core of special relativity. In all of Einstein's papers in which he presented derivations of the Lorentz transformation, he based it on these two principles. [7]

In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy. (It should be noted that this equivalence does not follow from the basic premises of special relativity. [8] The first of these was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905. In this and each of his subsequent three papers on this subject[9], Einstein augmented the two fundamental principles by postulating the relations involving momentum and energy of electromagnetic waves implied by Maxwell's equations (the assumption of which, of course, entails among other things the assumption of the constancy of the speed of light). He acknowledged in his 1907 survey paper on special relativity that it was problematic to rely on Maxwell's equations[10] for the heuristic mass-energy argument, and this is why he consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote

"The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events… The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws…"[11]

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime. [12][13]

## 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. A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher However, in the late 19th 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. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. In the late 19th century " luminiferous aether " (or " ether " meaning light-bearing aether, was the term used to describe a medium for the propagation 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. The Michelson–Morley experiment, one of the most important and famous experiments in the History of physics, was performed in 1887 by Albert Michelson and 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. Special relativity has several consequences that struck many people as counterintuitive among which are The time lapse between two events is not invariant from one observer In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

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. This article discusses a concept in physics For the concept in sociology see Time displacement. 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 or her twin sibling has aged much more). In physics the twin paradox is a thought experiment in Special Relativity, in which a person who makes a journey into space in a high-speed rocket will return home to find he
• 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). The relativity of simultaneity is the concept that simultaneity is not absolute but dependent on the observer In Physics, the concept of absolute time and absolute space are Hypothetical models in which time either runs at the same rate for all the observers in
• Lorentz contraction – the dimensions (e. Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in Length detected by an observer in objects that travel at any non-zero 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). The ladder paradox (or barn-pole paradox) is a Thought experiment in Special relativity.
• 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. The velocity-addition formula is one of two physics equations that relates the velocities of a moving object in different reference frames Galilean Addition Of Velocities (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 speed 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. The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product
• Equivalence of mass and energy, E = mc2 – The energy content of an object at rest with mass m equals mc2. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. 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

Diagram 1. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to Changing views of spacetime along the world line of a rapidly accelerating observer. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. 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. In Special relativity, a light cone (or null cone) is the pattern describing the temporal evolution of a flash of Light in Minkowski spacetime 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. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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'. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Then the Lorentz transformation specifies that these coordinates are related in the following way:

$\begin{cases}t' = \gamma \left(t - \frac{v x}{c^{2}} \right) \\x' = \gamma (x - v t) \\y' = y \\z' = z ,\end{cases}$

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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to The Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the

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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In Physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation.

## Simultaneity

Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame.
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 $\Delta t' = 0\,$). The relativity of simultaneity is the concept that simultaneity is not absolute but dependent on the observer 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

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

and

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

Suppose we have a clock at rest in the unprimed system S. Circadian Locomotor Output Cycles Kaput, or Clock is a gene which encodes proteins regulating Circadian rhythm. Two consecutive ticks of this clock are then characterized by Δx = 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 Δt' between the two ticks as seen in the 'moving' frame S' is larger than the time Δt between these ticks as measured in the rest frame of the clock. This phenomenon is called time dilation. This article discusses a concept in physics For the concept in sociology see Time displacement.

Similarly, suppose we have a measuring rod at rest in the unprimed system. A Measuring rod is a kind of Ruler. This phrase is often used without mention of a particular kind or length of ruler and has been used since ancient times In this system, the length of this rod is written as Δx. 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 Δt' = 0, which we can combine with the fourth equation to find the relation between the lengths Δx and Δx':

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

This shows that the length Δx' of the rod as measured in the 'moving' frame S' is shorter than the length Δx in its own rest frame. This phenomenon is called length contraction or Lorentz contraction. Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in Length detected by an observer in objects that travel at any non-zero

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.

See also the twin paradox. In physics the twin paradox is a thought experiment in Special Relativity, in which a person who makes a journey into space in a high-speed rocket will return home to find he

## Causality and prohibition of motion faster than light

Diagram 2. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence 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 were 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[14][15] 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. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence On the other hand, the logical situation is not as clear in the case of general relativity, so it is an open question whether there is some fundamental principle that preserves causality (and therefore prevents motion faster than light) in general relativity. The chronology protection conjecture is a Conjecture by the Physicist Professor Stephen Hawking that the Laws of physics are such as to prevent

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 = mγv 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.

See also the Tachyonic Antitelephone. The tachyonic antitelephone is a hypothetical device in Theoretical physics that can be used to send Signals into one's own Past.

## 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. The velocity-addition formula is one of two physics equations that relates the velocities of a moving object in different reference frames Galilean Addition Of Velocities 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. The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves 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. A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory 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 Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the The term γ occurs frequently in relativity, and comes from the Lorentz transformation equations. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The kinetic energy of an object is the extra Energy which it possesses due to its motion 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. [16]

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. In Nuclear physics, a nuclear reaction is the process in which two nuclei or nuclear particles collide to produce products different from the initial particles A nuclear weapon is an explosive device that derives its destructive force from Nuclear reactions either fission or a combination of fission and fusion. 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. The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with 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". This article discusses a concept in physics For the concept in sociology see Time displacement. Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in Length detected by an observer in objects that travel at any non-zero

## 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

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 space-time. In the Special theory of relativity four-force is a Four-vector that replaces the classical Force; the four-force is the four-vector defined as the change In Special relativity, four-acceleration is a Four-vector and is defined as the change in Four-velocity over the particle's Proper time: In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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 (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In differential calculus, a differential is traditionally an Infinitesimally small change in a Variable. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes:

$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2$

If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict . Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane In this case the above equation becomes symmetric:

$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$

This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime According to Misner (1971 §2. 3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate.

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. In General relativity, Geodesics generalize the notion of "straight lines" to curved Spacetime. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces If we extend this to three spatial dimensions, the null geodesics are the 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. A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth We are looking at an event $d = \sqrt{x_1^2+x_2^2+x_3^2}$ 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. The Minkowski diagram was developed in 1908 by Herman Minkowski and provides an illustration of the properties of space and time in the Special theory of relativity

## Physics in spacetime

Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally The position of an event in spacetime is given by a contravariant four vector whose components are:

$x^\nu=\left(t, x, y, z\right)$

That is, x0 = t and x1 = x and x2 = y and x3 = z. Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:

$\partial_0 \phi = \frac{\partial \phi}{\partial t}, \quad \partial_1 \phi = \frac{\partial \phi}{\partial x}, \quad \partial_2 \phi = \frac{\partial \phi}{\partial y}, \quad \partial_3 \phi = \frac{\partial \phi}{\partial z}.$

### Metric and transformations of coordinates

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

$\eta_{\alpha\beta} = \begin{pmatrix}-c^2 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}$

Its reciprocal is:

$\eta^{\alpha\beta} = \begin{pmatrix}-1/c^2 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}$

Then we recognize that co-ordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually For the special case of motion along the x-axis, we have:

$\Lambda^{\mu'}{}_\nu = \begin{pmatrix}\gamma & -\beta\gamma/c & 0 & 0\\-\beta\gamma c & \gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}$

which is simply the matrix of a boost (like a rotation) between the x and t coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:

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

More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:

$\eta_{\alpha\beta} = \eta_{\mu'\nu'} \Lambda^{\mu'}{}_\alpha \Lambda^{\nu'}{}_\beta \!$

where there is an implied summation of $\mu' \!$ and $\nu' \!$ from 0 to 3 on the right-hand side in accordance with the Einstein summation convention. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity

All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law

$T^{\left[i_1',i_2',...i_p'\right]}_{\left[j_1',j_2',...j_q'\right]} = \Lambda^{i_1'}{}_{i_1}\Lambda^{i_2'}{}_{i_2}...\Lambda^{i_p'}{}_{i_p}\Lambda_{j_1'}{}^{j_1}\Lambda_{j_2'}{}^{j_2}...\Lambda_{j_q'}{}^{j_q}T^{\left[i_1,i_2,...i_p\right]}_{\left[j_1,j_2,...j_q\right]}$

Where $\Lambda_{j_k'}{}^{j_k} \!$ is the reciprocal matrix of $\Lambda^{j_k'}{}_{j_k} \!$. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

To see how this is useful, we transform the position of an event from an unprimed co-ordinate system S to a primed system S', we calculate

$\begin{pmatrix}t'\\ x'\\ y'\\ z'\end{pmatrix} = x^{\mu'}=\Lambda^{\mu'}{}_\nu x^\nu=\begin{pmatrix}\gamma & -\beta\gamma/c & 0 & 0\\-\beta\gamma c & \gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}t\\ x\\ y\\ z\end{pmatrix} =\begin{pmatrix}\gamma t- \gamma\beta x/c\\\gamma x - \beta \gamma ct \\ y\\ z\end{pmatrix}$

which is the Lorentz transformation given above. All tensors transform by the same rule.

The squared length of the differential of the position four-vector $dx^\mu \!$ constructed using

$\mathbf{dx}^2 = \eta_{\mu\nu}dx^\mu dx^\nu = -(c \cdot dt)^2+(dx)^2+(dy)^2+(dz)^2\,$

is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line element $\mathbf{dx}^2$ is negative that $d\tau=\sqrt{-\mathbf{dx}^2} / c$ is the differential of proper time, while when $\mathbf{dx}^2$ is positive, $\sqrt{\mathbf{dx}^2}$ is differential of the proper distance. A line element in Mathematics can most generally be thought of as the square of the change in a position vector in an Affine space equated to the square of the change In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock In relativistic Physics, proper Length is an invariant quantity which is the rod Distance between Spacelike

The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.

### Velocity and acceleration in 4D

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

$U^\mu = \frac{dx^\mu}{d\tau} = \begin{pmatrix} \gamma \\ \gamma v_x \\ \gamma v_y \\ \gamma v_z \end{pmatrix}$

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. In Physics, in particular in Special relativity and General relativity, the four-velocity of an object is a Four-vector (vector in four-dimensional Uμ also has an invariant form:

${\mathbf U}^2 = \eta_{\nu\mu} U^\nu U^\mu = -c^2 .$

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 $A^\mu = d{\mathbf U^\mu}/d\tau$. In Special relativity, four-acceleration is a Four-vector and is defined as the change in Four-velocity over the particle's Proper time: Given this, differentiating the above equation by τ produces

$2\eta_{\mu\nu}A^\mu U^\nu = 0. \!$

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

### Momentum in 4D

The momentum and energy combine into a covariant 4-vector:

$p_\nu = m \cdot \eta_{\nu\mu} U^\mu = \begin{pmatrix}-E \\ p_x\\ p_y\\ p_z\end{pmatrix}.$

where m is the invariant mass.

The invariant magnitude of the momentum 4-vector is:

$\mathbf{p}^2 = \eta^{\mu\nu}p_\mu p_\nu = -(E/c)^2 + p^2 .$

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. In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime.

$\mathbf{p}^2 = - (E_{rest}/c)^2 = - (m \cdot c)^2 .$

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

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

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

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.

### Force in 4D

To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.

If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. In the Special theory of relativity four-force is a Four-vector that replaces the classical Force; the four-force is the four-vector defined as the change It is the rate of change of the above energy momentum four-vector with respect to proper time. In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space. The covariant version of the four-force is:

$F_\nu = \frac{d p_{\nu}}{d \tau} = \begin{pmatrix} -{d E}/{d \tau} \\ {d p_x}/{d \tau} \\ {d p_y}/{d \tau} \\ {d p_z}/{d \tau} \end{pmatrix}$

where $\tau \,$ is the proper time.

In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i. e. $\frac{d p}{d t}$ while the four force is defined by the rate of change of momentum with respect to proper time, i. e. $\frac{d p} {d \tau}$.

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

## Relativity and unifying electromagnetism

Main article: Classical electromagnetism and special relativity

Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. The theory of Special relativity plays an important role in the modern theory of Classical electromagnetism. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity. The Liénard-Wiechert potential describes the electromagnetic effect of a moving Electric charge.

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. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges 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. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric 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.

### Electromagnetism in 4D

Main article: Covariant formulation of classical electromagnetism

Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. The covariant formulation of Classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular Maxwell's equations In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric But we must rewrite them to make them manifestly invariant. [17]

The charge density $\rho \!$ and current density $[J_x,J_y,J_z] \!$ are unified into the current-charge 4-vector:

$J^\mu = \begin{pmatrix}\rho \\ J_x\\ J_y\\ J_z\end{pmatrix}$

The law of charge conservation, $\frac{\partial \rho} {\partial t} + \nabla \cdot \mathbf{J} = 0$, becomes:

$\partial_\mu J^\mu = 0. \!$

The electric field $[E_x,E_y,E_z] \!$ and the magnetic induction $[B_x,B_y,B_z] \!$ are now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor:

$F_{\mu\nu} = \begin{pmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0 \end{pmatrix}$

The density, $f_\mu \!$, of the Lorentz force, $\mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}$, exerted on matter by the electromagnetic field becomes:

$f_\mu = F_{\mu\nu}J^\nu .\!$

Faraday's law of induction, $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$, and Gauss's law for magnetism, $\nabla \cdot \mathbf{B} = 0$, combine to form:

$\partial_\lambda F_{\mu\nu}+ \partial _\mu F_{\nu \lambda}+ \partial_\nu F_{\lambda \mu} = 0. \!$

Although there appear to be 64 equations here, it actually reduces to just four independent equations. The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. Current density is a measure of the Density of flow of a conserved charge. In special and General relativity, the four-current is the Lorentz covariant Four-vector that replaces the Electromagnetic Current Charge conservation is the principle that Electric charge can neither be created nor destroyed In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

The electric displacement $[D_x,D_y,D_z] \!$ and the magnetic field $[H_x,H_y,H_z] \!$ are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor:

$\mathcal{D}^{\mu\nu} = \begin{pmatrix} 0 & D_x & D_y & D_z \\ -D_x & 0 & H_z & -H_y \\ -D_y & -H_z & 0 & H_x \\ -D_z & H_y & -H_x & 0 \end{pmatrix}$

Ampère's law, $\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}$, and Gauss's law, $\nabla \cdot \mathbf{D} = \rho$, combine to form:

$\partial_\nu \mathcal{D}^{\mu \nu} = J^{\mu}. \!$

In a vacuum, the constitutive equations are:

$\mu_0 \mathcal{D}^{\mu\nu} = \eta^{\mu\alpha} \eta^{\nu\beta} F_{\alpha\beta}.$

Antisymmetry reduces these 16 equations to just six independent equations. In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed In Structural analysis, constitutive relations connect applied stresses or Forces to strains or Deformations The constitutive relation

The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. Energy density is the amount of Energy stored in a given system or region of space per unit Volume, or per unit Mass, depending on the context although In Physics, the Poynting vector can be thought of as representing the Energy Flux (in W/m2 of an Electromagnetic field. The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor is used to calculate the stresses on objects in magnetic or electrical fields In Physics, the electromagnetic stress-energy tensor is the portion of the Stress-energy tensor due to the Electromagnetic field. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is:

$T_\alpha^\pi = F_{\alpha\beta} \mathcal{D}^{\pi\beta} - \frac{1}{4} \delta_\alpha^\pi F_{\mu\nu} \mathcal{D}^{\mu\nu}$

where $\delta_\alpha^\pi$ is the Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two When upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.

The conservation of linear momentum and energy by the electromagnetic field is expressed by:

$f_\mu + \partial_\nu T_\mu^\nu = 0\!$

where $f_\mu \!$ is again the density of the Lorentz force. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation This equation can be deduced from the equations above (with considerable effort).

## Status

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). See also Special relativity Special relativity (SR is usually concerned with the behaviour of objects and "observers" (inertial reference systems which remain at Potential energy can be thought of as Energy stored within a physical system General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. The Planck length, denoted by \scriptstyle\ell_P \, is the unit of Length approximately 1 Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10-20)[18] 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. In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation

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). In quantum field theory (QFT the forces between particles are mediated by other particles String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

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. See also Special relativity Special relativity (SR is usually concerned with the behaviour of objects and "observers" (inertial reference systems which remain at

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 gave further support to the postulate that detecting an absolute reference velocity was not achievable. The Trouton–Noble experiment attempted to detect motion of the Earth through the Luminiferous aether, and was conducted in 1901&ndash1903 by Frederick Thomas The Michelson–Morley experiment, one of the most important and famous experiments in the History of physics, was performed in 1887 by Albert Michelson and 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.

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

• Kaufmann-Bucherer-Neumann experiments – electron deflection in approximate agreement with Lorentz-Einstein prediction. Walter Kaufmann ( June 5, 1871, Elberfeld - January 1, 1947, Freiburg im Breisgau) was a German physicist
• Hammar 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. The Hammar experiment was an experiment designed by Gustaf Wilhelm Hammar to test the Aether drag hypothesis. The Kennedy–Thorndike experiment ('Experimental Establishment of the Relativity of Time' first conducted in 1932, is a modified form of the Michelson–Morley experimental Performed in 1940 at Echo Lake and Denver in Colorado, the Rossi -Hall experiment measured the relativistic decay of mesotrons ( Mesons Emission theory (also called "emitter theory" was a competing theory for the Special theory of relativity, explaining the results of the Michelson-Morley experiment

In addition, particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects These machines would simply not work if they were not engineered according to relativistic principles.

## References

1. ^ http://www.fourmilab.ch/etexts/einstein/specrel/www/ On the Electrodynamics of Moving Bodies, A. Einstein, Annalen der Physik, 17:891, June 30, 1905 (in English translation)
2. ^ Edwin F. Events 350 - Roman usurper Nepotianus, of the Constantinian dynasty, is defeated and killed by troops of the Usurper Year 1905 ( MCMV) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar (or a Common year starting Taylor and John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. W. H. Freeman. ISBN 0-7167-2327-1.
3. ^ Einstein, Autobiographical Notes, 1949.
4. ^ Einstein, On the Electrodynamics of Moving Bodies, 1905.
5. ^ Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920)
6. ^ For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990
7. ^ Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics, 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the Special and General Theory, 1916; The Principle Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949.
8. ^ Rindler, Essential Relativity, 1977
9. ^ Einstein, The Principle of Conservation of Motion of the Center of Gravity and The Inertia of Energy, 1906; On the Inertia of Energy Required by the Relativity Principle, 1907; Elementary Derivation of the Equivalence of Mass and Energy, 1946.
10. ^ In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity. ", Einstein letter to Carl Seelig, 1955.
11. ^ Einstein, Autobiographical Notes, 1949.
12. ^ Das, A. , The The Special Theory of Relativity, A Mathematical Exposition, Springer, 1993.
13. ^ Schutz, J. , Independent Axioms for Minkowski Spacetime, 1997.
14. ^ R. C. Tolman, The theory of the Relativity of Motion, (Berkeley 1917), p. 54
15. ^ G. A. Benford, D. L. Book, and W. A. Newcomb, The Tachyonic Antitelephone, Phys. Rev. D 2, 263 - 265 (1970) article
16. ^ Einstein on Newton 1927
17. ^ E. J. Post (1962). Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications Inc. . ISBN 0-486-65427-3.
18. ^ The number of works is vast, see as example:
Sidney Coleman, Sheldon L. Glashow, Cosmic Ray and Neutrino Tests of Special Relativity, Phys. Lett. B405 (1997) 249-252, online

### Textbooks

• Einstein, Albert. "Relativity: The Special and the General Theory".
• Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity. New York: Springer. ISBN 978-0-387-69199-2.
• Silberstein, Ludwik (1914) The Theory of Relativity. Optics Book of Optics
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed. ). W. H. Freeman Company. ISBN 0-7167-4345-0
• Schutz, Bernard F. A First Course in General Relativity, Cambridge University Press. ISBN 0-521-27703-5
• Taylor, Edwin, and Wheeler, John (1992). John Archibald Wheeler ( July 9, 1911 &ndash April 13, 2008) was an eminent American Theoretical physicist. Spacetime Physics (2nd ed. ). W. H. Freeman and Company. ISBN 0-7167-2327-1
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9
• Geroch, Robert (1981). General Relativity From A to B. University of Chicago Press. ISBN 0-226-28864-1
• 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.
• Misner, Charles W. ; Thorne, Kip S. , Wheeler, John Archibald (1971). Gravitation. San Francisco: W. H. Freeman & Co. . ISBN 0-7167-0334-3.
• Post, E. J. , Formal Structure of Electromagnetics: General Covariance and Electromagnetics, Dover Publications Inc. Mineola NY, 1962 reprinted 1997.
• Freund, Jűrgen (2008) Special Relativity for Beginners - A Textbook for Undergraduates World Scientific. ISBN-10 981-277-160-3

### 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. Events 350 - Roman usurper Nepotianus, of the Constantinian dynasty, is defeated and killed by troops of the Usurper Year 1905 ( MCMV) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar (or a Common year starting "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).
• Olivier Darrigol (2004) "The Mystery of the Poincaré-Einstein Connection", Isis 95 (4), 614 - 626.

## See also

People: Arthur Eddington | Albert Einstein | Hendrik Lorentz | Hermann Minkowski | Bernhard Riemann | Henri Poincaré | Alexander MacFarlane | Harry Bateman | Robert S. Shankland | Walter Ritz
Relativity: Theory of relativity | principle of relativity | general relativity | Fundamental Speed | frame of reference | inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijk-Putnam Argument
Physics: Newtonian Mechanics | spacetime | speed of light | simultaneity | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem
Maths: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number
Philosophy: actualism | conventionalism | formalism
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 108, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9200263023376465, "perplexity_flag": "head"}