I am trying to understand the physics of an interstellar race.
Say two ships are racing past a black hole as if it were a corner on a race track. One of the ships takes a hard line and comes very close to the black hole. The other ship takes a weak line, and does not get very close to the black hole.
The Ship that took a hard line will travel less distance, and likely use more fuel to escape the black hole's gravity (although I'm not certain of this).
Due to time-dilation, the hard-line ship will also age less. That is, due to the strong gravity near the black hole, time will go slower on the hard-line ship.
My question is: If the two ships converge after they pass the black hole (intersect at a point) which of these ships will actually arrive at this point first?
The hard-line ship will certainly be younger when it gets there, but maybe the weak-line ship will get there first?
Additionally: Who will use more fuel? While the hardline ship will have to deal with escaping the strong field of gravity, the weak-line ship will be aging much more and will be burning fuel for longer...