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 $1+\mu /R_E-3\mu /(2R_s)$.