- #1

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1st light year takes 1/2 hours in rocket's own frame

2nd light year takes 1/4 hours

3rd light year takes 1/8 hours

4th light year takes 1/16 hours

5th light year takes 1/32 hours

and generally, nth light year takes (1/2)^n hours in rocket's own frame

As it's seen, the sum of travel times form converging series that has limit 1. In the outside rest frame (the path frame), the rocket's speed approaches light speed and kinetic energy grows without bound, but light speed or infinite energy is never actually reached.

However, in rocket's own frame, the passengers (strong enough to survive the acceleration) expect that they reach infinity at 1 hour and eagerly wait to see what happens and where they end up to. Will they be disappointed? Is there something in relativity why, at 1 hour, the passengers see that they still are far away from infinity, and actually will never reach it? Thanks.