I’ve seen some articles recently that talk about this exact question (because it’s all related to the movie). I’m at work so can’t look any up now, but if you keep on accelerating at 1g that time dilation kicks in. You can reach Andromeda within something like 15 years and the edge of our observable universe in about 33 years. It’s wild to think that we could get that far in a human lifetime.
Of course, the longer you keep it up, the more magical a constant-acceleration drive needs to be. A constant 1 g from here to tau Ceti is just maybe plausible, with an antimatter-based fuel. To the Andromeda galaxy, though, even with pure antimatter fuel, the fuel ratio (amount of fuel needed to total payload) gets ludicrous, and for the edge of the observable Universe (as observed from Earth right now)? Fuhgettaboutit.
If you can maintain 1g acceleration for a year and stop accelerating, you’d buzz past Tau Ceti at roughly 0.77c, give or take. It would take roughly 16 years to get there, as measured from earth. Ship time around 10 and a half years, time dilation still being significant at that velocity.
Longer if you decide to decelerate and orbit, of course.
Well, assuming you can maintain that sort of acceleration indefinitely, sure, as least in ship time. But that’s in the same realm as unobtanium and spherical cows as far as we know.
The math is interesting regardless of the engineering 
One interesting option I’ve seen is that the starship drive opens a tiny wormhole into a newly expanding universe, so you can ‘borrow’ power from a Big Bang somewhere in a different cosmos.
This would give you enough power to accelerate at one g for an indefinite period; but itt doesn’t solve the problem of shielding. At any velocity faster than 0.85c the energy imparted by a gram of dust would be equivalent to a gram of antimatter hitting the ship. Interstellar protons become deadly radiation, etcetera.
Also all that energy in your rocket motor would heat the ship - even at 99% efficiency there would be waste heat, which would melt the rocket quite quickly at these energies. You’d need massive radiators - which would also need to be shielded from interstellar dust and gas.
I don’t know how PHM deals with these problems.
I once calculated travel at 1G for an sf story. IIRC, I found that the time in earth years was always 1 year off the number of light years traveled. It’s been decades so please confirm if this memory is correct.
Certainly not that simple, or it’d take 1 year to travel 0 distance. But it’d be a reasonable approximation for long distances.
Yes, obviously for star travel.
How does this transmission work when the earth is revolving so that half the day the transmitter is pointed the wrong way? I suppose this could be a pulsed energy transmission, 12 hours on, 12 hours off. But large energy generators seem to have difficulty turning on & off – look at the Fukshima nuclear plants, the 1960’s East Coast electrical blackout, even old coal-fired power plants seem to take a while to get going. And if you keep it going all the time, where does all that energy go the other 12 hours?
Possibly you site this earthbound ‘motor’ at the North or South pole, but then the transmitting antenna would have to revolve steadily for years on end. And there would be major logistics problems for any large installation at either pole.
I think the transmitter is supposed to be in orbit. Probably solar orbit.
Yes. Trying to transmit that kind of energy over interstellar distances would be hard enough without having to punch through the Earth’s atmosphere first, even ignoring all the complications doing it that way would cause. Also if it’s solar powered you want the transmitter as close to the Sun as possible, to take advantage of the more intense sunlight.
For that matter it might well be desirable to have it orbit the Sun opposite of Earth just so nobody can decide to change its aim and fry the planet.
Put three transmitters spaced more or less evenly around the Earth. What latitudes work best depends on which direction we’re beaming the power towards.
That’s exactly how our deep space tracking systems have worked for 60 years now.
Not only is the propulsion laser probably in orbit, but if it’s transmitting in microwaves, it’s not too tough to aim it using a phased array, so you never need to move any hardware to aim it.
Accelerate another probe going in the opposite direction.
I read an s-f story generations ago, so before hand calculators never mind computers. The ship was accelerating at 1g to the halfway point then decelerating and the navigator was grumbling about calculating trip time, both stationary and relativistic, with a slipstick. “It would be so much easier if 1g was 10 meters/second2.”
“How can that be?”
“Simple – define Earth’s gravity as a bit less than 1g.”
Much easier to define the “space meter” as the distance that causes 1G to be 10 space meters per second squared. Much like we have statute and nautical miles for measuring distances in different environments where different factors predominate.
Easier yet to simply accelerate at 10m/s/s and leave it at that. The people and cargo aren’t going to be bothered by the extra 0.19m/s/s ~= 2% of felt gravity. At least not for long.
Or invent a similar sounding name like with megabytes and mebibytes - so 10m/s/s is one ‘Gee Whiz’.
LOL. Good one @Mangetout