Consider a 4-dimensional space $\mathrm{S}$ in which all objects move at the same speed $c_S$ but in different directions. For the objects A and B that move in directions $\overrightarrow{u_A}$ and $\overrightarrow{u_B}$, respectively, if $\Delta x$ is the distance that B moves in the (3-dimensional) space perpendicular to $\overrightarrow{u_A}$ during a time interval $\Delta T$, the distance it moves in the direction of $\overrightarrow{u_A}$ will be \begin{equation*} \sqrt{c_S^2 \Delta T^2 - \Delta x^2} = \sqrt{c_S^2 - v^2} \Delta T, \end{equation*} where $ v = \Delta x / \Delta T$. Let $\Delta \tau$ be the quantity such that \begin{equation*} c_S^2 \Delta T ^2 = v^2 \Delta T^2 + c_S^2 \Delta \tau ^2. \end{equation*}
Thus, if $\overrightarrow{u_A}$ is interpreted as constituting the coordinate temporal axis and the 3-dimensional space perpendicular to $\overrightarrow{u_A}$ as constituting the coordinate space, and if $c_S=c$ is the speed of light, then the last equation describes the motion of the object B in a special relativity spacetime traveling with speed $v$ during the coordinate time interval $\Delta T$, and that $\Delta \tau$ is the proper time interval experienced for B: \begin{equation*} \Delta \tau = \sqrt{1 - \frac{v^2}{c^2}} \Delta T. \end{equation*}
Is this a correct interpretation?
