Assuming constant acceleration to the 1/2-way point, flip, deceleration, and using http://www.projectrho.com/public_html/rocket/slowerlight3.ph... :
Time elapsed (in starship's frame of reference, "Proper time")
T = (c/a) * ArcCosh[a*d/(c^2) + 1] (given acceleration and distance)
year = 365.25*24*3600; c = 3E8; a=9.8; d=1.25*1_000_000*(c * year)
from math import acosh
T = (c/a) * acosh(a*d/(c*c) + 1)
print(T/year)
=> 14.3 years each quarter
=> 57 years round trip
Assuming the fully loaded RV weighs 5 tons, this means at max velocity the rest of the ship weighs 10 megatons, which is 10M cubic meters of water, or a cube 200m on each side.
That sounds like plenty of material, right?
The same Project Rho page links to https://arxiv.org/ftp/physics/papers/0610/0610030.pdf which calculates that at the relatively slow 0.995c "the penetration depth of protons of this energy will be ~40 m in water and ~10 m in titanium".
For 99.99999999993978% c, even 10M cubic meters isn't going to be enough.
Basically people ran the EFE "backwards" to see what matter distribution makes the wanted curvature. You get either negative mass-energy or the bubble doesn't travel ftl iirc.
We can’t do that either for now, but is way easier on the feasibility scale.
The Alcubierre drive is only a thought experiment that requires "exotic matter" (aka fairy dust) to work.
> They are only theoretically possible if you allow for negative mass and energy--not an engineering problem so much as a "need to find exotic matter"
Basically people ran the EFE "backwards" to see what matter distribution makes the wanted curvature. You get either negative mass-energy or the bubble doesn't travel ftl iirc.