TheInfoList

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its
proper length Proper length or rest length is the length of an object in the object's rest frameIn special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest. The rest frame of compound obje ...
, which is the length as measured in the object's own
rest frame In special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entiti ...
.Extract of page 106
/ref> It is also known as Lorentz contraction or Lorentz–FitzGerald contraction (after
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical met ...

and
George Francis FitzGerald Prof Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "pers ...

) and is usually only noticeable at a substantial fraction of the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
. 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 (1889) and
Hendrik Antoon Lorentz File:H. A. Lorentz - rot B, rot E - La théorie electromagnétique de Maxwell et son application aux corps mouvants, Archives néerlandaises, 1892 - p 452 - Eq. IV & V.png, Lorentz' theory of electrons. Formulas for the Curl (mathematics), cur ...

(1892) to explain the negative outcome of the
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium Medium may refer to: Science and technolo ...
and to rescue the hypothesis of the stationary aether (
Lorentz–FitzGerald contraction hypothesis 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. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
). Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after
Oliver Heaviside Oliver Heaviside Fellow of the Royal Society, FRS (; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who brought complex numbers to Network analysis (electrical circuits), circuit analysis, invented a new technique ...
, who derived this deformation from electromagnetic theory in 1888), it was considered an
ad hoc hypothesis In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and prediction ...
, 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 Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical m ...
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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the s ...
(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,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

(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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
, which changed the notions of space, time, and simultaneity. Einstein's view was further elaborated by
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( an ...

, who demonstrated the geometrical interpretation of all relativistic effects by introducing his concept of four-dimensional
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
.

# Basis in relativity

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 In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a const ...
. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the
proper length Proper length or rest length is the length of an object in the object's rest frameIn special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest. The rest frame of compound obje ...
$L_0$ 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 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 Simultaneity is the relation between two events assumed to be happening at the same time in a given frame of reference In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, fr ...

is crucial for measuring the length of moving objects. Another method is to use a clock indicating its
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

$T_0$, 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 $L_ = T\cdot v$ in the rod's rest frame or $L = T_\cdot v$ in the clock's rest frame. 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

and
time dilation In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

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 In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

and time dilation (see
Derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the preced ...
). 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 :$L = \fracL_0$ 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 ''
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

'', defined as $\gamma (v) \equiv \frac$ 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 :$L = L_0\sqrt$ 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 In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to 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.

# Symmetry

The 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
Minkowski diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation In physics and Th ...

s, because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
.

# 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. In 1820,
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method o ...
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 Drift or Drifts may refer to: Geography * Drift or ford (crossing) A ford is a shallow place with good footing where a river A river is a natural flowing watercourse, usually freshwater, flowing towards an ocean, sea, lake or an ...
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 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 experimentThe Trouton–Rankine experiment was an experiment designed to measure if the Lorentz–FitzGerald contraction hypothesis, Lorentz–FitzGerald contraction of an object according to one frame (as defined by the luminiferous aether) produced a measura ...

). 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 The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium Medium may refer to: Science and technolo ...
(and later also the
Kennedy–Thorndike experiment The Kennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the lumini ...
). 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,
muon The muon (; from the Greek alphabet, Greek letter mu (letter), mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 ''e'' and a spin-½, spin of 1/2, but with a much greater ma ...

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
time dilation In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. However, in the muon's frame, the effect is explained by the atmosphere being contracted, shortening the trip. *Heavy
ion An ion () is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ...
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
ionization Ionization or ionisation is the process by which an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyda ...
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
synchrotron A synchrotron is a particular type of cyclic particle accelerator , a synchrotron collider type particle accelerator at Fermi National Accelerator Laboratory (Fermilab), Batavia, Illinois, USA. Shut down in 2011, until 2007 it was the most po ...

s and
free-electron laser A free-electron laser (FEL) is a (fourth generation) synchrotron light source A synchrotron light source is a source of electromagnetic radiation (EM) usually produced by a storage ring, for scientific and technical purposes. First observed i ...

s, relativistic electrons were injected into an
undulator An undulator is an insertion device An insertion device (ID) is a component in modern synchrotron light source crystal at the Daresbury Synchrotron Radiation Source, 1990 A synchrotron light source is a source of electromagnetic radiation ...

, so that
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physic ...
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 The relativistic Doppler effect is the change in frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of exi ...
. 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 contraction

In 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.

Due to superficial application of the contraction formula some paradoxes can occur. Examples are the
ladder paradoxThe ladder paradox (or barn-pole paradox) is a thought experiment in special relativity. It involves a ladder, parallel to the ground, travelling horizontally at relativistic speed (near the speed of light) and therefore undergoing a Lorentz contract ...
and
Bell's spaceship paradox Bell's spaceship paradox is a thought experiment in special relativity. It was designed by E. Dewan and M. Beran in 1959 and became more widely known when John Stewart Bell, J. S. Bell included a modified version.J. S. Bell: ''How to teach special ...
. However, those paradoxes can be solved by a correct application of relativity of simultaneity. Another famous paradox is the
Ehrenfest paradox The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity a ...

, which proves that the concept of
rigid bodies In physics, a rigid body (also known as a rigid object) is a solid Physical body, body in which deformation (engineering), deformation is zero or so small it can be neglected. The distance between any two given Point (geometry), points on a rigi ...
is not compatible with relativity, reducing the applicability of
Born rigidityBorn rigidity is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), kn ...
, and showing that for a co-rotating observer the geometry is in fact non-Euclidean.

# 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 Sir Roger Penrose (born 8 August 1931) is a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas an ...
and James Terrell that moving objects generally do not appear length contracted on a photograph. This result was popularized by
Victor Weisskopf Victor (aka Viktor) Frederick "Viki" Weisskopf (September 19, 1908 – April 22, 2002) was an Austrians, Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli and Niels Boh ...
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.

# Derivation

Length contraction can be derived in several ways:

## Known moving length

In an inertial reference frame S, $x_$ and $x_$ 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 $t_=t_$. 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: :$x\text{'}_=\gamma\left\left(x_-vt_\right\right)\quad\text\quad x\text{'}_=\gamma\left\left(x_-vt_\right\right)$ Since $t_1 = t_2$, and by setting $L=x_-x_$ and $L_^=x_^-x_^$, 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: :$\begin x_^ & =\gamma\left\left(x_-vt_\right\right) & \quad\mathrm\quad & & x_^ & =\gamma\left\left(x_-vt_\right\right)\\ t_^ & =\gamma\left\left(t_-vx_/c^\right\right) & \quad\mathrm\quad & & t_^ & =\gamma\left\left(t_-vx_/c^\right\right) \end$ Computing length interval $\Delta x\text{'}=x_^-x_^$ as well as assuming simultaneous time measurement $\Delta t\text{'}=t_^-t_^=0$, and by plugging in proper length $L_=x_-x_$, it follows: :$\begin\Delta x\text{'} & =\gamma\left\left(L_-v\Delta t\right\right) & \left(1\right)\\ \Delta t\text{'} & =\gamma\left\left(\Delta t-\frac\right\right)=0 & \left(2\right) \end$ Equation (2) gives :$\Delta t=\frac$ which, when plugged into (1), demonstrates that $\Delta x\text{'}$ becomes the contracted length $L\text{'}$: :$L\text{'}=L_/\gamma$. Likewise, the same method gives a symmetric result for an object at rest in S': :$L=L^_/\gamma$.

## Using time dilation

Length contraction can also be derived from
time dilation In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, according to which the rate of a single "moving" clock (indicating its
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

$T_0$) is lower with respect to two synchronized "resting" clocks (indicating $T$). Time dilation was experimentally confirmed multiple times, and is represented by the relation: :$T=T_\cdot\gamma$ Suppose a rod of proper length $L_0$ at rest in $S$ and a clock at rest in $S\text{'}$ are moving along each other with speed $v$. 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 $T=L_/v$ in $S$ and $T\text{'}_=L\text{'}/v$ in $S\text{'}$, thus $L_=Tv$ and $L\text{'}=T\text{'}_v$. By inserting the time dilation formula, the ratio between those lengths is: :$\frac=\frac=1/\gamma$. Therefore, the length measured in $S\text{'}$ is given by :$L\text{'}=L_/\gamma$ So since the clock's travel time across the rod is longer in $S$ than in $S\text{'}$ (time dilation in $S$), the rod's length is also longer in $S$ than in $S\text{'}$ (length contraction in $S\text{'}$). Likewise, if the clock were at rest in $S$ and the rod in $S\text{'}$, the above procedure would give :$L=L\text{'}_/\gamma$

## Geometrical considerations

Additional geometrical considerations show, that length contraction can be regarded as a ''trigonometric'' phenomenon, with analogy to parallel slices through a
cuboid In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to ...

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
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
s which can be characterized as the transformations between alternative on
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
corresponding to alternative states of inertial motion (and different choices of an
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
). Lorentz transformations are Poincaré transformations which are
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the
Lorentz group In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
forms the ''isotropy group'' of the self-isometries of the spacetime) which are played by
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

s in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean
trigonometry Trigonometry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

in Minkowski spacetime, as suggested by the following table: