Editing Length contraction (section)

=== 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>.