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I was intrigued by this question about minimizing the travel time between two worlds that are at rest with respect to each other, and disappointed that the answer turned out to be trivial. However, I would like to propose a modified version of the question that I feel ought to be less trivial.

To recall, the setting is this (I'll skip the fanciful sci-fi setup from the earlier question): there are two planets, A and B, that are a relativistic distance (some number of light-years) apart. A traveler sets out from A and wants to get to B in "the least amount of time".

As pointed out in the other question, if by that phrase we mean time as measured in the common inertial frame of reference of A and B, then the answer is trivial: the traveler should just go as fast as possible! Not so interesting.

My variation however concerns minimizing the proper time of the traveler: the traveler wants to get to B having herself aged as little as possible. How fast should she go?

Even this question can be broken into two separate variants:

  • In the first variant, we are looking for a speed $v$ such that the traveler should go from A to B at speed $v$, such that the elapsed time of the trip in her inertial frame of reference will be as small as possible. We neglect the effects of her accelerating from speed 0 to $v$ at the beginning of the trip and decelerating from $v$ to $0$ at the end. (For example, maybe she came from a third world C and just wants to fly by both A and B at her already-relativistic speed to take photos.)

  • In the second variant, we assume the traveler is setting out from A at speed 0, spends some time accelerating (not necessarily with a uniform acceleration), then coasts by at an inertial speed for a while, and then decelerates (again, with an arbitrary deceleration profile) to finally land at world B at speed 0. She wants to minimize the proper time that elapses between takeoff at A and landing at B (neglecting orbital maneuvers etc). What should her speed look like as a function of time? Assume she can withstand arbitrarily high acceleration forces, so if need be she can ramp her speed up/down as quickly as she wants. (In yet another variant, which I'm not sure is interesting enough to be worth thinking about, we can ask the same question but put a limit on the forces her body is able to withstand.)

I think the math behind the second variant will involve a relativistic rocket-style analysis, maybe with an interesting calculus of variations minimization problem, but I haven't tried working out the details.

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    $\begingroup$ It’s still trivial. Accelerate as fast as possible, to the fastest speed you can achieve, and decelerate as fast as possible at the end. Try to be a photon-like as possible. $\endgroup$ – G. Smith Jul 4 at 0:47
  • $\begingroup$ @G.Smith is there a simple way to see why that's true? $\endgroup$ – GenlyAi Jul 4 at 0:53
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    $\begingroup$ Because time dilation increases monotonically with speed, and does not depend on acceleration. $\endgroup$ – G. Smith Jul 4 at 0:58
  • $\begingroup$ @G.Smith for the first variant of my question I can see that what you’re saying makes sense (and suspected it was trivial when I posted the question). For the second variant, what may be confusing me is a feeling that since the equivalence principle says that accelerating feels locally like being in a uniform gravitational field, and being in a gravitational field does cause time dilation, that acceleration ought to have an effect on proper time that needs to be taken into account. Is that not true? And is there way to show this in a way that’s a bit more mathematical? (I am a mathematician.) $\endgroup$ – GenlyAi Jul 4 at 2:27
  • $\begingroup$ I don’t think it’s true. That would be a good question for a separate post. I don’t immediately know how to explain the apparent paradox. $\endgroup$ – G. Smith Jul 4 at 2:44

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