TL;DR

A simple relativity question.

Suppose you have a space craft that has an engine that can exert an acceleration of 1G. You fly this to Alpha Centauri accelerating, then halfway turn around and decelerate at 1G, so you have gravity for the whole trip.
Once you arrive, you turn around and fly back home the same way.

How long would appear to elapse for the traveler, and how long would appear to elapse for people on Earth? What is the formulea for different accelerations?

Note that if you use light years for distance and years for time then ##c=1## and 1g is within 5% of 1 light year per year squared, so ##s\approx\sqrt{1+t^2}-1## and each two light year phase of the journey takes about ##\sqrt{8}\approx 2.8## years. So 5.7 years all the way there, and 11.3 for the round trip.

Ship board time, ##\tau##, is related to Earth time, ##t##, by ##\tau=\frac ca\sinh^{-1}(at/c)##. Again setting ##c=1## and ##a\approx 1##, ##\sinh^{-1}(2.8)\approx 1.76## for each two light year phase, so 3.5 years to Alpha Centauri and 7 years for the round trip.

Another little point, there is often discussion about traveling faster than the speed of light - you can, but the people back home won't see it that way!

There are ways to calculate a "speed" that will give a result greater than ##c##, but given the above fact, describing any such calculation as showing that anything is "moving faster than light" is problematic.

Note that the fuel requirements even for the relatively modest trip we're talking about are absurd.

The key physical fact is that nothing ever overtakes a light pulse, no matter how you play with the numbers.

And yes, the practical problems in actually doing it are massive, and even if you are very close to a "perfect" engine, you need to carry insane amounts of fuel.

One idea I liked was to use a powerful solar powered laser sited near the sun, and a solar sail, which avoids carrying fuel, but requires a similar setup at the other end to slow down.

Our first interstellar travellers are likely to be AI robots.

By the way, this is basically why you'll very rarely find the word "celerity" outside a passing mention in some relativity textbooks. It's mixing one frame's distance measurement with another's time measurement.

The above calculation says the traveller will see 3.5 years pass when travelling 4 light years..

Consider that a light ray emitted from earth at the same time that the spaceship leaves earth will arrive at alpha centauri before the spaceship.

It is possible to do consistent calculations where either Alpha Centauri or the traveller have something-like-a-speed that exceeds ##c##, but those same calculations should then be applied to light and you will find its something-like-a-speed exceeding ##c## by an even greater margin. To put it another way you can, in some senses, go faster than ##c##, but you can never go faster than light.

As I said in #10, the traveller can certainly exploit time dilation to arrive at his destination with less time elapsed on his clocks than the four years Earth measurements would say light needed. But interpreting this as faster than light travel is an apples to oranges comparison, and it leads into a lot of common misunderstandings of relativity so we strongly recommend against it.

However, getting to such speeds (by sustaining a constant proper acceleration) requires so much fuel, that perhaps a more realistic scenario is where the elapsed time on the ship is not crucial. In that case, a journey could take hundreds or thousands or years according to Earth or onboard time-keeping.

In any case, IMO, the calculation of the onboard elapsed proper time is perfectly meaningful.

Aboard ship, if they look out their front window, A.centauri will only be 3.5 ly away (because everything ahead of them is longitudinally squished). They will have no dissonance wondering why the trip only takes them 3.5 years.

To be clear (and please PFati correct me), its not a matter of merely perceiving.

Aboard ship, if they look out their front window, A.centauri will only be 3.5 ly away (because everything ahead of them is longitudinally squished).

My two cents:
The travelers always think they are accelerating at 1g, whereas the Earthlings know that is impossible and the travelers are exerting more and more effort to accelerate less and less. That is why the travelers think they are going faster and take less time.

ADDED: After thinking more about it, my "two cents" may not have been worth that much. I will not comment further.

The point I'm trying to make for the OP, is that the experience of the ship's occupants is no less real - no more illusory, and no more paradoxical - than the experience of those on Earth.

Yes. I was oversimplifying. One could more accurately argue that, at the midpoint, A.centauri is only 1.75ly away, which is why the second leg of the journey only take 1.75y and not 2.15y.

Yes. It's not like they are ignorant of relativity. The point I'm making is their reduced journey is not merely a "perception" issue, as oracle might have been implying.

This is another perspective yes. But less useful, in my view.

The point I'm trying to make for the OP, is that the experience of the ship's occupants is no less real - no more illusory, and no more paradoxical - than the experience of those on Earth.

To be clear (and please PFati correct me), its not a matter of merely perceiving.

Aboard ship, if they look out their front window, A.centauri will only be 3.5 ly away (because everything ahead of them is longitudinally squished). They will have no dissonance wondering why the trip only takes them 3.5 years.

The reason you interpret Alpha Centauri as accelerating before you do is that the radar pulse flight time is initially constant, but starts to fall the instant you turn on your rockets. And you know the radar pulse returning as you hit the big red button echoed off Alpha C four years ago - ergo you interpret it as starting to move four years ago.

The maths for this isn't terribly complicated, but more than I have time for this morning.

To be clear (and please PFati correct me), its not a matter of merely perceiving.

Aboard ship, if they look out their front window, A.centauri will only be 3.5 ly away (because everything ahead of them is longitudinally squished). They will have no dissonance wondering why the trip only takes them 3.5 years.

The main point is that the initial distance divided by the on-board travel time is not a speed to be compared with the speed of light. As measured on Earth, it must take at least 4.4 light years travel time. That can then be compared with the speed of light from Earth to AC.

Squishing and dissonance are not what it's about! Those words belong in your literary endeavours!

If an occupant woke up from sleep and did not know they were travelling at .99c, they could still look at their destination and see that it is 1.76ly away and moving toward them at .99c. If they turn their rocket around and start accelerating (to match velocities with it) they could accurately determine that they will arrive in the correct time.

His journey aboard the spaceship is not merely "perceived" time.