= Relative speed from four-velocity contraction
{title2=$\gamma_{\rm rel}=U_1\cdot U_2/c^2$}
For future-directed unit timelike <four-velocities> in <metric signature> $(+---)$, $U_1\cdot U_2/c^2$ is the <Lorentz factor> of either observer measured in the <rest frame> of the other. In a common <inertial frame> it equals $\gamma_1\gamma_2(1-\boldsymbol\beta_1\cdot\boldsymbol\beta_2)$. Therefore
$$
\beta_{\rm rel}^2=1-\frac{(1-\beta_1^2)(1-\beta_2^2)}{(1-\boldsymbol\beta_1\cdot\boldsymbol\beta_2)^2}.
$$
Expanding the numerator converts this to the dot-and-cross-product form of relativistic velocity addition. The invariant construction explains why the relative speed is symmetric even when the spatial velocity components belong to different <rest frames>.
Back to article page
