Editing Length contraction (section)

==Derivation==

Length contraction can be derived in several ways:

=== Known moving length ===
In an inertial reference frame S, let <math>x_{1}</math> and <math>x_{2}</math> denote the endpoints of an object in motion. In this frame the object's length <math>L</math> is measured, according to the above conventions, by determining the simultaneous positions of its endpoints at <math>t_{1}=t_{2}</math>. Meanwhile, the proper length of this object, as measured in its rest frame S', can be calculated by using the Lorentz transformation. Transforming the time coordinates from S into S' results in different times, but this is not problematic, since 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:<ref name=born>{{Citation|author=Born, Max|author-link=Max Born|title=Einstein's Theory of Relativity|publisher=Dover Publications|year=1964|isbn=0-486-60769-0|url-access=registration|url=https://archive.org/details/einsteinstheoryo0000born}}</ref>

:<math>x'_{1}=\gamma\left(x_{1}-vt_{1}\right)\quad\text{and}\quad x'_{2}=\gamma\left(x_{2}-vt_{2}\right) \ \ .</math>

Since <math>t_1 = t_2</math>, and by setting <math>L=x_{2}-x_{1}</math> and <math>L_{0}^{'}=x_{2}^{'}-x_{1}^{'}</math>, the proper length in S' is given by

{{NumBlk|:|<math>L_{0}^{'}=L\cdot\gamma \ \ . </math>|{{EquationRef|1}}}}

Therefore, the object's length, measured in the frame S, is contracted by a factor <math>\gamma</math>:

{{NumBlk|:|<math>L=L_{0}^{'}/\gamma \ \ . </math>|{{EquationRef|2}}}}

Likewise, according to the principle of relativity, an object that is at rest in S will also be contracted in S'. By exchanging the above signs and [[Prime (symbol)|primes]] symmetrically, it follows that

{{NumBlk|:|<math>L_{0}=L'\cdot\gamma \ \ . </math>|{{EquationRef|3}}}}

Thus an object at rest in S, when measured in S', will have the contracted length

{{NumBlk|:|<math>L'=L_{0}/\gamma \ \ . </math>|{{EquationRef|4}}}}

=== 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:<ref>{{Cite book |author=Walter Greiner|title=Classical Mechanics: Point Particles and Relativity |publisher=Springer |year=2006 |isbn=9780387218519 |url={{Google books|plainurl=y|id=CynrBwAAQBAJ|text=length contraction|page=396}}}}; Equations 31.4 – 31.6</ref>

:<math>\begin{align}
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{align}</math>

Computing length interval <math>\Delta x'=x_{2}^{\prime}-x_{1}^{\prime}</math> as well as assuming simultaneous time measurement <math>\Delta t'=t_{2}^{\prime}-t_{1}^{\prime}=0</math>, and by plugging in proper length <math>L_{0}=x_{2}-x_{1}</math>, it follows:

:<math>\begin{align}\Delta x' & =\gamma\left(L_{0}-v\Delta t\right) & (1)\\
\Delta t' & =\gamma\left(\Delta t-\frac{vL_{0}}{c^{2}}\right)=0 & (2)
\end{align}
</math>

Equation (2) gives

:<math>\Delta t=\frac{vL_{0}}{c^{2}}</math>

which, when plugged into (1), demonstrates that <math>\Delta x'</math> becomes the contracted length <math>L'</math>:

:<math>L'=L_{0}/\gamma</math>.

Likewise, the same method gives a symmetric result for an object at rest in S':

:<math>L=L^{'}_{0}/\gamma</math>.

=== Using time dilation ===
<!--This section is linked from [[Derivations of the Lorentz transformations]]-->

Length contraction can also be derived from [[time dilation]],<ref>{{Citation|author=[[David Halliday (physicist)|David Halliday]], [[Robert Resnick]], [[Jearl Walker]]|title = Fundamentals of Physics, Chapters 33-37|publisher=John Wiley & Son|year =2010|isbn=978-0470547946|pages=1032f }}</ref> according to which the rate of a single "moving" clock (indicating its [[proper time]] <math>T_0</math>) is lower with respect to two synchronized "resting" clocks (indicating <math>T</math>). Time dilation was experimentally confirmed multiple times, and is represented by the relation:

:<math>T=T_{0}\cdot\gamma</math>

Suppose a rod of proper length <math>L_0</math> at rest in <math>S</math> and a clock at rest in <math>S'</math> are moving along each other with speed <math>v</math>. 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 <math>T=L_{0}/v</math> in <math>S</math> and <math>T'_{0}=L'/v</math> in <math>S'</math>, thus <math>L_{0}=Tv</math> and <math>L'=T'_{0}v</math>. By inserting the time dilation formula, the ratio between those lengths is:

:<math>\frac{L'}{L_{0}}=\frac{T'_{0}v}{Tv}=1/\gamma</math>.

Therefore, the length measured in <math>S'</math> is given by

:<math>L'=L_{0}/\gamma</math>

So since the clock's travel time across the rod is longer in <math>S</math> than in <math>S'</math> (time dilation in <math>S</math>), the rod's length is also longer in <math>S</math> than in <math>S'</math> (length contraction in <math>S'</math>). Likewise, if the clock were at rest in <math>S</math> and the rod in <math>S'</math>, the above procedure would give

:<math>L=L'_{0}/\gamma</math>

=== Geometrical considerations ===

[[File:Slabs.svg|thumb|250px|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 '''E'''<sup>3</sup> (see left half figure at the right).  This is the Euclidean analog of ''boosting'' a cuboid in '''E'''<sup>1,2</sup>.  In the latter case, however, we can interpret the boosted cuboid as the ''world slab'' of a moving plate.

''Image'': Left: a ''rotated cuboid'' in three-dimensional euclidean space '''E'''<sup>3</sup>. 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'''<sup>1,2</sup>, 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'''<sup>1,2</sup> at right, and in the sense of '''E'''<sup>3</sup> at left).

In special relativity, [[Poincaré group|Poincaré transformations]] are a class of [[affine transformation]]s which can be characterized as the transformations between alternative [[Cartesian coordinates|Cartesian coordinate charts]] on [[Minkowski spacetime]] corresponding to alternative states of [[inertial frame|inertial motion]] (and different choices of an [[origin (mathematics)|origin]]).  Lorentz transformations are Poincaré transformations which are [[linear transformation]]s (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 [[rotation]]s 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:

{| class="wikitable" style="text-align:center"
|+ Three plane trigonometries
|-
! Trigonometry 
! Circular !! Parabolic !! Hyperbolic
|-
! Kleinian Geometry 
| Euclidean plane      || Galilean plane        || Minkowski plane
|-
! Symbol 
| '''E'''<sup>2</sup>  || '''E'''<sup>0,1</sup> || '''E'''<sup>1,1</sup>
|-
! Quadratic form 
| Positive definite || Degenerate || Non-degenerate but indefinite
|- 
! Isometry group 
|  '''E'''(2) ||  '''E'''(0,1) ||  '''E'''(1,1)
|-
! Isotropy group 
| '''SO'''(2) || '''SO'''(0,1) || '''SO'''(1,1)
|-
! Type of isotropy
| Rotations || Shears || Boosts  
|- 
! Algebra over R 
| [[Complex number]]s || [[Dual number]]s || [[Split-complex number]]s 
|-
! ε<sup>2</sup> 
| −1 || 0 || 1
|-
! Spacetime interpretation 
| None || Newtonian spacetime || Minkowski spacetime
|-
! Slope 
| tan φ = m || tanp φ = u  || tanh φ = v
|-
! "cosine" 
| cos φ = (1 + m<sup>2</sup>)<sup>−1/2</sup> || cosp φ = 1 || cosh φ = (1 − v<sup>2</sup>)<sup>−1/2</sup>
|-
! "sine" 
| sin φ = m (1 + m<sup>2</sup>)<sup>−1/2</sup> || sinp φ = u || sinh φ = v (1 − v<sup>2</sup>)<sup>−1/2</sup> 
|- 
! "secant" 
| sec φ = (1 + m<sup>2</sup>)<sup>1/2</sup> || secp φ = 1 || sech φ = (1 − v<sup>2</sup>)<sup>1/2</sup>
|-
! "cosecant" 
| csc φ = m<sup>−1</sup> (1 + m<sup>2</sup>)<sup>1/2</sup> || cscp φ = u<sup>−1</sup> || csch φ = v<sup>−1</sup> (1 − v<sup>2</sup>)<sup>1/2</sup>
|}