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 , 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?