= Face-on frequency shift of a circular Schwarzschild source
{title2=$\nu_\infty/\nu_{\rm em}=\sqrt{1-3GM/(c^2r)}$}
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 $\nu_{\rm em}=U^t\nu_\infty$, where $U^t=[1-3GM/(c^2r)]^{-1/2}$, proving the formula.
Equivalently the <gravitational redshift> factor $\sqrt{1-2GM/(c^2r)}$ multiplies the transverse <Doppler effect> factor $1/\gamma$ measured by a local static observer. For $R_S=2GM/c^2\ll r$,
$$
\nu_{\rm em}=\nu_\infty\left[1+\frac{3R_S}{4r}+O\!\left(\frac{R_S^2}{r^2}\right)\right].
$$
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>.
Back to article page
