If you think about “a”’s journey in one way, he will not be one-fourth of the way to AC after a year, as part of his journey was spent at sub-light velocities - which is OK as the calculations are beyond my abilities.

If you think about his journey another way, again from Earth, the astronaut will arrive at AC in three more years - so he must have traveled one-fourth of the way the first year.

Maybe I don’t understand relativity ?

Now consider Graph 2 where, about 100000 years ago on a planet on the other side of the galaxy, an astronaut “b” climbed into his spaceship and accelerated at 1 G toward Earth and AC.

There is, of course, the question of how he knew he should do this - but maybe “b” just felt like making a journey to Earth and AC.

It will take light, and any knowledge of the events on the ancient, distance world of “b” to reach Earth. From such a great distance, however, for all practical purposes, the distance to our IS ( Imaginary Star) is the same and the light will arrive at the same time. I doubt that “a” in his spaceship would be aware of “b” until he appeared suddenly and streaked off toward AC.

I used the term “at the same time” above - but the light confirming that the observer on the planet near the IS exists will not reach Earth for four years. Still I can ask what he might see.

Where, relative to the Earth, will the observer near the IS first see “b” 's ship ? I would think it would be approaching the IS until it was halfway between Earth and AC - and would show ultraviolet, at that point, it would move away, becoming more and more infrared. Of course, the observer near the IS would be viewing this in “b” 's past.

It seems to me that from the IS, “b” and his ship would be getting more and more ultraviolet as he approached - but my math is not strong enough to verify this.

If this is true, there is another factor to consider. If the size of “b” increases and his time slows down as he accelerates away, might not “b” shrink and his time speed up as he accelerates toward the IS ? For an observer near the IS, “b” would shrink from sight and then begin to grow again.

At some point, if this is true, the observer near IS would find "b" would be too small to see - again, my mathematical abilities are insufficient to find this point.

I don't have proof for the following but believe it can be found. For "b", his world always feels normal. For an outside observer, for example, the one near IS, "b" ' s speed of light has changed.