= Solution
\b[<geodesic> integrals and affine normalization.] Put $F(r)=1-2\mu/r$. Choose the <affine parameter> to be <proper time>, so $g_{ab}\dot x^a\dot x^b=c^2$. Cyclic $t$ and $\phi$ in the <geodesic Lagrangian> give $F\dot t=k$ and $r^2\dot\phi=h$. Substituting into the <four-velocity> normalization gives
$$
c^2=\frac{c^2k^2}{F}-\frac{\dot r^2}{F}-\frac{h^2}{r^2},
$$
hence
$$
\boxed{\dot r^2-\frac{2\mu h^2}{r^3}
+\frac{h^2}{r^2}-\frac{2\mu c^2}{r}
=(k^2-1)c^2}.
$$
An arbitrary constant rescaling of <affine parameter> changes its normalization; the displayed unit constant assumes the proper-time choice.
\b[<circular orbit>.] Define $V(r)=h^2/r^2-2\mu h^2/r^3-2\mu c^2/r$. The radial <geodesic equation> is $\ddot r=-V'/2$. A circular <geodesic> therefore needs $V'(R)=0$, not merely $\dot r=0$ at an instant. Solving and substituting into the radial first integral gives
$$
\boxed{h^2=\frac{\mu c^2R^2}{R-3\mu},\qquad
k^2=\frac{(R-2\mu)^2}{R(R-3\mu)}}.
$$
Existence requires $R>3\mu$. <circular orbits> below $6\mu$ are radially unstable, although they still define the instantaneous comparison here.
\b[<orthogonal local velocities in Schwarzschild spacetime>.] Use Alice's static <orthonormal frame> at the meeting event. Her proper clock is $d\tau_A=\sqrt F\,dt$ and tangential proper distance is $R\,d\phi$. Bob's local tangential speed is
$$
v_{\rm AB}=\frac{R\,d\phi}{\sqrt F\,dt}
=\frac{h\sqrt F}{Rk},
\qquad
\boxed{v_{\rm AB}=c\sqrt{\frac{\mu}{R-2\mu}}}.
$$
Carol has $h_C=0$ and $k_C=1$, because she starts from rest at infinity. The radial integral gives $\dot r_C=-c\sqrt{2\mu/R}$. Her static-frame radial speed is $(dr/\sqrt F)/(\sqrt F\,dt)=dr/(Fdt)$, and $F\dot t_C=1$, so
$$
\boxed{v_{\rm AC}=c\sqrt{\frac{2\mu}{R}}}.
$$
This is inward radial motion, perpendicular to Bob's tangential motion in Alice's frame.
Their <Lorentz factors> relative to Alice are $\gamma_{\rm AB}=\sqrt{(R-2\mu)/(R-3\mu)}$ and $\gamma_{\rm AC}=\sqrt{R/(R-2\mu)}$. The <four-velocity> contraction therefore gives
$$
\gamma_{\rm BC}=\gamma_{\rm AB}\gamma_{\rm AC}
=\sqrt{\frac{R}{R-3\mu}},\qquad
\boxed{v_{\rm BC}=c\sqrt{\frac{3\mu}{R}}}.
$$
\b[Special-relativistic consistency.] Let Bob's tangential <velocity> in Alice's local frame be $u$ and Carol's inward radial speed be $w$. A boost to Bob's frame gives Carol components $v'_{\rm tang}=-u$ and $v'_{\rm rad}=-w/\gamma_u$. Consequently
$$
\frac{v_{\rm BC}^2}{c^2}
=\frac{u^2}{c^2}+\frac{w^2}{c^2}
-\frac{u^2w^2}{c^4}
=\frac{\mu}{R-2\mu}+\frac{2\mu}{R}
-\frac{2\mu^2}{R(R-2\mu)}
=\frac{3\mu}{R}.
$$
Thus the three measured relative speeds agree with local Lorentz <velocity> addition. Ordinary Euclidean subtraction of the two <velocity> vectors would omit the transverse-velocity transformation.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2014/paper-1-local-relative-velocities.png]
{title=Original local-frame velocity comparison for static, circular and infalling observers}
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