The key point you missing here is that you cannot catch up to a beam of light. No matter how fast you travel, you will always measure the beam to be traveling at C. To an observer stationary relative to a fast object, he/she will see time pass slower in that object (some pretty basic trigonometry can actually show you by what factor this occurs, and it is commonly called the Lorentz factor). This is true for all observers, the ship would see people outside appearing to travel slowly providing the ship was a non accelerating reference frame. Basically, this means that time appears slower for those travelling faster than about .4c relative to an observer.
Also If a light beam was released from earth traveling at c, and a ship was approaching earth at .9c, standard logic would suggest that the ship would read the lights speed as 1.9c, but this does not occur, the ship still measures the speed as c.
I would recommend reading up on special relativity, it's not to difficult to grasp (apart from simultaneity). We covered it in what is the Australian equivalent of junior year highschool physics.




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