When I wrote A Hierarchy of Gods, I needed relativistic flight to, then, Xi-Geminorum, both with and without turn-around at the halfway point.  The star has since changed as a consequence of what we know about exoplanets, but that is minor.  At the time, the Internet was much younger and one did not easily find such material on it, so, I had to do the integrals on the Lorentz transforms to figure it out myself.

As luck would have it, I made a mistake on the integral for ship-board proper time.  I had a curve that looked right, but it gave the wrong results, close enough not to notice right away.  I didn't find it for quite a while, which was a shame, because the whole time-line of several books depends upon how long that flight takes.  Anyway, it's all fixed now.  But it seems a good topic to place here, for posterity, and for myself the next time I want it.  I've long since lost the actual calculations, but I do have the results.

Definitions

It might do well to clarify a few terms.  It's fairly certain that any science fiction fan or writer has a fairly sound concept of relativity.  But even so, there are some terms that can easily lead to confusion.

Reference frame

A reference frame is the set of measurement standards, commonly called yardsticks and clocks, that an observer uses to measure length and time.  It is convenient, for formalitiy's sake, to include a balance or scale to measure mass, but I recall seeing those mentioned only rarely.

Event

In non-relativistic usage, it is common to think of an event as a specific place at a specific time.  But that only makes sense if every observer is in the same reference frame.  As soon as you have any appreciable velocity involved, there is sudden disagreement of where is where and when is when.  You have to actually have something occung there as a marker in both space and time.  Depending on the degree of resolution you need, it can be a supernova, discarding the first stage of a rocket, or the hatching of a flea egg.

Proper time

The proper time between two events is the time measured in the reference frame passing through both events.  Given a spaceship, for example, the time between when the engines shut off and the captain sneezes, as measured on the ship, is the proper time between those two events.  The proper time cannot be measured by an outside "stationary" observer, but it can be calculated, which is what we are doing in part 2.  Likewise, proper the time between when the outside observer develops a leak in his space suit and the time he dies cannot be directly measured from the ship.  The observer can measure it, but the information will be of no use to him.

The same goes for proper length (distance), proper mass, and proper acceleration, all of which vary from observer to observer.

The Calculations

So here they are, the formulas for calculating proper and "improper" time for a ship in relativistic flight to cover a given distance at constant acceleration.  I've had these equations saved as an appendix to A Hierarchy of Gods.  Perhaps I can take them out now.  In all cases, distance (d) and acceleration (a) are those measured by an outside observer at the ship's starting point.  c is, of course, the speed of light = 2.997925 m/s.  To work in gravities, one earth gravity = 9.80665 m/s²

Ship Proper Time t'

Remember, this is the time as measured aboard the ship.

Ship "Improper" time t

This is the time as measured from the starting point.

If I still have a mistake here, let me know.

Caveats

Turnaround

I shouldn't have to mention that this is for continuous constant acceleration.  Almost all the time, a ship would accelerate halfway, turn around, and decelerate the other half.  So in a real-world application, divide the distance by 2 before plugging it in, and multiply the result by 2 to yield the final time.

Changing mass

These equations are for constant acceleration, which is what I wanted and the terms in which most people think.  Normally, a ship's engines would run on some kind of fuel, and as the fuel is consumed the ship gets less massive.  Therefore, to maintain a constant acceleration, the thrust must be decreased.  There is also the possibility of constant thrust, in which case acceleration would change, and these equations won't work anymore.  I don't have the the correct ones for that case.  Sorry.  I might work them out or find them someday, but in the meantime they might be available somewhere else on the Internet.

Units

All units are in seconds, meters, and meters per second.  These units are convenient for life on Earth, but not so convenient for calculating interstellar relativistic flight.  Units of years and light-years are more convenient there.  So here are the conversion factors

1 year = 3.15567*10⁷ seconds

1 light-year = 9.46053*10¹⁵ meters

There is also the possibility of working in units of light-years/year² for acceleration, which requires fewer conversions.

1 gravity = 1.03226 light-years/year²

Notice that this value is conveniently close to 1.  So for approximate work, you can just specify acceleration in g's and do no conversions at all.

So there you are.  Enjoy doing your relativistic flight calculations, and let us know where you're headed.