Minkowski-space/Two observers/Relative velocity/Fact

Let ${}V$ be a Minkowski-space, and let ${}B$ and ${}C$ denote observers from the same time sheet with den four-velocities ${}v$ and ${}w$. Then the following statements hold.

1. The relative velocity vector ${}v_{BC}$ is orthogonal to ${}v$.
2. The relative velocity vector ${}v_{BC}$ is spacelike, and we have

   ${}\Vert {v_{BC}}\Vert =\Vert {v_{CB}}\Vert =\rho _{BC}=\rho \,.$

3. We have

   ${}\left\langle v,w\right\rangle =-{\frac {1}{\sqrt {1-\rho ^{2}}}}\,.$

4. We have

   ${}w={\frac {v}{\sqrt {1-\rho ^{2}}}}+{\frac {v_{BC}}{\sqrt {1-\rho ^{2}}}}\,,$

   and this is the decomposition of ${}w$ into the space component and the time component of ${}B$.

5. The time coefficient of ${}w$ with respect to ${}B$ is ${}{\frac {1}{\sqrt {1-\rho ^{2}}}}$.

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