Undergrad Relativistic Doppler Effect near a Black Hole

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Viewing a person rotating near a black hole involves both gravitational redshift and relativistic Doppler effect redshift. If the person moves towards the observer, the Doppler effect causes a blueshift while the gravitational redshift remains. It is possible for the blueshift and gravitational redshift to equalize, allowing the observer to see the light at its original frequency. This cancellation can occur naturally for objects in circular orbits. The discussion emphasizes the interplay between these effects in extreme gravitational environments.
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Relativistic Doppler Effect near a blackhole
Hey everyone, if I were to view a shining person rotating near a black hole at near the speed of light there would be 2 kinds of redshifts: gravitational redshift and relativistic doppler effect redshift. Right?

But, say at some point, the person is traveling towards me, then the doppler effect would result in a blueshift while the gravitational redshift would remain. Is it possible that if the doppler effect blueshift and the gravitational redshift were equal, would I view the light shining at its original frequency?
 
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Yes.
 
You can always adjust your speed relative to a nearby hovering observer so that your kinematic blueshift cancels the gravitational redshift, yes. It'll happen naturally at some point for an object in a circular orbit (and probably any other), if my algebra is correct.
 

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