(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
\[\operatorname{tick duration}:=\frac dc.\]
