Lets imagine in idealized situation in which we have an obsever in flat space-time, surrounded by distant static objects, which have no doppler shift (as they aren't moving and there isn't any gravity). Now, we introduce a massive object with a relativistic flyby. We expect the distant sources to doppler shift. In the Newtonian limit, the doppler shift would be closely related to the integral of the Newtonian gravitational acceleration over time, for instance. If we specify a particular distant object (which I will term a guide star) in the Newtonian case we can find an observer with no doppler shift during the flyby - such an observer would be "not moving relative to the guide star". The questions are: 1) In the special realtivistic case, is such a no-doppler relative to a guide star observer unique? I believe they are as long as one assumes one can measure second-order doppler effects. 1) In GR, if we specify a specific guide star can we find such a "no doppler" observer? Is his worldline unique? 2) If we pick a different distant object, do we get a different worldline? If the worldlines are not unique this question doesn't make a lot of sense, so we would ask instead "Is there any worldline that experiences no doppler shift from all distant object". My intuition is that in GR there is not such a "no doppler for all guide stars" observer, but I don't, at this point, have any proof that there isn't.