Special relativity and time dilation

Question

Im studying Relativity and there is something that my mind really cant proccess right now. That would be the Deformation of time on heavily gravitational bodies. For an example lets take the movie Interstellar, where the Cooper went to the water planet, and because of the heavy gravity and the proximity with the Black hole, time is 1h : 7 years on the spaceship orbiting the planet. Good! Lets say Cooper stayed 3 hours on that planet. Which took 21 years on the spaceship. Now, imagine both the spaceship and the water planet are static, and both are not that far away from each other, and we have an observer on the spaceship.

Now both questions are:

1- Will the observer on the spaceship see (after a long delay), the image of exact 3 hours of Cooper on that planet. Or will the observer keep seeing those 3 hours in slow motion, taking 21 years to see the whole travel to this planet?

2- If Cooper was still on his ship, not accelerating and far from the heavy gravity action of this planet, and we were observing him. Time Dilation effect would be pretty weak, right? Now he starts accelerating and entering the gravitational force of this planet. The continuous image to the observer would then be slowed down as Cooper descends to the planets surface?

Thanks in advance!

Answer 1

1. ... will the observer keep seeing those 3 hours in slow motion, taking 21 years to see the whole travel to this planet?

Yes. The 3 hours on the planet will be seen in slow motion by the occupant of the spaceship, spanning a total of 21 years from beginning to end.

2 ... The continuous image to the observer [who remains on the spaceship] would then be slowed down as Cooper descends to the planets surface?

Correct.

Keep in mind, though, that in order to get a time-dilation effect of that magnitude, the gravity of the planet itself won't be sufficient. Even if the "planet" is a neutron star, the time-dilation factor is $\lesssim 2$, and nobody would survive a visit to the surface of a neutron star. The effect can be larger if the planet is orbiting close to a black hole (because its orbital speed can be close to the speed of light), as long as the other spaceship that stays behind is orbiting much farther away.