Length contraction
Special relativity 


Foundations

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.^{[1]} 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.
History
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).^{[2]}^{[3]} Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("HeavisideEllipsoid" 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: nonelectric 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.^{[4]}
Eventually, Albert Einstein (1905) was the first^{[4]} 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.^{[5]} Einstein's view was further elaborated by Hermann Minkowski, who demonstrated the geometrical interpretation of all relativistic effects by introducing his concept of fourdimensional spacetime.^{[6]}
Basis in relativity
First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.^{[7]} 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 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:
The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the PoincaréEinstein 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 of the moving object.^{[7]} 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 , which is traveling from one endpoint of the rod to the other in time 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 in the rod's rest frame or in the clock's rest frame.^{[8]}
In Newtonian mechanics, simultaneity and time duration are absolute and therefore both methods lead to the equality of and . 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 and 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
where
 L is the length observed by an observer in motion relative to the object
 L_{0} is the proper length (the length of the object in its rest frame)
 γ(v) is the Lorentz factor, defined as
where
 v is the relative velocity between the observer and the moving object
 c is the speed of light
Replacing the Lorentz factor in the original formula leads to the relation
In this equation both L and L_{0} 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, 0.0447c) contracted length is 99.9% of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141c), 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 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 fourdimensional spacetime.^{[9]}^{[10]}
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 currentcarrying wire is a result of relativistic motion between electrons and protons.^{[11]}^{[12]}
In 1820, AndréMarie 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.
Experimental verifications
Any observer comoving 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 TroutonRankine 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 noncomoving 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 noncomoving frame:
 It was the negative result of a famous experiment, that required the introduction of length contraction: the MichelsonMorley 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 nonmoving frame thus making its travel time longer, the factor by which the longitudinal beam would be delayed by taking times L/(cv) & 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 twoway 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.^{[13]}
 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.^{[14]}^{[15]}^{[16]}
 The ionization ability of electrically charged particles with large relative velocities is higher than expected. In prerelativistic 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 higherthanexpected 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.^{[13]}^{[17]}
 In synchrotrons and freeelectron 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.^{[18]}^{[19]}
Reality of length contraction
In 1911 Vladimir Varićak asserted that length contraction is "real" according to Lorentz, while it is "apparent or subjective" according to Einstein.^{[20]} Einstein replied:
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 noncomoving observer.^{[21]}
— 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_{0}, 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.^{[21]}
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 simply be solved by a correct application of relativity of simultaneity. Another famous paradox is the Ehrenfest paradox, which proves that the concept of rigid bodies is not compatible with relativity, reducing the applicability of Born rigidity, and showing that for a corotating observer the geometry is in fact noneuclidean.
Visual effects
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.^{[22]} For instance, for a small angular diameter, a moving sphere remains circular and is rotated.^{[23]} This kind of visual rotation effect is called PenroseTerrell rotation.^{[24]}
Derivation
Using the Lorentz transformation
Length contraction can be derived from the Lorentz transformation in several ways:
Known moving length
In an inertial reference frame S, and shall denote the endpoints of an object in motion in this frame. There, its length was measured according to the above convention by determining the simultaneous positions of its endpoints at . Now, the proper length of this object in S' shall be calculated by using the Lorentz transformation. Transforming the time coordinates from S into S' results in different times, but this is not problematic, as 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:^{[7]}
Since , and by setting and , the proper length in S' is given by
with respect to which the measured length in S is contracted by
According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. By exchanging the above signs and primes symmetrically, it follows:
Thus the contracted length as measured in S' is given by:
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:^{[25]}
With and this results in nonsimultaneous differences:
In order to obtain the simultaneous positions of both endpoints, the second endpoint must be advanced by with the speed of S relative to S'. To obtain the length , the quantity must therefore be added to :
So the moving length in S' is contracted. Likewise, the preceding calculation gives a symmetric result for an object at rest in S':
Using time dilation
Length contraction can also be derived from time dilation,^{[26]} according to which the rate of a single "moving" clock (indicating its proper time ) is lower with respect to two synchronized "resting" clocks (indicating ). Time dilation was experimentally confirmed multiple times, and is represented by the relation:
Suppose a rod of proper length at rest in and a clock at rest in are moving along each other with speed . 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 in and in , thus and . By inserting the time dilation formula, the ratio between those lengths is:
 .
Therefore, the length measured in is given by
So since the clock's travel time across the rod is longer in than in (time dilation in ), the rod's length is also longer in than in (length contraction in ). Likewise, if the clock were at rest in and the rod in , the above procedure would give
Geometrical considerations
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^{3} (see left half figure at the right). This is the Euclidean analog of boosting a cuboid in E^{1,2}. 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 threedimensional euclidean space E^{3}. 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^{1,2}, 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^{1,2} at right, and in the sense of E^{3} at left).
In special relativity, Poincaré transformations are a class of affine transformations which can be characterized as the transformations between alternative Cartesian coordinate charts on Minkowski spacetime corresponding to alternative states of inertial motion (and different choices of an origin). Lorentz transformations are Poincaré transformations which are linear transformations (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the Lorentz group forms the isotropy group of the selfisometries of the spacetime) which are played by rotations 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:
Trigonometry  Circular  Parabolic  Hyperbolic 

Kleinian Geometry  Euclidean plane  Galilean plane  Minkowski plane 
Symbol  E^{2}  E^{0,1}  E^{1,1} 
Quadratic form  positive definite  degenerate  nondegenerate 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 numbers  dual numbers  splitcomplex numbers 
ε^{2}  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^{2})^{−1/2}  cosp φ = 1  cosh φ = (1v^{2})^{−1/2} 
"sine"  sin φ = m (1+m^{2})^{−1/2}  sinp φ = u  sinh φ = v (1v^{2})^{−1/2} 
"secant"  sec φ = (1+m^{2})^{1/2}  secp φ = 1  sech φ = (1v^{2})^{1/2} 
"cosecant"  csc φ = m^{−1} (1+m^{2})^{1/2}  cscp φ = u^{−1}  csch φ = v^{−1} (1v^{2})^{1/2} 
References
 ↑ Dalarsson, Mirjana; Dalarsson, Nils (2015). Tensors, Relativity, and Cosmology (2nd ed.). Academic Press. p. 106108. ISBN 9780128034019. Extract of page 106
 ↑ FitzGerald, George Francis (1889), "The Ether and the Earth's Atmosphere", Science, 13 (328): 390, Bibcode:1889Sci....13..390F, doi:10.1126/science.ns13.328.390, PMID 17819387
 ↑ Lorentz, Hendrik Antoon (1892), "The Relative Motion of the Earth and the Aether", Zittingsverlag Akad. V. Wet., 1: 74–79
 1 2 Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN 0195204387
 ↑ Einstein, Albert (1905a), "Zur Elektrodynamik bewegter Körper" (PDF), Annalen der Physik, 322 (10): 891–921, Bibcode:1905AnP...322..891E, doi:10.1002/andp.19053221004 . See also: English translation.
 ↑ Minkowski, Hermann (1909), "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88
 Various English translations on Wikisource: Space and Time
 1 2 3 Born, Max (1964), Einstein's Theory of Relativity, Dover Publications, ISBN 0486607690
 ↑ Edwin F. Taylor; John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. New York: W. H. Freeman. ISBN 0716723271.
 ↑ Albert Shadowitz (1988). Special relativity (Reprint of 1968 ed.). Courier Dover Publications. pp. 20–22. ISBN 0486657434.
 ↑ Leo Sartori (1996). Understanding Relativity: a simplified approach to Einstein's theories. University of California Press. pp. 151ff. ISBN 0520200292.
 ↑ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (20130101). he Feynman Lectures on Physics, Desktop Edition Volume II: The New Millennium Edition (illustrated ed.). Basic Books. p. 136. ISBN 9780465079988. Extract of page 136
 ↑ E M Lifshitz, L D Landau (1980). The classical theory of ields. Course of Theoretical Physics. Vol. 2 (Fourth ed.). Oxford UK: ButterworthHeinemann. ISBN 0750627689.
 1 2 Sexl, Roman; Schmidt, Herbert K. (1979), RaumZeitRelativität, Braunschweig: Vieweg, ISBN 3528172363
 ↑ Brookhaven National Laboratory. "The Physics of RHIC". Retrieved 20130101.
 ↑ Manuel Calderon de la Barca Sanchez. "Relativistic heavy ion collisions". Retrieved 20130101.
 ↑ Hands, Simon (2001). "The phase diagram of QCD". Contemporary Physics. 42 (4): 209–225. arXiv:physics/0105022. Bibcode:2001ConPh..42..209H. doi:10.1080/00107510110063843.
 ↑ Williams, E. J. (1931), "The Loss of Energy by β Particles and Its Distribution between Different Kinds of Collisions", Proceedings of the Royal Society of London. Series A, 130 (813): 328–346, Bibcode:1931RSPSA.130..328W, doi:10.1098/rspa.1931.0008
 ↑ DESY photon science. "What is SR, how is it generated and what are its properties?". Archived from the original on 20160603. Retrieved 20130101.
 ↑ DESY photon science. "FLASH The FreeElectron Laser in Hamburg (PDF 7,8 MB)" (PDF). Retrieved 20130101.
 ↑ Miller, A.I. (1981), "Varičak and Einstein", Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, pp. 249–253, ISBN 0201046792
 1 2 Einstein, Albert (1911). "Zum Ehrenfestschen Paradoxon. Eine Bemerkung zu V. Variĉaks Aufsatz". Physikalische Zeitschrift. 12: 509–510. ; Original: Der Verfasser hat mit Unrecht einen Unterschied der Lorentzschen Auffassung von der meinigen mit Bezug auf die physikalischen Tatsachen statuiert. Die Frage, ob die LorentzVerkürzung 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.
 ↑ Kraus, U. (2000). "Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited" (PDF). American Journal of Physics. 68 (1): 56–60. Bibcode:2000AmJPh..68...56K. doi:10.1119/1.19373.
 ↑ Penrose, Roger (2005). The Road to Reality. London: Vintage Books. pp. 430–431. ISBN 9780099440680.
 ↑ Can You See the LorentzFitzgerald Contraction? Or: PenroseTerrell Rotation
 ↑ Bernard Schutz (2009). "Lorentz contraction". A First Course in General Relativity. Cambridge University Press. p. 18. ISBN 0521887054. Archived from the original on 20111002.
 ↑ David Halliday, Robert Resnick, Jearl Walker (2010), Fundamentals of Physics, Chapters 3337, John Wiley & Son, pp. 1032f, ISBN 0470547944