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If, hypothetically, me and my rocket powered flashlight were falling straight toward the center of a black hole. The flashlight is a few kilometers behind me in our travels toward the center of the black hole, but since it is rocket powered, it manages to maintain the exact distance to me for a while.

The point is; The distance between me and my flashlight is constant as long as I am observing it.

The photons coming from the flashlight would obviously not be rocket powered - and they would be affected by the black holes gravitation.

Would the light I see from the flashlight be shifted towards red or blue, even though the distance between me and my dear flashlight is maintained?

If so; switching the positions of me and my flashlight, would it change the color I'd observe?

If we turn off the rocket on the flashlight, I assume it would be redshifted regardless of which were closer to the singularity, and the magnitude of redshift would appear to accelerate?

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A partial answer: The lower the gravitational potential the slower the clocks, see Gravitational time dilation. This means, as long as your flashlight is behind you in constant distance, it appears blue-shifted to you. With roles exchanged (light below you in constant distance) it appears red-shifted. By the heterogeneity of the gravitational field the shift grows with the time, while falling in constant distance.

This is called the Einstein shift, which is not the same as the doppler effect. The doppler effect is caused by an objects velocity relative to the observer.

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To clarify my understanding of your answer. There are two factors affecting the amount of redshift; the clock speed at the observer - a faster clock relative to the source means blueshift and vice versa, and the gravitational redshift. Due to the the steepness of the gravitation well growing exponentionally, I suspected we'd see even more shift then what is accounted for by gravitational time dilation alone? –  frodeborli Jan 6 at 19:21
    
A faster clock (due to higher potential) of the observer relative to the source means redshift. The amount of shift depends of the difference of the gravitational potential. As this difference is higher when closer to the singularity for objects of fixed distance the shift grows while falling. –  Gerald Jan 6 at 21:21
    
Thank you Gerald. And if both the flashlight and the observer were running at the same clock, while still at various distances from the black hole, there would be no shift in color? (For example if the flashlight itself was very heavy) –  frodeborli Jan 7 at 9:05
    
If the flashlight is exactly that heavy to produce the same gravitational potential as at the location of the observer, there wouldn't be a net reshift, and the clocks would run synchronous. –  Gerald Jan 7 at 15:03
    
But light itself is also affected by that clock rate? So being near a supermassive black hole, such as the one in the center of our galaxy, would make all light from sources further away look redshifted? Is "The Great Attractor" possibly such a source? –  frodeborli Jan 7 at 18:57
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