The key to triggering the relativistic effect of time dilation is acceleration, not velocity. If you're up to trillions upon trillions of G-forces, you could outlive the entire solar system on a voyage to Jupiter.
Except you'd still have passed Jupiter.
If you've got the power to do all that acceleration, then you could accelerate in the reverse direction for only double the cost. Not a problem when we're talking about forces so grand!
I'm done playing Valkyrie Profile for a bit, so I'll explain my reasoning.
I don't believe stop go travel would multiply the effect, but you're getting into general relativity and accelerated reference frames -- which is a bit beyond me at the moment (mostly because I mostly learned about this stuff from a college astronomy class that didn't involve any math and a little book by some Einstein guy). If you want to look at it that way, then this is how I'm pretty sure it'd work.
From your perspective on the ship, accelerate towards the earth, and Earth clocks run faster. Accelerate away from the earth, and Earth clocks run slower. This is because you're now in an accelerated reference frame with a sort of "pseudo-gravity" field covering the entire universe, in which the direction of acceleration can be used to determine a "top" and a "bottom" and time is faster above you and slower below.
Notice that those above you see you as slower while you'd see them as faster, instead of both of you seeing time slow or speed up as is the case with "time dilation" in special relativity.
Also, if you accelerate in one direction and then equally in the opposite, the effects pretty much cancel out, but then how does Earth experience more time than you do on your trip to Jupiter?
Gravitational time dilation is dependent on distance. The farther you are, the more pronounced the effect. When you leave and return from Earth, you accelerate in the same direction away from it. This makes time appear to slow down on Earth, but you're so close that the effect is minimal. When you reach Jupiter and decide to turn around though, the earth is hundreds of millions of kilometers away, and the effect is more pronounced. Only this time, the earth is way above you in your "pseudo-gravity" field, so Earth's time appears to advance incredibly fast from your perspective.
However, Jupiter just isn't far enough away to cause several more years to have elapsed on Earth than did for you. That's why I say that you must have passed Jupiter if you experience such a result.
Now, if you're willing to throw accelerated reference frames out the window and forget about gravitational time dilation, you can calculate what the time difference would be fairly easily based on the distance and velocity traveled.
The time elapsed for someone sitting on Earth would be distance * velocity * 2.
The time elapsed for you (your proper time) would be distance * velocity * 2 * sqrt(1-v^2/c^2).
Notice that your experienced time is always less than the time someone sitting on Earth sees you take, yet it's almost not different at all until you get pretty close to light speed.
Jupiter is only a few hours away at those speeds, so you'd only ever accomplish turning your 2 hour trip into ever smaller and smaller amounts of time.
- "Are we there yet?"
- "Yeah. Remember to set your watch ahead a few hours."
But damn, what do I know? I'm just a college drop-out.
