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I know that if one astronaut falls into a black hole, then a distant observer will see him take an infinite amount of time to reach the event horizon (provided the observer can see light of arbitrarily large wavelengths).

But the falling astronaut will only take a finite amount of time to reach the horizon.

My question is: What will the falling astronaut see if he "looks backwards" while falling. Will he see the distant observer growing old and all the stars dying by the time he reaches the horizon ?

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No. He would see that only if he stopped falling close to and above the horizon, by thrusting or going into orbit, say. (No stable orbits exist too close.) An observer falling straight in won't see time speed up above him.

A way to visualize this is with the river model of black holes (googleable). An observer who flows with the river, by falling straight inward toward a center of gravity, doesn't notice time speed up above him. To notice that the astronaut must "fight the current" somehow. By fighting the current the astronaut can receive information that is "flowing with the river" faster than normal.

We are able to do that just sitting in our chairs. It just requires fine instruments to measure. If we jump off a cliff (or even a curb) we can no longer notice it.

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