# Proof

(related to Proposition: Time Dilation, Lorentz Factor)

Let $$\mathcal I_1$$ and $$\mathcal I_2$$ be two inertial frames of reference. Assume that a light clock is installed in $$\mathcal I_2$$ with mirrors at a distance $$d_2$$. An observer $$o_2$$, who is situated in $$\mathcal I_2$$, will observe a tick duration of $$t_2={d_2}/c$$.

Assume that $$\mathcal I_1$$ and $$\mathcal I_2$$ are moving relatively to each other with the constant speed $$v$$ and that the light clock is positioned perpendicular to the direction of this movement.

From the perspective of another observer $$o_1$$ situated in $$\mathcal I_1$$, the light clock in $$\mathcal I_2$$ will cover the distance $$x$$ within the time of one "tick" of the light clock. Let the duration of this "tick" be $$t_1$$ from the perspective of the observer $$o_1$$. Note that from the perspective of the observer $$o_1$$ the light will cover the distance $$d_1$$, which is given by hypotenuse the right triangle $$\triangle {ABC}$$, as shown in the following figure: (c) bookofproofs own work

According to the Pythagorean theorem, we get $d_1^2=x^2+d_2^2,$ which is equivalent to

$\begin{array}{rcl}&&(ct_1)^2=(vt_1)^2+(ct_2)^2\\ &\Leftrightarrow&c^2t_1^2=v^2t_1^2+c^2t_2^2\\ &\Leftrightarrow&c^2t_1^2-v^2t_1^2=c^2t_2^2\\ &\Leftrightarrow&t_1^2-\frac{v^2}{c^2}t_1^2=t_2^2\\ &\Leftrightarrow&t_1^2\left(1-\frac{v^2}{c^2}\right)=t_2^2.\\ \end{array}\quad\quad ( * )$

By hypothesis, $$0\le v < c$$, from which it follows $0\le \frac{v^2}{c^2} < 1$ and $1\ge 1- \frac{v^2}{c^2} > 0.\quad$ Because the term $$1- v^2/c^2$$ is non negative, is possible to calculate the square root, leading to the inequation $1\ge \sqrt{1- v^2/c^2} > 0.\quad\quad ( * * )$

It follows together with $$( * )$$ that

$t_1\sqrt{\left(1-\frac{v^2}{c^2}\right)}=t_2.$

Because $$( * * )$$ is non zero, we can solve the equation for $$t_1$$ leading to the equation

$t_1=t_2\frac 1{\sqrt{\left(1-\frac{v^2}{c^2}\right)}}.\quad\quad(0\le v < c)$

Github: ### References

#### Bibliography

1. Weingärtner, Andreas: "Spezielle Relativitätstheorie - ganz einfach", Books On Demand, 2016