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 (more formally called 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 parallel to the direction in which the observed body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes. Only at greater speeds, or for electron motion, does it become significant. 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 dominant, as can be seen from the formula:^{[1]}
where
and the Lorentz factor, ?(v), is defined as
In this equation it is assumed that the object is parallel with its 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 viewing 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.
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 ("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.^{[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 four-dimensional spacetime.^{[6]}
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 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 after 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 in the line of motion, and can be represented by the following relation (where is the relative velocity and the speed of light)
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.^{[9]}^{[10]}
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.^{[]}
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.^{[11]}^{[12]}
First image: If a rod at rest in S' is given, then its endpoints are located upon the ct' axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x') positions of the endpoints are O and B, thus the proper length is given by OB. But in S the simultaneous (parallel to the axis of x) positions are O and A, thus the contracted length is given by OA.
On the other hand, if another rod is at rest in S, then its endpoints are located upon the ct axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x) positions of the endpoints are O and D, thus the proper length is given by OD. But in S' the simultaneous (parallel to the axis of x') positions are O and C, thus the contracted length is given by OC.
Second image: A train at rest in S and a station at rest in S' with relative velocity of are given. In S a rod with proper length is located, so its contracted length in S' is given by:
Then the rod will be thrown out of the train in S and will come to rest at the station in S'. Its length has to be measured again according to the methods given above, and now the proper length will be measured in S' (the rod has become larger in that system), while in S the rod is in motion and therefore its length is contracted (the rod has become smaller in that system):
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:
In 1911 Vladimir Vari?ak 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 non-comoving 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. Let them move in opposite directions at the same speed with respect to a resting coordinate x-axis. Endpoints A'A" meet at point A*, and B'B" meet at point B*, both points being marked on that axis. Einstein pointed out that length A*B* is shorter than A'B' or A"B", which can also be demonstrated by one of the rods when brought to rest with respect to that axis.^{[21]}
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 co-rotating observer the geometry is in fact non-euclidean.
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 Penrose-Terrell rotation.^{[24]}
Length contraction can be derived from the Lorentz transformation in several ways:
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:
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 non-simultaneous 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':
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
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 three-dimensional 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 self-isometries 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 | 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 numbers | dual numbers | split-complex numbers |
?^{2} | -1 | 0 | 1 |
Spacetime interpretation | none | Newtonian spacetime | Minkowski spacetime |
slope | tan ? = m | tanp ? = u | tanh ? = v |
"cosine" | cos ? = (1+m^{2})^{-1/2} | cosp ? = 1 | cosh ? = (1-v^{2})^{-1/2} |
"sine" | sin ? = m (1+m^{2})^{-1/2} | sinp ? = u | sinh ? = v (1-v^{2})^{-1/2} |
"secant" | sec ? = (1+m^{2})^{1/2} | secp ? = 1 | sech ? = (1-v^{2})^{1/2} |
"cosecant" | csc ? = m^{-1} (1+m^{2})^{1/2} | cscp ? = u^{-1} | csch ? = v^{-1} (1-v^{2})^{1/2} |
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