Past exam of the general relativity course of the University of Oxford / 2021 / 3 / d / Solution

The geodetic precession per orbit isThe orbital position and four-velocity repeat after a full orbit, so the direction to distant Earth in the instantaneous rest frame repeats. The freely transported gyroscope does not: its beam axis has turned through $\Delta \psi$ relative to the original aiming direction. This applies to any initial in-plane spin direction by linearity of parallel transport, not only the radial initial condition used in part (b). The beam's angular diameter is $1^{\circ }$, giving a half-angle of $0.5^{\circ }$. Thus Earth lies outside the beam after one orbit; comparing the offset with the diameter would already show a miss, but the correct acceptance angle is the half-angle.

The initial radial ray must be outward from the black hole for a direct path to Earth. A precisely inward radial null geodesic falls into the black hole and cannot provide the assumed initial communication. For outgoing near-radial Schwarzschild light deflection, the angular correction is especially small. If a static observer at $R$ measures a small emission angle $\psi$ from the outward radial direction, its impact parameter is $b=R\sin \psi /\sqrt{f(R)}$. The outgoing null geodesic givesRelative to flat propagation, curvature changes the small pointing angle and beam radius by the fraction $f(R{)}^{-1/2}-1\simeq 0.00251$. At an offset of $1.35^{\circ }$ this is only about $0.0034^{\circ }$; the beam's $0.5^{\circ }$ radius changes by about $0.0013^{\circ }$. The exactly radial central ray has zero angular deflection. This is not the geometry of the usual $4GM/(b{c}^2)$ scattering formula, which assumes a ray coming from infinity, passing the black hole, and returning to infinity.

Relativistic aberration also distinguishes a coordinate-radial light ray from a beam aimed radially in the moving crew's frame. Here $v/c=1/\sqrt{398}\simeq 0.0501$. Successful initial aiming includes that fixed correction; the same correction applies at the repeated orbital position. More precisely, an outgoing static-radial ray initially has crew-frame angle ${\psi }_0^{\prime }=-\arcsin (v/c)$ from the radial axis. The local small-angle aberration factor about that aim is $d\psi /d{\psi }^{\prime }=\gamma =\sqrt{398/397}$; including the outgoing propagation factor gives $\gamma /\sqrt{f}=1/q\simeq 1.00377$. Thus the asymptotic miss is about $1.358^{\circ }$ and the beam half-angle about $0.502^{\circ }$, with the same conclusion. The small distortion of the outward beam cannot correct the much larger accumulated geodetic precession.