How long for a 20 LY trip at 1 G.

I know I ought to be able to figure it out, but maybe one of you already knows the answer. Imagine you could build and launch a space ship with enough power and reaction mass to go to a star that is 20 LY away accelerating and then decelerating at 1G at halfway. How long would such a trip take as seen from earth? And how would it be to an occupant of the ship?

Spaceflight calculator says 21.85 years long for an outside observer, 6.04 years for a passenger, max speed of 0.996c.

The thing also computes fuel consumption based on a conversion rate (energy yield converting fuel into energy, and the energy converting into acceleration). Since the default conversion rate is the yield rate of H-H fusion (.008), it’s already pretty optimistic, and the calculator still says you’d have to start with 1.59 million tonnes of fuel. (Which is probably not factored into the mass used in the acceleration calculations, so not very realistic either. It’s handwaved with the following quote: “Also note that if the fuel mass is calculated to be more than the mass of your spacecraft, then your trip cannot be done unless you extract fuel from space. If your fuel mass is more than half the mass of your spacecraft, you’re probably on a one way trip, so take enough food, books and episodes of Star Trek to last the rest of your life.”)

I think you calculated at 1 g the whole way there. You forgot the flip and decelerate.
So 23.56 for the observer, 9.7 for the passenger.

I don’t know where you got your calculations, but that site is calculating a mid-course turnaround.

That is one magical rocket!

But you can solve this type of problem by applying a Lorentz transformation familiar from special relativity. Roughly, it takes the magic rocket 6 years to arrive at its destination, measured 22 years from the earth. The trick is that the 20 light years are not as long at ludicrous speed; in the same manner you could cross the galaxy in a reasonable amount of time. (As in, with respect to the earth it takes about however many years to travel that many light-years, but the proper time is logarithmic with respect to that.)

Thx. Exactly what I was looking for.

How is the discrepancy explained by the travelers? What would their maps/star charts and positioning algorithms say as they make their voyage? When they start their location would say 20 LY from target, what changes so that it says 10 LY from target half way through and after only 3 years?

From the travelers’ point of view, as their speed increases, the two stars get closer together, so they’d say that they didn’t actually travel so far.

Right, space would appear to shrink along their traveling vector, right? But wouldn’t that break certain things based on distance?

So let’s say they start up their space ship, and look at the holographic map. They’d see Earth, then zoom out and see their target star, and an arrow between the two says 20 light years.

As they go, the map would… indicate the distance (the arrow on the map) shrinking… but faster than what they would (classically) expect, right?

What if along the way there were a pair of binary stars. Their orbits and movement should be determined by the distance between them… however, if the travelers see space as compressing, wouldn’t the orbits look different to them? Wouldn’t the gravitational interactions appear all sorts of incorrect?

It would look pretty impressive.

ETA: Space Balls - Ludicrous Speed - YouTube

What “discrepancy” do you think there is? Circles would still look like circles, etc. Not sure what you mean by “(classically) expect”.

Actually, I think the confusion is between what would show up if the travellers pointed a camera out the porthole and took a picture, like in the videos, versus the “holographic map” you are presumably generating starting from the original map and applying a Lorentz transformation.

Your first question doesn’t have a definite answer as you need to come up with a way of extending the notion of distance for non-accelerating observers given by special relativity to accelerating observers. There are actually several sensible ways you could do this.

In addition a map requires some kind of coordinate system, I think with the turnaround included there isn’t really a good coordinate system to use for the observer on the spaceship.

Obviously regardless of the existence of a nice set of coordinates the observer on the spaceship will still definitely see visual effects due to relativity, but note length contraction is a coordinate effect and not a visual effect. There are visual effects associated with length contraction like Terell rotation though.

Calculating the dynamics of a binary star system from the point of view of a relativistically accelerating observer requires a relativistic theory of gravity, so unsurprisingly Newtonian gravitational theory would fail to describe the dynamics correctly.

How a circle is visually distorted by relative motion depends on the position of the points on the circle in 3D space relative to the observer.

In this case you can’t take a map and apply a Lorentz transformation as the relative velocity between the observer and the original map is constantly changing.

You’re right, I missed that. It’s not really clear unless you watch the little video.

The people on board would regard their map as inaccurate. I mean, they could calculate where they are on it, but it wouldn’t look like what they see out the window.

The passengers measure the travel distance prior to and after travel as 20 LY. They measure the time taken as less than 20 years. The conclusion they could come to is that they travelled faster than light, but this is not allowed, so this particular viewpoint is invalid. Can someone explain why this is not a valid way of measuring the speed of travel?

Before and after, the distance is 20 ly. While they’re actually traveling it, though, the distance is less.

Technically you would say in their own frame an observer does not travel, but you can easily turn it on its head and say from their point of view the passengers might conclude Earth has travelled FTL.

You can certainly define (change in distance to Earth between start and end inertial frames)/(proper time of the observer) as a speed, but it does not cause problems for causality for this speed to be greater than c.

Another way to think about this is that when stationary with respect to a reference frame we are moving through time a c over a distance c⋅t, so when we start to move at speeds approaching c in some direction, time t that the subject experiences has to reduce by a proportionate amount compared to the ‘proper time’ of the non-moving observer.

As a practical matter, accelerating a spacecraft at 1 g for a significant duration would require a colossal amount of propellant or a propulsion system with a specific impulse far beyond what is physically plausible (I[SUB]sp[/SUB]>1 million seconds), notwithstanding that a propulsion system capable of producing 9.81 m/s[SUP]2[/SUP] at the beginning of travel would have to be throttled to a tiny fraction of its thrust capability at end of operation due to the vast change in mass from lost propellant. The only way such a vessel would be plausible is if it used some kind of propulsion system that did not use carry propellant on board or dispensed with it whatsoever.