Minkowski-space/Motion/Realization by scaling/Example

In a four-dimensional standard-Minkowski-space, we want to move unformly from the point ${\displaystyle {}P={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\r\end{pmatrix}}}$ to the point ${\displaystyle {}Q={\begin{pmatrix}q_{1}\\q_{2}\\q_{3}\\s\end{pmatrix}}}$. In the classical framework, we would just take the vector

${\displaystyle {}{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}}:={\begin{pmatrix}q_{1}\\q_{2}\\q_{3}\\s\end{pmatrix}}-{\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\r\end{pmatrix}}\,.}$

However, this is in general not an observer vector, and then the looked-for motion is not realizable. If ${\displaystyle {}y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-t^{2}}$ is negative, which means that we have a timelike vector, then we can rescale this vector to obtain an observer vector

${\displaystyle {}{\begin{pmatrix}z_{1}\\z_{2}\\z_{3}\\u\end{pmatrix}}={\frac {1}{\sqrt {-\left\langle {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}},{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}}\right\rangle }}}{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}}\,.}$

Then, the mapping

${\displaystyle x\longmapsto {\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\r\end{pmatrix}}+x{\begin{pmatrix}z_{1}\\z_{2}\\z_{3}\\u\end{pmatrix}}}$

describes a motion, which, for ${\displaystyle {}x=0}$, starts at the point ${\displaystyle {}P}$, and ends, for ${\displaystyle {}x={\sqrt {-\left\langle {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}},{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\t\end{pmatrix}}\right\rangle }}}$, at the point ${\displaystyle {}Q}$, and which is realizable in the physics world.