View source for Length contraction

{{short description|Contraction of length in the direction of propagation in Minkowski space}}
{{Special relativity sidebar |consequences}}
[[File:Relativistic wheels.gif|thumb|300px|right|Wheels which travel at 9/10 the speed of light.  The speed of the top of a wheel is 0.994 c while the speed of the bottom is always zero. This is why the top is contracted relative to the bottom. This animation is made with the assumption that the spokes of a wheel are much more elastic than its circumference. Otherwise there could be a rupture of the spokes or of the circumference. In the rest frame of the center of a wheel, wheels are circular and their spokes are straight and equidistant, but their circumference is contracted and exerts a pressure on the spokes.]]

'''Length contraction''' is the phenomenon that a moving object's length is measured to be shorter than its [[proper length]], which is the length as measured in the object's own [[rest frame]].<ref name=Dalarsson>{{cite book |title=Tensors, Relativity, and Cosmology |edition=2nd |first1=Mirjana |last1=Dalarsson |first2=Nils |last2=Dalarsson |publisher=Academic Press |year=2015 |isbn=978-0-12-803401-9 |pages=106–108 |url=https://books.google.com/books?id=KZOZBgAAQBAJ}} [https://books.google.com/books?id=KZOZBgAAQBAJ&pg=PA106 Extract of page 106]</ref> It is also known as '''Lorentz contraction''' or '''Lorentz–FitzGerald contraction''' (after [[Hendrik Lorentz]] and [[George Francis FitzGerald]]) and is usually only noticeable at a substantial fraction of the [[speed of light]]. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

==History==
{{Main|History of special relativity}}
Length contraction was postulated by [[George Francis FitzGerald|George FitzGerald]] (1889) and [[Hendrik Antoon Lorentz]] (1892) to explain the negative outcome of the [[Michelson–Morley experiment]] and to rescue the hypothesis of the stationary aether ([[Lorentz ether theory#Length contraction|Lorentz–FitzGerald contraction hypothesis]]).<ref>{{Citation|author=FitzGerald, George Francis|year=1889|title=The Ether and the Earth's Atmosphere|journal=Science|volume=13|pages=390|doi=10.1126/science.ns-13.328.390|pmid=17819387|issue=328|bibcode = 1889Sci....13..390F |title-link=s:The Ether and the Earth's Atmosphere|s2cid=43610293}}</ref><ref>{{Citation|last=Lorentz|first=Hendrik Antoon|year=1892|title=The Relative Motion of the Earth and the Aether|journal=Zittingsverlag Akad. V. Wet.|volume=1|pages=74–79|title-link=s:Translation:The Relative Motion of the Earth and the Aether}}</ref>
Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after [[Oliver Heaviside]], who derived this deformation from electromagnetic theory in 1888), it was considered an [[ad hoc hypothesis]], because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones. In 1897 [[Joseph Larmor]] developed a model in which all forces are considered to be of electromagnetic origin, and length contraction appeared to be a direct consequence of this model. Yet it was shown by [[Henri Poincaré]] (1905) that electromagnetic forces alone cannot explain the electron's stability. So he had to introduce another ad hoc hypothesis: non-electric binding forces ([[Electromagnetic mass#Poincaré stresses and the 4/3 problem|Poincaré stresses]]) that ensure the electron's stability, give a dynamical explanation for length contraction, and thus hide the motion of the stationary aether.<ref name=pais>{{Citation
|author=Pais, Abraham
|author-link=Abraham Pais
|year=1982
|title= [[Subtle is the Lord: The Science and the Life of Albert Einstein]]
|location= New York
|publisher=Oxford University Press
|isbn=0-19-520438-7}}</ref>

Lorentz believed that length contraction represented a ''physical contraction'' of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.<ref name="Miller1998">{{cite book |last1=Miller |first1=Arthur I. |title=Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911) |date=1998 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-94870-6}}</ref>{{rp|62–68}}
Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation<ref name="LorentzPolarization">{{cite journal |last1=Lorentz |first1=H.A. |title=The rotation of the plane of polarization in moving media |journal=Huygens Institute – Royal Netherlands Academy of Arts and Sciences (KNAW) |date=1902 |volume=4 |pages=669–678 |url=http://www.dwc.knaw.nl/DL/publications/PU00014324.pdf |access-date=15 November 2018| bibcode=1901KNAB....4..669L }}</ref> and induction of double refraction,<ref name="LorentzElectromagnetic">{{cite journal |last1=Lorentz |first1=H. A. |title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light |journal=Huygens Institute – Royal Netherlands Academy of Arts and Sciences (KNAW) |date=1904 |volume=6 |pages=809–831 |url=http://www.dwc.knaw.nl/DL/publications/PU00014148.pdf |access-date=15 November 2018|bibcode=1903KNAB....6..809L }}</ref> and the induction of torques on charged condensers moving at an angle with respect to the aether.

Lorentz was perplexed by experiments such as the [[Trouton–Noble experiment]] and the [[experiments of Rayleigh and Brace]], which failed to validate his theoretical expectations.<ref name="Miller1998"/>

For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation {{nowrap|1={{prime|''t''}} = ''t'' − ''vx''/''c''<sup>2</sup>}}.<ref name="Lorentz1895">{{cite book |last1=Lorentz |first1=Hendrik |title=Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern) |date=1895 |chapter=Investigation of oscillations excited by oscillating ions |chapter-url=https://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies/Section_III |at=(subsection § 31) |publisher=E. J. Brill |location=Leiden |url=https://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies}}</ref> Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.<ref name="Bernstein2006">{{cite book |last1=Bernstein |first1=Jeremy |title=Secrets of the Old One: Einstein, 1905 |date=2006 |publisher=Copernicus Books (imprint of Springer Science + Business Media) |isbn=978-0387-26005-1}}</ref>{{rp|51,80}}

Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the ''apparent'' states of the field for a moving observer. ''True states'' remained those defined with respect to the ether.<ref name="Darrigol2005">{{cite journal |last1=Darrigol |first1=Olivier |title=The Genesis of the Theory of Relativity |journal=Séminaire Poincaré |date=2005 |volume=1 |pages=1–22 |url=http://www.bourbaphy.fr/darrigol2.pdf |access-date=15 November 2018 |bibcode=2006eins.book....1D }}</ref>

[[Albert Einstein]] (1905) is credited<ref name=pais /> with removing the ad hoc character from the contraction hypothesis, by deriving this contraction from his postulates instead of experimental data.<ref>{{Citation
|doi=10.1002/andp.19053221004
|author=Einstein, Albert
|year=1905a
|title=Zur Elektrodynamik bewegter Körper
|journal=Annalen der Physik
|volume=322
|issue=10
|pages=891–921|bibcode = 1905AnP...322..891E
|url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf|doi-access=free
}}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].</ref> [[Hermann Minkowski]] gave the geometrical interpretation of all relativistic effects by introducing his concept of four-dimensional [[spacetime]].<ref>{{Citation|author=Minkowski, Hermann|year=1909|title=Raum und Zeit|journal=Physikalische Zeitschrift|volume=10|pages=75–88|title-link=s:de:Raum und Zeit (Minkowski)}}
:*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]</ref>

==Basis in relativity==
[[File:Observer in special relativity.svg|thumb|In special relativity, the observer measures events against an infinite latticework of synchronized clocks.]]

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.<ref name=born /> Here, "object" simply means a distance with endpoints that are always mutually at rest, ''i.e.'', that are at rest in the same [[inertial frame of reference]]. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the [[proper length]] <math>L_0</math> of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity is greater than zero, then one can proceed as follows:

[[File:Lorentzkontraktion.svg|thumb|300px|right|''Length contraction'': Three blue rods are at rest in S, and three red rods in S'. At the instant when the left ends of A and D attain the same position on the axis of x, the lengths of the rods shall be compared. In S the simultaneous positions of the left side of A and the right side of C are more distant than those of D and F, while in S' the simultaneous positions of the left side of D and the right side of F are more distant than those of A and C.]]
The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the [[Einstein synchronization|Poincaré–Einstein synchronization]], or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of [[zero of a function|vanishing]] transport velocity. Now, when the synchronization process is finished, the object is moved along the clock row and every clock stores the exact time when the left or the right end of the object passes by. After that, the observer only has to look at the position of a clock A that stored the time when the left end of the object was passing by, and a clock B at which the right end of the object was passing by ''at the same time''. It's clear that distance AB is equal to length <math>L</math> of the moving object.<ref name=born /> Using this method, the definition of [[Relativity of simultaneity|simultaneity]] is crucial for measuring the length of moving objects.

Another method is to use a clock indicating its [[proper time]] <math>T_0</math>, which is traveling from one endpoint of the rod to the other in time <math>T</math> as measured by clocks in the rod's rest frame. The length of the rod can be computed by multiplying its travel time by its velocity, thus <math>L_{0} = T\cdot v</math> in the rod's rest frame or <math>L = T_{0}\cdot v</math> in the clock's rest frame.<ref>{{cite book|author1=Edwin F. Taylor| author2=John Archibald Wheeler| title=Spacetime Physics: Introduction to Special Relativity| year=1992| publisher=W. H. Freeman| location=New York| isbn=0-7167-2327-1| url=https://archive.org/details/spacetimephysics00edwi_0}}</ref>

In Newtonian mechanics, simultaneity and time duration are absolute and therefore both methods lead to the equality of <math>L</math> and <math>L_0</math>. Yet in relativity theory the constancy of light velocity in all inertial frames in connection with [[relativity of simultaneity]] and [[time dilation]] destroys this equality. In the first method an observer in one frame claims to have measured the object's endpoints simultaneously, but the observers in all other inertial frames will argue that the object's endpoints were ''not'' measured simultaneously. In the second method, times <math>T</math> and <math>T_0</math> are not equal due to time dilation, resulting in different lengths too.

The deviation between the measurements in all inertial frames is given by the formulas for [[Lorentz transformation]] and time dilation (see [[#Derivation|Derivation]]). It turns out that the proper length remains unchanged and always denotes the greatest length of an object, and the length of the same object measured in another inertial reference frame is shorter than the proper length. This contraction only occurs along the line of motion, and can be represented by the relation
:<math>L = \frac{1}{\gamma(v)}L_0</math>

where
*<math>L</math> is the length observed by an observer in motion relative to the object
*<math>L_0
</math> is the proper length (the length of the object in its rest frame)
*<math>\gamma(v)</math> is the ''[[Lorentz factor]]'',  defined as <math display="block">\gamma (v) \equiv \frac{1}{\sqrt{1 - v^2/c^2}} </math> where
**<math>v</math> is the relative velocity between the observer and the moving object
**<math>c</math> is the speed of light

Replacing the Lorentz factor in the original formula leads to the relation
:<math>L = L_0\sqrt{1 - v^2/c^2}</math>

In this equation both <math>L</math> and <math>L_0
</math> are measured parallel to the object's line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object. For more general conversions, see the [[Lorentz transformations]]. An observer at rest observing an object travelling very close to the speed of light would observe the length of the object in the direction of motion as very near zero.

Then, at a speed of {{val|13,400,000|u=m/s}} (30 million mph, 0.0447{{math|''c''}}) contracted length is 99.9% of the length at rest; at a speed of {{val|42,300,000|u=m/s}} (95 million mph, 0.141{{math|''c''}}), the length is still 99%. As the magnitude of the velocity approaches the speed of light, the effect becomes prominent.

== Symmetry ==
The principle of relativity (according to which the laws of nature are invariant across inertial reference frames) requires that length contraction is symmetrical: If a rod is at rest in an inertial frame <math>S</math>, it has its proper length in <math>S</math> and its length is contracted in <math>S'
</math>. However, if a rod rests in <math>S'
</math>, it has its proper length in <math>S'
</math> and its length is contracted in <math>S</math>. This can be vividly illustrated using symmetric [[Minkowski diagram]]s, because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional [[spacetime]].<ref>{{cite book |author=Albert Shadowitz  |title=Special relativity |url=https://archive.org/details/specialrelativit0000shad  |url-access=registration  |isbn=0-486-65743-4 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988|pages=[https://archive.org/details/specialrelativit0000shad/page/20 20–22]}}</ref><ref name=sartori>{{cite book |author=Leo Sartori |title=Understanding Relativity: a simplified approach to Einstein's theories |isbn=0-520-20029-2 |publisher=University of California Press |year=1996|pages=151ff}}</ref>

== Magnetic forces ==
{{Main|Relativistic electromagnetism}}
Magnetic forces are caused by relativistic contraction when electrons are moving relative to atomic nuclei. The magnetic force on a moving charge next to a current-carrying wire is a result of relativistic motion between electrons and protons.<ref>{{Cite web|url=https://www.feynmanlectures.caltech.edu/II_13.html|title=The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics|website=www.feynmanlectures.caltech.edu}}</ref><ref name=Landau>{{cite book |author=E M Lifshitz, L D Landau |title=The classical theory of ields |volume=2 |series=[[Course of Theoretical Physics]] |edition=Fourth |publisher= Butterworth-Heinemann |location=Oxford UK |year=1980 |isbn=0-7506-2768-9 |url=http://worldcat.org/isbn/0750627689}}</ref>

In 1820, [[André-Marie Ampère]] showed that parallel wires having currents in the same direction attract one another. In the electrons' frame of reference, the moving wire contracts slightly, causing the protons of the opposite wire to be locally ''denser''.  As the electrons in the opposite wire are moving as well, they do not contract (as much). This results in an apparent local imbalance between electrons and protons; the moving electrons in one wire are attracted to the extra protons in the other. The reverse can also be considered.  To the static proton's frame of reference, the electrons are moving and contracted, resulting in the same imbalance. The electron [[drift velocity]] is relatively very slow, on the order of a meter an hour but the force between an electron and proton is so enormous that even at this very slow speed the relativistic contraction causes significant effects.

This effect also applies to magnetic particles without current, with current being replaced with electron spin.{{citation needed|date=December 2016}}

==Experimental verifications==
{{See also|Tests of special relativity}}

Any observer co-moving with the observed object cannot measure the object's contraction, because he can judge himself and the object as at rest in the same inertial frame in accordance with the principle of relativity (as it was demonstrated by the [[Trouton–Rankine experiment]]). So length contraction cannot be measured in the object's rest frame, but only in a frame in which the observed object is in motion. In addition, even in such a non-co-moving frame, ''direct'' experimental confirmations of length contraction are hard to achieve, because (a) at the current state of technology, objects of considerable extension cannot be accelerated to relativistic speeds, and (b) the only objects traveling with the speed required are atomic particles, whose spatial extensions are too small to allow a direct measurement of contraction.

However, there are ''indirect'' confirmations of this effect in a non-co-moving frame:
*It was the negative result of a famous experiment, that required the introduction of length contraction: the [[Michelson–Morley experiment]] (and later also the [[Kennedy–Thorndike experiment]]). In special relativity its explanation is as follows: In its rest frame the interferometer can be regarded as at rest in accordance with the relativity principle, so the propagation time of light is the same in all directions. Although in a frame in which the interferometer is in motion, the transverse beam must traverse a longer, diagonal path with respect to the non-moving frame thus making its travel time longer, the factor by which the longitudinal beam would be delayed by taking times ''L''/(''c''−''v'') and ''L''/(''c''+''v'') for the forward and reverse trips respectively is even longer. Therefore, in the longitudinal direction the interferometer is supposed to be contracted, in order to restore the equality of both travel times in accordance with the negative experimental result(s). Thus the two-way speed of light remains constant and the round trip propagation time along perpendicular arms of the interferometer is independent of its motion & orientation.
* Given the thickness of the atmosphere as measured in Earth's reference frame, [[muon]]s' extremely short lifespan shouldn't allow them to make the trip to the surface, even at the speed of light, but they do nonetheless. From the Earth reference frame, however, this is made possible only by the muon's time being slowed down by time dilation. However, in the muon's frame, the effect is explained by the atmosphere being contracted, shortening the trip.<ref name=sexl />
*Heavy [[ion]]s that are spherical when at rest should assume the form of "pancakes" or flat disks when traveling nearly at the speed of light{{px2}}{{mdash}}{{hsp}}and in fact, the results obtained from particle collisions can only be explained when the increased nucleon density due to length contraction is considered.<ref>{{cite web|author=Brookhaven National Laboratory|url=http://www.bnl.gov/rhic/physics.asp|title=The Physics of RHIC|access-date=2013-01-01}}</ref><ref>{{cite web|author=Manuel Calderon de la Barca Sanchez|url=http://nuclear.ucdavis.edu/~calderon/Research/physicsResearch.html|title=Relativistic heavy ion collisions|access-date=2013-01-01}}</ref><ref>{{Cite journal|author=Hands, Simon|title=The phase diagram of QCD|year=2001|journal=Contemporary Physics|volume=42|issue=4|pages=209–225|doi=10.1080/00107510110063843|arxiv=physics/0105022|bibcode = 2001ConPh..42..209H |s2cid=16835076}}</ref>
* The [[ionization]] ability of electrically charged particles with large relative velocities is higher than expected. In pre-relativistic physics the ability should decrease at high velocities, because the time in which ionizing particles in motion can interact with the electrons of other atoms or molecules is diminished; however, in relativity, the higher-than-expected ionization ability can be explained by length contraction of the [[Coulomb's law|Coulomb field]] in frames in which the ionizing particles are moving, which increases their electrical field strength normal to the line of motion.<ref name=sexl>{{Citation|author1=Sexl, Roman |author2=Schmidt, Herbert K.|title=Raum-Zeit-Relativität|year=1979|publisher=Vieweg|location=Braunschweig| bibcode=1979raum.book.....S| isbn=3-528-17236-3}}</ref><ref>{{Citation|author=Williams, E. J.|title=The Loss of Energy by β -Particles and Its Distribution between Different Kinds of Collisions|year=1931|journal=Proceedings of the Royal Society of London. Series A|volume=130|issue=813|pages=328–346|doi=10.1098/rspa.1931.0008|bibcode = 1931RSPSA.130..328W |doi-access=free}}</ref> 
* In [[synchrotron]]s and [[free-electron laser]]s, relativistic electrons were injected into an [[undulator]], so that [[synchrotron radiation]] is generated. In the proper frame of the electrons, the undulator is contracted which leads to an increased radiation frequency. Additionally, to find out the frequency as measured in the laboratory frame, one has to apply the [[relativistic Doppler effect]]. So, only with the aid of length contraction and the relativistic Doppler effect, the extremely small wavelength of undulator radiation can be explained.<ref>{{cite web|author=DESY photon science|url=http://photon-science.desy.de/research/studentsteaching/primers/synchrotron_radiation/index_eng.html|title=What is SR, how is it generated and what are its properties?|access-date=2013-01-01|url-status=dead|archive-url=https://web.archive.org/web/20160603161837/http://photon-science.desy.de/research/studentsteaching/primers/synchrotron_radiation/index_eng.html|archive-date=2016-06-03}}</ref><ref>{{cite web|author=DESY photon science|url=http://flash.desy.de/sites2009/site_vuvfel/content/e395/e2188/FLASH-Broschrefrs_web.pdf|title=FLASH The Free-Electron Laser in Hamburg (PDF 7,8 MB)|access-date=2013-01-01}}</ref>

==Reality of length contraction==
[[File:EinsteinContraction.svg|thumb|300px|Minkowski diagram of Einstein's 1911 [[thought experiment]] on length contraction. Two rods of rest length <math>A' B' = A'' B'' = L_0</math> are moving with <math> 0.6 c </math> in opposite directions, resulting in <math>A^\ast B^\ast <L_0</math>.]]
In 1911 [[Vladimir Varićak]] asserted that one sees the length contraction in an objective way, according to Lorentz, while it is "only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement", according to Einstein.<ref>{{Cite book|url=https://en.wikisource.org/wiki/Translation:On_Ehrenfest's_Paradox|title=On Ehrenfest's Paradox|first=Vladimir|last=Varićak|via=Wikisource}}</ref><ref>{{Citation | author= Miller, A.I. | year= 1981 | title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) | chapter= Varičak and Einstein | place= Reading | publisher= Addison–Wesley | pages= [https://archive.org/details/alberteinsteinss0000mill/page/249 249–253] | isbn= 0-201-04679-2 | chapter-url= https://archive.org/details/alberteinsteinss0000mill/page/249 }}</ref> Einstein published a rebuttal:

{{Blockquote|text=The author unjustifiably stated a difference of Lorentz's view and that of mine ''concerning the physical facts''. The question as to whether length contraction ''really'' exists or not is misleading. It doesn't "really" exist, in so far as it doesn't exist for a comoving observer; though it "really" exists, ''i.e.'' in such a way that it could be demonstrated in principle by physical means by a non-comoving observer.<ref name=einst1911>{{cite journal|author=Einstein, Albert|title=Zum Ehrenfestschen Paradoxon. Eine Bemerkung zu V. Variĉaks Aufsatz|journal=Physikalische Zeitschrift|volume=12|pages=509–510|year=1911}}; Original: Der Verfasser hat mit Unrecht einen Unterschied der ''Lorentz''schen Auffassung von der meinigen ''mit Bezug auf die physikalischen Tatsachen'' statuiert. Die Frage, ob die ''Lorentz''-Verkürzung ''wirklich'' besteht oder nicht, ist irreführend. Sie besteht nämlich nicht "wirklich", insofern sie für einen mitbewegten Beobachter nicht existiert; sie besteht aber "wirklich", d. h. in solcher Weise, daß sie prinzipiell durch physikalische Mittel nachgewiesen werden könnte, für einen nicht mitbewegten Beobachter.</ref>|sign=Albert Einstein, 1911}}

Einstein also argued in that paper, that length contraction is not simply the product of ''arbitrary'' definitions concerning the way clock regulations and length measurements are performed. He presented the following thought experiment: Let A'B' and A"B" be the endpoints of two rods of the same proper length ''L''<sub>0</sub>, as measured on x' and x" respectively. Let them move in opposite directions along the x* axis, considered at rest, at the same speed with respect to it. Endpoints A'A" then meet at point A*, and B'B" meet at point B*. Einstein pointed out that length A*B* is shorter than A'B' or A"B", which can also be demonstrated by bringing one of the rods to rest with respect to that axis.<ref name=einst1911 />

==Paradoxes==
Due to superficial application of the contraction formula, some paradoxes can occur. Examples are the [[ladder paradox]] and [[Bell's spaceship paradox]]. However, those paradoxes can be solved by a correct application of the relativity of simultaneity. Another famous paradox is the [[Ehrenfest paradox]], which proves that the concept of [[Rigid body|rigid bodies]] is not compatible with relativity, reducing the applicability of [[Born rigidity]], and showing that for a co-rotating observer the geometry is in fact [[Non-Euclidean Geometry|non-Euclidean]].

==Visual effects==
{{Main|Terrell rotation}}
[[File:Lorentz contractie - mural Leiden, 2017.jpg|upright=1.35|thumb|Formula on a wall in Leiden, Netherlands. Lorentz was chair of theoretical physics at the [[University of Leiden]] (1877–1910).]]
Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could suggest that if one could take a picture of a fast moving object, that the image would show the object contracted in the direction of motion. However, such visual effects are completely different measurements, as such a photograph is taken from a distance, while length contraction can only directly be measured at the exact location of the object's endpoints. It was shown by several authors such as [[Roger Penrose]] and James Terrell that moving objects generally do not appear length contracted on a photograph.<ref>{{cite journal |author=Kraus, U. |year=2000 |title=Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited |journal=American Journal of Physics |volume=68 |issue=1 |pages=56–60 |doi=10.1119/1.19373|url = http://www.tempolimit-lichtgeschwindigkeit.de/sphere/sphere.pdf |bibcode = 2000AmJPh..68...56K }}</ref> This result was popularized by [[Victor Weisskopf]] in a Physics Today article.<ref>{{Cite journal |doi = 10.1063/1.3057105|title = The visual appearance of rapidly moving objects|journal = Physics Today|volume = 13|issue = 9|pages = 24–27|year = 1960|last1 = Weisskopf|first1 = Victor F.|bibcode = 1960PhT....13i..24W|s2cid = 36707809}}</ref> For instance, for a small angular diameter, a moving sphere remains circular and is rotated.<ref>{{cite book|last=Penrose|first=Roger|title=The Road to Reality|year=2005|publisher=Vintage Books|location=London|isbn=978-0-09-944068-0|pages=430–431}}</ref> This kind of visual rotation effect is called Penrose-Terrell rotation.<ref>{{Cite web|url=https://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html|title=Can You See the Lorentz–Fitzgerald Contraction?|website=math.ucr.edu}}</ref>

==Derivation==

Length contraction can be derived in several ways:

=== Known moving length ===
In an inertial reference frame S, let <math>x_{1}</math> and <math>x_{2}</math> denote the endpoints of an object in motion. In this frame the object's length <math>L</math> is measured, according to the above conventions, by determining the simultaneous positions of its endpoints at <math>t_{1}=t_{2}</math>. Meanwhile, the proper length of this object, as measured in its rest frame S', can be calculated by using the Lorentz transformation. Transforming the time coordinates from S into S' results in different times, but this is not problematic, since the object is at rest in S' where it does not matter when the endpoints are measured. Therefore, the transformation of the spatial coordinates suffices, which gives:<ref name=born>{{Citation|author=Born, Max|author-link=Max Born|title=Einstein's Theory of Relativity|publisher=Dover Publications|year=1964|isbn=0-486-60769-0|url-access=registration|url=https://archive.org/details/einsteinstheoryo0000born}}</ref>

:<math>x'_{1}=\gamma\left(x_{1}-vt_{1}\right)\quad\text{and}\quad x'_{2}=\gamma\left(x_{2}-vt_{2}\right) \ \ .</math>

Since <math>t_1 = t_2</math>, and by setting <math>L=x_{2}-x_{1}</math> and <math>L_{0}^{'}=x_{2}^{'}-x_{1}^{'}</math>, the proper length in S' is given by

{{NumBlk|:|<math>L_{0}^{'}=L\cdot\gamma \ \ . </math>|{{EquationRef|1}}}}

Therefore, the object's length, measured in the frame S, is contracted by a factor <math>\gamma</math>:

{{NumBlk|:|<math>L=L_{0}^{'}/\gamma \ \ . </math>|{{EquationRef|2}}}}

Likewise, according to the principle of relativity, an object that is at rest in S will also be contracted in S'. By exchanging the above signs and [[Prime (symbol)|primes]] symmetrically, it follows that

{{NumBlk|:|<math>L_{0}=L'\cdot\gamma \ \ . </math>|{{EquationRef|3}}}}

Thus an object at rest in S, when measured in S', will have the contracted length

{{NumBlk|:|<math>L'=L_{0}/\gamma \ \ . </math>|{{EquationRef|4}}}}

=== Known proper length ===
Conversely, if the object rests in S and its proper length is known, the simultaneity of the measurements at the object's endpoints has to be considered in another frame S', as the object constantly changes its position there. Therefore, both spatial and temporal coordinates must be transformed:<ref>{{Cite book |author=Walter Greiner|title=Classical Mechanics: Point Particles and Relativity |publisher=Springer |year=2006 |isbn=9780387218519 |url={{Google books|plainurl=y|id=CynrBwAAQBAJ|text=length contraction|page=396}}}}; Equations 31.4 – 31.6</ref>

:<math>\begin{align}
x_{1}^{'} & =\gamma\left(x_{1}-vt_{1}\right) & \quad\mathrm{and}\quad &  & x_{2}^{'} & =\gamma\left(x_{2}-vt_{2}\right)\\
t_{1}^{'} & =\gamma\left(t_{1}-vx_{1}/c^{2}\right) & \quad\mathrm{and}\quad &  & t_{2}^{'} & =\gamma\left(t_{2}-vx_{2}/c^{2}\right)
\end{align}</math>

Computing length interval <math>\Delta x'=x_{2}^{\prime}-x_{1}^{\prime}</math> as well as assuming simultaneous time measurement <math>\Delta t'=t_{2}^{\prime}-t_{1}^{\prime}=0</math>, and by plugging in proper length <math>L_{0}=x_{2}-x_{1}</math>, it follows:

:<math>\begin{align}\Delta x' & =\gamma\left(L_{0}-v\Delta t\right) & (1)\\
\Delta t' & =\gamma\left(\Delta t-\frac{vL_{0}}{c^{2}}\right)=0 & (2)
\end{align}
</math>

Equation (2) gives

:<math>\Delta t=\frac{vL_{0}}{c^{2}}</math>

which, when plugged into (1), demonstrates that <math>\Delta x'</math> becomes the contracted length <math>L'</math>:

:<math>L'=L_{0}/\gamma</math>.

Likewise, the same method gives a symmetric result for an object at rest in S':

:<math>L=L^{'}_{0}/\gamma</math>.

=== Using time dilation ===
<!--This section is linked from [[Derivations of the Lorentz transformations]]-->

Length contraction can also be derived from [[time dilation]],<ref>{{Citation|author=[[David Halliday (physicist)|David Halliday]], [[Robert Resnick]], [[Jearl Walker]]|title = Fundamentals of Physics, Chapters 33-37|publisher=John Wiley & Son|year =2010|isbn=978-0470547946|pages=1032f }}</ref> according to which the rate of a single "moving" clock (indicating its [[proper time]] <math>T_0</math>) is lower with respect to two synchronized "resting" clocks (indicating <math>T</math>). Time dilation was experimentally confirmed multiple times, and is represented by the relation:

:<math>T=T_{0}\cdot\gamma</math>

Suppose a rod of proper length <math>L_0</math> at rest in <math>S</math> and a clock at rest in <math>S'</math> are moving along each other with speed <math>v</math>. Since, according to the principle of relativity, the magnitude of relative velocity is the same in either reference frame, the respective travel times of the clock between the rod's endpoints are given by <math>T=L_{0}/v</math> in <math>S</math> and <math>T'_{0}=L'/v</math> in <math>S'</math>, thus <math>L_{0}=Tv</math> and <math>L'=T'_{0}v</math>. By inserting the time dilation formula, the ratio between those lengths is:

:<math>\frac{L'}{L_{0}}=\frac{T'_{0}v}{Tv}=1/\gamma</math>.

Therefore, the length measured in <math>S'</math> is given by

:<math>L'=L_{0}/\gamma</math>

So since the clock's travel time across the rod is longer in <math>S</math> than in <math>S'</math> (time dilation in <math>S</math>), the rod's length is also longer in <math>S</math> than in <math>S'</math> (length contraction in <math>S'</math>). Likewise, if the clock were at rest in <math>S</math> and the rod in <math>S'</math>, the above procedure would give

:<math>L=L'_{0}/\gamma</math>

=== Geometrical considerations ===

[[File:Slabs.svg|thumb|250px|Cuboids in Euclidean and Minkowski spacetime]]

Additional geometrical considerations show that length contraction can be regarded as a ''trigonometric'' phenomenon, with analogy to  parallel slices through a [[cuboid]] before and after a ''rotation'' in '''E'''<sup>3</sup> (see left half figure at the right).  This is the Euclidean analog of ''boosting'' a cuboid in '''E'''<sup>1,2</sup>.  In the latter case, however, we can interpret the boosted cuboid as the ''world slab'' of a moving plate.

''Image'': Left: a ''rotated cuboid'' in three-dimensional euclidean space '''E'''<sup>3</sup>. The cross section is ''longer'' in the direction of the rotation than it was before the rotation. Right: the ''world slab'' of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed) '''E'''<sup>1,2</sup>, which is a ''boosted cuboid''. The cross section is ''thinner'' in the direction of the boost than it was before the boost. In both cases, the transverse directions are unaffected and the three planes meeting at each corner of the cuboids are ''mutually orthogonal'' (in the sense of '''E'''<sup>1,2</sup> at right, and in the sense of '''E'''<sup>3</sup> at left).

In special relativity, [[Poincaré group|Poincaré transformations]] are a class of [[affine transformation]]s which can be characterized as the transformations between alternative [[Cartesian coordinates|Cartesian coordinate charts]] on [[Minkowski spacetime]] corresponding to alternative states of [[inertial frame|inertial motion]] (and different choices of an [[origin (mathematics)|origin]]).  Lorentz transformations are Poincaré transformations which are [[linear transformation]]s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the [[Lorentz group]] forms the ''isotropy group'' of the self-isometries of the spacetime) which are played by [[rotation]]s in euclidean geometry.  Indeed, special relativity largely comes down to studying a kind of noneuclidean [[trigonometry]] in Minkowski spacetime, as suggested by the following table:

{| class="wikitable" style="text-align:center"
|+ Three plane trigonometries
|-
! Trigonometry 
! Circular !! Parabolic !! Hyperbolic
|-
! Kleinian Geometry 
| Euclidean plane      || Galilean plane        || Minkowski plane
|-
! Symbol 
| '''E'''<sup>2</sup>  || '''E'''<sup>0,1</sup> || '''E'''<sup>1,1</sup>
|-
! Quadratic form 
| Positive definite || Degenerate || Non-degenerate but indefinite
|- 
! Isometry group 
|  '''E'''(2) ||  '''E'''(0,1) ||  '''E'''(1,1)
|-
! Isotropy group 
| '''SO'''(2) || '''SO'''(0,1) || '''SO'''(1,1)
|-
! Type of isotropy
| Rotations || Shears || Boosts  
|- 
! Algebra over R 
| [[Complex number]]s || [[Dual number]]s || [[Split-complex number]]s 
|-
! ε<sup>2</sup> 
| −1 || 0 || 1
|-
! Spacetime interpretation 
| None || Newtonian spacetime || Minkowski spacetime
|-
! Slope 
| tan φ = m || tanp φ = u  || tanh φ = v
|-
! "cosine" 
| cos φ = (1 + m<sup>2</sup>)<sup>−1/2</sup> || cosp φ = 1 || cosh φ = (1 − v<sup>2</sup>)<sup>−1/2</sup>
|-
! "sine" 
| sin φ = m (1 + m<sup>2</sup>)<sup>−1/2</sup> || sinp φ = u || sinh φ = v (1 − v<sup>2</sup>)<sup>−1/2</sup> 
|- 
! "secant" 
| sec φ = (1 + m<sup>2</sup>)<sup>1/2</sup> || secp φ = 1 || sech φ = (1 − v<sup>2</sup>)<sup>1/2</sup>
|-
! "cosecant" 
| csc φ = m<sup>−1</sup> (1 + m<sup>2</sup>)<sup>1/2</sup> || cscp φ = u<sup>−1</sup> || csch φ = v<sup>−1</sup> (1 − v<sup>2</sup>)<sup>1/2</sup>
|}

==References==
{{Reflist}}

== External links ==
*Physics FAQ: [http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html Can You See the Lorentz–Fitzgerald Contraction? Or: Penrose-Terrell Rotation]; [http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html The Barn and the Pole]

{{Relativity}}

[[Category:Special relativity]]
[[Category:Length]]
[[Category:Hendrik Lorentz]]