Controls
The ship holds a constant proper acceleration, the push its own crew feel, for the first half of the distance, then turns round and brakes at the same rate for the second half. Left: a spacetime diagram of the trip as seen from home. Right: the bill.
Things to notice
Anywhere in the galaxy in a working lifetime
Pick the galactic centre at 1 g: about 20 years by the ship's clock. Pick Andromeda: under 30. The ship never crosses the light line on the diagram, and it never needs to.
Why the logYou can go, but you cannot come home
Watch the home bar, not the ship bar. To the galactic centre it reads 26,000 years, only a couple of years more than light itself would take. Everyone you knew is history before you land.
Where the years goThe fuel is the real bill
Slide the thrust down to 0.1 g for Proxima. The crew's trip stretches to about 13 years but the fuel per tonne falls from 39 to under 4. Push it to 3 g and the bill jumps to about 230, to save under two ship years.
Register LOR-1Time dilation is drawn on the ship's line
The dots along the ship's worldline are its own years. Near the turnaround they spread out: one ship year covers thousands of home years there. Try Vega, where you can still count them.
The dotsHaldeman and Le Guin did this sum
Set a custom distance of about 30 ly at 1 g and the home clock reads within a couple of years of 33, which is what Anderson's Tau Zero planned for Beta Virginis. Haldeman's soldiers and Le Guin's NAFAL crews pay the same arithmetic, novel after novel.
The dossiersNo paradox, at any setting
Push thrust to 3 g for any destination. The ship's line hugs the light line and never touches it, so it stays inside the light cone. That is why this diagram cannot be turned into an antitelephone.
Register CAU-1What is going on
This is the control group for the whole catalogue. Every other entry is a scheme to beat this ship. The ship itself is plain special relativity: hold a steady push, and your speed creeps toward light without ever reaching it, while your own clock runs slower and slower compared with the clocks at home. With c set to one light year per year and one g equal to 1.032 light years per year squared, each half of a trip of distance d takes the crew (c/a) arccosh(1 + a(d/2)/c²) of their own time and takes home sqrt((d/2)² / c² + 2(d/2)/a). The peak Lorentz factor at the turnaround is 1 + a(d/2)/c², and a perfect photon rocket needs a mass ratio of exp(aτ/c) for the one-way trip, where τ is the crew's total time. Coming back squares it.
Crew time grows only as the logarithm of the distance, which is why the galaxy fits in a career. Home time grows as the distance itself, which is why nobody waits up. The exponential fuel bill is LOR-1 in the register, the rule that a massive body cannot be pushed to c because its energy runs away first; in the register's own words the rocket satisfies LOR-1 exactly and its bill is B-mass. The reason the diagram never produces a paradox is CAU-1: the worldline is timelike, always inside the light cone, so every observer agrees on the order of departure and arrival. The full physics dossier is the relativistic rocket; the novels that keep this account are The Forever War, Le Guin's Hainish cycle and Tau Zero.