Length contraction

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Length contraction
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{{Special relativity sidebar |consequences}}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.BOOK, Tensors, Relativity, and Cosmology, 2nd, Mirjana, Dalarsson, Nils, Dalarsson, Academic Press, 2015, 978-0-12-803401-9, 106-108,weblink Extract of page 106 This contraction (also known as Lorentz contraction or Lorentz–FitzGerald contraction after Hendrik Lorentz and George Francis FitzGerald) 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.


Length contraction was postulated by 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–FitzGerald contraction hypothesis).{{Citation|author=FitzGerald, George Francis|year=1889|title=s:The Ether and the Earth's Atmosphere|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 }}{{Citation|last=Lorentz|first=Hendrik Antoon|year=1892|title=s:Translat|journal=Zittingsverlag Akad. V. Wet.|volume=1|pages=74–79}}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 (Poincaré stresses) that ensure the electron's stability, give a dynamical explanation for length contraction, and thus hide the motion of the stationary aether.{{Citation|author=Pais, Abraham|authorlink=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}}Eventually, Albert Einstein (1905) was the first to completely remove the ad hoc character from the contraction hypothesis, by demonstrating that this contraction did not require motion through a supposed aether, but could be explained using special relativity, which changed our notions of space, time, and simultaneity.{{Citation|doi=10.1002/andp.19053221004|author=Einstein, Albert|year=1905a|title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik|volume=322|issue=10
bibcode = 1905AnP...322..891EEnglish translation. Einstein's view was further elaborated by Hermann Minkowski, who demonstrated the geometrical interpretation of all relativistic effects by introducing his concept of four-dimensional spacetime.{{Citation>author=Minkowski, Hermanntitle=s:>journal=Physikalische Zeitschriftpages=75–88}}
*Various English translations on Wikisource: s:Space and Time|Space and Time]]

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. 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 L_0 of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity > 0, then one can proceed as follows:(File:Lorentzkontraktion.png|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 Poincaré-Einstein synchronization, or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of 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 L of the moving object. Using this method, the definition of simultaneity is crucial for measuring the length of moving objects.Another method is to use a clock indicating its proper time T_0, which is traveling from one endpoint of the rod to the other in time T 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 L_{0}=Tcdot v in the rod's rest frame or L=T_{0}cdot v in the clock's rest frame.BOOK, Edwin F. Taylor, John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity, 1992, W. H. Freeman, New York, 0-7167-2327-1, In Newtonian mechanics, simultaneity and time duration are absolute and therefore both methods lead to the equality of L and L_0. 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 T and T_0 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). 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}} is the length observed by an observer in motion relative to the object {{math|L0}} is the proper length (the length of the object in its rest frame) {{math|γ(v)}} is the Lorentz factor, defined as gamma (v) equiv frac{1}{sqrt{1-v^2/c^2}}
{{math|v}} is the relative velocity between the observer and the moving object {{math|c}} is the speed of light
Replacing the Lorentz factor in the original formula leads to the relation
L =L_{0}sqrt{1-v^{2}/c^{2}}
In this equation both L and L0 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 13,400,000 m/s (30 million mph, 0.0447{{math|c}}) contracted length is 99.9% of the length at rest; at a speed of 42,300,000 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.


The principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames) requires that length contraction is symmetrical: If a rod rests in inertial frame S, it has its proper length in S and its length is contracted in S'. However, if a rod rests in S', it has its proper length in S' and its length is contracted in S. This can be vividly illustrated using symmetric Minkowski diagrams (or Loedel diagrams), because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime.BOOK, Albert Shadowitz, Special relativity, 0-486-65743-4, Courier Dover Publications, Reprint of 1968, 1988, 20–22, BOOK, Leo Sartori, Understanding Relativity: a simplified approach to Einstein's theories, 0-520-20029-2, University of California Press, 1996, 151ff,

Magnetic forces

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.BOOK, he Feynman Lectures on Physics, Desktop Edition Volume II: The New Millennium Edition, illustrated, Richard P., Feynman, Robert B., Leighton, Matthew, Sands, Basic Books, 2013-01-01, 978-0-465-07998-8, 13-6,weblink Extract of page 13-6BOOK, E M Lifshitz, L D Landau, The classical theory of ields, Vol. 2, Course of Theoretical Physics, Fourth, Butterworth-Heinemann, Oxford UK, 1980, 0-7506-2768-9,weblink In 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. To the electrons, the 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.{{cn|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 at the current state of technology, objects of considerable extension cannot be accelerated to relativistic speeds. And the only objects traveling with the speed required are atomic particles, yet 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) & 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, muons' 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.
  • Heavy ions that are spherical when at rest should assume the form of "pancakes" or flat disks when traveling nearly at the speed of light. And in fact, the results obtained from particle collisions can only be explained when the increased nucleon density due to length contraction is considered.WEB, Brookhaven National Laboratory,weblink The Physics of RHIC, 2013-01-01, WEB, Manuel Calderon de la Barca Sanchez,weblink Relativistic heavy ion collisions, 2013-01-01, JOURNAL, Hands, Simon, The phase diagram of QCD, 2001, Contemporary Physics, 42, 4, 209–225, 10.1080/00107510110063843, physics/0105022, 2001ConPh..42..209H,
  • 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. Though in relativity, the higher-than-expected ionization ability can be explained by length contraction of the Coulomb field in frames in which the ionizing particles are moving, which increases their electrical field strength normal to the line of motion.{{Citation|author1=Sexl, Roman |author2=Schmidt, Herbert K.|title=Raum-Zeit-Relativität|year=1979|publisher=Vieweg|location=Braunschweig|isbn=3-528-17236-3}}{{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 }}
  • In synchrotrons and free-electron lasers, 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.WEB, DESY photon science,weblink What is SR, how is it generated and what are its properties?, 2013-01-01, yes,weblink" title="">weblink 2016-06-03, WEB, DESY photon science,weblink FLASH The Free-Electron Laser in Hamburg (PDF 7,8 MB), 2013-01-01,

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 A'B'=AB=L_0 are moving with 0.6c in opposite directions, resulting in A^ast B^ast

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