Suppose we send a spaceship to Betelgeuse at 0.99 c and then, after it has traveled for a hundred years, both Earth and the spaceship transmit signals to one another. When will the signals arrive? (Erm, 100 years on Earth, so that's, what, 14.1 years for the ship?)
The signal the spaceship sends seems simple enough: The ship is 99 light-years away so it should take 99 years to reach us, right? The signal from Earth also seems simple. It takes 99 years for the signal to reach where the ship is currently. But since, by that point, the ship had moved away from that spot, it takes the signal a total of 9,900 years because it has to catch up.
But, I've never really studied relativity. I understand some cool stuff, but I don't know about this idea of light playing "catch-up." From what relativity I do know, I think that the two signals should be arriving simultaneously? Furthermore, after the hundred years are up, if you launched a second ship at .99 c then the two ships will be stationary relative to one another, meaning I don't have to deal with all this relativistic business. Signal delay would be 99 years, and that's the end of the story. But I do have to deal with relativity.
One more thing. At the destination, Betelgeuse (642 ly away), am I correct in thinking that there would be 91 years between Betelgeusian observers seeing the spaceship launch from sol and the ship arriving? I.e. 642ly / .99c / 7.09γ
PS. This is for a story, if you are wondering. Basically, humanity launches grey goo at .99 c and then try to hit the abort button. Not a critical detail, so it doesn't really matter if physics complicates this problem or not. ("Not critical," Sil said. "They'll be fiiine," Sil said. Famous last words before the stars go out.)