 Solution

Put $v=r\omega$, $\beta =v/c<1$ and $\gamma =(1-{\beta }^2{)}^{-1/2}$. For a photon emitted in laboratory direction $\mathbf{n}$ toward the stationary receiver, its frequency there isSince the receiver lies outside the circle, two emission points have a line of sight tangent to the circle. At those points the source velocity is directly toward or away from the receiver, so $\mathbf{n}\cdot {\mathbf{v}}_e=\pm v$. Thus the doppler extrema for a circularly moving source areThey do not depend on $d$, provided $d>r$; changing $d$ changes the emission and reception phases of the extrema. Closest and farthest source positions instead have transverse velocity relative to that ray, and are not the extrema.

For the corotating receiver, velocities must be evaluated at the actual emission and reception events. An unobstructed vacuum ray satisfies ${\mathbf{x}}_r-{\mathbf{x}}_e=c\Delta t\mathbf{n}$, while rigid steady rotation gives ${\mathbf{v}}_r-{\mathbf{v}}_e=\mathbf{\omega }\times ({\mathbf{x}}_r-{\mathbf{x}}_e)$. Therefore $\mathbf{n}\cdot {\mathbf{v}}_r=\mathbf{n}\cdot {\mathbf{v}}_e$. The emitter and receiver have equal speed magnitudes, hence equal Lorentz factors. The frequency ratio of corotating clocks is consequentlyThus the rim receiver measures ${\nu }_0$, independently of its angular separation. Different transit times or a Sagnac effect do not by themselves imply a frequency change between these identical-radius clocks. The model treats ${\nu }_0$ as a fixed proper clock frequency and assumes a signal path is available; it does not specify the disc's electromagnetic opacity.