Coordinates in general relativity: Frequency shift

It is easily shown that the theoretical calculations for the frequency shift in the Schwarzschild spacetime under the standard, isotropic, and harmonic coordinates are different. Since the relations between these coordinates and the observer's reference frame are not clear, it remains an unsolved problem to make the coordinate-independent prediction for the frequency shift when the light comes from the extremely strong gravitational fields, for example, the light emitter is near to the horizon of black holes or the surface of neutron stars.