The paradox I envision involves two objects that exist in very different gravitational potentials -- one very high (eg: close to a black hole) and one very low (eg: far away from a black hole).
Now, imagine these objects are actually little spaceships that are blasting their engines so that they remain a constant distance away from the black hole. Also, imagine they are lined up with each other (the black hole, the near spaceship, and the far spaceship form a line).
Would it be possible to lower a string from the further spaceship to the spaceship which is closer? That is, will the distance between the two spaceships be measured to be a constant?
If it's not.. please explain this! It seems odd that the distance between the two ships wouldn't be constant.
If it is, then it seems like there would be a paradox. First, let's label the spaceship nearer to the blackhole "A" and the one further "B". Next, let's say the string's length is exactly 1 light second.
Now, let's start a light clock between "A" and "B" -- "A" will send a light pulse to "B", and upon receiving this, "B" will send a light pulse to "A", and so forth. Since both "A" and "B" know that their distance is 1 light second apart, each time they receive the pulse, they will add 2 seconds to their clock. While we're at it, they might as well send along their recorded time with the light pulse.
The paradox is that from the perspective of B viewing A, A should appear to be moving in "slow motion" -- everything will be redshifted, and everything will transpire slower. From the perspective of A viewing B, B will be blueshifted and moving in "fast motion". Yet, this cannot be the case, because both will be receiving ticks at 2 second intervals, and each tick will represent 2 seconds of elapsed time on the other party.
So, what gives? I'm assuming you cannot have a constant distance between two objects if their gravitational potentials are different. But that just seems very strange.
Edit: It would be very helpful if the answers could give an example of what it would be like to be onboard A and onboard B. If you're on A, and you send the pulse, how much time will you measure before you hear back from B? If you're on B, and you send the pulse, how much time will you measure before you hear back from A?