Comparison of circular-orbit and static Schwarzschild clocks (${\tau }_s/{\tau }_E=\sqrt{(1-3\mu /R_s)/(1-2\mu /R_E)}$)

A clock following a timelike circular Schwarzschild geodesic of radius $R_s$ advances at $\sqrt{1-3\mu /R_s}$ per unit Schwarzschild coordinate time. A static clock at radius $R_E>2\mu$ advances at $\sqrt{1-2\mu /R_E}$. Their ratio over the same coordinate-time interval gives the displayed result. For periodic signal comparisons, successive corresponding signals must also have the same propagation delay, as happens for repeated overhead passages in a fixed circular orbit observed from a static pole. To first order in $\mu /R$, the ratio is $1+\mu /R_E-3\mu /(2R_s)$.

For the receding edge of a circular emitter ring, the direct-projection model gives $1+z=(1-3r_g/r{)}^{-1/2}[1+\sqrt{r_g/(r-2r_g)}\sin i]$, with $r_g=GM/c^2$. Gravitational and transverse shifts combine with the line-of-sight Doppler factor. Mapping the distant inclination directly to a local photon direction neglects gravitational ray bending.

At $r=6r_g$, the local circular speed is $c/2$, so the receding direct-projection shift is $\sqrt{2}(1+\sin i/2)$. The upper envelope over inclinations is $3/\sqrt{2}$. It is not the shift for a face-on or arbitrary azimuthal emitter, and it excludes plunging-region emission and ray-tracing corrections.