Length contraction

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Foundations and formulation Relative motion Inertial frame of reference Speed of light Maxwell's equations Lorentz transformation
Consequences Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disk Bell's spaceship paradox Ehrenfest paradox
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In physics, length contraction is the phenomenon of a decrease in length measured by an observer of objects which are traveling at any non-zero velocity relative to the observer. This contraction (more formally called Lorentz contraction or Lorentz–FitzGerald contraction after Hendrik Lorentz and George 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. This effect is negligible at everyday speeds, and can be ignored for all regular purposes. Only at greater speeds does it become relevant. At a speed of 13,400,000 m/s (30 million mph, 0.0447 $c$ ), the contracted length is 99.9% of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141 $c$ ), 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:

$L={\frac {L_{{0}}}{\gamma (v)}}=L_{{0}}{\sqrt {1-v^{{2}}/c^{{2}}}}$

where

L 0

is the proper length (the length of the object in its rest frame),

L

is the length observed by an observer in relative motion with respect to the object,

v

is the relative velocity between the observer and the moving object,

c

is the speed of light,

and the Lorentz factor, $γ(v)$ , is defined as

$\gamma (v)\equiv {\frac {1}{{\sqrt {1-v^{2}/c^{2}}}}}\$ .

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.

History

Main article: History of special relativity

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).^[1]^[2] 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.^[3]

Eventually, Albert Einstein (1905) was the first^[3] who completely removed the ad hoc character from the contraction hypothesis, by demonstrating that this contraction was not due to motion through a supposed aether, but was due to the change in the notions of space, time and simultaneity brought about by special relativity.^[4] Einstein's view was further elaborated by Hermann Minkowski and others, who demonstrated the geometrical meaning of all relativistic effects in spacetime.^[5]

Basis in relativity

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.

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.^[6] 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:

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 $L$ of the moving object.^[6] 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 $L_{{0}}=T\cdot v$ in the rod's rest frame or $L=T_{{0}}\cdot v$ in the clock's rest frame.^[7]

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, yet the length of the same object as measured in another inertial 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 $v$ is the relative velocity and $c$ the speed of light)

$L=L_{{0}}/\gamma .$

Symmetry

Minkowski diagram

In S all events parallel to the axis of x are simultaneous, while in S' all events parallel to the axis of x' are simultaneous.

A rod is transported from S to S'

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.^[8]^[9]

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 $v=0{.}8c$ are given. In S a rod with proper length $L_{0}={\mathrm {AB}}=30\ {\mathrm {cm}}$ is located, so its contracted length $L'$ in S' is given by:

$L'={\mathrm {AC}}=L_{{0}}/\gamma =18\ {\mathrm {cm}}.$

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 $L'_{0}={\mathrm {EF}}=30\ {\mathrm {cm}}$ 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):

$L={\mathrm {DE}}=L'_{{0}}/\gamma =18\ {\mathrm {cm}}.$

Experimental verifications

Reality of length contraction

Minkowski diagram of Einstein's 1911 thought experiment on length contraction. Two rods of rest length $A'B'=A''B''=L_{0}$ are moving with 0.6c in opposite direction, resulting in $A^{\ast }B^{\ast }<L_{0}$ .

In 1911 Vladimir Varićak asserted that length contraction is "real" according to Lorentz, while it is "apparent or subjective" according to Einstein.^[17] 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.^[18]
—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 same proper length. Let them move in opposite directions with 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.^[18]

Paradoxes

Due to superficial application of the contraction formula some paradoxes can occur. For examples see the Ladder paradox or 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, and reduces the applicability of Born rigidity. It also shows that for a co-rotating observer the geometry is in fact non-euclidean.

Visual effects

Main article: Terrell rotation

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 do not appear length contracted on a photograph.^[19] For instance, for a small angular diameter, a moving sphere remains circular and is rotated.^[20] This kind of visual rotation effect is called Penrose-Terrell rotation.^[21]

Derivation

Lorentz transformation

Length contraction can be derived from the Lorentz transformation in several ways:

${\begin{aligned}x'&=\gamma \left(x-vt\right),\\t'&=\gamma \left(t-vx/c^{{2}}\right).\end{aligned}}$

Moving length is known

In an inertial reference frame S, $x_{{1}}$ and $x_{{2}}$ shall denote the endpoints of an object in motion in this frame. There, its length $L$ was measured according to the above convention by determining the simultaneous positions of its endpoints at $t_{{1}}=t_{{2}}\,$ . 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:^[6]

$x'_{{1}}=\gamma \left(x_{{1}}-vt_{{1}}\right)\quad {\mathrm {and}}\quad x'_{{2}}=\gamma \left(x_{{2}}-vt_{{2}}\right).$

Since $t_{{1}}=t_{{2}}\,$ , and by setting $L=x_{{2}}-x_{{1}}\,$ and $L_{{0}}^{{'}}=x_{{2}}^{{'}}-x_{{1}}^{{'}}$ , the proper length in S' is given by

$L_{{0}}^{{'}}=L\cdot \gamma .\qquad \qquad {\text{(1)}},$

with respect to which the measured length in S is contracted by

$L=L_{{0}}^{{'}}/\gamma .\qquad \qquad {\text{(2)}}$

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:

$L_{{0}}=L'\cdot \gamma .\qquad \qquad {\text{(3)}}$

Thus the contracted length as measured in S' is given by:

$L'=L_{{0}}/\gamma .\qquad \qquad {\text{(4)}}$

Proper length is known

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: ^[22]

${\begin{aligned}x_{{1}}^{{'}}&=\gamma \left(x_{{1}}-vt_{{1}}\right)&\quad {\mathrm {and}}\quad &&x_{{2}}^{{'}}&=\gamma \left(x_{{2}}-vt_{{2}}\right)\\t_{{1}}^{{'}}&=\gamma \left(t_{{1}}-vx_{{1}}/c^{{2}}\right)&\quad {\mathrm {and}}\quad &&t_{{2}}^{{'}}&=\gamma \left(t_{{2}}-vx_{{2}}/c^{{2}}\right).\end{aligned}}$

With $t_{{1}}=t_{{2}}$ and $L_{{0}}=x_{{2}}-x_{{1}}$ this results in non-simultaneous differences:

${\begin{aligned}\Delta x'&=\gamma L_{{0}}\\\Delta t'&=\gamma vL_{{0}}/c^{{2}}\end{aligned}}$

In order to obtain the simultaneous positions of both endpoints, the distance traveled by the second endpoint with $v$ during $\Delta t'$ must be subtracted from $\Delta x'$ :

${\begin{aligned}L'&=\Delta x'-v\Delta t'\\&=\gamma L_{{0}}-\gamma v^{{2}}L_{{0}}/c^{{2}}\\&=L_{{0}}/\gamma \end{aligned}}$

So the moving length in S' is contracted. Likewise, the preceding calculation gives a symmetric result for an object at rest in S':

$L=L_{{0}}^{{'}}/\gamma$ .

Time dilation

Length contraction can also be derived from time dilation,^[23] according to which the rate of a single "moving" clock (indicating its proper time $T_{0}$ ) is lower with respect to two synchronized "resting" clocks (indicating $T$ ). Time dilation was experimentally confirmed multiple times, and is represented by the relation:

$T=T_{{0}}\cdot \gamma$ .

Suppose a rod of proper length $L_{0}$ at rest in $S$ and a clock at rest in $S'$ are moving along each other. The respective travel times of the clock between the rod's endpoints are given by $T=L_{{0}}/v$ in $S$ and $T'_{{0}}=L'/v$ in $S'$ , thus $L_{{0}}=Tv$ and $L'=T'_{{0}}v$ . By inserting the time dilation formula, the ratio between those lengths is:

${\frac {L'}{L_{{0}}}}={\frac {T'_{{0}}v}{Tv}}=1/\gamma$ .

Therefore, the length measured in $S'$ is given by

$L'=L_{{0}}/\gamma$ .

So the effect that the moving clock indicates a lower travel time in $S$ due to time dilation, is interpreted in $S'$ as due to length contraction of the moving rod. Likewise, if the clock were at rest in $S$ and the rod in $S'$ , the above procedure would give

$L=L'_{{0}}/\gamma$ .

Geometrical considerations

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 E³ (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 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) 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 E³ 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:

**Three plane trigonometries**
Trigonometry	Circular	Parabolic	Hyperbolic
Kleinian Geometry	euclidean plane	Galilean plane	Minkowski plane
Symbol	E²	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
Cayley algebra	complex numbers	dual numbers	split-complex numbers
ε²	-1	0	1
Spacetime interpretation	none	Newtonian spacetime	Minkowski spacetime
slope	tan φ = m	tanp φ = u	tanh φ = v
"cosine"	cos φ = (1+m²)^-1/2	cosp φ = 1	cosh φ = (1-v²)^-1/2
"sine"	sin φ = m (1+m²)^-1/2	sinp φ = u	sinh φ = v (1-v²)^-1/2
"secant"	sec φ = (1+m²)^1/2	secp φ = 1	sech φ = (1-v²)^1/2
"cosecant"	csc φ = m⁻¹ (1+m²)^1/2	cscp φ = u⁻¹	csch φ = v⁻¹ (1-v²)^1/2

References

↑ FitzGerald, George Francis (1889), "The Ether and the Earth's Atmosphere", Science 13 (328): 390, Bibcode:1889Sci....13..390F, doi:10.1126/science.ns-13.328.390, PMID 17819387
↑ Lorentz, Hendrik Antoon (1892), "The Relative Motion of the Earth and the Aether", Zittingsverlag Akad. V. Wet. 1: 74–79
↑ 3.0 3.1 Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN 0-19-520438-7
↑ Einstein, Albert (1905a), "Zur Elektrodynamik bewegter Körper", 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
↑ 6.0 6.1 6.2 Born, Max (1964), Einstein's Theory of Relativity, Dover Publications, ISBN 0-486-60769-0
↑ Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. New York: W. H. Freeman. ISBN 0-7167-2327-1.
↑ Albert Shadowitz (1988). Special relativity (Reprint of 1968 edition ed.). Courier Dover Publications. pp. 20–22. ISBN 0-486-65743-4.
↑ Leo Sartori (1996). Understanding Relativity: a simplified approach to Einstein's theories. University of California Press. pp. 151ff. ISBN 0-520-20029-2.
↑ 10.0 10.1 Sexl, Roman & Schmidt, Herbert K. (1979), Raum-Zeit-Relativität, Braunschweig: Vieweg, ISBN 3-528-17236-3
↑ Brookhaven National Laboratory. "The Physics of RHIC". Retrieved 2013.
↑ Manuel Calderon de la Barca Sanchez. "Relativistic heavy ion collisions". Retrieved 2013.
↑ Hands, Simon (2001). "The phase diagram of QCD". Contemporary Physics 42 (4): 209–225. arXiv:physics/0105022. 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, doi:10.1098/rspa.1931.0008
↑ DESY photon science. "What is SR, how is it generated and what are its properties?". Retrieved 2013.
↑ DESY photon science. "FLASH The Free-Electron Laser in Hamburg (PDF 7,8 MB)". Retrieved 2013.
↑ 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 0-201-04679-2
↑ 18.0 18.1 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 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.
↑ Kraus, U. (2000). "Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited". American Journal of Physics 68 (1): 56–60. doi:10.1119/1.19373.
↑ Penrose, Roger (2005). The Road to Reality. London: Vintage Books. pp. 430–431. ISBN 978-0-09-944068-0.
↑ Can You See the Lorentz-Fitzgerald Contraction? Or: Penrose-Terrell Rotation
↑ Bernard Schutz (2009). "Lorentz contraction". A First Course in General Relativity. Cambridge University Press. p. 18. ISBN 0521887054. (GBS, p. 18.)
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Relativity

Special
relativity

Background	Principle of relativity Introduction to special relativity Special relativity

Foundations	Relative motion Frame of reference Speed of light Maxwell's equations

Formulation	Galilean relativity Galilean transformation Lorentz transformation

Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks

Spacetime	Minkowski spacetime World line Spacetime diagrams Light cone

General
relativity

Background	Introduction Mathematical formulation Resources

Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Spacetime diagrams Spacetime in General relativity

Phenomena	Kepler problem Lenses Waves Frame-dragging Geodetic effect Event horizon Singularity Black hole

Equations	Linearized gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism Hamilton–Jacobi–Einstein equation

Advanced theories	Kaluza–Klein Quantum gravity

Solutions	Schwarzschild Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub-NUT Milne Robertson–Walker pp-wave van Stockum dust

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