# Lorentz Factor from Minkowski's Original Paper 'Space and Time'

Consider the following 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.

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.