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Red-shifting is determined by the difference between the space-time curvature at the light source and the observer's space-time curvature.

For the light source, all that matters is the distance from the center of gravity -- I think the motion of the source is irrelevant.

For the observer, if he is free-falling, then his curvature varies with distance from the center of gravity, increasing from flat at infinity to infinite at the singularity.

But if the observer is hovering at rest relative to the center of mass that means that his rocket thrust is unbending the curvature of gravity so that his space-time is made flat -- the same as the space-time of the free-faller at infinity. The distance of the hoverer doesn't matter. Whether he hovers just above the event horizon or at infinity or anywhere in between, his space-time will be flat, because his rockets are exactly cancelling out the gravity-caused curvature wherever he is.

Flat space-time means that there is no net acceleration relative to the center of mass, however it feels like acceleration to the hovering observer -- his reference frame is not inertial. The free-faller at infinity has no net acceleration relative to the center of mass and he feels no acceleration -- his reference frame is inertial.

The difference in curvature between the source at the event horizon and an observer in flat space-time is just enough so that the observer sees time stopped at the event horizon.

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# Hovering in gravity makes curved space flat?

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