Length contraction

# Length contraction

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, length contraction – according to Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

– is the physical phenomenon of a decrease in length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

detected by an observer of objects that travel at any non-zero velocity relative to that observer. This contraction (more formally called Lorentz contraction or Lorentz–Fitzgerald
George FitzGerald
George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....

contraction
) is usually only noticeable at a substantial fraction of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

; the contraction is only in the direction parallel to the direction in which the observed body is travelling.
Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, length contraction – according to Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

– is the physical phenomenon of a decrease in length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

detected by an observer of objects that travel at any non-zero velocity relative to that observer. This contraction (more formally called Lorentz contraction or Lorentz–Fitzgerald
George FitzGerald
George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....

contraction
) is usually only noticeable at a substantial fraction of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

; the 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 important. At a speed of 13,400,000 m/s (30 million mph, .0447c), the length is 99.9% of the length at rest; at a speed of 42,300,000 m/s (95 million mph, .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:

where is the proper length
Proper length
In relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....

(the length of the object in its rest frame), is the length observed by an observer in relative motion with respect to the object, is the relative velocity between the observer and the moving object, is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

,

and the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...

is defined as
.

Note that in this equation it is assumed that the object is parallel with its line of movement. Also note that 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

Length contraction was postulated by George Francis FitzGerald (1889) and Hendrik Antoon Lorentz (1892) to explain the negative outcome of the Michelson-Morley experiment
Michelson-Morley experiment
The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special...

and to rescue the hypothesis of the stationary aether (Lorentz–FitzGerald contraction hypothesis). Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...

, who derived this deformation from electromagnetic theory in 1888), it was considered an Ad hoc hypothesis
In science and philosophy, an ad hoc hypothesis is a hypothesis added to a theory in order to save it from being falsified. Ad hoc hypothesizing is compensating for anomalies not anticipated by the theory in its unmodified form....

, 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
Joseph Larmor
Sir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...

developed a model in which all forces are considered as of electromagnetic origin, and length contraction appeared to be a direct consequence of this model. Yet it was shown by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

(1905) that electromagnetic forces alone cannot explain the electron's stability, and he had to introduce non-electric binding forces to ensure stability and to give a dynamical explanation for length contraction. But this model was subject to the same problem as the original hypotheses: Length contraction (and the non-electromagnetic forces) were only invented to hide the motion of the preferred reference frame, i.e., the stationary aether. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

(1905) was the first who completely removed the ad-hoc character from this hypothesis, by demonstrating that length contraction was no dynamical effect in the aether, but rather a kinematic effect due to the change in the notions of space, time and simultaneity brought about by special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. Einstein's view was further elaborated by Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

and others, who demonstrated the geometrical meaning of all relativistic effects in spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. So length contraction is not of kinetic
Kinetics (physics)
In physics and engineering, kinetics is a term for the branch of classical mechanics that is concerned with the relationship between the motion of bodies and its causes, namely forces and torques...

, but kinematic
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

origin.

## Basis in relativity

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects, where "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
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the proper length
Proper length
In relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....

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.
Thus the definition of simultaneity is crucial for measuring the length of moving objects. In Newtonian mechanics, simultaneity
Simultaneity
Simultaneity is the property of two events happening at the same time in at least one frame of reference. The word derives from the Latin simul, at the same time plus the suffix -taneous, abstracted from spontaneous .The noun simult means a supernatural coincidence, two or more divinely...

is absolute and therefore and are always equal. Yet in relativity theory the constancy of light velocity in all inertial frames in connection with the relativity of simultaneity
Relativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...

destroys this equality. So if an observer in one frame claims to have measured the object's endpoints simultaneously, the observers in all other inertial frames will argue that the object's endpoints were not measured simultaneously. The deviation between the measurements in all inertial frames is given by the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

. As the result of this transformation (see Derivation), 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 is the relative velocity and the speed of light)

For example, 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):

Thus, as it is required by the principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames), length contraction is symmetrical: If the rod is at rest in the train, it has its proper length in S' and its length is contracted in S. However, if the rod comes to rest relative to the station, it has its proper length in S and its length is contracted in S'.

## Derivation

Length contraction can simply be derived from the Lorentz transformation as it was shown, among many others, by Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

:

In an inertial reference frame S', and shall denote the endpoints for an object of length at rest in this system. The coordinates in S' are connected to those in S by the Lorentz transformations as follows:
and

As this object is moving in S, its length has to be measured according to the above convention by determining the simultaneous positions of its endpoints, so we have to put . Because and , we obtain

Thus the length as measured in S is given by

According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. For this case the Lorentz transformation is as follows:
and

By the requirement of simultaneity and by putting and , we actually obtain:

Thus its length as measured in S' is given by:

So (1), (3) give the proper length when the contracted length is known, and (2), (4) give the contracted length when the proper length is known.

## Geometrical representation

The Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, and it can be illustrated by a Minkowski diagram: 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.

Additional geometrical considerations show, that length contraction can be regarded as a trigonometric phenomenon, with analogy to parallel slices through a cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

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.

In special relativity, Poincaré transformations
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

are a class of affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

s 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
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

). Lorentz transformations are Poincaré transformations which are linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...

forms the isotropy group of the self-isometries of the spacetime) which are played by rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

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 E0,1 E1,1
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

## Experimental verifications

Since the occurrence of length contraction depends on the inertial frame chosen, it can only be measured by an observer not at rest in the same inertial frame, i.e., it exists only in a non-co-moving frame. This is because the effect vanishes after a Lorentz transformation into the rest frame of the object, where a co-moving observer can judge himself and the object as at rest in the same inertial frame in accordance with the relativity principle (as it was demonstrated by the Trouton-Rankine experiment
Trouton-Rankine experiment
The Trouton–Rankine experiment was an experiment designed to measure if the Lorentz–FitzGerald contraction of an object according to one frame produced a measurable effect in the rest frame of the object, so that the ether would act as a "preferred frame"...

). In addition, even in a non-co-moving frame, direct experimental confirmations of Lorentz contraction are hard to achieve, because at the current sate 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 in fact the negative result of a famous experiment, that required the introduction of Lorentz contraction: the Michelson-Morley experiment
Michelson-Morley experiment
The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special...

(and later also the Kennedy–Thorndike experiment). In special relativity its explanation is as follows: In its rest frame the interferometer can be regarded as at rest in accordance with the relativity principle, so the propagation time of light is the same in all directions. However, in a frame in which the interferometer is in motion, the propagation time of the transverse beam is time dilated
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

, while in the longitudinal direction the interferometer is also contracted, so that speed of light is constant and the propagation time in both directions is the same in this frame as well.

Other indirect confirmations are: Heavy ion
Heavy ion
Heavy ion refers to an ionized atom which is usually heavier than helium. Heavy-ion physics is devoted to the study of extremely hot nuclear matter and the collective effects appearing in such systems, differing from particle physics, which studies the interactions between elementary particles...

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 Lorentz contraction is considered. Another confirmation is the increased ionization
Ionization
Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...

ability of electrically charged particles in motion. According to pre-relativistic physics the ability should decrease at high speed, however, the Lorentz contraction of the Coulomb field
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

leads to an increase of the electrical field strength normal to the line of motion, which leads to the actually observed increase of the ionization ability. Lorentz contraction is also necessary to understand the function of free-electron lasers. Relativistic electrons were injected into an undulator
Undulator
An undulator is an insertion device from high-energy physics and usually part of a largerinstallation, a synchrotron storage ring. It consists of a periodic structure of dipole magnets . The static magnetic field is alternating along the length of the undulator with a wavelength \lambda_u...

The electromagnetic radiation emitted when charged particles are accelerated radially is called synchrotron radiation. It is produced in synchrotrons using bending magnets, undulators and/or wigglers...

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
Relativistic Doppler effect
The relativistic Doppler effect is the change in frequency of light, caused by the relative motion of the source and the observer , when taking into account effects described by the special theory of relativity.The relativistic Doppler effect is different from the non-relativistic Doppler effect...

. So, only with the aid of Lorentz contraction and the rel. Doppler effect, the extremely small wavelength of undulator radiation can be explained. Another example is the observed lifetime of muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

s in motion and thus their range of action, which is much higher than that of muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

s at low velocities. In the proper frame of the atmosphere, this is explained by the time dilation of the moving muons. However, in the proper frame of the muons their lifetime is unchanged, but the atmosphere is contracted so that even their small range is sufficient to reach the surface of earth.

## Reality of Lorentz contraction

Another issue that is sometimes discussed concerns the question whether this contraction is "real" or "apparent". However, this problem only stems from terminology, as our common language attributes different meanings to both of them. On one side, the word "real" is used for things that we can measure without considerable observational error
Observational error
Observational error is the difference between a measured value of quantity and its true value. In statistics, an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process.-Science and experiments:...

s, and "apparent" therefore denotes to the products of observational error, optical distortions, or displaced images like a Fata Morgana
Fata Morgana (mirage)
A Fata Morgana is an unusual and very complex form of mirage, a form of superior mirage, which, like many other kinds of superior mirages, is seen in a narrow band right above the horizon...

. If this definition is chosen, length contraction would be "real" since it principally can be detected by error free measurements of the simultaneous positions of the object's endpoints, and also by measuring its consequences (see the section "experimental verifications"). On the other side, "real" is also used in connection with "absolute", and "apparent" is thus "relative". This is related to the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....

, according to which any inertially moving observer can consider himself as at rest, and attribute the motion to the other observers. If this definition is chosen, length contraction would be "apparent" since it depends on the inertial motion of bodies. Yet, whatever terminology is chosen, in physics the measurement and the consequences of length contraction with respect to any reference frame are clearly and unambiguously defined in the way stated above.

Due to superficial application of the contraction formula some paradoxes can occur. For examples see the Ladder paradox
The ladder paradox is a thought experiment in special relativity. It involves a ladder travelling horizontally and undergoing a length contraction, the result of which being that it can fit into a much smaller garage...

Bell's spaceship paradox is a thought experiment in special relativity involving accelerated spaceships and strings. The results of this thought experiment are for many people paradoxical. While J. S. Bell's 1976 version of the paradox is the most widely known, it was first designed by E. Dewan and M...

. However, those paradoxes can simply 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.In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry...

, which proves that the concept of rigid bodies
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

is not compatible with relativity. It also shows that for a co-rotating observer the geometry is in fact non-euclidean
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

.

## Visual effects

Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could naively lead to a thinking 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, it is important to realize that such visual effects are completely different measurements, as such a photography is taken from a distance, while length contraction can only directly be measured at the exact location of the object's endpoints. In 1959 Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

and James Terrell
Terrell rotation
Terrell rotation is the name of a mathematical and physical effect. Specifically, Terrell rotation is the distortion that a passing object would appear to undergo, according to the special theory of relativity if it were travelling a significant fraction of the speed of light...

published papers demonstrating that length contraction instead actually show up as elongation or even a rotation in an image of photography. This kind of visual rotation effect is called Penrose-Terrell rotation.

• Time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry...