Editing Length contraction (section)

=== 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>