Editing Length contraction (section)

=== 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}}}}