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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between
space and time Space and Time or Time and Space, or ''variation'', may refer to: * '' Space and time'' or ''time and space'' or ''spacetime'', any mathematical model that combines space and time into a single interwoven continuum * Philosophy of space and time S ...
. In
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's original treatment, the theory is based on two
postulate An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s: # The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is,
frames of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
with no
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
). # The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer.


Origins and significance

Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "
On the Electrodynamics of Moving Bodies The ''annus mirabilis'' papers (from Latin '' annus mīrābilis'', "miracle year") are the four papers that Albert Einstein published in '' Annalen der Physik'' (''Annals of Physics''), a scientific journal, in 1905. These four papers were major ...
".
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
(1905)
''Zur Elektrodynamik bewegter Körper''
, ''Annalen der Physik'' 17: 891; English translatio
On the Electrodynamics of Moving Bodies
by George Barker Jeffery and Wilfrid Perrett (1923); Another English translation
On the Electrodynamics of Moving Bodies The ''annus mirabilis'' papers (from Latin '' annus mīrābilis'', "miracle year") are the four papers that Albert Einstein published in '' Annalen der Physik'' (''Annals of Physics''), a scientific journal, in 1905. These four papers were major ...
by
Megh Nad Saha Meghnad Saha (6 October 1893 – 16 February 1956) was an Indian astrophysicist who developed the Saha ionization equation, used to describe chemical and physical conditions in stars. His work allowed astronomers to accurately relate the spe ...
(1920).
The incompatibility of Newtonian mechanics with Maxwell's equations of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and, experimentally, the Michelson–Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized
luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
did not exist. This led to Einstein's development of special relativity, which corrects mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as '). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible. Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth. Special relativity has a wide range of consequences that have been experimentally verified. They include the relativity of simultaneity, length contraction,
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession. It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant
spacetime interval In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, as expressed in the mass–energy equivalence formula E = mc^2, where c is the speed of light in a vacuum.The Feynman Lectures on Physics Vol. I Ch. 15-9: Equivalence of mass and energy
/ref> It also explains how the phenomena of electricity and magnetism are related. A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into a single continuum known as "spacetime". Events that occur at the same time for one observer can occur at different times for another. Until several years later when Einstein developed
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, which introduced a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case"."Science and Common Sense", P. W. Bridgman, ''The Scientific Monthly'', Vol. 79, No. 1 (Jul. 1954), pp. 32–39.The Electromagnetic Mass and Momentum of a Spinning Electron, G. Breit, Proceedings of the National Academy of Sciences, Vol. 12, p.451, 1926Kinematics of an electron with an axis. Phil. Mag. 3:1-22. L. H. Thomas.]Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes "The word 'special' is meant to intimate that the principle is restricted to the case ...". See p. 111 of The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, Dover reprint of 1923 translation by Methuen and Company.] Some of the work of Albert Einstein in special relativity is built on the earlier work by Hendrik Lorentz and Henri Poincaré. The theory became essentially complete in 1907. The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, where the curvature of spacetime (a consequence of the energy–momentum tensor and representing
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
) is negligible.Wald, General Relativity, p. 60: "... the special theory of relativity asserts that spacetime is the manifold \mathbb^4 with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..." In order to correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate
accelerations In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
as well as accelerating frames of reference. Just as
Galilean relativity Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his '' Dialogue Concerning the Two Chief World Systems'' using t ...
is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, that is, at a sufficiently small scale (e.g., when
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
s are negligible) and in conditions of
free fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
. General relativity, however, incorporates
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
in order to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
. As long as the universe can be modeled as a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He wa ...
had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
.


Traditional "two postulates" approach to special relativity

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in a vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as: * The principle of relativity – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other. * The principle of invariant light speed – "... light is always propagated in empty space with a definite velocity peed''c'' which is independent of the state of motion of the emitting body" (from the preface). That is, light in vacuum propagates with the speed ''c'' (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source. The constancy of the speed of light was motivated by
Maxwell's theory of electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and the lack of evidence for the
luminiferous ether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
. In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance. The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions ( made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920 Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations. However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is: Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
of symmetry transformations. Einstein later derived these transformations from his axioms. Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.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 Principal 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.


Principle of relativity


Reference frames and relative motion

Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity). An event is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. Since the speed of light is constant in relativity irrespective of the reference frame, pulses of light can be used to unambiguously measure distances and refer back to the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame ''S''. In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called ''transformation equations''.


Standard configuration

To gain insight into how the spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a ''standard configuration.'' With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two Galilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′. * The ''x'', ''y'', ''z'' axes of frame S are oriented parallel to the respective primed axes of frame S′. * Frame S′ moves, for simplicity, in a single direction: the ''x''-direction of frame S with a constant velocity ''v'' as measured in frame S. * The origins of frames S and S′ are coincident when time ''t'' = 0 for frame S and ''t''′ = 0 for frame S′. Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be ''comoving''. Therefore, ''S'' and ''S''′ are not ''comoving''.


Lack of an absolute reference frame

The principle of relativity, which states that physical laws have the same form in each
inertial reference frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, dates back to Galileo, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance they called " aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an
absolute reference frame Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a preferr ...
against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist. Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be ''c'', even when measured by multiple systems that are moving at different (but constant) velocities.


Relativity without the second postulate

From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity) it can be shown that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.David Morin (2007) ''Introduction to Classical Mechanics'', Cambridge University Press, Cambridge, chapter 11, Appendix I, .


Lorentz invariance as the essential core of special relativity


Alternative approaches to special relativity

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote: 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.Das, A. (1993) ''The Special Theory of Relativity, A Mathematical Exposition'', Springer, .Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, . Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations. Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler and by Callahan. This is also the approach followed by the Wikipedia articles
Spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
and
Minkowski diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
.


Lorentz transformation and its inverse

Define an event to have spacetime coordinates in system ''S'' and in a reference frame moving at a velocity v with respect to that frame, ''S''′. Then the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
specifies that these coordinates are related in the following way: \begin t' &= \gamma \ (t - vx/c^2) \\ x' &= \gamma \ (x - v t) \\ y' &= y \\ z' &= z , \end where \gamma = \frac is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
and ''c'' is the speed of light in vacuum, and the velocity ''v'' of ''S''′, relative to ''S'', is parallel to the ''x''-axis. For simplicity, the ''y'' and ''z'' coordinates are unaffected; only the ''x'' and ''t'' coordinates are transformed. These Lorentz transformations form a one-parameter group of
linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s, that parameter being called rapidity. Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation: \begin t &= \gamma ( t' + v x'/c^2) \\ x &= \gamma ( x' + v t') \\ y &= y' \\ z &= z'. \end Enforcing this ''inverse'' Lorentz transformation to coincide with the Lorentz transformation from the primed to the unprimed system, shows the unprimed frame as moving with the velocity , as measured in the primed frame. There is nothing special about the ''x''-axis. The transformation can apply to the ''y''- or ''z''-axis, or indeed in any direction parallel to the motion (which are warped by the ''γ'' factor) and perpendicular; see the article
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
for details. A quantity invariant under Lorentz transformations is known as a Lorentz scalar. Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates and , another event has coordinates and , and the differences are defined as *    \Delta x' = x'_2-x'_1 \ , \ \Delta t' = t'_2-t'_1 \ . *    \Delta x = x_2-x_1 \ , \ \ \Delta t = t_2-t_1 \ . we get *    \Delta x' = \gamma \ (\Delta x - v \,\Delta t) \ ,\ \ \Delta t' = \gamma \ \left(\Delta t - v \ \Delta x / c^ \right) \ . *    \Delta x = \gamma \ (\Delta x' + v \,\Delta t') \ , \ \Delta t = \gamma \ \left(\Delta t' + v \ \Delta x' / c^ \right) \ . If we take differentials instead of taking differences, we get *    dx' = \gamma \ (dx - v \, dt) \ ,\ \ dt' = \gamma \ \left( dt - v \ dx / c^ \right) \ . *    dx = \gamma \ (dx' + v \,dt') \ , \ dt = \gamma \ \left(dt' + v \ dx' / c^ \right) \ .


Graphical representation of the Lorentz transformation

Spacetime diagrams (
Minkowski diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
s) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario. To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1. Fig. 3-1a. Draw the x and t axes of frame S. The x axis is horizontal and the t (actually ct) axis is vertical, which is the opposite of the usual convention in kinematics. The ct axis is scaled by a factor of c so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the ''worldlines'' of two photons passing through the origin at time t = 0. The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, \text and \text, have been plotted on this graph so that their coordinates may be compared in the S and S' frames. Fig. 3-1b. Draw the x' and ct' axes of frame S'. The ct' axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, v = c/2. Both the ct' and x' axes are tilted from the unprimed axes by an angle \alpha = \tan^(\beta), where \beta = v/c. The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that t=0 when t'=0. Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that (x', ct') coordinates of (0, 1) in the primed coordinate system transform to (\beta \gamma, \gamma) in the unprimed coordinate system. Likewise, (x', ct') coordinates of (1, 0) in the primed coordinate system transform to (\gamma, \beta \gamma) in the unprimed system. Draw gridlines parallel with the ct' axis through points (k \gamma, k \beta \gamma) as measured in the unprimed frame, where k is an integer. Likewise, draw gridlines parallel with the x' axis through (k \beta \gamma, k \gamma) as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between ct' units equals \sqrt times the spacing between ct units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as \beta \to 1. Fig. 3-1d. Since the speed of light is an invariant, the ''worldlines'' of two photons passing through the origin at time t' = 0 still plot as 45° diagonal lines. The primed coordinates of \text and \text are related to the unprimed coordinates through the Lorentz transformations and ''could'' be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space. While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, but the frames are actually equivalent.


Consequences derived from the Lorentz transformation

The consequences of special relativity can be derived from the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
equations. These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially
counterintuitive A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
.


Invariant interval

In Galilean relativity, length (\Delta r) and temporal separation between two events (\Delta t) are independent invariants, the values of which do not change when observed from different frames of reference. In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume ''negative'' values (is not positive definite for non-coinciding events), and that the ''square''-denotation is misleading. This ''negative square'' lead to, now not broadly used, concepts of ''imaginary time''. It is immediate that the negative of \Delta s^2 is also an invariant, generated by a variant of the
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
of spacetime.
\Delta s^2 \; \overset\text \; c^2 \Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames. The form of \Delta s^2 , being the ''difference'' of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. The invariance of this interval is a property of the ''general'' Lorentz transform (also called the Poincaré transformation), making it an isometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, Lorentz boosts, in the x-direction) with all other
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
, reflections, and rotations between any Cartesian inertial frame. In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed: \Delta s^2 \, = \, c^2 \Delta t^2 - \Delta x^2 Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration: \begin c^2 \Delta t^2 - \Delta x^2 &= c^2 \gamma ^2 \left(\Delta t' + \dfrac \right)^2 - \gamma ^2 \ (\Delta x' + v \Delta t')^2 \\ &= \gamma ^2 \left( c^2 \Delta t' ^ + 2 v \Delta x' \Delta t' + \dfrac \right) - \gamma ^2 \ (\Delta x' ^ + 2 v \Delta x' \Delta t' + v^2 \Delta t' ^ ) \\ &= \gamma ^2 c^2 \Delta t' ^ - \gamma ^2 v^2 \Delta t' ^ - \gamma ^2 \Delta x' ^ + \gamma ^2 \dfrac \\ &= \gamma ^2 c^2 \Delta t' ^ \left( 1 - \dfrac \right) - \gamma ^2 \Delta x' ^ \left( 1 - \dfrac \right) \\ &= c^2 \Delta t' ^ - \Delta x' ^ \end The value of \Delta s^2 is hence independent of the frame in which it is measured. In considering the physical significance of \Delta s^2, there are three cases to note: *Δs2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be ''timelike'' separated. This implies that , \Delta x / \Delta t , < c , and given the Lorentz transformation \Delta x' = \gamma \ (\Delta x - v \,\Delta t) , it is evident that there exists a v less than c for which \Delta x' = 0 (in particular, v = \Delta x / \Delta t). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, \Delta s / c, is called the ''proper time''. *Δs2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be ''spacelike'' separated. This implies that , \Delta x / \Delta t , > c , and given the Lorentz transformation \Delta t' = \gamma \ (\Delta t - v \Delta x / c^2) , there exists a v less than c for which \Delta t' = 0 (in particular, v = c^2 \Delta t / \Delta x). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, \sqrt , is called the ''proper distance'', or ''proper length''. For values of v greater than and less than c^2 \Delta t / \Delta x , the sign of \Delta t' changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. The temporal order of timelike-separated events, however, is absolute, since the only way that v could be greater than c^2 \Delta t / \Delta x would be if v > c . *Δs2 = 0: In this case, the two events are said to be ''lightlike'' separated. This implies that , \Delta x / \Delta t , = c , and this relationship is frame independent due to the invariance of s^2 . From this, we observe that the speed of light is c in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.


Relativity of simultaneity

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity). From (the forward Lorentz transformation in terms of coordinate differences) \Delta t' = \gamma \left(\Delta t - \frac \right) It is clear that the two events that are simultaneous in frame ''S'' (satisfying ), are not necessarily simultaneous in another inertial frame ''S''′ (satisfying ). Only if these events are additionally co-local in frame ''S'' (satisfying ), will they be simultaneous in another frame ''S''′. The Sagnac effect can be considered a manifestation of the relativity of simultaneity. Since relativity of simultaneity is a first order effect in v, instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity.


Time dilation

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (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 the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels). Suppose a
clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and t ...
is at rest in the unprimed system ''S''. The location of the clock on two different ticks is then characterized by . To find the relation between the times between these ticks as measured in both systems, can be used to find: :\Delta t' = \gamma\, \Delta t for events satisfying\Delta x = 0 \ . This shows that the time (Δ''t''′) between the two ticks as seen in the frame in which the clock is moving (''S''′), is ''longer'' than the time (Δ''t'') between these ticks as measured in the rest frame of the clock (''S''). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.


Length 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 The ladder paradox (or barn-pole paradox) is a thought experiment in special relativity. It involves a ladder, parallel to the ground, travelling horizontally at relativistic speed (near the speed of light) and therefore undergoing a Lorentz lengt ...
involves a long ladder traveling near the speed of light and being contained within a smaller garage). Similarly, suppose a
measuring rod A measuring rod is a tool used to physically measure lengths and survey areas of various sizes. Most measuring rods are round or square sectioned; however, they can also be flat boards. Some have markings at regular intervals. It is likely th ...
is at rest and aligned along the ''x''-axis in the unprimed system ''S''. In this system, the length of this rod is written as Δ''x''. To measure the length of this rod in the system ''S''′, in which the rod is moving, the distances ''x''′ to the end points of the rod must be measured simultaneously in that system ''S''′. In other words, the measurement is characterized by , which can be combined with to find the relation between the lengths Δ''x'' and Δ''x''′: :\Delta x' = \frac for events satisfying\Delta t' = 0 \ . This shows that the length (Δ''x''′) of the rod as measured in the frame in which it is moving (''S''′), is ''shorter'' than its length (Δ''x'') in its own rest frame (''S''). Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring ''time intervals'' between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are ''different'' in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will ''not'' occur at the same ''spatial distance'' from each other when seen from another moving coordinate system.


Lorentz transformation of velocities

Consider two frames ''S'' and ''S′ '' in standard configuration. A particle in ''S'' moves in the x direction with velocity vector \mathbf. What is its velocity \mathbf in frame ''S′ ''? We can write Substituting expressions for dx' and dt' from into , followed by straightforward mathematical manipulations and back-substitution from yields the Lorentz transformation of the speed u to u': The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing v with -v \ . For \mathbf not aligned along the x-axis, we write: The forward and inverse transformations for this case are: and can be interpreted as giving the ''resultant'' \mathbf of the two velocities \mathbf and \mathbf, and they replace the formula \mathbf which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the ''relativistic velocity addition (or composition) formulas'', valid for the three axes of ''S'' and ''S′ '' being aligned with each other (although not necessarily in standard configuration). We note the following points: * If an object (e.g., a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
) were moving at the speed of light in one frame then it would also be moving at the speed of light in any other frame, moving at . * The resultant speed of two velocities with magnitude less than ''c'' is always a velocity with magnitude less than ''c''. * If both , ''u'', and , ''v'', (and then also , ''u′'', and , ''v′'', ) are small with respect to the speed of light (that is, then the intuitive Galilean transformations are recovered from the transformation equations for special relativity * Attaching a frame to a photon (''riding a light beam'' like Einstein considers) requires special treatment of the transformations. There is nothing special about the ''x'' direction in the standard configuration. The above formalism applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.


Thomas rotation

The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. Thomas rotation results from the relativity of simultaneity. In Fig. 4-2a, a rod of length L in its rest frame (i.e., having a
proper length Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...
of L) rises vertically along the y-axis in the ground frame. In Fig. 4-2b, the same rod is observed from the frame of a rocket moving at speed v to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized ''in the frame of the rod,'' relativity of simultaneity causes the observer in the rocket frame to observe (not ''see'') the clock at the right end of the rod as being advanced in time by Lv/c^2 , and the rod is correspondingly observed as tilted. Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion. Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".


Causality and prohibition of motion faster than light

In Fig. 4-3, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; that is, there is a frame of reference in which events A and B occur at the ''same location in space'', separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect). The interval AC in the diagram is 'space-like'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. However, there are no frames accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams). A variety of causal paradoxes could then be constructed. Consider the spacetime diagrams in Fig. 4-4. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical (''ct''), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (''ct′''), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground. # Fig. 4-4a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the -x' axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives ''earlier'' than it was sent. # Fig. 4-4b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the +x axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality. It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the x' axis (and the signal from A to B slightly steeper than the x axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the ct' and x' axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only ''slightly'' faster than the speed of light will result in causality violation. Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel
faster than light Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with zero ...
in vacuum. This is not to say that ''all'' faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light. For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).


Optical effects


Dragging effects

In 1850,
Hippolyte Fizeau Armand Hippolyte Louis Fizeau FRS FRSE MIF (; 23 September 181918 September 1896) was a French physicist, best known for measuring the speed of light in the namesake Fizeau experiment. Biography Fizeau was born in Paris to Louis and Beatrice Fi ...
and
Léon Foucault Jean Bernard Léon Foucault (, ; ; 18 September 1819 – 11 February 1868) was a French physicist best known for his demonstration of the Foucault pendulum, a device demonstrating the effect of Earth's rotation. He also made an early measurement ...
independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory. The speed of light was measured in still water. What would be the speed of light in flowing water? In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light. According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed ''through'' the medium plus the speed ''of'' the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If u' = c/n is the speed of light in still water, and v is the speed of the water, and u_ is the water-bourne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then u_ =\frac \pm v\left(1-\frac\right) \ . Fizeau's results, although consistent with Fresnel's earlier hypothesis of partial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since n depends on wavelength, ''the aether must be capable of sustaining different motions at the same time.'' A variety of theoretical explanations were proposed to explain Fresnel's dragging coefficient that were completely at odds with each other. Even before the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies. From the point of view of special relativity, Fizeau's result is nothing but an approximation to , the relativistic formula for composition of velocities. : u_ = \frac = \frac \approx c \left( \frac \pm \frac \right) \left( 1 \mp \frac \right) \approx \frac \pm v \left( 1 - \frac \right)


Relativistic aberration of light

Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the
aberration of light In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of t ...
. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light. (2) If the source is in motion, the displacement would be the consequence of light-time correction. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver. The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light, and in 1870,
George Airy Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope. A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag, but was incompatible with the results of the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
, which apparently demanded ''complete'' aether-drag. Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include : \cos \theta ' = \frac   OR   \sin \theta ' = \frac   OR   \tan \frac = \left( \frac \right)^ \tan \frac


Relativistic Doppler effect


Relativistic longitudinal Doppler effect

The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
term, and that is the treatment described here. Assume the receiver and the source are moving ''away'' from each other with a relative speed v\, as measured by an observer on the receiver or the source (The sign convention adopted here is that v is ''negative'' if the receiver and the source are moving ''towards'' each other). Assume that the source is stationary in the medium. Then f_ = \left(1 - \frac v \right) f_s where c_s is the speed of sound. For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be f_r = \gamma\left(1 - \beta\right) f_s = \sqrt\,f_s. where *\beta = v/c   and *\gamma = \frac is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the ''receiver'' with a moving source.


Transverse Doppler effect

The transverse Doppler effect is one of the main novel predictions of the special theory of relativity. Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments. In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of \gamma. In Fig. 5-3b, the light is redshifted by the same factor.


Measurement versus visual appearance

Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer. Scientists make a fundamental distinction between ''measurement'' or ''observation'' on the one hand, versus ''visual appearance'', or what one ''sees''. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. The visual appearance of an object, however, is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye. For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be ''seen'' as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would ''appear'' contracted, an approaching object would ''appear'' elongated, and a passing object would have a skew appearance that has been likened to a rotation. A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although the surface of the sphere and the images on it will appear distorted. Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig. 5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen. This illusion has come to be known as '' Terrell rotation'' or the ''Terrell–Penrose effect''.Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in ''Einstein's Cosmos'' (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length." Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various
radio galaxies A radio galaxy is a galaxy with giant regions of radio emission extending well beyond its visible structure. These energetic radio lobes are powered by jets from its active galactic nucleus. They have luminosities up to 1039  W at radio wav ...
, BL Lac objects,
quasars A quasar is an extremely luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a mass rangi ...
, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel. In Fig. 5-6, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5-4 has been stretched out.


Dynamics

Section ''Consequences derived from the Lorentz transformation'' dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.


Equivalence of mass and energy

As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference. The energy content of an object at rest with mass ''m'' equals ''mc''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. In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for . Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is : it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes . The momentum is equal to the energy multiplied by the velocity divided by ''c''2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to ''E''/''c''2. The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations. The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.Does the inertia of a body depend upon its energy content?
A. Einstein, ''Annalen der Physik.'' 18:639, 1905 (English translation by W. Perrett and G.B. Jeffery)
Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong. Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions. Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.
A. Einstein, Annalen der Physik 23 (1907): 371–384
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.


How far can you travel from the Earth?

Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems which exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant 1''g'', it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by: v(t) = \frac where ''v''(''t'') is the velocity at a time ''t'', ''a'' is the acceleration of the spaceship and ''t'' is the coordinate time as measured by people on Earth. Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at ''v'' = 0.712''c'' and 0.946''c'' after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth. During their journey, people on Earth will experience more time than they do - since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years. A full 40-year trip at 1''g'' will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1''g'' will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1''g'' acceleration could reach 2,000,000 light-years to the Andromeda Galaxy. This same time dilation is why a muon travelling close to ''c'' is observed to travel much farther than ''c'' times its half-life (when at rest).


Relativity and unifying electromagnetism

Theoretical investigation in
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
led to the discovery of wave propagation. 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 Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the ''magnetic'' field generated by a moving charge disappears and becomes a purely ''electrostatic'' field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of ''electromagnetic'' fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame. Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, that is, in the language of
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
calculus.


Theories of relativity and quantum mechanics

''Special'' relativity can be combined with
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
to form
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
and
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. How ''general'' relativity and quantum mechanics can be unified is one of the unsolved problems in physics; quantum gravity and a "
theory of everything A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...
", which require a unification including general relativity too, are active and ongoing areas in theoretical research. The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
of the time. In 1928, Paul Dirac constructed an influential
relativistic wave equation In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
, now known as the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
in his honour, that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called '' spin'', it also led to the prediction of the
antiparticle In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
of the electron (the positron), and fine structure could only be fully explained with special relativity. It was the first foundation of ''
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
''. On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called '' quantum field theory'', becomes necessary; in which particles can be created and destroyed throughout space and time.


Status

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
is much less than ''c''2 in the region of interest. In a strong gravitational field, one must use
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. 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) 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. Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields). Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
for a more detailed discussion. Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905, and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory. * The
Fizeau experiment The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light. ...
(1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities. * The famous
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
(1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times. * The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame. * The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.
Particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples: * Tests of relativistic energy and momentum – testing the limiting speed of particles * Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation *
Experimental testing of time dilation Time dilation as predicted by special relativity is often verified by means of particle lifetime experiments. According to special relativity, the rate of a clock C traveling between two synchronized laboratory clocks A and B, as seen by a laborat ...
– relativistic effects on a fast-moving particle's half-life * Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations * Hughes–Drever experiment – testing isotropy of space and mass * Modern searches for Lorentz violation – various modern tests * Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter. * Experiments to test the
aether drag hypothesis In the 19th century, the theory of the luminiferous aether as the hypothetical medium for the propagation of light waves was widely discussed. The aether hypothesis arose because physicists of that era could not conceive of light waves propagating ...
– no "aether flow obstruction".


Technical discussion of spacetime


Geometry of spacetime


Comparison between flat Euclidean space and Minkowski space

Special relativity uses a 'flat' 4-dimensional Minkowski space – an example of a
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. Minkowski spacetime appears to be very similar to the standard 3-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, but there is a crucial difference with respect to time. In 3D space, the differential of distance (line element) ''ds'' is defined by ds^2 = d\mathbf \cdot d\mathbf = dx_1^2 + dx_2^2 + dx_3^2, where are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate ''X''0 derived from time, such that the distance differential fulfills ds^2 = -dX_0^2 + dX_1^2 + dX_2^2 + dX_3^2, where are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1). Just as Euclidean space uses a
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
, so spacetime uses a
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
. Basically, special relativity can be stated as the ''invariance of any spacetime interval'' (that is the 4D distance between any two events) when viewed from ''any inertial reference frame''. All equations and effects of special relativity can be derived from this rotational symmetry (the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
) of Minkowski spacetime. The actual form of ''ds'' above depends on the metric and on the choices for the ''X''0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: (this is called a
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that s ...
). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take , rather than a "disguised" Euclidean metric using ''ict'' as the time coordinate. Some authors use , with factors of ''c'' elsewhere to compensate; for instance, spatial coordinates are divided by ''c'' or factors of ''c''±2 are included in the metric tensor. These numerous conventions can be superseded by using
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
where . Then space and time have equivalent units, and no factors of ''c'' appear anywhere.


3D spacetime

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2, we see that the
null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value * Null character, the zero-valued ASCII character, also designated by , often use ...
geodesics lie along a dual-cone (see Fig. 10-2) defined by the equation; ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2 or simply dx_1^2 + dx_2^2 = c^2 dt^2,  which is the equation of a circle of radius ''c dt''.


4D spacetime

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 so dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2. As illustrated in Fig. 10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = ''c dt''. This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance d = \sqrt away and a time ''d/c'' 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 Fig. 10-2 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 diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
s, which are useful also in understanding many of the thought experiments in special relativity. Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see Center of mass (relativistic).


Physics in spacetime


Transformations of physical quantities between reference frames

Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation. The Lorentz transformation in standard configuration above, that is, for a boost in the ''x''-direction, can be recast into matrix form as follows: \begin ct'\\ x'\\ y'\\ z' \end = \begin \gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end \begin ct\\ x\\ y\\ z \end = \begin \gamma ct- \gamma\beta x\\ \gamma x - \beta \gamma ct \\ y\\ z \end. In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "
four vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
s", in Minkowski spacetime. The components of vectors are written using
tensor index notation In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be c ...
, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, thus bypassing the tedious calculations to check this fact. 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. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used. The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ''ct'' and spacelike component , in a contravariant position
four vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
with components: X^\nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z) = (ct, \mathbf ). where we define so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally. Now the transformation of the contravariant components of the position 4-vector can be compactly written as: X^=\Lambda^_\nu X^\nu where there is an implied summation on \nu from 0 to 3, and \Lambda^_ is a matrix. More generally, all contravariant components of a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
T^\nu transform from one frame to another frame by a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
: T^ = \Lambda^_ T^\nu Examples of other 4-vectors include the
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
U^\mu, defined as the derivative of the position 4-vector with respect to
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
: U^\mu = \frac = \gamma(v)( c , v_x , v_y, v_z ) = \gamma(v) (c, \mathbf ). where the Lorentz factor is: \gamma(v)= \frac \qquad v^2 = v_x^2 + v_y^2 + v_z^2. The relativistic energy E = \gamma(v)mc^2 and
relativistic momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
\mathbf = \gamma(v)m \mathbf of an object are respectively the timelike and spacelike components of a contravariant four momentum vector: P^\mu = m U^\mu = m\gamma(v)(c,v_x,v_y,v_z)= \left (\frac,p_x,p_y,p_z \right ) = \left (\frac, \mathbf \right ). where ''m'' is the invariant mass. The four-acceleration is the proper time derivative of 4-velocity: A^\mu = \frac. The transformation rules for ''three''-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of ''four''-velocity and ''four''-acceleration are simpler by means of the Lorentz transformation matrix. The four-gradient of a scalar field φ transforms covariantly rather than contravariantly: \begin \dfrac \dfrac & \dfrac & \dfrac & \dfrac \end = \begin \dfrac \dfrac & \dfrac & \dfrac & \dfrac \end \begin \gamma & +\beta\gamma & 0 & 0\\ +\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end. which is the transpose of: (\partial_ \phi) = \Lambda_^ (\partial_\nu \phi)\qquad \partial_ \equiv \frac. only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates. More generally, the ''co''variant components of a 4-vector transform according to the ''inverse'' Lorentz transformation: T_ = \Lambda_^ T_\nu, where \Lambda_^ is the reciprocal matrix of \Lambda^_. The postulates of special relativity constrain the exact form the Lorentz transformation matrices take. More generally, most physical quantities are best described as (components of)
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s. So to transform from one frame to another, we use the well-known
tensor transformation law In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
T^_ = \Lambda^_ \Lambda^_ \cdots \Lambda^_ \Lambda_^ \Lambda_^ \cdots \Lambda_^ T^_ where \Lambda_^ is the reciprocal matrix of \Lambda^_. All tensors transform by this rule. An example of a four-dimensional second order
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ' ...
is the
relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
, which has six components: three are the classical
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ' ...
. The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.


Metric

The metric tensor allows one to define the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
''η'' has components (valid with suitably chosen coordinates) which can be arranged in a matrix: \eta_ = \begin -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end which is equal to its reciprocal, \eta^, in those frames. Throughout we use the signs as above, different authors use different conventions – see
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
alternative signs. The
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
is the most general group of transformations which preserves the Minkowski metric: \eta_ = \eta_ \Lambda^_\alpha \Lambda^_\beta and this is the physical symmetry underlying special relativity. The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector ''T'' with another 4-vector ''S'' is: T^S_=T^\eta_S^ = T_\eta^S_ = \text Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no appears in its trivial transformation. The magnitude of the 4-vector ''T'' is the positive square root of the inner product with itself: , \mathbf, = \sqrt One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants: T^_,T^_T^_,T^_T^_T^_ = \text, similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.


Relativistic kinematics and invariance

The coordinate differentials transform also contravariantly: dX^=\Lambda^_\nu dX^\nu so the squared length of the differential of the position four-vector ''dXμ'' constructed using d\mathbf^2 = dX^\mu \,dX_\mu = \eta_\,dX^\mu \,dX^\nu = -(c dt)^2+(dx)^2+(dy)^2+(dz)^2 is an invariant. Notice that when the
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
''d''X2 is negative that is the differential of
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
, while when ''d''X2 is positive, is differential of the proper distance. The 4-velocity ''U''μ has an invariant form: \mathbf U^2 = \eta_ U^\nu U^\mu = -c^2 \,, which means 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. Differentiating the above equation by ''τ'' produces: 2\eta_A^\mu U^\nu = 0. So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.


Relativistic dynamics and invariance

The invariant magnitude of the momentum 4-vector generates the energy–momentum relation: \mathbf^2 = \eta^P_\mu P_\nu = -\left (\frac \right )^2 + p^2 . We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero. \mathbf^2 = - \left (\frac \right )^2 = - (m 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_\text = m c^2. 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. To use
Newton's third law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. 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. It is the rate of change of the above energy momentum
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
with respect to proper time. The covariant version of the four-force is: F_\nu = \frac = m A_\nu 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 (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times ''c''. 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, that is, ''dp''/''dt'' while the four force is defined by the rate of change of momentum with respect to proper time, that is, ''dp''/''dτ''. 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 −1/''c'' times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.


See also

*People: **
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
** Hermann Minkowski ** Max von Laue ** Arnold Sommerfeld ** Max Born *Relativity: **
History of special relativity The history of special relativity consists of many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity proposed by Albert Eins ...
** Doubly special relativity **
Bondi k-calculus Bondi ''k''-calculus is a method of teaching special relativity popularised by Sir Hermann Bondi, that has been used in university-level physics classes (e.g. at The University of Oxford), and in some relativity textbooks. The usefulness of the ...
** Einstein synchronisation ** Rietdijk–Putnam argument ** Special relativity (alternative formulations) ** Relativity priority dispute *Physics: **
Einstein's thought experiments A hallmark of Albert Einstein's career was his use of visualized thought experiments (german: Gedankenexperiment) as a fundamental tool for understanding physical issues and for elucidating his concepts to others. Einstein's thought experiments too ...
**
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
** Relativistic Euler equations ** Lorentz ether theory **
Moving magnet and conductor problem The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
** Shape waves **
Relativistic heat conduction Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conduct ...
** Relativistic disk ** Born rigidity ** Born coordinates *Mathematics: **
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
** Relativity in the APS formalism *Philosophy: **
actualism In analytic philosophy, actualism is the view that everything there ''is'' (i.e., everything that has ''being'', in the broadest sense) is wiktionary:actual, actual. Another phrasing of the thesis is that the domain of discourse, domain of Range o ...
**
conventionalism Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicit or implicit) agreements in society, rather than on external reality. Unspoken rules play a key role in the philosophy's structur ...
*Paradoxes: ** Ehrenfest paradox **
Bell's spaceship paradox Bell's spaceship paradox is a thought experiment in special relativity. It was designed by E. Dewan and M. Beran in 1959 and became more widely known when J. S. Bell included a modified version.J. S. Bell: ''How to teach special relativity'', Prog ...
** Velocity composition paradox ** Lighthouse paradox


Notes


Primary sources


References


Further reading


Textbooks

* Einstein, Albert (1920). '' Relativity: The Special and General Theory''. * Einstein, Albert (1996). ''The Meaning of Relativity''. Fine Communications. * Logunov, Anatoly A. (2005).
Henri Poincaré and the Relativity Theory
' (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow. *
Charles Misner Charles W. Misner (; born June 13, 1932) is an American physicist and one of the authors of '' Gravitation''. His specialties include general relativity and cosmology. His work has also provided early foundations for studies of quantum gravity ...
,
Kip Thorne Kip Stephen Thorne (born June 1, 1940) is an American theoretical physicist known for his contributions in gravitational physics and astrophysics. A longtime friend and colleague of Stephen Hawking and Carl Sagan, he was the Richard P. F ...
, and
John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in ...
(1971) ''Gravitation''. W. H. Freeman & Co. * Post, E.J., 1997 (1962) ''Formal Structure of Electromagnetics: General Covariance and Electromagnetics''. Dover Publications. *
Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity where he is known for introducing the term "event horizon", Rindler coordinates, and (in collaboration with Roger Penrose) for the use of ...
(1991). ''Introduction to Special Relativity'' (2nd ed.), Oxford University Press. ; * Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, ; * * * Silberstein, Ludwik (1914). '' The Theory of Relativity''. * * * * Taylor, Edwin, and
John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in ...
(1992). ''Spacetime Physics'' (2nd ed.). W. H. Freeman & Co. . * Tipler, Paul, and Llewellyn, Ralph (2002). ''Modern Physics'' (4th ed.). W. H. Freeman & Co. .


Journal articles

* * *
Special Relativity
Scholarpedia *


External links


Original works


''Zur Elektrodynamik bewegter Körper''
Einstein's original work in German,
Annalen der Physik ''Annalen der Physik'' (English: ''Annals of Physics'') is one of the oldest scientific journals on physics; it has been published since 1799. The journal publishes original, peer-reviewed papers on experimental, theoretical, applied, and mathe ...
, Bern 1905
''On the Electrodynamics of Moving Bodies''
English Translation as published in the 1923 book ''The Principle of Relativity''.


Special relativity for a general audience (no mathematical knowledge required)


Einstein Light
A
award
winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.

Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics. * Audio: Cain/Gay (2006) �
Astronomy Cast
Einstein's Theory of Special Relativity


Special relativity explained (using simple or more advanced mathematics)



– A simple introduction to the special theory of relativity.


The Hogg Notes on Special Relativity
A good introduction to special relativity at the undergraduate level, using calculus.
Relativity Calculator: Special Relativity
– An algebraic and integral calculus derivation for .

A complete online book on relativity with an extensive bibliography.
Special Relativity
An introduction to special relativity at the undergraduate level. * , by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

Special Relativity Lecture Notes
is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.

The theory of special relativity in an easily understandable way.
An Introduction to the Special Theory of Relativity
(1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).
Lecture Notes on Special Relativity
by J D Cresser Department of Physics Macquarie University.
SpecialRelativity.net
– An overview with visualizations and minimal mathematics.
Relativity 4-ever?
The problem of superluminal motion is discussed in an entertaining manner.


Visualization



Software visualizing several scenarios under the influence of special relativity.
Real Time Relativity
The Australian National University. Relativistic visual effects experienced through an interactive program.
Spacetime travel
A variety of visualizations of relativistic effects, from relativistic motion to black holes.
Through Einstein's Eyes
The Australian National University. Relativistic visual effects explained with movies and images.
Warp Special Relativity Simulator
A computer program to show the effects of traveling close to the speed of light. * visualizing the Lorentz transformation.

from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
lightspeed
An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.

showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed. {{DEFAULTSORT:Special Relativity Albert Einstein