A video conference between earth and a space shuttle I have just started looking into special relativity and I have come up with an intriguing gedanke, as Einstein himself called such theoretical thought experiments.
Imagine a space shuttle traveling through space at a constant velocity close to $c$. As the shuttle passes earth, a previously set-up camera starts broadcasting from earth to the shuttle. Since radio waves travel at the speed of light, the shuttle is receiving a constant transmission feed, assuming the camera is broadcasting 24/7.
Now, from what I have understood of special relativity so far, time will flow slower for  the astronaut than for the earthlings. Hence, assuming $v=0.8c$, the astronaut will after 30 years have received a video transmission 50 years long!
Is my reasoning correct, that even though the transmission is live, the astronaut would actually be watching things that happened many years ago, while still receiving the "live" feed, which would be stored/buffered in the shuttles memory, thus making it possible for the astronaut to fast-forward the clip to see what happened more than 30 years after passing the earth?
My second question is, what happens when we consider the space shuttle to be at rest and the earth to be moving instead? If that would imply that it has been 50 years from the astronauts point of view, while only 30 years have passed on earth, then the astronaut would run out of video material after the first 30 years of watching the broadcast. Then what?
I hope it makes sense. Thank you!
 A: The astronaut will not have received 50 years worth of transmissions at the time you specify. 
Let's start by stating precisely what time dilation means in this instance. Consider two "events": 


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*The people on Earth have a party to celebrate 50 years of radio broadcasting.

*The astronaut on board the ship has a party to celebrate 30 years of travel.


In a reference frame in which the Earth is at rest, those two events are simultaneous. That's what we mean when we say that the astronaut's clocks run slow.
But the astronaut won't receive the 50th year of radio broadcasting until much later -- to be precise, 200 more years of Earth-time will elapse before he receives this signal. (Let's check this: in 200 years at $0.8c$, he will travel 160 light-years. Add that to the 40 light-years he's already gone, and you find that he's 200 light-years from Earth at the time. The radio signal will just be reaching that distance at that time.)
Of course, because of time dilation, that's only 120 additional years of astronaut time. Still, it means the astronaut can't see into the future.
To say it another way, at the moment in question (when the astronaut throws his 30-year party), the radio signal he is receiving is one that left Earth merely 10 years after he passed Earth. In Earth's reference frame, that radio signal traveled a distance of 40 light-years (since that's how far away the astronaut is at that moment), and took 40 years to do it (50 years minus 10 years).
