HistoryLength contraction was postulated by (1889) and (1892) to explain the negative outcome of the and to rescue the hypothesis of the stationary aether ( ). Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after , who derived this deformation from electromagnetic theory in 1888), it was considered an , because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones. In 1897 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 (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. Eventually, (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 , which changed the notions of space, time, and simultaneity. Einstein's view was further elaborated by , who demonstrated the geometrical interpretation of all relativistic effects by introducing his concept of four-dimensional .
Basis in relativityFirst 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 . If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the 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 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 of the moving object. Using this method, the definition of is crucial for measuring the length of moving objects. Another method is to use a clock indicating its , 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. 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 and 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 and time dilation (see ). 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 * is the length observed by an observer in motion relative to the object * is the proper length (the length of the object in its rest frame) * is the '' '', defined as where ** is the relative velocity between the observer and the moving object ** 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 . 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 (30 million mph, 0.0447) contracted length is 99.9% of the length at rest; at a speed of (95 million mph, 0.141), the length is still 99%. As the magnitude of the velocity approaches the speed of light, the effect becomes prominent.
SymmetryThe 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 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 s, because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional .
Magnetic forcesMagnetic 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. In 1820, 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 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 verificationsAny 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 ). 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 (and later also the ). 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, 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 . However, in the muon's frame, the effect is explained by the atmosphere being contracted, shortening the trip. *Heavy s 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. * The 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. * In s and s, relativistic electrons were injected into an , so that 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 . So, only with the aid of length contraction and the relativistic Doppler effect, the extremely small wavelength of undulator radiation can be explained.
Reality of length contractionIn 1911 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. Einstein published a rebuttal: 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.
ParadoxesDue to superficial application of the contraction formula some paradoxes can occur. Examples are the and . However, those paradoxes can be solved by a correct application of relativity of simultaneity. Another famous paradox is the , which proves that the concept of is not compatible with relativity, reducing the applicability of , and showing that for a co-rotating observer the geometry is in fact non-Euclidean.
Visual effectsLength 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 and James Terrell that moving objects generally do not appear length contracted on a photograph. This result was popularized by in a Physics Today article. For instance, for a small angular diameter, a moving sphere remains circular and is rotated. This kind of visual rotation effect is called Penrose-Terrell rotation.
DerivationLength contraction can be derived in several ways:
Known moving lengthIn 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: : 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 lengthConversely, 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: : Computing length interval as well as assuming simultaneous time measurement , and by plugging in proper length , it follows: : Equation (2) gives : which, when plugged into (1), demonstrates that becomes the contracted length : :. Likewise, the same method gives a symmetric result for an object at rest in S': :.
Using time dilationLength contraction can also be derived from , according to which the rate of a single "moving" clock (indicating its ) 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 considerationsAdditional geometrical considerations show, that length contraction can be regarded as a ''trigonometric'' phenomenon, with analogy to parallel slices through a before and after a ''rotation'' in E3 (see left half figure at the right). This is the Euclidean analog of ''boosting'' a cuboid in E1,2. 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 three-dimensional euclidean space E3. 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) E1,2, 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 E1,2 at right, and in the sense of E3 at left). In special relativity, Poincaré transformations are a class of s which can be characterized as the transformations between alternative on corresponding to alternative states of inertial motion (and different choices of an ). Lorentz transformations are Poincaré transformations which are s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the forms the ''isotropy group'' of the self-isometries of the spacetime) which are played by s in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean in Minkowski spacetime, as suggested by the following table:
External links*Physics FAQ