Minkowski-space/Two observers/Relative velocity (vector)/Definition

Relative velocity

Let ${}V$ denote a Minkowski space, and let ${}B$ and ${}C$ denote observers with the four-velocities ${}v$ and ${}w$. The vector

${}v_{BC}=-v-{\frac {1}{\left\langle v,w\right\rangle }}w\,$

is called the velocity vector of ${}C$ relative to ${}B$. The number

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

is called the relative velocity of the two observers.