Length contraction

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Length contraction, according to Albert Einstein's special theory of relativity, is the decrease in length observed in objects traveling, relative to an observer, at a substantial fraction of the speed of light. The effect is observed parallel to the direction in which the observed body is travelling.

It is important to note that this effect is negligible at everyday speeds, and can be ignored for all regular purposes. It is only when an object approaches speeds on the order of 30,000,000 m/s, i.e. 1/10 of the speed of light, that it becomes important. As the magnitude of the velocity approaches the speed of light, the effect becomes dominant, as we can see from the formula:

$L_1 = \frac{L_0}{\gamma}$

where

L₀ is the proper length (the length of the object in its rest frame),

L₁ is the length observed by an observer,

$\gamma \ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}$ is the Lorentz factor,

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

c is the speed of light.

Note that this equation assumes that the two ends of an object (the length) are measured at the same time, and that the +x direction of their coordinate systems are the same. For more general conversions, see the Lorentz transformations.

An observer at rest viewing an object travelling at the speed of light would observe the length of the object in the direction of motion as zero. Among other reasons, this suggests that objects with mass cannot travel at the speed of light.

[edit] Physical origin of length contraction?

Length contraction as a physical effect on bodies composed of atoms held together by electromagnetic forces was proposed by George Fitzgerald based on a paper by Oliver Heaviside in the 1880s and worked out in more detail by Hendrik Lorentz (1895, 1899) and Joseph Larmor (1897, 1900). The following quote from Larmor is indicative of the pre-Relativity view of the effect as a consequence of James Clerk Maxwell's electromagnetic theory:

"... if the internal forces of a material system arise wholly from electromagnetic actions between the system of electrons which constitute the atoms, then the effect of imparting to a steady material system a uniform velocity of translation is to produce a uniform contraction of the system in the direction of motion, of amount (1-v²/c²)^1/2" Joseph Larmor (1900) Aether and Matter (Cambridge University Press)

The extension of this specific result to a general result was (and is) considered "ad hoc" by many who prefer Einstein's deduction of it from the Principle of Relativity without reference to any physical mechanism. Apparently Lorentz did not agree as this draft of a letter to Einstein in 1915 shows (Janssen 1995, see Lorentz-FitzGerald contraction hypothesis)

"... the interpretation given by me and FitzGerald was not artificial. It was more so that it was the only possible one, and I added the comment that one arrives at the hypothesis if one extends to other forces what one could already say about the influence of a translation on electrostatic forces. Had I emphasized this more, the hypothesis would have created less of an impression of being invented ad hoc." (emphasis added)

The Trouton-Rankine experiment in 1908 showed that length contraction of an object according to one frame, did not cause changes in the resistance of the object in its rest frame. This is in agreement with some current theories at the time (Special Relativity and Lorentz ether theory) but in disagreement with Trouton and Rankine's ideas on length contraction.

[edit] A trigonometric effect?

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 supressed) 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).

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 supressed) 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).

The modern view ^{[citation needed]} is that the so-called "Lorentz contraction" is essentially a geometric, in fact a trigonometric phenomenon, which is analogous to something which happens when we consider 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.

Special relativity concerns relativistic kinematics. 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	Minkowski 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