Well-known physics establishes that, compared to some reference frame (I'll denote as "stationary" here), one can do several things to start at rest, end at rest, and experience less time than objects in the stationary reference frame do, although one can not experience more time given the start and end at rest condition. The former can be accomplished by the following:
- Take a trip at relativistic speeds
- Approach to an object in a deep gravity well, approaching an event horizon potential
Question: Can either of these be done for an object with mass $m$ while expending a small amount of energy to do it, formally using an amount of energy $E\ll m c^2$? To be sufficiently clear, the requirement for traveling to the "distant future" is satisfied if $\Delta t_m \ll \Delta t$, meaning that the time experienced for the object in question is much less than that experienced by the stationary reference frame.
Conventional thinking is that #1 is simply not possible given my requirement. Any physical ship that let you planet-hop while living a single human lifetime would require spaceships that probably expend much more energy in the propellent than what the rest mass of the spaceship is. This is the relativistic rocket problem.
Thinking about #2 is what caused me to ask this question. Imagine a large black hole and a spaceship that starts off at a great distance from it. It then uses a small amount of energy to enter into a very highly elliptical orbit that comes close to the event horizon. I think (although I am not sure) it would seem to the stationary reference frame that the flyby takes a small amount of time. The spaceship would be going very fast (large time dilatation), but only for a short time and thus could not travel to the distant future and make it back out without using huge amounts of energy. You can introduce other possibilities, like a stationary planet-like surface where a traveler could sit to pass the time, but the escape potential renders this useless too.
Is it a fundamental reality that we can not use time dilation to go to the distance future without using extraordinary energy to do so, or do I simply lack the creativity to think of how?