Relative speed from four-velocity contraction (${\gamma }_{\mathrm{rel}}=U_1\cdot U_2/c^2$)

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 ${\gamma }_1{\gamma }_2(1-{\mathbf{\beta }}_1\cdot {\mathbf{\beta }}_2)$. ThereforeExpanding 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.

Compare two observers at one event using a common static orthonormal frame outside a Schwarzschild horizon. If one observer moves tangentially at speed $u$ and another radially at speed $w$, their four-velocity contraction is ${\gamma }_u{\gamma }_w{c}^2$. Their relative speed therefore has the displayed form. The radial component measured by the tangential observer is $w/{\gamma }_u$, so Euclidean subtraction of the two original three-velocities fails. For a circular timelike geodesic and radial fall from rest at infinity at radius $R$, the relative speed is $c\sqrt{3\mu /R}$, where $\mu =GM/c^2$ and $R>3\mu$.