Every Circle Looks the Same: A Metric-Independent Invariant for Relativistic Circular Motion

We consider light emitted from a circular source orbiting a compact central mass in a stationary, axisymmetric spacetime (e.g.\ Schwarzschild or Kerr). For an observer at infinity in the equatorial plane, we show that the time-domain signal --- the observed frequency as a function of coordinate time, up to overall scaling --- is observationally equivalent to that of a circular orbiter in flat spacetime at a different radius and velocity. In this sense, curved-space circular orbiters fall into equivalence classes of signals that admit flat-space representatives. All metric dependence and gravitational lensing effects are absorbed into a small set of effective parameters in an affine reparametrization of the standard relativistic Doppler law.