@Franko
I just have to say, with respect to general relativity, it stopped being a "theory" in "well it's just a theory, isn't it?" sense when time dilation and bending of light by gravity were experimentally demonstrated. Both phenomena are now well recognised. Indeed, even GPS positioning satellites have to take these effects into consideration.
So, the objection that you can't make an object travel faster than the speed of light in vacuum just by pushing it harder in the direction you are trying to accelerate seems a perfectly solid one.
I suspect that if travel over such distances in a manageable time-scale is to be accomplished, it will be by "cheating" in some way.