Timelike circular Schwarzschild geodesic ($h^2=\mu c^2{R}^2/(R-3\mu ),k^2=(R-2\mu {)}^2/[R(R-3\mu )]$)

For the Schwarzschild metric with $\mu =GM/c^2$, let $k=(1-2\mu /r)\dot{t}$ and $h=r^2\dot{\varphi }$ on an equatorial timelike geodesic. The normalization of the four-velocity gives ${\dot{r}}^2+V(r)=c^2(k^2-1)$, where $V(r)=(1-2\mu /r)h^2/r^2-2\mu c^2/r$. A circular geodesic obeys $V^{\prime }(R)=0$, giving the displayed constants and requiring $R>3\mu$. Its coordinate angular velocity is ${\Omega }^2=GM/R^3$, and its proper clock rate is $d\tau /dt=\sqrt{1-3\mu /R}$. Differentiating $V$ twice at the circular solution gives stability for $R>6\mu$.

A source on a timelike circular Schwarzschild geodesic at radius $r>3GM/c^2$ emits light to a static observer at infinity on the orbital symmetry axis. Such a ray has zero conserved axial angular momentum. The locally measured photon energy is the contraction of its four-momentum with the observer's four-velocity. Comparing the emitter with the observer at infinity therefore gives ${\nu }_{\mathrm{em}}=U^t{\nu }_{\infty }$, where $U^t=[1-3GM/(c^{2}r){]}^{-1/2}$, proving the formula.

Equivalently the gravitational redshift factor $\sqrt{1-2GM/(c^{2}r)}$ multiplies the transverse Doppler effect factor $1/\gamma$ measured by a local static observer. For $R_S=2GM/c^2\ll r$,The coefficient contains both gravitational and orbital time dilation. It assumes an observer at infinity on the symmetry axis; other viewing directions generally introduce a first-order longitudinal Doppler effect.

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