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

For a timelike circular Schwarzschild geodesic, the coordinate orbital period and onboard proper time per orbit areUsing $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the Schwarzschild radius and orbital areal radius are $R_s=2GM/c^2\simeq 1.21789\times 10^{10}\mathrm{m}$ and $R=200R_s\simeq 2.43578\times 10^{12}\mathrm{m}$. ThereforeThe crew accumulates about $3836$ fewer seconds, or $63.9$ fewer minutes, per orbit. Both gravitational time dilation and kinematic time dilation contribute: a static clock at $R$ has $d{T}_{\mathrm{static}}=\sqrt{1-R_s/R}dt$, while the orbiting clock additionally has $d\tau =d{T}_{\mathrm{static}}/\gamma$, where $\gamma =\sqrt{398/397}$. A distant observer measures the coordinate period; comparing arrivals emitted at the same orbital phase also removes the repeated light propagation delay.