= Orthogonal local velocities in Schwarzschild spacetime
{title2=$v_{\rm rel}^2=u^2+w^2-u^2w^2/c^2$}
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 $R>3\mu$.
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