Transit time for a round trip to alpha centuri...

[oracle99]

TL;DR

A simple relativity question.

Just a simple little question as my first post here..

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?

 

[Ibix]

Using Earth time and starting from rest, distance travelled in time $t$ at constant acceleration $a$ is $s=\frac 1a(\sqrt{1+a^2t^2/c^2}-1)$.

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.

 

[FactChecker]

Using Earth time and starting from rest, distance travelled in time $t$ at constant acceleration $a$ is $s=\frac 1a(\sqrt{1+a^2t^2/c^2}-1)$.

I assume the acceleration here is measured in the spaceship time and distance, since the OP wants it to feel like Earth's 1g to the travelers.

 

[Ibix]

I assume the acceleration here is measured in the spaceship time and distance, since the OP wants it to feel like Earth's 1g to the travelers.

Yeah, $a$ is proper acceleration here.

 

[Ibix]

You can't maintain a Minkowski coordinate acceleration of 1g for more than a year(ish) anyway, even in principle, since you'd reach lightspeed at that point.

 

[oracle99]

So if you travel 3 years you get
sqrt(3*3+1)-1= 2.16 light years, about half way to Alpha Centauri, so decelerate is the same, so approx 6 years out, 6 years back, 12 years, from the travelers perspective?
oh I see you updated it, that would be earth dwellers time. Thanks!

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!

 

[phyzguy]

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!

No, you can't.

 

[PeterDonis]

you can, but the people back home won't see it that way!

You can't outrun a light ray that's moving in the same direction as you are. That's not a matter of who is "seeing" it; it's an invariant fact.

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.

 

[Sagittarius A-Star]

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!

No, you can't. With reference to the traveler's restframe, the distance between Earth and Alpha Centauri is length-contracted, which explains for him the short proper travel-time.

 

[Ibix]

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!

If you divide the distance to Alpha Centauri (4ly) by the shipboard elapsed time (3.5y) you will get a value greater than $c$, yes, and you can use this fact to get to distant places in your lifetime (although not that of your friends at home). However, this isn't a speed - it's a (rather useless) quantity called celerity (or average celerity in this case, since the speed isn't constant). The celerity of light is infinite, so your celerity is always less than that of light.

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.

 

[oracle99]

I understand that the traveller is not literally travelling faster than light, just that you would percieve that, in that time frame.

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.

 

[Matterwave]

just that you would percieve that, in that time frame.

How?

 

[Ibix]

I understand that the traveller is not literally travelling faster than light, just that you would percieve that, in that time frame.

Not if you do the maths consistently, I'm afraid. I think you're comparing the ship's celerity to light's velocity, which is a similar mistake to thinking two feet must be longer than one meter because 2>1.

 

[Ibix]

Just to add, accidentally mixing measurements made using different frames is one of the more common reasons people end up confused about relativity. So it's something we recommend against, even half-jokingly.

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.

 

[oracle99]

In the original question I made a point if returning the traveller to the original frame of reference, earth.
Obviously i bow to your superior knowledge, but maybe there is a misunderstanding? The above calculation says the traveller will see 3.5 years pass when travelling 4 light years..

 

[neilparker62]

Time and fuel problems aside - great discussion! Gives me goose bumps just to see us starting to think seriously about interstellar travel. And that it's 'do-able' at least from a traveler's time perspective.

 

[Matterwave]

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

Not every calculation is physically meaningful.

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.

 

[Ibix]

In the original question I made a point if returning the traveller to the original frame of reference, earth.
Obviously i bow to your superior knowledge, but maybe there is a misunderstanding? The above calculation says the traveller will see 3.5 years pass when travelling 4 light years..

The frame where the traveller sees 3.5 years pass is his rest frame, so he doesn't travel any distance at all (his destination comes to him). The frame where he's travelled 4ly is not his rest frame, so he shouldn't be using his moving clocks to calculate his speed (if he does, he's calculating his celerity, not his speed).

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.

 

[oracle99]

Yes, I understand it is only percieved time elapsed, the traveller cannot actually travel faster than light.

 

[PeroK]

In the original question I made a point if returning the traveller to the original frame of reference, earth.
Obviously i bow to your superior knowledge, but maybe there is a misunderstanding? The above calculation says the traveller will see 3.5 years pass when travelling 4 light years..

This is an important point if we consider speeds close to the speed of light. In principle, humans could make an interstellar trip and back in less of their own proper time than the distance divided by $c$. There are several places online where the calculations associated with constant proper acceleration (and deceleration) at $g$ are presented.

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.