If the Alcubierre Warp Drive was used to travel to Alpha Centauri, how much time would pass on Earth? Per this article, it would take merely days to travel there. Say a ship travels there, spends a few months and then returns to Earth, how much time would have passed for the folks that did not travel?
The Alcubierre bubble could get to Alpha Centauri in less than 4 years according to outside observers, not just observers riding inside the bubble--the exact time depends on the exact 'shape' of the warped spacetime in the bubble, but the theory allows you to fine-tune the bubble to give as short a time as desired. The fact that observers outside the bubble also see it travel faster than light is stated on p. 117 of the book Time Travel and Warp Drives by physicists Allen Everett and Thomas Roman:
The bubble and its contents could travel through spacetime at a speed faster than light, as seen by observers outside the bubble.
And pages 118-119 elaborate:
So essentially what we've done here is "speed up" light inside the bubble relative to observers outside the bubble. Observers inside the bubble can thus travel at faster than light speeds relative to observers outside, but still slower than the local light speed inside the bubble.
However, there is a complication--the person inside the bubble is essentially riding a sort of "track" of distorted spacetime from Earth to Alpha Centauri, and although once the track is "laid down" the person can then travel from Earth to Alpha Centauri as fast as desired, work on laying down this track must begin more than 4 years before the date they want to arrive. This is explained on p. 121 of the same book:
[Y]ou can't create and control warp bubbles on demand, they have to be prepared in advance before you can use them, as also pointed out by Krasnikov. Suppose you want to create a warp bubble that will get you from Earth to Alpha Centauri 4 light-years away (it's actually about 4.2, but let's use 4 to make the arithmetic simpler) in one day, say, on January 1, 2200. You can't do this by starting on December 31, 2199. By then the spacetime point on Alpha Centauri on January 1, 2200 is far outside your future light cone, and the earliest time at which you can affect anything happening on the star is December 31, 2203. Remember from chapter 4, you can't do anything to affect what happens outside your future light cone. If you want a warp bubble to arrive at the star on January 1, 2200, the latest date at which you can arrange for this is Jan. 1, 2196. Starting then, you can in principle arrange for a warp bubble to leave your location, on December 31, 2199, and arrive at the star on January 1, 2200. If you wanted to, you could then arrange for a daily warp bubble service to Alpha Centauri arriving every day after January 1, 2200.
I believe it would not take any more time than a normal ship. The bending of space-time will negate the relativistic effects, so in essence the ship, while moving quite fast, doesn't have the relativity effects based on the bending of space-time. Bottom line, ship time and Earth time should be approximately the same. There might be some differences at most at the hours range (Or more likely minutes or seconds), depending on things like correcting for the speed of Alpha Centauri relative to Earth, but these would be minor.
I depends where on the ship you are. In the front time would appear to slow relative to sounding space and vise versa in the rear. I believe that this can be done with a magnetic field. Nonmagnetic matter would not feel the space time dilation. Magnetic fields exist on a different set principles and the space can be stressed with magnetic friction against the space fabric.
