TheInfoList

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between
space and time In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. The fabric of space-time is a conceptual model combining the ...
. 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 physicists of all time. Einstein is known for developing the theory of relativity The theo ...

's original treatment, the theory is based on two
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s: # The
laws of physics Scientific laws or laws of science are statements, based on repeated experiment An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrat ...
are invariant (that is, identical) in all
inertial frames of reference In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a c ...
(that is,
frames of reference In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

with no
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

). # The
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
in
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gr ...

is the same for all observers, regardless of the motion of the light source or 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 200px, Einstein in 1904 or 1905, about the time he wrote the ''Annus Mirabilis'' papers The ''Annus mirabilis'' papers (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European l ...
".
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

(1905)
''Zur Elektrodynamik bewegter Körper''
, ''Annalen der Physik'' 17: 891; English translatio
On the Electrodynamics of Moving Bodies
by
George Barker Jeffery George Barker Jeffery FRS (9 May 1891 – 27 April 1957) was a leading mathematical physicist in the early twentieth century. He is probably best known to the scientifically literate public as the translator of papers by Albert Einstein, Hendrik ...
and Wilfrid Perrett (1923); Another English translation
On the Electrodynamics of Moving Bodies 200px, Einstein in 1904 or 1905, about the time he wrote the ''Annus Mirabilis'' papers The ''Annus mirabilis'' papers (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European l ...
The incompatibility of
Newtonian mechanics Newton's laws of motion are three Scientific law, 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: ''Law 1''. A body continues ...
with
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
of
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

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 Medium may refer to: Science and technology Aviation *Medium bomber, a class of war plane *Tecma Medium, a French hang glider design Communic ...
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 In physics, the relativity of simultaneity is the concept that ''distant simultaneity'' – whether two spatially separated events occur at the same Time in physics, time – is not absolute time and space, absolute, but depends on th ...

,
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its 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 compl ...
,
time dilation In physics and Theory of relativity, relativity, time dilation is the difference in the elapsed Time in physics, time as measured by two clocks. It is either due to a relative velocity between them (special relativity, special relativistic "kine ...

, the relativistic velocity addition formula, the relativistic
Doppler effect The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the ...

,
relativistic mass The word ''mass'' has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the ''relativistic mass'' is dependent on the velocity ...
, a universal speed limit,
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, the speed of causality and the
Thomas precession In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
. 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 position. Rather than an invariant time interval between two events, there is an invariant
spacetime interval In physics, spacetime is any mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sys ...
. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
and
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

, as expressed in the
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
formula $E = mc^2$, where $c$ is the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
in a vacuum. It also explains how the phenomena of electricity and magnetism are related. A defining feature of special relativity is the replacement of the
Galilean transformation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
s 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 coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

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 Einstein developed
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, introducing 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 Lorentz' theory of electrons. Formulas for the curl of the magnetic field (IV) and the electrical field E (V), ''La théorie electromagnétique de Maxwell et son application aux corps mouvants'', 1892, p. 452. Hendrik Antoon Lorentz (; 18 Ju ...

and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
. The theory became essentially complete in 1907. The theory is "special" in that it only applies in the
special case In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
where the spacetime is "flat", that is, the
curvature of spacetime General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern p ...
, described by the
energy–momentum tensorEnergy–momentum may refer to: *Four-momentum *Stress–energy tensor *Energy–momentum relation {{dab ...
and causing
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

, 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 (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ( ...
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. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using Galileo's ship, ...
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 field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

s, 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 In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It ...

s are negligible) and in conditions of
free fall #REDIRECT Free fall #REDIRECT Free fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—inc ...

. 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 geome ...
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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
. As long as the universe can be modeled as a
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...
, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this
curved spacetime Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cur ...
.
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific q ...

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 Scientific laws or laws of science are statements, based on repeated experiment An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrat ...
, including both the laws of mechanics and of
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
.

# 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
– 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 and the lack of evidence for the
luminiferous ether upright=1.25, The luminiferous aether: it was hypothesised that the Earth moves through a "medium" of aether that carries light Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of ...
. 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 upright=1.25, The luminiferous aether: it was hypothesised that the Earth moves through a "medium" of aether that carries light Luminiferous aet ...
. 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 Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by t ...
and
homogeneity Homogeneity and heterogeneity are concepts often used in the sciences Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts ...
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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
provided the mathematical framework for relativity theory by proving that
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...
are a subset of his
Poincaré group The Poincaré group, named after Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to ...
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 In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame . ...

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 Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed in a sequestered place and according to a set sequence. Rit ...
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 any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
. 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 A firecracker (cracker, noise maker, banger,) is a small explosive An explosive (or explosive material) is a reactive substance that contains a great amount of potential energy that can produce an explosion if released suddenly, usually acco ...
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 In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame . ...
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 frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matt ...
s (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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
, which states that physical laws have the same form in each
inertial reference frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies mat ...
, dates back to
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a in the field of who focuses their studies on a specific question or field o ...

, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of
electromagnetic waves In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

led some physicists to suggest that the universe was filled with a substance they called "
aetherAether, æther or ether may refer to: Metaphysics and mythology * Aether (classical element), the material supposed to fill the region of the universe above the terrestrial sphere * Aether (mythology), the personification of the "upper sky", spac ...
", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an absolute reference frame against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether upright=1.25, The luminiferous aether: it was hypothesised that the Earth moves through a "medium" of aether that carries light Luminiferous aet ...
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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
.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 any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
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 contraction ...

.

## Lorentz transformation and its inverse

Define an
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed in a sequestered place and according to a set sequence. Rit ...
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 coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

specifies that these coordinates are related in the following way: :$\begin t\text{'} &= \gamma \ \left(t - vx/c^2\right) \\ x\text{'} &= \gamma \ \left(x - v t\right) \\ y\text{'} &= y \\ z\text{'} &= z , \end$ where :$\gamma = \frac$ is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

and ''c'' is the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, that parameter being called
rapidity In Theory of relativity, relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being a ...
. Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation: :$\begin t &= \gamma \left( t\text{'} + v x\text{'}/c^2\right) \\ x &= \gamma \left( x\text{'} + v t\text{'}\right) \\ y &= y\text{'} \\ z &= z\text{'}. \end$ 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 ''v′'' = −''v'', as measured in the primed frame. There is nothing special about the ''x''-axis. The transformation can apply to the ''y''- or ''z''-axis, or indeed in any direction parallel to the motion (which are warped by the ''γ'' factor) and perpendicular; see the article
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

for details. A quantity invariant under
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...
is known as a
Lorentz scalar In a Theory of relativity, relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar (physics), scalar, invariant (physics), invariant under any Lorentz transformation. A Lorentz ...
. 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 :    $\Delta x\text{'} = x\text{'}_2-x\text{'}_1 \ , \ \Delta t\text{'} = t\text{'}_2-t\text{'}_1 \ .$ :    $\Delta x = x_2-x_1 \ , \ \ \Delta t = t_2-t_1 \ .$ we get :    $\Delta x\text{'} = \gamma \ \left(\Delta x - v \,\Delta t\right) \ ,\ \$ $\Delta t\text{'} = \gamma \ \left\left(\Delta t - v \ \Delta x / c^ \right\right) \ .$ :    $\Delta x = \gamma \ \left(\Delta x\text{'} + v \,\Delta t\text{'}\right) \ , \$ $\Delta t = \gamma \ \left\left(\Delta t\text{'} + v \ \Delta x\text{'} / c^ \right\right) \ .$ If we take differentials instead of taking differences, we get :    $dx\text{'} = \gamma \ \left(dx - v \, dt\right) \ ,\ \$ $dt\text{'} = \gamma \ \left\left( dt - v \ dx / c^ \right\right) \ .$ :    $dx = \gamma \ \left(dx\text{'} + v \,dt\text{'}\right) \ , \$ $dt = \gamma \ \left\left(dt\text{'} + v \ dx\text{'} / c^ \right\right) \ .$

## 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 contraction ...

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 $x\text{'}$ and $ct\text{'}$ axes of frame S'. The $ct\text{'}$ 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 $ct\text{'}$ and $x\text{'}$ axes are tilted from the unprimed axes by an angle $\alpha = \tan^\left(\beta\right),$ 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 $t\text{'}=0.$ 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 $\left(x\text{'}, ct\text{'}\right)$ coordinates of $\left(0, 1\right)$ in the primed coordinate system transform to $\left(\beta \gamma, \gamma\right)$ in the unprimed coordinate system. Likewise, $\left(x\text{'}, ct\text{'}\right)$ coordinates of $\left(1, 0\right)$ in the primed coordinate system transform to $\left(\gamma, \beta \gamma\right)$ in the unprimed system. Draw gridlines parallel with the $ct\text{'}$ axis through points $\left(k \gamma, k \beta \gamma\right)$ as measured in the unprimed frame, where $k$ is an integer. Likewise, draw gridlines parallel with the $x\text{'}$ axis through $\left(k \beta \gamma, k \gamma\right)$ as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between $ct\text{'}$ 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 $\beta \rightarrow 1.$ Fig. 3-1d. Since the speed of light is an invariant, the ''worldlines'' of two photons passing through the origin at time $t\text{'} = 0$ 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 Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

, 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 coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

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 {{Short pages monitor Theories by Albert Einstein