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Consider the following figure:

Figure

Minkowski, in his paper 'Space and Time', derives the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ from considerations of this figure.

He establishes that:

  • $PP = l \cdot OC$, where $OC$ is the unit of mesaure of the $x$-axis. Also $OC'$ is the unit of measure of the $x'$-axis.
  • $Q'Q'= l \cdot OC'$ and $QQ = l \cdot OD'$.

Suppose we can write $v =\frac{dx}{dt}$ for the second strip.

The question is: how does he conclude that $OD' = OC \sqrt{1 - \frac{v^2}{c^2}}$ from the figure alone?


P.S. The article can be accessed here: https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf

For the figure go to page 41 and for the equation go to page 45.

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You can see plenty of books for these spacetime diagrams. I sujest **DAVID MORIN-CLASSICAL MECHANICS ** you have to properly identify the two end points in the other frame. You know the angel is $tan^{-1}(v/c)$ use pythagorous theorem.

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