Minkowski-space/Motion/Realization by scaling/Example

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

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

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

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

Then, the mapping

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

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