# Proof

(related to Proposition: Construction of a Light Clock)

We will prove that the construction of a light clock is (theoretically) possible in any inertial reference frame.

Let $$\mathcal I$$ be a given inertial reference frame. By the principle of relativity, the time unit of one second $$s$$ is well-defined in $$\mathcal I$$.

By the principle of the constancy of light speed, the speed of light in vacuum in $$\mathcal I$$ is constant. Let this constant be denoted by $$c$$.

By the definition of $$s$$, and making use of the principle of the constancy of light speed once again, the definition of a distance unit of one meter $$m$$ is also well-defined in $$\mathcal I$$, from which it follows that $c=299\,792\,458\frac ms.$

Since $$\mathcal I$$ has a metric $$||~||$$ induced by the $$3$$-dimensional normed vector space $$(V_{\mathbb R},||~||)$$ over the field of real numbers $$\mathbb R$$, let $$d$$ be the number of meters (i.e. the distance), at which we position two mirrors apart in $$\mathcal I$$. Since any light particle would bounce in $$\mathcal I$$ between these to mirrors at a constant speed $$c$$, it follows that the "tick duration" of such a light clock in $$\mathcal I$$ equals

$\operatorname{tick duration}:=\frac dc.$

Github: ### References

#### Bibliography

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