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{{short description|Theory of interwoven space and time by Albert Einstein}}
[[File:Einstein patentoffice.jpg|thumb|[[Albert Einstein]] around 1905, the year his "[[Annus mirabilis papers|''annus mirabilis'' papers]]" were published. These included ''Zur Elektrodynamik bewegter Körper''<!-- ("On the Electrodynamics of Moving Bodies")-->, the paper founding special relativity.]]
{{Special relativity sidebar}}

In [[physics]], the '''special theory of relativity''', or '''special relativity''' for short, is a scientific theory of the relationship between [[Spacetime|space and time]]. In [[Albert Einstein]]'s 1905 paper, [[Annus Mirabilis papers#Special relativity|
"On the Electrodynamics of Moving Bodies"]], the theory is presented as being based on just [[Postulates of special relativity|two postulates]]:<ref name="electro" group="p"/><ref name="Griffiths-2013">{{Cite book | last = Griffiths | first = David J. | title = Introduction to Electrodynamics | title-link = Introduction to Electrodynamics | publisher = Pearson | year = 2013 | isbn = 978-0-321-85656-2 | edition = 4th | at = Chapter 12 | chapter = Electrodynamics and Relativity | author-link = David J. Griffiths}}</ref><ref name="Jackson-1999">{{Cite book | last = Jackson | first = John D. | title = Classical Electrodynamics | title-link = Classical Electrodynamics (book) | publisher = John Wiley & Sons | year = 1999 | isbn = 0-471-30932-X | edition = 3rd | at = Chapter 11 |chapter = Special Theory of Relativity | author-link = John David Jackson (physicist)}}</ref>
# The [[laws of physics]] are [[Invariant (physics)|invariant]] (identical) in all [[Inertial frame of reference|inertial frames of reference]] (that is, [[Frame of reference|frames of reference]] with no [[acceleration]]). This is known as the [[principle of relativity]].
# The [[speed of light]] in [[vacuum]] is the same for all observers, regardless of the motion of light source or observer. This is known as the principle of light constancy, or the principle of light speed invariance.
The first postulate was first formulated by [[Galileo Galilei]] (see ''[[Galilean invariance]]'').

== Overview ==
Relativity is a theory that accurately describes objects moving at speeds far beyond normal experience. Relativity replaces the idea that time flows equally everywhere in the universe with a new concept that time flows differently for every independent object. The flow of time can be expressed by counting ticks on a clock. Moving clocks run slower. At speeds encountered in normal experience, the slow down cannot be observed. Two events measured at the same time on a stationary clock occur at different times if measured on moving clocks. Near the speed of light many physical effects can only be understood by including the effects of special relativity.<ref name="Mermin-2009"/> 
===Basis===
Unusual among modern topics in physics, the theory of special relativity needs only mathematics at high school level and yet it fundamentally alters our understanding, especially our understanding of the concept of [[time]].<ref name=Mermin-2009/>{{rp|ix}} Built on just two postulates or assumptions, many interesting consequences follow. 

The two postulates both concern observers moving at a constant speed relative to each other. The first postulate, the {{slink|#principle of relativity}}, says the laws of physics do not depend on objects being at absolute rest: for example, an observer on a train sees natural phenomena on that train that look the same whether the train is moving or not.<ref name=Mermin-2009/>{{rp|5}} The second postulate, constant speed of light, says observers in a train station see light travel at the same speed whether they measure light from within the station or light from a moving train. A light signal from the station to the train has the same speed, no matter how fast a train goes.<ref name=Mermin-2009/>{{rp|25}}

In the theory of special relativity, the two postulates combine to change the definition of "relative speed". Rather than the simple concept of distance traveled divided by time spent, the new theory incorporates the speed of light as the maximum possible speed. In special relativity, covering ten times more distance on the ground in the same amount of time according to a moving watch does not result in a speed up as seen from the ground by a factor of ten.<ref name=Mermin-2009/>{{rp|28}}

===Consequences===
Special relativity has a wide range of consequences that have been experimentally verified.<ref>{{cite web |url=http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html |title=What is the experimental basis of Special Relativity? |access-date=2008-09-17 |author1=Tom Roberts |author2=Siegmar Schleif |name-list-style=amp |date=October 2007 |work=Usenet Physics FAQ}}</ref><ref>Will, C. M. (2005). Special relativity: a centenary perspective. In Einstein, 1905–2005: Poincaré Seminar 2005 (pp. 33-58). Basel: Birkhäuser Basel.</ref> The conceptual effects include:
* The [[#Relativity of simultaneity|relativity of simultaneity]]{{snd}} events that appear simultaneous to one observer may not be simultaneous to an observer in motion<ref name=Mermin-2009/>{{rp|49}}
* {{slink|#Time dilation}}{{snd}} time measured between two events by observers in motion differ
* {{slink|#Length contraction}}{{snd}} distances between two events by observers in motion differ
* The {{slink|#Lorentz transformation of velocities}}{{snd}} velocities no longer simply add
Combined with other laws of physics, the two postulates of special relativity predict the equivalence of [[mass]] and [[energy]], as expressed in the [[mass–energy equivalence]] formula {{tmath|1= E = mc^2 }}, where <math>c</math> is the [[speed of light]] in vacuum.<ref name="relativity">{{cite book |author=Albert Einstein |title=Relativity: The Special and the General Theory |page= 48 |url=https://books.google.com/books?id=idb7wJiB6SsC&pg=PA50 |isbn=978-0-415-25384-0 |publisher=Routledge |date=2001 |edition=Reprint of 1920 translation by Robert W. Lawson}}</ref><ref name="Feynman">[https://feynmanlectures.caltech.edu/I_15.html#Ch15-S9 The Feynman Lectures on Physics Vol. I Ch. 15-9: Equivalence of mass and energy]</ref>
Special relativity replaced the conventional notion of an absolute, universal time with the notion of a time that is local to each observer.<ref>{{Cite book |last=Hawking |first=Stephen W. |title=The illustrated a brief history of time |date=1996 |publisher=Bantam Books |isbn=978-0-553-10374-8 |edition=Updated and expanded |location=New York}}</ref>{{rp|33}} Information about distant objects can arrive no faster than the speed of light so visual observations always report events that have happened in the past. This effect makes visual descriptions of the effects of special relativity especially prone to mistakes.<ref>{{Cite journal |last1=Hughes |first1=Theo |last2=Kersting |first2=Magdalena |date=2021-03-01 |title=The invisibility of time dilation |journal=Physics Education |volume=56 |issue=2 |page=025011 |doi=10.1088/1361-6552/abce02 |bibcode=2021PhyEd..56b5011H |issn=0031-9120|doi-access=free }}</ref> 

Special relativity also has profound technical consequences.
A defining feature of special relativity is the replacement of [[Euclidean geometry]] with [[Lorentzian geometry]].<ref name="Taylor1992"/>{{rp|8}} Distances in Euclidean geometry are calculated with the [[Pythagorean theorem]] and only involved spatial coordinates. In Lorentzian geometry, 'distances' become 'intervals' and include a time coordinate with a minus sign. Unlike spatial distances, the interval between two events has the same value for all observers independent of their relative velocity.  When comparing two sets of coordinates in relative motion the [[Lorentz transformation]] replaces the [[Galilean transformation]] of Newtonian mechanics.<ref name="Taylor1992"/>{{rp|98}} 
Other effects include the relativistic corrections to the [[Doppler effect]] and the [[Thomas precession]].<ref name="Griffiths-2013" /><ref name="Jackson-1999" /> 
It also explains how electricity and magnetism are related.<ref name="Griffiths-2013" /><ref name="Jackson-1999" />

== History ==
{{Main|History of special relativity}}

The principle of relativity, forming one of the two postulates of special relativity, was described by [[Galileo Galilei]] in 1632 using a thought experiment involving observing natural phenomena on a moving ship.<ref>{{cite book| title=[[Dialogue Concerning the Two Chief World Systems]]| last=Galilei| first=Galileo| date= 1632| pages=216–217}}</ref> His conclusions were summarized as [[Galilean relativity]] and used as the basis of [[Newtonian mechanics]].<ref name=Mermin-2009/>{{rp|1}} This principle can be expressed as a [[coordinate transformation]], between two coordinate systems. [[Isaac Newton]] noted that many transformations, such as those involving rotation or acceleration, will not preserve the observation of physical phenomena.  Newton considered only those transformations involving motion with respect to an immovable absolute space, now called transformations between inertial frames.<ref name=Weinberg-1972>{{cite book |last=Weinberg |first=Steven |url=https://archive.org/details/gravitationcosmo00stev_0 |title=Gravitation and cosmology |date=1972 |publisher=John Wiley & Sons |isbn=978-0-471-92567-5 |author-link=Steven Weinberg |url-access=registration}}.</ref>{{rp|17}}

In 1864 [[James Clerk Maxwell]] presented a theory of [[Electromagnetism|electromagnetism]] which did not obey Galilean relativity. The theory specifically predicted a constant speed of light in vacuum, no matter the motion (velocity, acceleration, etc.) of the light emitter or receiver or its frequency, wavelength, direction, polarization, or phase. This, as yet untested theory, was thought at the time to be only valid in inertial frames fixed in an [[luminiferous aether|aether]]. Numerous experiments followed, attempting to measure the speed of light as Earth moved through the proposed fixed aether, culminating in the 1887 [[Michelson–Morley experiment]] which only confirmed the constant speed of light.<ref name=Weinberg-1972/>{{rp|18}}

Several fixes to the aether theory were proposed, with those of [[George Francis FitzGerald]], [[Hendrik Antoon Lorentz]], and [[Jules Henri Poincare]] all pointing in the direction of a result similar to the theory of special relativity. The final important step was taken by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".<ref name="electro" group="p">[[Albert Einstein]] (1905) "[https://web.archive.org/web/20050220050316/http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf ''Zur Elektrodynamik bewegter Körper'']", ''Annalen der Physik'' 17: 891; English translation [http://www.fourmilab.ch/etexts/einstein/specrel/www/ On the Electrodynamics of Moving Bodies] by [[George Barker Jeffery]] and Wilfrid Perrett (1923); Another English translation [[s:On the Electrodynamics of Moving Bodies|On the Electrodynamics of Moving Bodies]] by [[Megh Nad Saha]] (1920).</ref> Einstein applied the [[Lorentz transformations]] known to be compatible with [[Maxwell's equations]] for electrodynamics to the classical laws of mechanics. This changed Newton's mechanics situations involving all motions, especially velocities close to that of light<ref name=Weinberg-1972/>{{rp|18}} (known as ''{{vanchor|relativistic velocities|relativistic velocity}}''). 

Another way to describe the advance made by the special theory is to say Einstein extended the Galilean principle so that it accounted for the constant speed of light,<ref name="Taylor1992"/> 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]].<ref name="Rindler0">{{cite book |title=Essential Relativity: Special, General, and Cosmological |first1= Wolfgang |edition=illustrated |last1=Rindler |page= §1,11 p. 7 |url=https://books.google.com/books?id=0J_dwCmQThgC&pg=PT148 |isbn=978-3-540-07970-5 |publisher=Springer Science & Business Media|date=1977 }}</ref>
The theory became essentially complete in 1907, with [[Hermann Minkowski]]'s papers on spacetime.<ref name="Lanczos-1970" />

Special relativity has proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible.<ref>{{cite book |last=Goldstein |first=Herbert |title=Classical Mechanics |title-link=Classical Mechanics (Goldstein book) |publisher=Addison-Wesley Publishing Company |year=1980 |isbn=0-201-02918-9 |edition=2nd |chapter=Chapter 7: Special Relativity in Classical Mechanics |author-link=Herbert Goldstein }}</ref><ref name="Lanczos-1970">{{Cite book|last=Lanczos |first=Cornelius |title=The Variational Principles of Mechanics |publisher=Dover Publications |year=1970 |isbn=978-0-486-65067-8 |edition=4th |chapter=Chapter IX: Relativistic Mechanics |author-link=Cornelius Lanczos }}</ref> Even so, the Newtonian model remains accurate at low velocities relative to the speed of light, for example, everyday motion on Earth.

When updating his 1911 book on relativity, to include [[general relativity]] in 1920, [[Robert Daniel Carmichael]] called the earlier work the "restricted theory" as a "special case" of the new general theory; he also used the phrase "special theory of relativity".<ref>{{Cite book |last=Carmichael |first=R.D. |title=The Theory of Relativity |date=1920 |publisher=John Wiley & Sons, Incorporated. |location=United Kingdom}}</ref> In comparing to the general theory in 1923 Einstein specifically called his earlier work "the special theory of relativity", saying he meant a restriction to frames uniform motion.<ref>{{Cite book |last=Einstein |first=Albert |title=The principle of relativity: a collection of orig. memoirs on the special and general theory of relativity |date=1970 |publisher=Dover |isbn=978-0-486-60081-9 |editor-last=Lorentz |editor-first=Hendrik A. |edition=Nachdr. d. Ausg. 1923 |location=New York |chapter=The Foundation of the General Theory of Relativity |editor-last2=Einstein |editor-first2=Albert}}</ref>{{rp|111}}
Just as [[Galilean invariance|Galilean relativity]] is accepted as 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]]s, that is, at a sufficiently small scale (e.g., when [[tidal force]]s are negligible) and in conditions of [[free fall]]. But general relativity incorporates [[non-Euclidean geometry]] to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as [[Minkowski space]]. As long as the universe can be modeled as a [[pseudo-Riemannian manifold]], a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this [[curved spacetime]].

== Terminology ==
Special relativity builds upon important physics ideas. Among the most basic of these are the following:
* [[speed]] or [[velocity]], how fast an object moves relative to a reference point.<ref name="Mermin-2009">{{Cite book |last=Mermin |first=N. David |title=It's about time: understanding Einstein's relativity |date=2009 |publisher=Princeton University Press |isbn=978-0-691-14127-5 |edition=Sixth printing, and first paperback printing |location=Princeton, NJ and Oxford}}</ref>{{rp|25}}
* [[speed of light]], the maximum speed of information, independent of the speed of the source and receiver,<ref name="Taylor1992"/>{{rp|39}} 
* [[clock]], a device to measure differences in [[time]]; in relativity every object is imagined to have its own [[Proper time|proper clock]]<ref name="Taylor1992" />{{rp|3}} and moving clocks run slower.<ref name="Mermin-2009"/>{{rp|180}}
* [[Event (relativity)|event]]: something that happens at a definite place and time. For example, an explosion or a flash of light from an atom;<ref name="Taylor1992"/>{{rp|10}} a generalization of a point in geometrical space,<ref name=Mermin-2009/>{{rp|43}} 
Two observers in relative motion receive information about two events via light signals traveling at constant speed, independent of either observer's speed. Their motion during the transit time causes them to get the information at different times on their local clock.

The more technical background ideas include:
* [[spacetime]]: geometrical space and time considered together.<ref name="Taylor1992" />{{rp|18}}
* [[Spacetime#Spacetime_interval|spacetime interval]] between two events: a measure of separation between events that incorporates both the spatial distance between them and the duration of time separating them:<ref name="Taylor1992">{{cite book|url=https://archive.org/details/spacetime_physics/|title=Spacetime Physics|last1=Taylor|first1=Edwin F.|last2=Wheeler|first2=John Archibald|date=1992|publisher=W. H. Freeman|isbn=0-7167-2327-1|edition=2nd}}</ref>{{rp|9}}
<math display="block">(\text{interval})^2 = \left[ \text{event separation in time} \right]^2 - \left[ \text{event separation in space} \right]^2  </math>
* [[coordinate system]] or [[reference frame]]: a way to locate events in spacetime. Events have coordinates ''x'', ''y'', ''z'' for space and ''t'' for time. The coordinates of the event are different in a different reference frame.<ref name=":0">{{Cite book |last=Williams |first=William S. C. |title=Introducing special relativity |date=2002 |publisher=Taylor & Francis |isbn=978-0-415-27762-4 |edition=1. publ |location=London}}</ref>{{Reference page|page=67}}
* [[inertial reference frame]]: a region of a reference frame where objects at rest with respect to the frame stay as rest, or if in uniform motion, stay in motion; also called a ''free-float frame''.<ref name="Taylor1992"/>{{rp|31}}
* [[Prime (symbol)|prime]] system, frame, or coordinate. To emphasize the relationship between two systems of coordinates, both use the same x,y,z axes but one will be marked with a prime (') symbol.
* [[coordinate transformation]]: changing how an event is described from one reference frame to another.<ref name=":0" />{{Reference page|page=67}}
* [[invariance (physics)|invariance]]: when physical laws or quantities do not change in different inertial frames. The speed of light is invariant in special relativity: it is always the same.<ref name=":0" />{{Reference page|page=67}}

== Traditional "two postulates" approach to special relativity <span class="anchor" id="Postulates"></span> ==
{{Quote box
| quote = "Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results&nbsp;... How, then, could such a universal principle be found?"
| source = Albert Einstein: ''Autobiographical Notes''<ref name="autogenerated1" group=p>Einstein, Autobiographical Notes, 1949.</ref>
| align = right
| width = 40%
}}{{Main|Postulates of 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 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:<ref name=electro group=p/><ref>{{cite book |title=The Special Theory of Relativity: A Mathematical Approach |author1=Farook Rahaman |edition=2nd |publisher=Springer Nature |year=2022 |isbn=978-981-19-0497-4 |page=22 |url=https://books.google.com/books?id=GL9pEAAAQBAJ}} [https://books.google.com/books?id=GL9pEAAAQBAJ&pg=PA22 Extract of page 22]</ref>
* 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.<ref name=electro group=p/>
* The principle of invariant light speed – "...&nbsp;light is always propagated in empty space with a definite velocity [speed] ''c'' which is independent of the state of motion of the emitting body" (from the preface).<ref name=electro group=p/> 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]]<ref>{{Cite web |date=2006-12-01 |title=James Clerk Maxwell: a force for physics |url=https://physicsworld.com/james-clerk-maxwell-a-force-for-physics/ |access-date=2024-03-22 |website=Physics World |language=en-GB}}</ref> and the lack of evidence for the [[luminiferous ether]].<ref>{{Cite web |title=November 1887: Michelson and Morley report their failure to detect the luminiferous ether |url=http://www.aps.org/publications/apsnews/200711/physicshistory.cfm |access-date=2024-03-22 |website=www.aps.org |language=en}}</ref> There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.<ref>[[Michael Polanyi]] (1974) ''Personal Knowledge: Towards a Post-Critical Philosophy'', {{isbn|0-226-67288-3}}, footnote page 10–11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morley experiment had no role in the foundation of the theory." and "...&nbsp;the theory of relativity was not founded to explain its outcome at all". [https://books.google.com/books?id=0Rtu8kCpvz4C&lpg=PP1&pg=PT19]</ref><ref name="mM1905"/> 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, including the [[isotropy]] and [[homogeneity (physics)|homogeneity]] of space and the independence of measuring rods and clocks from their past history.<ref group=p>Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920</ref>

== Principle of relativity ==
{{main|Principle of relativity}}

=== Reference frames and relative motion ===

[[File:Frames of reference in relative motion.svg|thumb|Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity ''v'' only along the ''x''-axis, from the perspective of an observer stationary in the unprimed system. By the [[principle of relativity]], an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −''v''. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.]]

[[Frame of reference|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 (relativity)|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]]. 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 [[Inertial frame of reference|reference frames]] compare with each other, it is useful to work with a simplified setup with frames in a ''standard configuration''.<ref name="Collier">{{cite book|title=A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity|last1=Collier|first1=Peter|publisher=Incomprehensible Books|year=2017|isbn=978-0-9573894-6-5|edition=3rd}}</ref>{{rp|107}} With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig.&nbsp;2-1, two [[Galilean reference frame]]s (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer ''O'', and frame {{prime|''S''}} (pronounced "S&nbsp;prime" or "S dash") belongs to a second observer {{prime|''O''}}.
* The ''x'', ''y'', ''z'' axes of frame S are oriented parallel to the respective primed axes of frame {{prime|''S''}}.
* Frame {{prime|''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 {{prime|S}} are coincident when time {{nowrap|1=''t'' = 0}} for frame S and {{nowrap|1={{prime|''t''}} = 0}} for frame {{prime|''S''}}.

Since there is no absolute reference frame in relativity theory, a concept of "moving" does not 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 {{prime|''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]], dates back to [[Galileo Galilei|Galileo]], and was incorporated into Newtonian physics. But in the late 19th century the existence of [[electromagnetic radiation|electromagnetic waves]] led some physicists to suggest that the universe was filled with a substance they called "[[Luminiferous aether|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 [[preferred frame|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 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.<ref>Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift", ''Einstein's generation. The origins of the relativity revolution'', Chicago: University of Chicago Press, {{isbn|0-226-77057-5}}</ref> 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) [[Derivations of the Lorentz transformations#From group postulates|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.<ref name=Friedman group=p>{{cite book|author=Yaakov Friedman|title=Physical Applications of Homogeneous Balls|series=Progress in Mathematical Physics|volume=40|date=2004|pages=1–21|isbn=978-0-8176-3339-4}}</ref><ref name="Morin2007">{{Cite book |last=Morin |first=David |url=https://www.cambridge.org/core/product/identifier/9780511808951/type/book |title=Introduction to Classical Mechanics: With Problems and Solutions |date=2012-06-05 |publisher=Cambridge University Press |isbn=978-0-521-87622-3 |edition=1 |doi=10.1017/cbo9780511808951}}</ref>{{rp|511}}

== Lorentz transformation <span class="anchor" id="Lorentz transformation"></span> ==
{{Main|Lorentz transformation}}

=== Two- vs one- postulate approaches ===
{{main|Derivations of the Lorentz transformations}}

Einstein combined the two postulates – of relativity – and of the invariance of the speed of light, into a single postulate, the Lorentz transformation:
{{blockquote|The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events&nbsp;... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws&nbsp;...<ref name="autogenerated1" group=p/>}}

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations,<ref>For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990</ref> but Einstein stuck to his approach throughout work.<ref group=p>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.</ref>

[[Henri Poincaré]] provided the mathematical framework for relativity theory by proving that [[Lorentz transformations]] are a subset of his [[Poincaré group]] of symmetry transformations. Einstein later derived these transformations from his axioms.

While the traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations,<ref name="Miller2009">{{cite journal |last1=Miller |first1=D. J. |title=A constructive approach to the special theory of relativity |journal=American Journal of Physics |volume=78 |issue=6 |pages=633–638 |arxiv=0907.0902 |doi=10.1119/1.3298908 |year=2010 |bibcode=2010AmJPh..78..633M |s2cid=20444859 }}</ref> other treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of [[Minkowski spacetime]].<ref group=p>Das, A. (1993) ''The Special Theory of Relativity, A Mathematical Exposition'', Springer, {{isbn|0-387-94042-1}}.</ref><ref group=p>Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, {{isbn|0-582-31760-6}}.</ref> Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler<ref name="Taylor1992"/> and by Callahan.<ref name="Callahan">{{cite book |last1=Callahan |first1=James J. |title=The Geometry of Spacetime: An Introduction to Special and General Relativity |date=2011 |publisher=Springer |location=New York |isbn=978-1-4419-3142-9}}</ref>

=== Lorentz transformation and its inverse ===

Define an [[Spacetime#Basic concepts|event]] to have spacetime coordinates {{nowrap|(''t'', ''x'', ''y'', ''z'')}} in system ''S'' and {{nowrap|({{prime|''t''}}, {{prime|''x''}}, {{prime|''y''}}, {{prime|''z''}})}} in a reference frame {{prime|''S''}} moving at a velocity ''v'' along the ''x''-axis. Then the [[Lorentz transformation]] specifies that these coordinates are related in the following way:
<math display="block">\begin{align}
t' &= \gamma \ (t - vx/c^2) \\
x' &= \gamma \ (x - v t) \\
y' &= y \\
z' &= z ,
\end{align}</math>
where <math display="block">\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}</math> is the [[Lorentz factor]] and ''c'' is the [[speed of light]] in vacuum, and the velocity ''v'' of {{prime|''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]]s, that parameter being called [[rapidity]].

Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:
<math display="block">\begin{align}
t &= \gamma ( t' + v x'/c^2) \\
x &= \gamma ( x' + v t') \\
y &= y' \\
z &= z'.
\end{align}</math>

This shows that the unprimed frame is moving with the velocity −''v'', as measured in the primed frame.<ref>P. G. Bergmann (1976) ''Introduction to the Theory of Relativity'', Dover edition, Chapter IV, page 36 {{isbn|0-486-63282-2}}.</ref>

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]] for details.

A quantity that is 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 {{nowrap|(''x''<sub>1</sub>, ''t''<sub>1</sub>)}} and {{nowrap|({{prime|''x''}}<sub>1</sub>, {{prime|''t''}}<sub>1</sub>)}}, another event has coordinates {{nowrap|(''x''<sub>2</sub>, ''t''<sub>2</sub>)}} and {{nowrap|({{prime|''x''}}<sub>2</sub>, {{prime|''t''}}<sub>2</sub>)}}, and the differences are defined as
* {{EquationRef|1|Eq. 1:}} &nbsp;&nbsp; <math>\Delta x' = x'_2-x'_1 \ , \ \Delta t' = t'_2-t'_1 \ .</math>
* {{EquationRef|2|Eq. 2:}} &nbsp;&nbsp; <math>\Delta x = x_2-x_1 \ , \ \ \Delta t = t_2-t_1 \ .</math>
we get
* {{EquationRef|3|Eq. 3:}} &nbsp;&nbsp; <math>\Delta x' = \gamma \ (\Delta x - v \,\Delta t) \ ,\ \ </math> <math>\Delta t' = \gamma \ \left(\Delta t - v \ \Delta x / c^{2} \right) \ . </math>
* {{EquationRef|4|Eq. 4:}} &nbsp;&nbsp; <math>\Delta x = \gamma \ (\Delta x' + v \,\Delta t') \ , \ </math> <math>\Delta t = \gamma \ \left(\Delta t' + v \ \Delta x' / c^{2} \right) \ . </math>

If we take differentials instead of taking differences, we get
* {{EquationRef|5|Eq. 5:}} &nbsp;&nbsp; <math>dx' = \gamma \ (dx - v\,dt) \ ,\ \ </math> <math>dt' = \gamma \ \left( dt - v \ dx / c^{2} \right) \ . </math>
* {{EquationRef|6|Eq. 6:}} &nbsp;&nbsp; <math>dx = \gamma \ (dx' + v\,dt') \ , \ </math> <math>dt = \gamma \ \left(dt' + v \ dx' / c^{2} \right) \ . </math>

=== Graphical representation of the Lorentz transformation ===
{{multiple image|perrow = 2|total_width=400
| image1 = Spacetime diagram development A.svg |width1=535|height1=535
| image2 = Spacetime diagram development B.svg |width2=535|height2=535
| image3 = Spacetime diagram development C.svg |width3=535|height3=535
| image4 = Spacetime diagram development D.svg|width4=535|height4=535
| footer = Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.}}

Spacetime diagrams (also called [[Minkowski diagram]]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.<ref name=Morin2007/>{{rp|536}}
To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig.&nbsp;2-1.<ref name="Mermin1968">{{cite book |last1=Mermin |first1=N. David |title=Space and Time in Special Relativity |url=https://archive.org/details/spacetimeinspeci0000merm |url-access=registration |date=1968 |publisher=McGraw-Hill |isbn=978-0-88133-420-3}}</ref>{{rp|155–199}}

'''Fig.&nbsp;3-1a'''. Draw the <math>x</math> and <math>t</math> axes of frame S. The <math>x</math> axis is horizontal and the <math>t</math> (actually <math>ct</math>) axis is vertical, which is the opposite of the usual convention in kinematics. The <math>ct</math> axis is scaled by a factor of <math>c</math> 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 <math>t = 0.</math> The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, <math>\text{A}</math> and <math>\text{B},</math> have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

'''Fig.&nbsp;3-1b'''. Draw the <math>x'</math> and <math>ct'</math> axes of frame S'. The <math>ct'</math> axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, <math>v = c/2.</math> Both the <math>ct'</math> and <math>x'</math> axes are tilted from the unprimed axes by an angle <math>\alpha = \tan^{-1}(\beta),</math> where <math>\beta = v/c.</math> The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that <math>t=0</math> when <math>t'=0.</math>

'''Fig.&nbsp;3-1c'''. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that <math>(x', ct')</math> coordinates of <math>(0, 1)</math> in the primed coordinate system transform to <math> (\beta \gamma, \gamma)</math> in the unprimed coordinate system. Likewise, <math>(x', ct')</math> coordinates of <math>(1, 0)</math> in the primed coordinate system transform to <math>(\gamma, \beta \gamma)</math> in the unprimed system. Draw gridlines parallel with the <math>ct'</math> axis through points <math>(k \gamma, k \beta \gamma)</math> as measured in the unprimed frame, where <math> k </math> is an integer. Likewise, draw gridlines parallel with the <math>x'</math> axis through <math>(k \beta \gamma, k \gamma)</math> as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between <math>ct'</math> units equals <math display=inline>\sqrt{(1 + \beta ^2)/(1 - \beta ^2)}</math> times the spacing between <math>ct</math> units, as measured in frame S. This ratio is always greater than 1, and  approaches infinity as <math>\beta \to 1.</math>

'''Fig.&nbsp;3-1d'''. Since the speed of light is an invariant, the ''worldlines'' of two photons passing through the origin at time <math>t' = 0</math> still plot as 45° diagonal lines. The primed coordinates of <math>\text{A}</math> and <math>\text{B}</math> 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]], but the frames are actually equivalent.

== Consequences derived from the Lorentz transformation ==
{{See also|Twin paradox|Relativistic mechanics}}

The consequences of special relativity can be derived from the [[Lorentz transformation]] equations.<ref>{{cite book |title=Introduction to special relativity |author=Robert Resnick |publisher=Wiley |date=1968|pages=62–63 |isbn=978-0-471-71724-9 |url=https://books.google.com/books?id=fsIRAQAAIAAJ}}</ref> 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]].

=== Invariant interval ===
In Galilean relativity, the spatial separation, ({{tmath|1= \Delta r }}), and the temporal separation, ({{tmath|1= \Delta t }}), between two events 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 {{tmath|1= \Delta s^2 }}:
<math display="block"> \Delta s^2 \; \overset\text{def}{=} \; c^2 \Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) </math>
In considering the physical significance of {{tmath|1= \Delta s^2 }}, there are three cases:<ref name=Morin2007/>{{rp|533}}<ref name="Taylor1992"/>{{rp|25–39}}
* '''Δs<sup>2</sup> > 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 {{tmath|1= \vert \Delta x / \Delta t \vert < c }}, and given the Lorentz transformation {{tmath|1= \Delta x' = \gamma \ (\Delta x - v \ \Delta t) }}, it is evident that there exists a <math>v</math> less than <math>c</math> for which <math>\Delta x' = 0</math> (in particular, {{tmath|1= 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, {{tmath|1= \Delta s / c }}, is called the ''proper time''.
* '''Δs<sup>2</sup> < 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 {{tmath|1= \vert \Delta x / \Delta t \vert > c }}, and given the Lorentz transformation {{tmath|1= \Delta t' = \gamma \ (\Delta t - v \Delta x / c^2) }}, there exists a <math>v</math> less than <math>c</math> for which <math>\Delta t' = 0</math> (in particular, {{tmath|1= 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, {{tmath|1= \textstyle \sqrt { - \Delta s^2 } }}, is called the ''proper distance'', or ''proper length''. For values of <math>v</math> greater than and less than {{tmath|1= c^2 \Delta t / \Delta x }}, the sign of <math>\Delta t'</math> changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that <math>v</math> could be greater than <math> c^2 \Delta t / \Delta x</math> would be if {{tmath|1= v > c }}.
* '''Δs<sup>2</sup> = 0:''' In this case, the two events are said to be ''lightlike'' separated. This implies that {{tmath|1= \vert \Delta x / \Delta t \vert = c }}, and this relationship is frame independent due to the invariance of {{tmath|1= s^2 }}. From this, we observe that the speed of light is <math>c</math> 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.

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

The form of {{tmath|1= \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 Δ''s''<sup>2</sup> under standard Lorentz transformation is analogous to the invariance of squared distances Δ''r''<sup>2</sup> under rotations in Euclidean space.{{cn|date=July 2025}} Although space and time have an equal ''footing'' in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, [[Minkowski space]] differs from [[four-dimensional Euclidean space]]. 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 boost]]s, in the x-direction) with all other [[translation (geometry)|translations]], [[Reflection (mathematics)|reflections]], and [[rotation (mathematics)|rotations]] between any Cartesian inertial frame.<ref name="Rindler1977">{{cite book |last1=Rindler |first1=Wolfgang |title=Essential Relativity |date=1977 | publisher=Springer-Verlag |location=New York |isbn=978-0-387-10090-6 |edition=2nd}}</ref>{{rp|33–34}}

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:
<math display="block">\Delta s^2 \, = \, c^2 \Delta t^2 - \Delta x^2</math>

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:<ref name=Morin2007/>
<math display="block">\begin{align}
 c^2 \Delta t^2 - \Delta x^2 &= c^2 \gamma ^2 \left(\Delta t' + \dfrac{v \Delta x'}{c^2} \right)^2 - \gamma ^2 \ (\Delta x' + v \Delta t')^2 \\
&= \gamma ^2 \left( c^2 \Delta t' ^ {\, 2} + 2 v \Delta x' \Delta t' + \dfrac{v^2 \Delta x' ^ {\, 2}}{c^2} \right) - \gamma ^2 \ (\Delta x' ^ {\, 2} + 2 v \Delta x' \Delta t' + v^2 \Delta t' ^ {\, 2}) \\
&= \gamma ^2 c^2 \Delta t' ^ {\, 2} - \gamma ^2 v^2 \Delta t' ^{\, 2} - \gamma ^2 \Delta x' ^ {\, 2} + \gamma ^2 \dfrac{v^2 \Delta x' ^ {\, 2}}{c^2} \\
&= \gamma ^2 c^2 \Delta t' ^ {\, 2} \left( 1 - \dfrac{v^2}{c^2} \right) - \gamma ^2 \Delta x' ^{\, 2} \left( 1 - \dfrac{v^2}{c^2} \right) \\
&= c^2 \Delta t' ^{\, 2} - \Delta x' ^{\, 2}
\end{align}</math>

The value of <math>\Delta s^2</math> is hence independent of the frame in which it is measured.

=== Relativity of simultaneity ===
{{See also|Relativity of simultaneity|Ladder paradox}}
[[File:Relativity of Simultaneity Animation.gif|thumb|Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer '''O'''. In a reference frame moving at ''v'' = 0.3''c'', as measured by '''O''', the events occur in the order C, B, A. In a reference frame moving at {{nowrap|1=''v'' = −0.5''c''}} with respect to '''O''', the events occur in the order A, B, C. The white lines, the ''lines of simultaneity'', move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the [[light cone]] with respect to the origin of all considered frames.]]

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 {{EquationNote|3|Equation&nbsp;3}} (the forward Lorentz transformation in terms of coordinate differences)
<math display="block">\Delta t' = \gamma \left(\Delta t - \frac{v \,\Delta x}{c^{2}} \right)</math>

It is clear that the two events that are simultaneous in frame ''S'' (satisfying {{nowrap|1=Δ''t'' = 0}}), are not necessarily simultaneous in another inertial frame {{prime|''S''}} (satisfying {{nowrap|1=Δ{{prime|''t''}} = 0}}). Only if these events are additionally co-local in frame ''S'' (satisfying {{nowrap|1=Δ''x'' = 0}}), will they be simultaneous in another frame {{prime|''S''}}.

The [[Sagnac effect]] can be considered a manifestation of the relativity of simultaneity for local inertial frames comoving with a rotating Earth.<ref name="Ashby2003">{{cite journal |last1=Ashby |first1=Neil |title=Relativity in the Global Positioning System |journal=Living Reviews in Relativity |volume=6 |issue=1 |article-number=1 |doi=10.12942/lrr-2003-1 |pmid=28163638 |pmc=5253894 |year=2003 |doi-access=free |bibcode=2003LRR.....6....1A }}</ref> Instruments based on the Sagnac effect for their operation, such as [[ring laser gyroscope]]s and [[fiber optic gyroscope]]s, are capable of extreme levels of sensitivity.<ref name="Lin1979" group="p">{{cite journal |last1=Lin |first1=Shih-Chun |last2=Giallorenzi |first2=Thomas G. |title=Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer |journal=Applied Optics |date=1979 |volume=18 |issue=6 |pages=915–931 |doi=10.1364/AO.18.000915|pmid=20208844 |bibcode=1979ApOpt..18..915L |s2cid=5343180 }}</ref>

=== Time dilation ===
{{See also|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.

Suppose a [[clock]] is at rest in the unprimed system ''S''. The location of the clock on two different ticks is then characterized by {{nowrap|1=Δ''x'' = 0}}. To find the relation between the times between these ticks as measured in both systems, {{EquationNote|3|Equation&nbsp;3}} can be used to find:
: <math>\Delta t' = \gamma\, \Delta t </math>{{pad|4}}for events satisfying{{pad|4}}<math>\Delta x = 0 \ .</math>

This shows that the time (Δ{{prime|''t''}}) between the two ticks as seen in the frame in which the clock is moving ({{prime|''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 [[muon]]s 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.<ref>{{cite book|author1=Daniel Kleppner|author2=David Kolenkow|name-list-style=amp|title=An Introduction to Mechanics|date=1973|pages=[https://archive.org/details/introductiontome00dani/page/468 468–70]|publisher=McGraw-Hill |isbn=978-0-07-035048-9|url=https://archive.org/details/introductiontome00dani/page/468}}</ref>

[[File:Observer_in_special_relativity.svg|thumb|Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame]]
Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.<ref name="French_1968"/>{{rp|149–152}}

The measurements that we would get if we actually ''looked'' at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the [[Doppler effect]]. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.<ref name="French_1968">{{cite book |last1=French |first1=A. P. |title=Special Relativity |date=1968 |publisher=W. W. Norton & Company |location=New York |isbn=0-393-09793-5}}</ref>{{rp|149–152}}

==== Langevin's light-clock ====
{{anchor|Langevin's Light-Clock}}
[[File:Langevin Light Clock.gif|thumb|Figure 4–3. Thought experiment using a light-clock to explain time dilation]]
[[Paul Langevin]], an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909<ref name="Lewis_Tolman_1909">{{cite journal |last1=Lewis |first1=Gilbert Newton |author-link1=Gilbert N. Lewis|last2=Tolman |first2=Richard Chase| author-link2=Richard Chase Tolman| title=The Principle of Relativity, and Non-Newtonian Mechanics |journal=Proceedings of the American Academy of Arts and Sciences |date=1909 |volume=44 |issue=25 |pages=709–726 |doi=10.2307/20022495 |jstor=20022495 |url=https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics |access-date=22 August 2023}}</ref>), which he used to perform a novel derivation of the Lorentz transformation.<ref name="Cuvaj_1971">{{cite journal |last1=Cuvaj |first1=Camillo |title=Paul Langeyin and the Theory of Relativity |journal=Japanese Studies in the History of Science |date=1971 |volume=10 |pages=113–142 |url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf |access-date=12 June 2023}}</ref>

A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a  light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.<ref>{{Cite book |last1=Cassidy |first1=David C. |url=https://books.google.com/books?id=rpQo7f9F1xUC&pg=PA422 |title=Understanding Physics |last2=Holton |first2=Gerald James |last3=Rutherford |first3=Floyd James |publisher=[[Springer-Verlag]] |year=2002 |isbn=978-0-387-98756-9 |page=422}}</ref><ref>{{Cite book |last=Cutner |first=Mark Leslie |url=https://books.google.com/books?id=2QVmiMW0O0MC&pg=PA128 |title=Astronomy, A Physical Perspective |publisher=[[Cambridge University Press]] |year=2003 |isbn=978-0-521-82196-4 |page=128}}</ref><ref>{{Cite book |last1=Ellis |first1=George F. R. |url=https://books.google.com/books?id=Hos31wty5WIC&pg=PA28 |title=Flat and Curved Space-times |last2=Williams |first2=Ruth M. |publisher=[[Oxford University Press]] |year=2000 |isbn=978-0-19-850657-7 |edition=2n |pages=28–29}}</ref><ref name="Feynman_Lectures_1">{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The feynman lectures on physics; vol I: The new millennium edition |date=2011 |publisher=Basic Books |isbn=978-0-465-02414-8 |page=15-5 |url=https://www.feynmanlectures.caltech.edu/I_15.html |access-date=12 June 2023}}</ref> (Langevin himself made use of a light-clock oriented parallel to its line of motion.<ref name="Cuvaj_1971"/>)

Consider the scenario illustrated in {{nowrap|Fig. 4-3A.}} Observer A holds a light-clock of length <math>L</math> as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds {{tmath|1= t_\text{A} = 2 L/c }}. A time interval measured using a single clock that is motionless in a particular reference frame is called a ''[[proper time interval]]''.<ref name="Halliday_1988">{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |title=Fundamental Physics: Extended Third Edition |date=1988 |publisher=John Wiley & sons |location=New York |isbn=0-471-81995-6 |pages=958–959}}</ref>

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of {{tmath|1= v }}. Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same <math>c</math> that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as <math display=inline>\pm \sqrt{c^2 - v^2},</math> so that the total round-trip time of the pulses is <math display=inline>t_\text{B} = 2L \big/ \sqrt{ c^2 - v^2 } = {}</math>{{tmath|1= \textstyle t_\text{A} \big/ \sqrt {1 - v^2 / c^2} }}. Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore ''not'' a proper time interval because he did not measure it with a single resting clock.<ref name="Halliday_1988"/>

==== Reciprocal time dilation ====
In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of <math>v</math> to the left, in which case observer A would perceive B's clock as running slower than her local clock.

There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with ''both'' A's and B's measurement of time.<ref>{{Cite book |last=Adams |first=Steve |url=https://books.google.com/books?id=1RV0AysEN4oC&pg=PA54 |title=Relativity: An introduction to space-time physics |publisher=[[CRC Press]] |year=1997 |isbn=978-0-7484-0621-0 |page=54}}</ref>

=== Twin paradox ===
{{See also|Twin paradox}}
The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called [[twin paradox]], articulated in its present form by Langevin in 1911.<ref name="Langevin_1911">{{cite journal |last1=Langevin |first1=Paul |title=L'Évolution de l'espace et du temps |journal=Scientia |date=1911 |volume=10 |pages=31–54 |url=https://en.wikisource.org/wiki/Translation:The_Evolution_of_Space_and_Time |access-date=20 June 2023}}</ref> Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older.

This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in order for the two observers to perform side-by-side comparisons of their elapsed proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.<ref name='Debs_Redhead'>{{cite journal |author1=Debs, Talal A. |author2=Redhead, Michael L.G. |title=The twin "paradox" and the conventionality of simultaneity |journal=American Journal of Physics |volume=64 |issue=4 |year=1996 |pages=384–392 |doi=10.1119/1.18252 |bibcode=1996AmJPh..64..384D}}</ref>

[[File:Twin paradox Doppler analysis.svg|thumb|Figure 4-4. Doppler analysis of twin paradox]]
Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the [[Twin paradox]] article. We will examine in the following one such solution to the paradox.

Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences. {{nowrap|Fig 4-4}} illustrates a scenario where the traveling twin flies at {{nowrap|0.6 c}} to and from a star {{nowrap|3 ly}} distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of {{tmath|1= \textstyle f' = f \sqrt{(1-\beta)/(1+\beta)} }}. Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of {{tmath|1= \textstyle f' ' = f \sqrt{(1+\beta)/(1-\beta)} }}. The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.<ref name="French_1968"/>{{rp|152–159}}

{| class="wikitable"
|-
! Item!! Measured by the{{br}}stay-at-home !! Fig 4-4 !! Measured by{{br}}the traveler !! Fig 4-4
|-
| Total time of trip
| <math>T = \frac{2L}{v}</math>
| {{nowrap|10 yr}}
| <math>T' =  \frac{2L}{\gamma v} </math>
| {{nowrap|8 yr}}
|-
| Total number of pulses sent
| <math>fT = \frac{2fL}{v}</math>
| 10
| <math>fT' = \frac{2fL}{\gamma v}</math>
| 8
|-
| Time when traveler's turnaround is '''detected'''
| <math>t_1 = \frac{L}{v} + \frac{L}{c}</math>
| {{nowrap|8 yr}}
| <math>t_1' = \frac{L}{\gamma v}</math>
| {{nowrap|4 yr}}
|-
| Number of pulses received at initial <math>f'</math> rate
| <math>f't_1</math> <math>= \frac{fL}{v}(1 + \beta)\left(\frac{1-\beta}{1+\beta}\right)^{1/2}</math>{{br}}<math>= \frac{fL}{v}(1 - \beta ^2)^{1/2}</math>
| 4
| <math>f't_1'</math> <math>= \frac{fL}{v}(1 - \beta^2)^{1/2}\left(\frac{1-\beta}{1+\beta}\right)^{1/2}</math>{{br}}<math>= \frac{fL}{v}(1 - \beta )</math>
| 2
|-
| Time for remainder of trip
| <math>t_2 = \frac{L}{v} - \frac{L}{c}</math>
| {{nowrap|2 yr}}
| <math>t_2' = \frac{L}{\gamma v}</math>
| {{nowrap|4 yr}}
|-
| Number of signals received at final <math>f''</math> rate
| <math>f''t_2</math> <math>= \frac{fL}{v}(1 - \beta)\left( \frac{1 + \beta}{1 - \beta} \right)^{1/2}</math> <math>= \frac{fL}{v}(1 - \beta ^2)^{1/2}</math>
| 4
| <math>f''t_2'</math> <math>= \frac{fL}{v}(1 - \beta^2)^{1/2}\left( \frac{1 + \beta}{1 - \beta} \right)^{1/2}</math> <math>= \frac{fL}{v}(1 + \beta)</math>
| 8
|-
| Total number of received pulses
| <math>\frac{2fL}{v}(1 - \beta ^2)^{1/2}</math> <math>= \frac{2fL}{\gamma v}</math>
| 8
| <math>\frac{2fL}{v}</math>
| 10
|-
| Twin's calculation as to how much the '''''other''''' twin should have aged
| <math>T' = \frac{2L}{\gamma v}</math>
| {{nowrap|8 yr}}
| <math>T = \frac{2L}{v}</math>
| {{nowrap|10 yr}}
|}

=== Length contraction ===
{{See also|Lorentz contraction}}
The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the [[ladder paradox]] involves a long ladder traveling near the speed of light and being contained within a smaller garage).

Similarly, suppose a [[measuring rod]] 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 {{prime|''S''}}, in which the rod is moving, the distances ''{{prime|x}}'' to the end points of the rod must be measured simultaneously in that system {{prime|''S''}}. In other words, the measurement is characterized by {{nowrap|1=Δ{{prime|''t''}} = 0}}, which can be combined with {{EquationNote|4|Equation&nbsp;4}} to find the relation between the lengths Δ''x'' and Δ{{prime|''x''}}:
: <math>\Delta x' = \frac{\Delta x}{\gamma} </math>{{pad|4}}{{pad|4}}for events satisfying{{pad|4}}<math>\Delta t' = 0 \ .</math>
This shows that the length (Δ{{prime|''x''}}) of the rod as measured in the frame in which it is moving ({{prime|''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 ===
{{See also|Velocity-addition formula}}

Consider two frames ''S'' and {{prime|''S''}} in standard configuration. A particle in ''S'' moves in the x direction with velocity vector {{tmath|1= \mathbf{u} }}. What is its velocity <math>\mathbf{u'}</math> in frame {{prime|''S''}}?

We can write
{{NumBlk2||<math display="block"> \mathbf{|u|} = u = dx / dt \, . </math>|7}}
{{NumBlk2||<math display="block"> \mathbf{|u'|} = u' = dx' / dt' \, . </math>|8}}

Substituting expressions for <math>dx'</math> and <math>dt'</math> from {{EquationNote|5|Equation&nbsp;5}} into {{EquationNote|8|Equation&nbsp;8}}, followed by straightforward mathematical manipulations and back-substitution from {{EquationNote|7|Equation&nbsp;7}} yields the Lorentz transformation of the speed <math>u</math> to {{tmath|1= u' }}:
{{NumBlk2||<math display="block">u' = \frac{dx'}{dt'}=\frac{\gamma(dx-v\,dt)}{\gamma \left(dt-\dfrac{v\,dx}{c^2} \right)} = \frac{\dfrac{dx}{dt}-v}{1 - \dfrac{v}{c^2} \, \dfrac{dx}{dt} } =\frac{u-v}{1- \dfrac{uv}{c^2}}. </math>|9}}

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing <math> v </math> with {{tmath|1= -v }}.
{{NumBlk2||<math display="block">u=\frac{u'+v}{1+ u'v / c^2}.</math>|10}}

For <math>\mathbf{u}</math> not aligned along the x-axis, we write:<ref name="Rindler0"/>{{rp|47–49}}
{{NumBlk2||<math display="block"> \mathbf{u} = (u_1, \ u_2,\ u_3 ) = ( dx / dt, \ dy/dt, \ dz/dt) \ . </math>|11}}
{{NumBlk2||<math display="block"> \mathbf{u'} = (u_1', \ u_2', \ u_3') = ( dx' / dt', \ dy'/dt', \ dz'/dt') \ . </math>|12}}

The forward and inverse transformations for this case are:
{{NumBlk2||<math display="block">u_1'=\frac{u_1 -v}{1-u_1 v / c^2 } \ , \qquad u_2'=\frac{u_2}{\gamma \left( 1-u_1 v / c^2 \right) } \ , \qquad u_3'=\frac{u_3}{\gamma \left( 1- u_1 v / c^2 \right) } \ . </math>|13}}
{{NumBlk2||<math display="block">u_1=\frac{u_1' +v}{1+ u_1' v / c^2 } \ , \qquad u_2=\frac{u_2'}{ \gamma \left( 1+ u_1' v / c^2 \right) } \ , \qquad u_3=\frac{u_3'}{\gamma \left( 1+ u_1' v / c^2 \right)} \ . </math>|14}}

{{EquationNote|10|Equation&nbsp;10}} and {{EquationNote|14|Equation&nbsp;14}} can be interpreted as giving the ''resultant'' <math> \mathbf{u} </math> of the two velocities <math> \mathbf{v} </math> and {{tmath|1= \mathbf{u'} }}, and they replace the formula {{tmath|1= \mathbf{u = u' + v} }}. 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 {{prime|''S''}} being aligned with each other (although not necessarily in standard configuration).<ref name="Rindler0"/>{{rp|47–49}}

We note the following points:
* If an object (e.g., a [[photon]]) were moving at the speed of light in one frame {{nowrap|1=(i.e., ''u'' = ±''c''}} {{nowrap|1=or {{prime|''u''}} = ±''c'')}}, then it would also be moving at the speed of light in any other frame, moving at {{nowrap|{{abs|''v''}} < ''c''}}.
* The resultant speed of two velocities with magnitude less than ''c'' is always a velocity with magnitude less than ''c''.
* If both {{abs|''u''}} and {{abs|''v''}} (and then also {{abs|{{prime|''u''}}}} and {{abs|{{prime|''v''}}}}) are small with respect to the speed of light (that is, e.g., {{nowrap|{{abs|{{sfrac|''u''|''c''}}}} ≪ {{math|1}})}}, 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 (mathematics)|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 ===
{{See also|Thomas rotation}}
{{multiple image
 | direction = vertical
 | width = 220
 | footer = Figure 4-5. Thomas–Wigner rotation
 | image1 = Thomas-Wigner Rotation 1.svg
 | image2 = Thomas-Wigner Rotation 2.svg
}}

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.&nbsp;4-5a, a rod of length <math>L</math> in its rest frame (i.e., having a [[proper length]] of {{tmath|1= L }}) rises vertically along the y-axis in the ground frame.

In Fig.&nbsp;4-5b, the same rod is observed from the frame of a rocket moving at speed <math>v</math> 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 [[#Measurement_versus_visual_appearance|''see'']]) the clock at the right end of the rod as being advanced in time by {{tmath|1= Lv/c^2 }}, and the rod is correspondingly observed as tilted.<ref name="Taylor1992"/>{{rp|98–99}}

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–orbit interaction|spin of moving particles]], where [[Thomas precession]] is a relativistic correction that applies to the [[Spin (physics)|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.<ref name="Taylor1992"/>{{rp|169–174}}

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".<ref name="Shaw" group=p>{{cite journal |last1=Shaw |first1=R. |title=Length Contraction Paradox |journal=American Journal of Physics |date=1962 |volume=30 |issue=1 |page=72 |doi=10.1119/1.1941907 |bibcode=1962AmJPh..30...72S |s2cid=119855914 }}</ref><ref name="Taylor1992"/>{{rp|98–99}}

=== Causality and prohibition of motion faster than light ===
{{See also|Causality (physics)|Tachyonic antitelephone}}
[[File:Simple light cone diagram.svg|thumb|Figure 4–6. [[Light cone]]]]
In Fig.&nbsp;4-6, the time interval between the events A (the "cause") and B (the "effect") is 'timelike'; 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 'spacelike'; 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. But no frames are 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, 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).<ref>{{cite book | first = Richard C.|last = Tolman|title =The Theory of the Relativity of Motion |location=Berkeley|publisher = University of California Press|date = 1917|page = 54|url = https://books.google.com/books?id=8yodAAAAMAAJ&q=54}}</ref><ref group=p>{{cite journal|author1=G. A. Benford |author2=D. L. Book |author3=W. A. Newcomb |name-list-style=amp |doi=10.1103/PhysRevD.2.263|title=The Tachyonic Antitelephone|date=1970|journal=Physical Review D|volume=2|issue=2|pages=263–265|bibcode = 1970PhRvD...2..263B |s2cid=121124132 }}</ref> A variety of causal paradoxes could then be constructed.

{{multiple image
 | width = 160
 | image_gap = 4
 | image1 = Causality violation 1.svg
 | image2 = Causality violation 2.svg
 | alt1 = Causality violation: Beginning of scenario resulting from use of a fictitious instantaneous communicator   
 | alt2 = Causality violation: B receives the message before having sent it. 
 | footer_align = center
 | footer = Figure 4-7. Causality violation by the use of fictitious<br/>"instantaneous communicators"
}}
Consider the spacetime diagrams in Fig.&nbsp;4-7. 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 (''{{prime|ct}}''), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.
# Fig.&nbsp;4-7a. 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 <math>-x'</math> 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.&nbsp;4-7b. 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 <math>+x</math> axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, in the primed frames of C and D, B will receive the message before it was sent out, a violation of causality.<ref name="Takeuchi">{{cite web |last1=Takeuchi |first1=Tatsu |title=Special Relativity Lecture Notes – Section 10 |url=https://www1.phys.vt.edu/~takeuchi/relativity/notes/section10.html |publisher=Virginia Tech |access-date=31 October 2018}}</ref>

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 <math>x'</math> axis (and the signal from A to B slightly steeper than the <math>x</math> 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 <math>ct'</math> and <math>x'</math> 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.<ref name="Morin2017">{{cite book|last1=Morin|first1=David|title=Special Relativity for the Enthusiastic Beginner|date=2017|publisher=CreateSpace Independent Publishing Platform|pages=90–92|isbn=978-1-5423-2351-2}}</ref>

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]] 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.<ref>{{cite web |last1=Gibbs |first1=Philip |title=Is Faster-Than-Light Travel or Communication Possible? |url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html |website=Physics FAQ |publisher=Department of Mathematics, University of California, Riverside |access-date=31 October 2018}}</ref> 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).<ref>{{cite book |title=Applications of Electrodynamics in Theoretical Physics and Astrophysics |edition=illustrated |first1=David |last1=Ginsburg |publisher=CRC Press |year=1989 |isbn=978-2-88124-719-4 |page=206 |url=https://books.google.com/books?id=Lh0tjaBNzg0C|bibcode=1989aetp.book.....G }} [https://books.google.com/books?id=Lh0tjaBNzg0C&pg=PA206 Extract of page 206]</ref><ref>{{cite book |title=Four Decades of Scientific Explanation |author=Wesley C. Salmon |publisher=University of Pittsburgh |date=2006 |isbn=978-0-8229-5926-7 |page=107 |url=https://books.google.com/books?id=FHqOXCd06e8C}}, [https://books.google.com/books?id=FHqOXCd06e8C&pg=PA107 Section 3.7 page 107]</ref><!-- a pair of diagrams, with x–t and x'–t' coordinates would help here -->

== Optical effects ==
=== Dragging effects ===
{{main|Fizeau experiment}}
[[File:Fizeau experiment schematic.svg|thumb|Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.]]
In 1850, [[Hippolyte Fizeau]] and [[Léon Foucault]] independently established that light travels more slowly in water than in air, thus validating a prediction of [[Augustin-Jean Fresnel|Fresnel's]] [[wave theory of light]] and invalidating the corresponding prediction of Newton's [[Corpuscular theory of light|corpuscular theory]].<ref name=Lauginie2004>{{cite journal |last1=Lauginie |first1=P. |title=Measuring Speed of Light: Why? Speed of what? |journal=Proceedings of the Fifth International Conference for History of Science in Science Education |date=2004 |url=http://sci-ed.org/documents/Lauginie-M.pdf|access-date=3 July 2015 |archive-url=https://web.archive.org/web/20150704043700/http://sci-ed.org/documents/Lauginie-M.pdf |archive-date=4 July 2015}}</ref> 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.&nbsp;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 <math>u' = c/n</math> is the speed of light in still water, and <math>v</math> is the speed of the water, and <math> u_{\pm} </math> is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then
<math display="block">u_{\pm} =\frac{c}{n} \pm v\left(1-\frac{1}{n^2}\right) \ . </math>

Fizeau's results, although consistent with Fresnel's earlier hypothesis of [[Aether drag hypothesis|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 <math>n</math> depends on wavelength, ''the aether must be capable of sustaining different motions at the same time''.{{refn|group=note|The refractive index dependence of the presumed partial aether-drag was eventually confirmed by [[Pieter Zeeman]] in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to [[Amsterdam]]'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).<ref name=zee1 group=p>{{Cite journal|author=Zeeman, Pieter |title=Fresnel's coefficient for light of different colours. (First part) |journal=Proc. Kon. Acad. Van Weten.|volume=17|year=1914|pages=445–451|url=https://archive.org/details/p1proceedingsofs17akad|bibcode=1914KNAB...17..445Z}}</ref><ref name=zee2 group=p>{{Cite journal|author=Zeeman, Pieter |title=Fresnel's coefficient for light of different colours. (Second part)|journal=Proc. Kon. Acad. Van Weten.|volume=18|year=1915 |pages=398–408 |url=https://archive.org/details/proceedingsofsec181koni|bibcode=1915KNAB...18..398Z}}</ref>}} A variety of theoretical explanations were proposed to explain [[Fizeau experiment#Fresnel drag coefficient|Fresnel's dragging coefficient]], that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.<ref name=Stachel2005>{{cite book |last=Stachel |first=J. |title=The universe of general relativity |year=2005 |publisher=Birkhäuser |location=Boston |isbn=978-0-8176-4380-5 |pages=1–13 |chapter-url=https://books.google.com/books?id=-KlBhDwUKF8C&pg=PA1 |editor=Kox, A.J. |editor2=Eisenstaedt, J |access-date=17 April 2012 |chapter=Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies}}</ref>

From the point of view of special relativity, Fizeau's result is nothing but an approximation to {{EquationNote|10|Equation&nbsp;10}}, the relativistic formula for composition of velocities.<ref name=Rindler1977/>
: <math>u_{\pm} = \frac{u' \pm v}{ 1 \pm u'v/c^2 } =</math> <math> \frac {c/n \pm v}{ 1 \pm v/cn } \approx</math> <math> c \left( \frac{1}{n} \pm \frac{v}{c} \right) \left( 1 \mp \frac{v}{cn} \right) \approx </math> <math> \frac{c}{n} \pm v \left( 1 - \frac{1}{n^2} \right) </math>

=== Relativistic aberration of light ===
{{main|Aberration of light|Light-time correction}}

[[File:Stellar aberration illustration.svg|thumb|Figure 5–2. Illustration of stellar aberration]]
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]]. 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.<ref name=Mould>{{cite book |title=Basic Relativity |page=8 |url=https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA8 |isbn=978-0-387-95210-9 |date=2001 |publisher=Springer |author=Richard A. Mould |edition=2nd}}</ref> (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.<ref name="Seidelmann">{{cite book |editor1-last=Seidelmann |editor1-first=P. Kenneth |title=Explanatory Supplement to the Astronomical Almanac |date=1992 |publisher=University Science Books |location=ill Valley, Calif. |isbn=978-0-935702-68-2 |page=393 |url=https://archive.org/details/131123ExplanatorySupplementAstronomicalAlmanac/page/n209}}</ref>

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, [[François Arago|Arago]] used this expected phenomenon in a failed attempt to measure the speed of light,<ref name="Ferraro">{{cite journal |last1=Ferraro |first1=Rafael |last2=Sforza |first2=Daniel M. |title=European Physical Society logo Arago (1810): the first experimental result against the ether |journal=European Journal of Physics |volume=26 |pages=195–204 |doi=10.1088/0143-0807/26/1/020 |arxiv=physics/0412055 |year=2005 |issue=1 |bibcode=2005EJPh...26..195F |s2cid=119528074 }}</ref> and in 1870, [[George Airy]] 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.<ref name="Dolan">{{cite web |last1=Dolan |first1=Graham |title=Airy's Water Telescope (1870) |url=http://www.royalobservatorygreenwich.org/articles.php?article=1069 |publisher=The Royal Observatory Greenwich |access-date=20 November 2018}}</ref> A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,<ref name="Hollis">{{cite journal |last1=Hollis |first1=H. P. |title=Airy's water telescope |journal=The Observatory |date=1937 |volume=60 |pages=103–107 |url=http://adsbit.harvard.edu//full/1937Obs....60..103H/0000105.000.html |access-date=20 November 2018|bibcode=1937Obs....60..103H }}</ref> but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded ''complete'' aether-drag.<ref>{{cite book |author1=Janssen, Michel |author2=Stachel, John |editor1-last=Stachel |editor1-first=John |title=Going Critical |date=2004 |publisher=Springer |isbn=978-1-4020-1308-9 |chapter=The Optics and Electrodynamics of Moving Bodies |chapter-url=https://www.mpiwg-berlin.mpg.de/Preprints/P265.PDF}}</ref>

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.&nbsp;5-2, these include<ref name="Rindler1977"/>{{rp|57–60}}
: <math>\cos \theta ' = \frac{ \cos \theta + v/c}{ 1 + (v/c)\cos \theta}</math> &nbsp;&nbsp;'''OR'''&nbsp;&nbsp; <math> \sin \theta ' = \frac{\sin \theta}{\gamma [ 1 + (v/c) \cos \theta ]}</math> &nbsp;&nbsp;'''OR'''&nbsp;&nbsp; <math> \tan \frac{\theta '}{2} = \left( \frac{c - v}{c + v} \right)^{1/2} \tan \frac {\theta}{2}</math>

=== Relativistic Doppler effect ===
{{main|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]] term, and that is the treatment described here.<ref>{{cite journal |last1=Sher |first1=D. |title=The Relativistic Doppler Effect |journal=Journal of the Royal Astronomical Society of Canada |date=1968 |volume=62 |pages=105–111 |bibcode=1968JRASC..62..105S |url=http://adsbit.harvard.edu//full/1968JRASC..62..105S/0000105.000.html |access-date=11 October 2018}}</ref><ref name="Gill">{{cite book |last1=Gill |first1=T. P. |title=The Doppler Effect |date=1965 |publisher=Logos Press Limited |location=London |pages=6–9 |ol=5947329M }}</ref>

Assume the receiver and the source are moving ''away'' from each other with a relative speed <math>v</math> as measured by an observer on the receiver or the source (The sign convention adopted here is that <math>v</math> is ''negative'' if the receiver and the source are moving ''towards'' each other). Assume that the source is stationary in the medium. Then
<math display="block">f_{r} = \left(1 - \frac v {c_s} \right) f_s</math>
where <math>c_s</math> is the speed of sound.

For light, and with the receiver moving at relativistic speeds, clocks on the receiver are [[time dilation|time dilated]] relative to clocks at the source. The receiver will measure the received frequency to be
<math display="block">f_r = \gamma\left(1 - \beta\right) f_s = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_s.</math>
where
* <math>\beta = v/c </math>&nbsp; and
* <math>\gamma = \frac{1}{\sqrt{1 - \beta^2}}</math> is the [[Lorentz factor]].

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the ''receiver'' with a moving source.<ref name=Feynman1977>{{cite book| title=The Feynman Lectures on Physics: Volume 1 | publisher=[[Addison-Wesley]] | location=Reading, Massachusetts |date=February 1977 | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | lccn=2010938208 | isbn=978-0-201-02116-5 | pages=34–7 f |chapter-url=https://feynmanlectures.caltech.edu/I_34.html |chapter=Relativistic Effects in Radiation}}</ref><ref name=Morin2007/>{{rp|540}}

==== Transverse Doppler effect ====
[[File:Transverse Doppler effect scenarios 5.svg|thumb|Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.]]

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.&nbsp;5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.<ref name=Morin2007/>{{rp|541}} In Fig.&nbsp;5-3a, the receiver observes light from the source as being blueshifted by a factor of {{tmath|1= \gamma }}. In Fig.&nbsp;5-3b, the light is redshifted by the same factor.

=== Measurement versus visual appearance ===
{{main|Terrell rotation}}
[[File:Animated Terrell Rotation - Cube.gif|thumb|Figure 5–4. Comparison of the measured length contraction of a cube versus its 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. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

[[File:Terrell Rotation Sphere.gif|thumb|Figure 5–5. Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross.]]
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.<ref name="Terrell" group=p>{{cite journal |last1=Terrell |first1=James |title=Invisibility of the Lorentz Contraction |journal=[[Physical Review]] |date=15 November 1959 |volume=116 |issue=4 |pages=1041–1045 |doi=10.1103/PhysRev.116.1041 |bibcode=1959PhRv..116.1041T }}</ref><ref name="Penrose" group=p>{{cite journal |last1=Penrose |first1=Roger | author-link1=Roger Penrose |title=The Apparent Shape of a Relativistically Moving Sphere |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |date=24 October 2008 |volume=55 |issue=1 |pages=137–139 |doi=10.1017/S0305004100033776|bibcode=1959PCPS...55..137P |s2cid=123023118 }}</ref><ref>{{cite web |last1=Cook|first1=Helen |title=Relativistic Distortion |url=http://www.math.ubc.ca/~cass/courses/m309-01a/cook/ |publisher=Mathematics Department, University of British Columbia |access-date=12 April 2017 }}</ref><ref>{{cite web |last1=Signell |first1=Peter |title=Appearances at Relativistic Speeds |url=https://stuff.mit.edu/afs/athena/course/8/8.20/www/m44.pdf |website=Project PHYSNET |publisher=Michigan State University, East Lansing, MI |access-date=12 April 2017 |archive-url=https://web.archive.org/web/20170413153459/https://stuff.mit.edu/afs/athena/course/8/8.20/www/m44.pdf |archive-date=13 April 2017 }}</ref> 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.<ref>{{cite web |last1=Kraus |first1=Ute |title=The Ball is Round |url=http://www.spacetimetravel.org/fussball/fussball.html |website=Space Time Travel: Relativity visualized |publisher=Institut für Physik Universität Hildesheim |access-date=16 April 2017 |archive-url=https://web.archive.org/web/20170512165032/http://www.spacetimetravel.org/fussball/fussball.html |archive-date=12 May 2017 }}</ref><ref name="Boas_1961">{{cite journal |last1=Boas |first1=Mary L. |title=Apparent Shape of Large Objects at Relativistic Speeds |journal=American Journal of Physics |date=1961 |volume=29 |issue=5 |page=283 |doi=10.1119/1.1937751|bibcode=1961AmJPh..29..283B }}</ref>

[[File:M87 jet (1).jpg|thumb|Figure 5–6. Galaxy [[Messier 87|M87]] sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.]]
Both Fig.&nbsp;5-4 and Fig.&nbsp;5-5 illustrate objects moving transversely to the line of sight. In Fig.&nbsp;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.&nbsp;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.<ref name="Muller_2014">{{cite journal |last1=Müller |first1=Thomas |last2=Boblest |first2=Sebastian |title=Visual appearance of wireframe objects in special relativity |journal=European Journal of Physics |date=2014 |volume=35 |issue=6 |article-number=065025 |doi=10.1088/0143-0807/35/6/065025|arxiv=1410.4583 |bibcode=2014EJPh...35f5025M |s2cid=118498333 }}</ref> This illusion has come to be known as ''[[Terrell rotation]]'' or the ''Terrell–Penrose effect''.

Another example where visual appearance is at odds with measurement comes from the observation of apparent [[superluminal motion]] in various [[radio galaxies]], [[BL Lac objects]], [[quasars]], and other astronomical objects that eject [[astrophysical jet|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.<ref>{{cite book|last1=Zensus|first1=J. Anton|last2=Pearson|first2=Timothy J.|title=Superluminal Radio Sources|date=1987|publisher=Cambridge University Press|location=Cambridge, New York|isbn=978-0-521-34560-6|page=3|edition=1st}}</ref><ref>{{cite web|last1=Chase|first1=Scott I.|title=Apparent Superluminal Velocity of Galaxies|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/Superluminal/superluminal.html|website=The Original Usenet Physics FAQ|publisher=Department of Mathematics, University of California, Riverside|access-date=12 April 2017}}</ref><ref>{{cite web|last1=Richmond|first1=Michael|title="Superluminal" motions in astronomical sources|url=http://spiff.rit.edu/classes/phys200/lectures/superlum/superlum.html|website=Physics 200 Lecture Notes|publisher=School of Physics and Astronomy, Rochester Institute of Technology|access-date=20 April 2017|archive-url=https://web.archive.org/web/20170216155045/http://spiff.rit.edu/classes/phys200/lectures/superlum/superlum.html|archive-date=16 February 2017}}</ref> In Fig.&nbsp;5-6, galaxy [[Messier 87|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.&nbsp;5-4 has been stretched out.<ref>{{cite web|last1=Keel|first1=Bill|title=Jets, Superluminal Motion, and Gamma-Ray Bursts|url=http://pages.astronomy.ua.edu/keel/galaxies/jets.html|website=Galaxies and the Universe – WWW Course Notes|publisher=Department of Physics and Astronomy, University of Alabama|access-date=29 April 2017|archive-url=https://web.archive.org/web/20170301030027/http://pages.astronomy.ua.edu/keel/galaxies/jets.html|archive-date=1 March 2017}}</ref>

== Dynamics ==
Section ''{{slink|#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 ===
{{Main|Mass–energy equivalence}}

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a [[four-vector]] 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 {{nowrap|(''E''/''c'', 0, 0, 0)}}: 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 {{nowrap|(''E''/''c'', ''Ev''/''c''<sup>2</sup>, 0, 0)}}. The momentum is equal to the energy multiplied by the velocity divided by ''c''<sup>2</sup>. 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''<sup>2</sup>.

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 do not 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.<ref name=electro group=p/> The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.<ref name=inertia group=p>[http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ 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)</ref> 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.<ref name=Jammer>{{cite book |title=Concepts of Mass in Classical and Modern Physics |author=[[Max Jammer]] |pages=177–178 |url=https://books.google.com/books?id=lYvz0_8aGsMC&pg=PA177 |isbn=978-0-486-29998-3 |publisher=Courier Dover Publications |date=1997 }}</ref> Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.<ref name= Stachel>{{cite book |title=Einstein from ''B'' to ''Z'' |page= 221 |author=John J. Stachel |url=https://books.google.com/books?id=OAsQ_hFjhrAC&pg=PA215 |isbn=978-0-8176-4143-6 |publisher=Springer |date=2002}}</ref>

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.<ref name=survey group=p>[https://archive.today/20240524161707/https://www.webcitation.org/5knwYbqwK?url=http://www.geocities.com/physics_world/abstracts/Einstein_1907A_abstract.htm ''On the Inertia of Energy Required by the Relativity Principle''], A. Einstein, Annalen der Physik 23 (1907): 371–384</ref>

=== Einstein's 1905 demonstration of ''E'' = ''mc''<sup>2</sup> ===
In his fourth of his 1905 [[Annus mirabilis papers]],<ref name=inertia group=p/> Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time.

Einstein took as starting assumptions his recently discovered formula for [[relativistic Doppler shift]], the laws of [[conservation of energy]] and [[conservation of momentum]], and the relationship between the frequency of light and its energy as implied by [[Maxwell's equations]].

{{multiple image
 | direction         = vertical
 | width             = 221
 | image1            = Einstein's derivation of E=mc2 Part 1.svg
 <!-- | caption1          = -->
 | image2            = Einstein's derivation of E=mc2.svg
 | caption2          = Figure 6-1. Einstein's 1905 derivation of ''E'' = ''mc''<sup>2</sup>
}}
Fig. 6-1 (top). Consider a system of plane waves of light having frequency <math>f</math> traveling in direction <math>\phi</math> relative to the x-axis of reference frame ''S''. The frequency (and hence energy) of the waves as measured in frame {{prime|''S''}} that is moving along the x-axis at velocity <math>v</math> is given by the relativistic Doppler shift formula that Einstein had developed in his 1905 paper on special relativity:<ref name="electro" group="p"/>
: <math> \frac{f'}{f} = \frac{1 - (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}} </math>

Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame ''S''. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle <math>\phi</math> with respect to the x-axis. Each pulse has energy {{tmath|1= L/2 }}. Because of conservation of momentum, the body remains stationary in ''S'' after emission of the two pulses. Let <math>E_0</math> be the energy of the body before emission of the two pulses and <math>E_1</math> after their emission.

Next, consider the same system observed from frame {{prime|''S''}} that is moving along the x-axis at speed <math>v</math> relative to frame ''S''. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let <math>H_0</math> be the energy of the body measured in reference frame {{prime|''S''}} before emission of the two pulses and <math>H_1</math> after their emission. We obtain the following relationships:<ref name=inertia group=p/>
: <math>\begin{align}
E_0 &= E_1 + \tfrac{1}{2}L + \tfrac{1}{2}L = E_1 + L \\[5mu]
H_0 &= H_1 + \tfrac12 L \frac{1 - (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}} + \tfrac12 L \frac{1 + (v/c) \cos{\phi}}{\sqrt{1 - v^2/c^2}}   =  H_1 + \frac{L}{{\sqrt{1 - v^2/c^2}}}
\end{align}</math>

From the above equations, we obtain the following:
{{NumBlk2||<math>\quad\quad (H_0 - E_0) - (H_1 - E_1)  = L \left( \frac{1}{\sqrt{1 - v^2/c^2}}  - 1 \right) </math>  |6-1}}

The two differences of form <math> H - E </math> seen in the above equation have a straightforward physical interpretation. Since <math>H</math> and <math>E</math> are the energies of the arbitrary body in the moving and stationary frames, <math> H_0 - E_0 </math> and <math>H_1 - E_1</math> represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence,
{{NumBlk2||<math>\quad\quad K_0 - K_1  = L \left( \frac{1}{\sqrt{1 - v^2/c^2}}  - 1 \right) </math>  |6-2}}

Taking a Taylor series expansion and neglecting higher order terms, he obtained
{{NumBlk2||<math>\quad\quad K_0 - K_1 = \frac{1}{2} \frac{ L}{c^2} v^2 </math>  |6-3}}

Comparing the above expression with the classical expression for kinetic energy, ''K.E.''&nbsp;=&nbsp;{{sfrac|1|2}}''mv''<sup>2</sup>, Einstein then noted: "If a body gives off the energy ''L'' in the form of radiation, its mass diminishes by ''L''/''c''<sup>2</sup>."

Rindler has observed that Einstein's heuristic argument suggested merely that energy ''contributes'' to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that ''all'' inertial mass represented a reserve of energy. "To equate ''all'' mass with energy required an act of aesthetic faith, very characteristic of Einstein."<ref name="Rindler0"/>{{rp|81–84}} Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal.

For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.<ref name="Fernflores_2018">{{cite book |last1=Fernflores |first1=Francisco |title=Einstein's Mass–Energy Equation, Volume I: Early History and Philosophical Foundations |date=2018 |publisher=Momentum Pres |location=New York |isbn=978-1-60650-857-2}}</ref>

=== How far can you travel from the Earth? ===
{{See also|Space travel under constant acceleration}}
Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~&nbsp;100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems that 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 [[Gravity of Earth|1''g'']], it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by:
<math display="block">v(t) = \frac{at}{\sqrt{1+ a^2t^2/c^2}} ,</math>
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.<ref group=p>{{cite web|url = http://www.normalesup.org/~baglio/physique/acceleration.pdf|title = Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?|date = 26 May 2007|access-date = 22 January 2016|publisher = Physics Department, ENS Cachan|last = Baglio|first = Julien}}</ref> Therefore, after one year of accelerating at 9.81&nbsp;m/s<sup>2</sup>, the spaceship will be travelling at {{nowrap|1=''v'' = 0.712 ''c''}} and {{nowrap|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.<ref name=gibbskoks>{{cite web|author1=Philip Gibbs |author2=Don Koks |name-list-style=amp |title=The Relativistic Rocket |url=http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html |access-date=30 August 2012}}</ref> 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 {{nowrap|1.1 ''g''}} will take {{val|148,000}} years and cover about {{val|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.<ref name=gibbskoks /> 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).<ref>[http://library.thinkquest.org/C0116043/specialtheorytext.htm The special theory of relativity shows that time and space are affected by motion] {{Webarchive|url=https://web.archive.org/web/20121021183616/http://library.thinkquest.org/C0116043/specialtheorytext.htm |date=2012-10-21 }}. Library.thinkquest.org. Retrieved on 2013-04-24.</ref>

=== Elastic collisions ===
Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.<ref name="Idema_2022">{{cite web |last1=Idema |first1=Timon |title=Mechanics and Relativity. Chapter 14: Relativistic Collisions |url=https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Mechanics_and_Relativity_(Idema)/14%3A_Relativistic_Collisions |website=LibreTexts Physics |date=17 April 2019 |publisher=California State University Affordable Learning Solutions Program |access-date=2 January 2023}}</ref>

In Newtonian mechanics, analysis of collisions involves use of the [[conservation of mass|conservation laws for mass]], [[conservation of momentum|momentum]] and [[conservation of energy|energy]]. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. (''Inelastic'' collisions are discussed in [[Spacetime#Conservation laws]]. Radioactive decay may be considered a sort of time-reversed inelastic collision.<ref name="Idema_2022"/>)

Elastic scattering of charged elementary particles deviates from ideality due to the production of [[Bremsstrahlung]] radiation.<ref name="Nakel_1994">{{cite journal |last1=Nakel |first1=Werner |title=The elementary process of bremsstrahlung |journal=Physics Reports |date=1994 |volume=243 |issue=6 |pages=317–353 |doi=10.1016/0370-1573(94)00068-9|bibcode=1994PhR...243..317N }}</ref><ref>{{cite book |last1=Halbert |first1=M.L. |editor1-last=Austin |editor1-first=S.M. |editor2-last=Crawley |editor2-first=G.M. |title=The Two-Body Force in Nuclei |date=1972 |publisher=Springer |location=Boston, MA. |chapter=Review of Experiments on Nucleon-Nucleon Bremsstrahlung}}</ref>

==== Newtonian analysis ====
[[File:Elastic collision of moving particle with equal mass stationary particle.svg|thumb|Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle]]
Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls  will subtend a right angle. (a) In the stationary frame, an incident sphere traveling at 2'''v''' strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±'''v'''. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±'''u'''. Energy conservation requires that {{abs|'''u'''}} = {{abs|'''v'''}}. (c) Reverting to the stationary frame, the rebound velocities are {{nowrap|'''v''' ± '''u'''}}. The dot product {{nowrap|1=('''v''' + '''u''') ⋅ ('''v''' − '''u''') = '''v'''<sup>2</sup> − '''u'''<sup>2</sup> = 0}}, indicating that the vectors are orthogonal.<ref name="Rindler0"/>{{rp|26–27}}

==== Relativistic analysis ====
[[File:Relativistic elastic collision of equal mass particles.svg|thumb|Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle]]
Consider the elastic collision scenario in Fig.&nbsp;6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light:

The relativistic momentum and total relativistic energy of a particle are given by
{{NumBlk2||<math>\quad\quad \vec{p} = \gamma m \vec{v} \quad \text{and} \quad E = \gamma m c^2 </math>  |6-4}}

Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles:
{{NumBlk2||<math>\quad\quad \gamma_1 m \vec{v_1} + 0 = \gamma_2 m \vec{v_2}  +  \gamma_3 m \vec{v_3} </math>  |6-5}}

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc<sup>2</sup>) equals the sum of the total energies of the emergent particles:
{{NumBlk2||<math>\quad\quad \gamma_1 m c^2 + m c^2 = \gamma_2 m c^2 + \gamma_3 m c^2 </math>  |6-6}}

Breaking down ({{EquationNote|6-5}}) into its components, replacing <math>v</math> with the dimensionless {{tmath|1= \beta }}, and factoring out common terms from ({{EquationNote|6-5}}) and ({{EquationNote|6-6}}) yields the following:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \beta_1 \gamma_1 = \beta_2 \gamma_2 \cos{\theta} +  \beta_3 \gamma_3 \cos{\phi} </math>  |6-7}}
{{NumBlk2||<math>\quad\quad \beta_2 \gamma_2 \sin{\theta} = \beta_3 \gamma_3 \sin{\phi} </math>  |6-8}}
{{NumBlk2||<math>\quad\quad \gamma_1 + 1 = \gamma_2 + \gamma_3 </math>  |6-9}}

From these we obtain the following relationships:<ref name="Champion_1932" group="p">{{cite journal |last1=Champion |first1=Frank Clive |title=On some close collisions of fast β-particles with electrons, photographed by the expansion method. |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |year=1932 |volume=136 |issue=830 |pages=630–637 |publisher=The Royal Society Publishing |doi=10.1098/rspa.1932.0108 |bibcode=1932RSPSA.136..630C |s2cid=123018629 |doi-access=free }}</ref>
{{NumBlk2||<math>\quad\quad \beta_2 = \frac{\beta_1 \sin{\phi}}{ \{ \beta_1^2 \sin^2{\phi} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-10}}
{{NumBlk2||<math>\quad\quad \beta_3 = \frac{\beta_1 \sin{\theta}}{ \{ \beta_1^2 \sin^2{\theta} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-11}}
{{NumBlk2||<math>\quad\quad \cos{(\phi + \theta)} = \frac{ (\gamma_1 - 1) \sin{\theta} \cos{\theta} }{ \{ (\gamma_1 + 1)^2 \sin^2 \theta + 4 \cos^2 \theta  \}^{1/2} }    </math>  |6-12}}
For the symmetrical case in which <math> \phi = \theta</math> and {{tmath|1= \beta_2 = \beta_3 }}, ({{EquationNote|6-12}}) takes on the simpler form:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \cos{\theta} = \frac{\beta_1}{ \{ 2/\gamma_1 + 3 \beta_1^2 - 2 \}^{1/2} }    </math>  |6-13}}

== Rapidity ==
{{Main|Rapidity}}
{{multiple image
 | direction         = horizontal
 | width1            = 135
 | image1            = Trig functions (sine and cosine).svg
 | caption1          = Figure 7-1a.  A ray through the [[unit circle]] {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cos ''a'', sin ''a'')}}, where ''a'' is twice the area between the ray, the circle, and the ''x''-axis.
 | width2            = 190
 | image2            = Hyperbolic functions-2.svg
 | caption2          = Figure 7-1b. A ray through the [[unit hyperbola]] {{nowrap|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cosh ''a'', sinh ''a'')}}, where ''a'' is twice the area between the ray, the hyperbola, and the ''x''-axis.
}}
[[File:Sinh+cosh+tanh.svg|thumb|Figure 7–2. Plot of the three basic [[Hyperbolic function]]s: hyperbolic sine ([[:File:Hyperbolic Sine.svg|sinh]]), hyperbolic cosine ([[:File:Hyperbolic Cosine.svg|cosh]]) and hyperbolic tangent ([[:File:Hyperbolic Tangent.svg|tanh]]). Sinh is red, cosh is blue and tanh is green.]]
Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.

This nonlinearity is an artifact of our choice of parameters.<ref name="Taylor1992" />{{rp|47–59}} We have previously noted that in an {{nowrap|1=''x''–''ct''}} spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

The natural functions for expressing these relationships are the [[Hyperbolic functions|hyperbolic analogs of the trigonometric functions]]. Fig.&nbsp;7-1a shows a [[unit circle]] with sin(''a'') and cos(''a''), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that ''a'' is interpreted, not as the angle between the ray and the {{nowrap|1=''x''-axis}}, but as twice the area of the sector swept out by the ray from the {{nowrap|1=''x''-axis}}. Numerically, the angle and {{nowrap|1=2 × area}} measures for the unit circle are identical. Fig.&nbsp;7-1b shows a [[unit hyperbola]] with sinh(''a'') and cosh(''a''), where ''a'' is likewise interpreted as twice the tinted area.<ref>{{cite book|last1=Thomas|first1=George B.|last2=Weir|first2=Maurice D.|last3=Hass|first3=Joel|last4=Giordano|first4=Frank R.|title=Thomas' Calculus: Early Transcendentals|date=2008|publisher=Pearson Education, Inc.|location=Boston|isbn=978-0-321-49575-4|page=533|edition=Eleventh}}</ref> Fig.&nbsp;7-2 presents plots of the sinh, cosh, and tanh functions.

For the unit circle, the slope of the ray is given by
: <math>\text{slope} = \tan a  = \frac{\sin a }{\cos a }.</math>

In the Cartesian plane, rotation of point {{nowrap|1=(''x'', ''y'')}} into point {{nowrap|1=(''x''{{'}}, ''y''{{'}})}} by angle ''&theta;'' is given by
: <math>
\begin{pmatrix}
x' \\
y' \\
\end{pmatrix} = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{pmatrix}\begin{pmatrix}
x \\
y \\
\end{pmatrix}.</math>

In a spacetime diagram, the velocity parameter <math>\beta \equiv \frac{v}{c}</math> is the analog of slope. The ''rapidity'', ''&phi;'', is defined by<ref name="Morin2007" />{{rp|543}}
: <math>\beta \equiv \tanh \phi ,</math>
where
: <math>\tanh \phi = \frac{\sinh \phi}{\cosh \phi} = \frac{e^\phi-e^{-\phi}}{e^\phi+e^{-\phi}}.</math>

The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;<ref name="Morin2007" />{{rp|544}}
: <math>\beta = \frac{\beta_1 + \beta_2}{1 + \beta_1 \beta_2} = </math> <math>\frac{\tanh \phi_1 + \tanh \phi_2}{1 + \tanh \phi_1 \tanh \phi_2} =</math> <math>\tanh(\phi_1 + \phi_2),</math>
or in other words, {{tmath|1= \phi = \phi_1 + \phi_2 }}.

The Lorentz transformations take a simple form when expressed in terms of rapidity. The ''&gamma;'' factor can be written as
: <math>\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \cosh \phi,</math>
: <math>\gamma \beta = \frac{\beta}{\sqrt{1 - \beta^2}} = \frac{\tanh \phi}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \sinh \phi.</math>

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called ''boosts''.

Substituting ''&gamma;'' and ''&gamma;β'' into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & -\sinh \phi \\
    -\sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    ct \\
    x
  \end{pmatrix},</math>
and the inverse Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t \\
    x
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & \sinh \phi \\
    \sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}.</math>

In other words, Lorentz boosts represent [[hyperbolic rotation]]s in Minkowski spacetime.{{cn|date=July 2025}}

The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.<ref name="Taylor1992">{{cite book|url=https://archive.org/details/spacetime_physics/|title=Spacetime Physics|last1=Taylor|first1=Edwin F.|last2=Wheeler|first2=John Archibald|date=1992|publisher=W. H. Freeman|isbn=0-7167-2327-1|edition=2nd}}</ref>

== Minkowski spacetime ==
{{Main|Minkowski space}}

[[File:Orthogonality and rotation.svg|thumb|Figure 10–1. Orthogonality and rotation of coordinate systems compared between '''left:''' [[Euclidean space]] through circular [[angle]] ''φ'', '''right:''' in [[Minkowski spacetime]] through [[hyperbolic angle]] ''φ'' (red lines labelled ''c'' denote the [[worldline]]s of a light signal, a vector is orthogonal to itself if it lies on this line).<ref>{{cite book|title=Gravitation|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|page=58|date=1973|isbn=978-0-7167-0344-0}}</ref>]]

The physical theory of special relativity was recast by [[Hermann Minkowski]] in a 4-dimensional geometry now called Minkowski space. Minkowski spacetime appears to be very similar to the standard 3-dimensional [[Euclidean space]], but there is a crucial difference with respect to time.
In 3D space, the [[differential (infinitesimal)|differential]] of distance (line element) ''ds'' is defined by
<math display="block"> ds^2 = d\mathbf{x} \cdot d\mathbf{x} = dx_1^2 + dx_2^2 + dx_3^2, </math>
where {{nowrap|1=''d'''''x''' = (''dx''<sub>1</sub>, ''dx''<sub>2</sub>, ''dx''<sub>3</sub>)}} are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate ''X''<sup>0</sup> derived from time, such that the distance differential fulfills
<math display="block"> ds^2 = -dX_0^2 + dX_1^2 + dX_2^2 + dX_3^2, </math>
where {{nowrap|1=''d'''''X''' = (''dX''<sub>0</sub>, ''dX''<sub>1</sub>, ''dX''<sub>2</sub>, ''dX''<sub>3</sub>)}} 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.&nbsp;10-1).<ref>{{cite book|title=Dynamics and Relativity|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley|page=247|date=2009|isbn=978-0-470-01460-8}}</ref> Just as Euclidean space uses a [[Euclidean metric]], so spacetime uses a [[Minkowski metric]]. {{Anchor|interval}}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]]) of Minkowski spacetime.

The actual form of ''ds'' above depends on the metric and on the choices for the ''X''<sup>0</sup> coordinate.
To make the time coordinate look like the space coordinates, it can be treated as [[imaginary number|imaginary]]: {{nowrap|1=''X''<sub>0</sub> = ''ict''}} (this is called a [[Wick rotation]]).
According to [[Gravitation (book)|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 {{nowrap|1=''X''<sup>0</sup> = ''ct''}}, rather than a "disguised" Euclidean metric using ''ict'' as the time coordinate.

Some authors use {{nowrap|1=''X''<sup>0</sup> = ''t''}}, with factors of ''c'' elsewhere to compensate; for instance, spatial coordinates are divided by ''c'' or factors of ''c''<sup>±2</sup> are included in the metric tensor.<ref>{{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| date=2007 | isbn=978-0-679-77631-4| title-link=The Road to Reality}}</ref>
These numerous conventions can be superseded by using [[natural units]] where {{nowrap|1=''c'' = 1}}. Then space and time have equivalent units, and no factors of ''c'' appear anywhere.

A four dimensional space has four-dimensional vectors, or "four-vectors". The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ''ct'' and spacelike component {{nowrap|1='''x''' = (''x'', ''y'', ''z'')}}, in a [[Covariance and contravariance of vectors|contravariant]] [[position vector|position]] [[four-vector]] with components:
<math display="block">X^\nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z) = (ct, \mathbf{x} ).</math>
where we define {{nowrap|1=''X''<sup>0</sup> = ''ct''}} so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.<ref>Jean-Bernard Zuber & Claude Itzykson, ''Quantum Field Theory'', pg 5, {{isbn|0-07-032071-3}}</ref><ref>[[Charles W. Misner]], [[Kip S. Thorne]] & [[John A. Wheeler]], ''Gravitation'', pg 51, {{isbn|0-7167-0344-0}}</ref><ref>[[George Sterman]], ''An Introduction to Quantum Field Theory'', pg 4, {{isbn|0-521-31132-2}}</ref>

=== 4‑vectors ===
{{Main|Four-vector}}
{{nowrap|1=4‑vectors}}, and more generally [[tensors]], simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are ''manifestly'' relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of [[Maxwell's equations]] in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the [[field strength tensor]] formulation.<ref name=Post_1962>{{cite book|title=Formal Structure of Electromagnetics: General Covariance and Electromagnetics|date=1962|publisher=Dover Publications Inc.|isbn=978-0-486-65427-0|author=E. J. Post}}</ref>

==== Definition of 4-vectors ====
A 4-tuple, {{tmath|1=A=\left(A_{0}, A_{1}, A_{2}, A_{3}\right)}} is a "4-vector" if its component ''A<sub>i</sub>'' transform between frames according to the Lorentz transformation.

If using {{tmath|1=(ct, x, y, z)}} coordinates, ''A'' is a {{nowrap|1=4–vector}} if it transforms (in the {{nowrap|1=''x''-direction}}) according to
: <math>\begin{align}
  A_0' &= \gamma \left( A_0 - (v/c) A_1 \right)  \\ 
  A_1' &= \gamma \left( A_1 - (v/c) A_0 \right)\\
  A_2' &= A_2 \\ 
  A_3' &= A_3
\end{align} ,</math>
which comes from simply replacing ''ct'' with ''A''<sub>0</sub> and ''x'' with ''A''<sub>1</sub> in the earlier presentation of the [[#Lorentz transformations|'''Lorentz transformation.''']]

As usual, when we write ''x'', ''t'', etc. we generally mean Δ''x'', Δ''t'' etc.

The last three components of a {{nowrap|1=4–vector}} must be a standard vector in three-dimensional space. Therefore, a {{nowrap|1=4–vector}} must transform like {{tmath|1=(c \Delta t, \Delta x, \Delta y, \Delta z)}} under Lorentz transformations as well as rotations.<ref name="Schutz1985">{{cite book| last1=Schutz |first1= Bernard F. |title=A first course in general relativity|date=1985|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0-521-27703-5|page=26}}</ref>{{rp|36–59}}

==== Properties of 4-vectors ====
* '''Closure under linear combination:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then {{tmath|1=C = aA + aB}} is also a {{nowrap|1=4-vector}}.
* '''Inner-product invariance:''' If ''A'' and ''B'' are {{nowrap|1=4-vectors}}, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a {{nowrap|1=3-vector}}. In the following, <math>\vec{A}</math> and <math>\vec{B}</math> are {{nowrap|1=3-vectors}}:
*: <math>A \cdot B \equiv </math> <math>A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 \equiv </math> <math>A_0 B_0 - \vec{A} \cdot \vec{B}</math>
: In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in {{nowrap|1=3-space}}.
: Two vectors are said to be ''orthogonal'' if {{tmath|1= A \cdot B = 0 }}. Unlike the case with {{nowrap|1=3-vectors}}, orthogonal {{nowrap|1=4-vectors}} are not necessarily at right angles to each other. The rule is that two {{nowrap|1=4-vectors}} are orthogonal if they are offset by equal and opposite angles from the 45° line, which is the world line of a light ray. This implies that a lightlike {{nowrap|1=4-vector}} is orthogonal to ''itself''.
* '''Invariance of the magnitude of a vector:''' The magnitude of a vector is the inner product of a {{nowrap|1=4-vector}} with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which {{tmath|1= A \cdot A = 0 }}, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval <math>c^2 t^2 - x^2</math> and the invariant length of the relativistic momentum vector {{tmath|1= E^2 - p^2 c^2 }}.<ref name="Morin2007" />{{rp|639}}<ref name="Schutz1985" />{{rp|36–59}}

==== Examples of 4-vectors ====
* '''Displacement 4-vector:''' Otherwise known as the ''spacetime separation'', this is {{nowrap|1=(''&Delta;t, &Delta;x, &Delta;y, &Delta;z''),}} or for infinitesimal separations, {{nowrap|1=(''dt'', ''dx'', ''dy'', ''dz'')}}.
*: <math>dS \equiv (dt, dx, dy, dz)</math>
* '''Velocity 4-vector:''' This results when the displacement {{nowrap|1=4-vector}} is divided by <math>d \tau</math>, where <math>d \tau</math> is the proper time between the two events that yield ''dt'', ''dx'', ''dy'', and ''dz''.
*: <math>V \equiv \frac{dS}{d \tau} = \frac{(dt, dx, dy, dz)}{dt/\gamma} = </math> <math>\gamma \left(1, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) = </math> <math>(\gamma, \gamma \vec{v} ) </math>
{{multiple image
 | direction         = horizontal
 | width1            = 150
 | image1            = Momentarily Comoving Reference Frame.gif
 | caption1          = Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
 | width2            = 150
 | image2            = Lorentz transform of world line.gif
 | caption2          = Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
}}
: The {{nowrap|1=4-velocity}} is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.
: An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found that is momentarily comoving with the particle. This frame, the ''momentarily comoving reference frame'' (MCRF), enables application of special relativity to the analysis of accelerated particles.
: Since photons move on null lines, <math>d \tau = 0</math> for a photon, and a {{nowrap|1=4-velocity}} cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.
* '''Energy–momentum 4-vector:'''
*: <math>P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)</math>
: As indicated before, there are varying treatments for the energy–momentum {{nowrap|1=4-vector}} so that one may also see it expressed as <math>(E, \vec{p})</math> or {{tmath|1= (E, \vec{p}c) }}. The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy–momentum {{nowrap|1=4-vector}} is a conserved quantity.
* '''Acceleration 4-vector:''' This results from taking the derivative of the velocity {{nowrap|1=4-vector}} with respect to {{tmath|1= \tau }}.
*: <math>A \equiv \frac{dV}{d \tau} = </math> <math>\frac{d}{d \tau} (\gamma, \gamma \vec{v}) = </math> <math>\gamma \left( \frac{d \gamma}{dt}, \frac{d(\gamma \vec{v})}{dt} \right)</math>
* '''Force 4-vector:''' This is the derivative of the momentum {{nowrap|1=4-vector}} with respect to <math>\tau .</math>
*: <math>F \equiv \frac{dP}{d \tau} = </math> <math>\gamma \left(\frac{dE}{dt}, \frac{d \vec{p}}{dt} \right) = </math> <math> \gamma \left( \frac{dE}{dt},\vec{f} \right) </math>

As expected, the final components of the above {{nowrap|1=4-vectors}} are all standard {{nowrap|1=3-vectors}} corresponding to spatial {{nowrap|1=3-momentum}}, {{nowrap|1=3-force}} etc.<ref name="Schutz1985" />{{rp|36–59}}

==== 4-vectors and physical law ====
The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving {{nowrap|1=4-vectors}} rather than give up on conservation of momentum.

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving {{nowrap|1=4-vectors}} require the use of tensors with appropriate rank, which themselves can be thought of as being built up from {{nowrap|1=4-vectors}}.<ref name="Morin2007" />{{rp|644}}
{{anchor|Acceleration}} General relativity from the outset relies heavily on {{nowrap|1=4‑vectors}}, and more generally tensors, representing physically relevant entities.

== Acceleration ==
{{Further|Acceleration (special relativity)}}

Special relativity does accommodate [[acceleration (special relativity)|accelerations]] as well as [[Rindler coordinates|accelerating frames of reference]].<ref>{{cite book |title=Relativity Made Relatively Easy |edition=illustrated |first1=Andrew M. |last1=Steane |publisher=OUP Oxford |year=2012 |isbn=978-0-19-966286-9 |page=226 |url=https://books.google.com/books?id=75rCErZkh7EC}} [https://books.google.com/books?id=75rCErZkh7EC&pg=PA226 Extract of page 226]</ref>
It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames.<ref>{{cite book |title=Explorations in Mathematical Physics: The Concepts Behind an Elegant Language |edition=illustrated |first1=Don |last1=Koks |publisher=Springer Science & Business Media |year=2006 |isbn=978-0-387-32793-8 |page=234 |url=https://books.google.com/books?id=ObMb7l9-9loC}} [https://books.google.com/books?id=ObMb7l9-9loC&pg=PA234 Extract of page 234]</ref> It is only when gravitation is significant that general relativity is required.<ref name="PhysicsFAQ">{{cite web|last1=Gibbs|first1=Philip|title=Can Special Relativity Handle Acceleration?|url=http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html|website=The Physics and Relativity FAQ|publisher=math.ucr.edu|access-date=28 May 2017|archive-date=7 June 2017|archive-url=https://web.archive.org/web/20170607102331/http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html|url-status=live}}</ref>

Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.<ref name="PhysicsFAQ" />

In this section, we analyze several scenarios involving accelerated reference frames.

{{anchor|Dewan–Beran–Bell spaceship paradox}}

=== Dewan–Beran–Bell spaceship paradox ===
{{Main|Bell's spaceship paradox}}

The Dewan–Beran–Bell spaceship paradox ([[Bell's spaceship paradox]]) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.{{cn|date=September 2025}}

[[File:Bell's spaceship paradox - two spaceships - initial setup.png|thumb|Figure 7–4. Dewan–Beran–Bell spaceship paradox]]
In Fig.&nbsp;7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string that is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration. In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Will the string break?

When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.
# To observers in the rest frame, the spaceships start a distance ''L'' apart and remain the same distance apart during acceleration. During acceleration, ''L'' is a length contracted distance of the distance {{nowrap|1=''L{{'}} = &gamma;L''}} in the frame of the accelerating spaceships. After a sufficiently long time, ''&gamma;'' will increase to a sufficiently large factor that the string must break.
# Let ''A'' and ''B'' be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. ''A'' says that ''B'' has the same acceleration that he has, and ''B'' sees that ''A'' matches her every move. So the spaceships stay the same distance apart, and the string does not break.

The problem with the first argument is that there is no "frame of the spaceships". There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.
[[File:Bell spaceship paradox.svg|thumb|Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.]]
A spacetime diagram (Fig.&nbsp;7-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude <math>k</math> acceleration for proper time <math>\sigma</math> (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity <math>A'B''</math> turns out to be greater than the length along the line of simultaneity {{tmath|1= AB }}.

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig.&nbsp;7-5, the acceleration is finished, the ships will remain at a constant offset in some frame {{tmath|1= S' }}. If <math>x_{A}</math> and <math>x_{B}=x_{A}+L</math> are the ships' positions in {{tmath|1= S }}, the positions in frame <math>S'</math> are:<ref name="Franklin">{{cite journal |author=Franklin, Jerrold |title=Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity |journal=European Journal of Physics |volume=31 |year=2010 |pages=291–298 |doi=10.1088/0143-0807/31/2/006 |bibcode = 2010EJPh...31..291F |issue=2 |arxiv = 0906.1919|s2cid=18059490 }}</ref>
: <math>\begin{align}
x'_{A}& = \gamma\left(x_{A}-vt\right)\\
x'_{B}& = \gamma\left(x_{A}+L-vt\right)\\
L'& = x'_{B}-x'_{A} =\gamma L
\end{align}</math>

The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame {{tmath|1= S }}. As shown in Fig.&nbsp;7-5, Bell's example asserts the moving lengths <math>AB</math> and <math>A'B'</math> measured in frame <math>S</math> to be fixed, thereby forcing the rest frame length <math>A'B''</math> in frame <math>S'</math> to increase.

{{anchor|Accelerated observer with horizon}}

==== Accelerated observer with horizon ====
{{Main|Event horizon#Apparent horizon of an accelerated particle|Rindler coordinates}}

Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as [[event horizons]]. In the text accompanying [[Spacetime#Invariant hyperbola|Section "Invariant hyperbola" of the article Spacetime]], the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just ''approaches'' the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

[[File:Accelerated relativistic observer with horizon.png|thumb|Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed [[:File:ConstantAcceleration02.jpg|'''here''']]. ]]
Fig.&nbsp;7-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter <math>\beta</math> approaches a limit of one as <math>ct</math> increases. Likewise, <math>\gamma</math> approaches infinity.

The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:
# We remember that {{tmath|1= \beta = ct/x }}.
# Since {{tmath|1= c^2 t^2 - x^2 = s^2 }}, we conclude that {{tmath|1= \beta (ct) = ct/ \sqrt{c^2 t^2 - s^2} }}.
# <math>\gamma = 1/\sqrt{1 - \beta ^2} = </math> <math>\sqrt{c^2 t^2 - s^2}/s</math>
# From the relativistic force law, <math>F = dp/dt = </math>{{tmath|1= dpc/d(ct) = d(\beta \gamma m c^2)/d(ct) }}.
# Substituting <math>\beta(ct)</math> from step 2 and the expression for <math>\gamma</math> from step 3 yields {{tmath|1= F = mc^2 / s }}, which is a constant expression.<ref name="Bais">{{cite book|last1=Bais|first1=Sander|title=Very Special Relativity: An Illustrated Guide|url=https://archive.org/details/veryspecialrelat0000bais|url-access=registration|date=2007|publisher=Harvard University Press|location=Cambridge, Massachusetts|isbn=978-0-674-02611-7}}</ref>{{rp|110–113}}

Fig.&nbsp;7-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light&nbsp;hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01&nbsp;''c'' per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she ''never'' receives any communications from Terence after 100 hours on his clock (dashed green lines).<ref name="Bais" />{{rp|110–113}}

After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to '''receive''' Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an ''apparent'' event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.<ref name="Bais" />{{rp|110–113}}

== Relativity and unifying electromagnetism ==
{{Main|Classical electromagnetism and special relativity|Covariant formulation of classical electromagnetism}}
Theoretical investigation in [[classical electromagnetism]] 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]] calculus.<ref name=Post_1962/>

== Theories of relativity and quantum mechanics ==
''Special'' relativity can be combined with [[quantum mechanics]] to form [[relativistic quantum mechanics]] and [[quantum electrodynamics]]. How [[general relativity|''general'' relativity]] and quantum mechanics can be [[Unification of theories in physics|unified]] is [[List of unsolved problems in physics|one of the unsolved problems in physics]]; [[quantum gravity]] and a "[[theory of everything]]", which require a unification including general relativity too, are active and ongoing areas in theoretical research.

The early [[Bohr model#Refinements|Bohr–Sommerfeld atomic model]] explained the [[fine structure]] of [[alkali metal]] atoms using both special relativity and the preliminary knowledge on [[quantum mechanics]] of the time.<ref>{{cite book|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles|edition=2nd|pages=[https://archive.org/details/quantumphysicsof00eisb/page/114 114–116]|author1=R. Resnick|author2=R. Eisberg|publisher=John Wiley & Sons|date=1985|isbn=978-0-471-87373-0|url=https://archive.org/details/quantumphysicsof00eisb/page/114}}</ref>

In 1928, [[Paul Dirac]] constructed an influential [[relativistic wave equation]], now known as the [[Dirac equation]] in his honour,<ref name="Dirac" group=p>{{cite journal | author = P.A.M. Dirac | author-link = P.A.M. Dirac | date =1930 | title = A Theory of Electrons and Protons | journal =Proceedings of the Royal Society | volume = A126 |bibcode=1930RSPSA.126..360D | jstor=95359 | doi=10.1098/rspa.1930.0013 | issue = 801| pages = 360–365 | doi-access = free }}</ref> 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 (physics)|spin]]'', it also led to the prediction of the [[antiparticle]] of the electron (the [[positron]]),<ref name="Dirac" group=p /><ref group=p>{{cite journal|author=C.D. Anderson|author-link=Carl David Anderson |title=The Positive Electron|journal=Phys. Rev. |volume=43|pages=491–494 |date=1933|doi=10.1103/PhysRev.43.491|issue=6|bibcode = 1933PhRv...43..491A |doi-access=free}}</ref> and [[fine structure]] could only be fully explained with special relativity. It was the first foundation of ''[[relativistic quantum mechanics]]''.

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.{{cn|date=September 2025}} Instead, a theory of particles interpreted as quantized fields, called ''[[quantum field theory]]'', becomes necessary; in which particles can be [[annihilation|created and destroyed]] throughout space and time.

== Status ==
{{Main|Tests of special relativity|Criticism of the theory of relativity}}

Special relativity in its [[Minkowski spacetime]] is accurate only when the [[absolute value]] of the [[gravitational potential]] is much less than ''c''<sup>2</sup> in the region of interest.<ref>
{{cite book
 |title=Einstein's general theory of relativity: with modern applications in cosmology
 |author1=Øyvind Grøn |author2=Sigbjørn Hervik
 |name-list-style=amp |publisher=Springer
 |date=2007
 |isbn=978-0-387-69199-2
 |page=195
 |url=https://books.google.com/books?id=IyJhCHAryuUC
}} [https://books.google.com/books?id=IyJhCHAryuUC&pg=PA195 Extract of page 195 (with units where ''c'' = 1)]
</ref> In a strong gravitational field, one must use [[general relativity]]. 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]]. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10<sup>−20</sup>)<ref>The number of works is vast, see as example:<br />
{{cite journal|author1=Sidney Coleman |author2=Sheldon L. Glashow |title=Cosmic Ray and Neutrino Tests of Special Relativity|journal= Physics Letters B |volume=405 |date=1997|pages= 249–252|arxiv=hep-ph/9703240|doi=10.1016/S0370-2693(97)00638-2|issue=3–4|bibcode = 1997PhLB..405..249C |s2cid=17286330 }}<br />
An overview can be found on [http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html this page]</ref>
and thus accepted by the physics community. Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.<ref name="Roberts_2007">{{cite web |last1=Roberts |first1=Tom |last2=Schleif |first2=Siegmar |title=Experiments that Apparently are NOT Consistent with SR/GR |url=https://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Experiments_not_consistent_with_SR |website=What is the experimental basis of Special Relativity? |publisher=University of California at Riverside |access-date=10 July 2024}}</ref>

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]]'' 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,<ref>{{cite journal|author=John D. Norton|date=2004|first1=John D.|journal=Archive for History of Exact Sciences|title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905|pages= 45–105|volume=59|issue=1|url=http://philsci-archive.pitt.edu/archive/00001743/|doi=10.1007/s00407-004-0085-6|bibcode=2004AHES...59...45N|s2cid=17459755}}</ref> 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.<ref name="mM1905">{{cite journal|author= Jeroen van Dongen|title=On the role of the Michelson–Morley experiment: Einstein in Chicago|date= 2009|journal=Archive for History of Exact Sciences |volume=63 |issue=6|pages=655–663 |arxiv=0908.1545|bibcode=2009arXiv0908.1545V|doi=10.1007/s00407-009-0050-5|s2cid=119220040}}</ref>
* The [[Fizeau experiment]] (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 (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]]s accelerate and measure the properties of particles moving at near the speed of light, where their behavior is 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]] – 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 (relativity)|emission theory]] demonstrated that the speed of light is independent of the speed of the emitter.
* Experiments to test the [[aether drag hypothesis]] – no "aether flow obstruction".

== See also ==
; People :
* [[Arnold Sommerfeld]]
* [[Hermann Minkowski]]
* [[Max Born]]
* [[Max Planck]]
* [[Max von Laue]]
* [[Mileva Marić]]

; Relativity :
* [[Bondi k-calculus]]
* [[Doubly special relativity]]
* [[Einstein synchronisation]]
* [[History of special relativity]]
* [[Relativity priority dispute]]
* [[Rietdijk–Putnam argument]]
* [[Special relativity (alternative formulations)]]

; Physics :
* [[Born coordinates]]
* [[Born rigidity]]
* [[Einstein's thought experiments]]
* [[Lorentz ether theory]]
* [[Moving magnet and conductor problem]]
* [[physical cosmology]]
* [[Relativistic disk]]
* [[Relativistic Euler equations]]
* [[Relativistic heat conduction]]
* [[Shape waves]]

; Mathematics :
* [[Lorentz group]]
* [[Algebra of physical space|Relativity in the APS formalism]]

; Philosophy :
* [[actualism]]
* [[conventionalism]]

; Paradoxes :
* [[Bell's spaceship paradox]]
* [[Ehrenfest paradox]]
* [[Lighthouse paradox]]
* [[Mocanu's velocity composition paradox|Velocity composition paradox]]

== Notes ==
{{reflist|group=note|35em}}

== Primary sources ==
{{reflist|group=p|35em}}

== References ==
{{reflist|35em}}

== Further reading ==
=== Texts by Einstein and text about history of special relativity ===
* Einstein, Albert (1920). ''[[s:Relativity (1931)|Relativity: The Special and General Theory]]''.
* Einstein, Albert (1923). ''[https://gutenberg.org/ebooks/36276 The Meaning of Relativity]''. Princeton University Press.
* Logunov, Anatoly A. (2005). ''[https://arxiv.org/abs/physics/0408077 Henri Poincaré and the Relativity Theory]'' (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow.

=== Textbooks ===
* Timon Idema, (2018). ''[https://books.open.tudelft.nl/home/catalog/book/198 Mechanics and Relativity]'', TU Delft OPEN Publishing. {{isbn|978-94-6366-085-3}}
* Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, {{isbn|0-19-927583-1}}; {{isbn|978-0-19-927583-0}}
* {{cite book |title=Space, Time and Spacetime |author=Lawrence Sklar |url=https://books.google.com/books?id=cPLXqV3QwuMC&pg=PA206 |isbn=978-0-520-03174-6 |publisher=University of California Press |date=1977 }}
* {{cite book |title=Relativistic World |author=Sergey Stepanov |url= https://books.google.com/books?id=Rc6PtAEACAAJ | isbn = 978-3-11-051587-9 | publisher=De Gruyter |date=2018 }}
* Tipler, Paul, and Llewellyn, Ralph (2002). ''Modern Physics'' (4th ed.). W. H. Freeman & Co. {{isbn|0-7167-4345-0}}.

=== Journal articles ===
* {{cite journal | author1 = Alvager, T. | author2 = Farley, F. J. M. | date = 1964 | author3 = Kjellman, J. | author4 = Wallin, L. | title = Test of the Second Postulate of Special Relativity in the GeV region | journal = Physics Letters | volume = 12 | issue = 3| pages = 260–262 | doi = 10.1016/0031-9163(64)91095-9 | bibcode = 1964PhL....12..260A |display-authors=etal}}
* {{cite journal | doi = 10.1086/430652 | last1 = Darrigol | first1 = Olivier | date = 2004 | title = The Mystery of the Poincaré–Einstein Connection | journal = Isis | volume = 95 | issue = 4| pages = 614–26 | pmid = 16011297 | s2cid = 26997100 }}
* {{cite journal | doi = 10.1103/PhysRevA.56.4405 | last1 = Wolf | first1 = Peter | last2 = Petit | first2 = Gerard | date = 1997 | title = Satellite test of Special Relativity using the Global Positioning System | journal = Physical Review A | volume = 56 | issue = 6| pages = 4405–09 |bibcode = 1997PhRvA..56.4405W }}
* [http://scholarpedia.org/article/Special_relativity Special Relativity] [[Scholarpedia]]
* {{cite journal | last1 = Rindler | first1 = Wolfgang | author-link = Wolfgang Rindler | year = 2011| title = Special relativity: Kinematics | journal = [[Scholarpedia]] | volume = 6 | issue = 2| page = 8520 | doi = 10.4249/scholarpedia.8520 | bibcode = 2011SchpJ...6.8520R | doi-access = free }}

== External links ==
{{Wikisource|Relativity: The Special and General Theory}}
{{Wikisource portal|Relativity}}
{{Wikibooks|Special Relativity}}
{{Wikiversity|Special Relativity}}
{{Wiktionary|special relativity}}

=== Original works ===
* [http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf ''Zur Elektrodynamik bewegter Körper''] Einstein's original work in German, [[Annalen der Physik]], [[Bern]] 1905
* [https://fourmilab.ch/etexts/einstein/specrel/specrel.pdf ''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) ===
* [http://www.phys.unsw.edu.au/einsteinlight Einstein Light] An [http://www.sciam.com/article.cfm?chanID=sa004&articleID=0005CFF9-524F-1340-924F83414B7F0000 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.
* [http://www.einstein-online.info/en/elementary/index.html Einstein Online] {{Webarchive|url=https://web.archive.org/web/20100201234156/http://www.einstein-online.info/en/elementary/index.html |date=2010-02-01 }} Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
* Audio: Cain/Gay (2006) – [http://www.astronomycast.com/astronomy/einsteins-theory-of-special-relativity/ Astronomy Cast]. Einstein's Theory of Special Relativity

=== Special relativity explained (using simple or more advanced mathematics) ===
* [https://web.archive.org/web/20040206110957/http://www.geocities.com/autotheist/Bondi/intro.htm Bondi K-Calculus] – A simple introduction to the special theory of relativity.
* [http://gregegan.customer.netspace.net.au/FOUNDATIONS/01/found01.html Greg Egan's ''Foundations''] {{Webarchive|url=https://web.archive.org/web/20130425091908/http://gregegan.customer.netspace.net.au/FOUNDATIONS/01/found01.html |date=2013-04-25 }}.
* [http://cosmo.nyu.edu/hogg/sr/ The Hogg Notes on Special Relativity] A good introduction to special relativity at the undergraduate level, using calculus.
* [http://www.relativitycalculator.com/E=mc2.shtml Relativity Calculator: Special Relativity] {{Webarchive|url=https://web.archive.org/web/20130321052504/http://www.relativitycalculator.com/E=mc2.shtml |date=2013-03-21 }} – An algebraic and integral calculus derivation for {{nowrap|1=''E'' = ''mc''<sup>2</sup>}}.
* [http://www.mathpages.com/rr/rrtoc.htm MathPages – Reflections on Relativity] A complete online book on relativity with an extensive bibliography.
* [http://lightandmatter.com/sr/ Special Relativity] An introduction to special relativity at the undergraduate level.
* {{gutenberg|no=5001|name=Relativity: the Special and General Theory}}, by [[Albert Einstein]]
* [http://www.phys.vt.edu/~takeuchi/relativity/notes 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.
* [http://www.rafimoor.com/english/SRE.htm Understanding Special Relativity] The theory of special relativity in an easily understandable way.
* [http://digitalcommons.unl.edu/physicskatz/49/ 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).
* [http://www.physics.mq.edu.au/~jcresser/Phys378/LectureNotes/VectorsTensorsSR.pdf Lecture Notes on Special Relativity] by J D Cresser Department of Physics Macquarie University.
* [http://specialrelativity.net/ SpecialRelativity.net] – An overview with visualizations and minimal mathematics.
* [https://www.mdpi.com/2624-8174/4/2/28 Relativity 4-ever?] The problem of superluminal motion is discussed in an entertaining manner.

=== Visualization ===
* [http://www.hakenberg.de/diffgeo/special_relativity.htm Raytracing Special Relativity] Software visualizing several scenarios under the influence of special relativity.
* [http://www.anu.edu.au/Physics/Savage/RTR/ Real Time Relativity] {{Webarchive|url=https://web.archive.org/web/20130508021027/http://www.anu.edu.au/Physics/Savage/RTR/ |date=2013-05-08 }} The Australian National University. Relativistic visual effects experienced through an interactive program.
* [http://www.spacetimetravel.org/ Spacetime travel] A variety of visualizations of relativistic effects, from relativistic motion to black holes.
* [http://www.anu.edu.au/Physics/Savage/TEE/ Through Einstein's Eyes] {{Webarchive|url=https://web.archive.org/web/20130514175026/http://www.anu.edu.au/Physics/Savage/TEE/ |date=2013-05-14 }} The Australian National University. Relativistic visual effects explained with movies and images.
* [http://www.adamauton.com/warp/ Warp Special Relativity Simulator] A computer program to show the effects of traveling close to the speed of light.
* {{YouTube|C2VMO7pcWhg|Animation clip}} visualizing the Lorentz transformation.
* [http://math.ucr.edu/~jdp/Relativity/SpecialRelativity.html Original interactive FLASH Animations] from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
* [https://lightspeed.sourceforge.net/ lightspeed] An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.
* [http://specialrelativity.net/animations/starfield/starfield.html?beta=0.8&color=on&circles=on&avgstellardensity=0.11&starpopulation=yalebsc&limitingMag=5&projection=stereographic&anim=on&runningTime=8 Animation] showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.

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