Since I like to play in a happy wishful thinking little imaginary universe where an FTL drive becomes available in the relatively near future(at least sometime somewhat familiar and comprehensible), I tend not to use relativistic travel much in my fiction. But I do postulate an ancient Precursor intelligence that seeded the universe with wormholes transported in very fast negative-matter driven ships. Much as described in Robert L. Forward’s novel Timemaster. This required me to think some about the difference between ship times and “Earth” times of the various wormhole mouths. I also have some more advanced alien races who have been puttering around in interstellar space at slower than light speeds for, in some cases hundreds of millennia.
So, in order to figure out the timeline of pre-Human(in the same sense as “pre-Columbian”) space exploration and settlement, I need to know how long it takes for more or less realistic spacecraft to travel between the stars.
Below, I’ll start by giving the formulae. Then I will describe how they are used in a few more complex, but still very simple cases.
t: Time in unaccelerated frame of reference(Earth time).
d: Distance travelled(as measured in an unaccelerated frame of reference).
v: Final velocity(as measured in an unaccelerated frame of reference).
T: Proper time(as measured in an accelerated frame of reference)(Ship time)
γ: Gamma. Ratio of time dilation or length contraction at the final velocity.
These formulae assume constant acceleration all the way to the destination, flying past a point at distance,d, at time, t, or ship time, T, at velocity, v.
For a trajectory of accelerating out to the halfway point, flipping and decelerating the rest of the way, use equation 2 with d being set to half the distance to the target to determine half the Earth time of the trip. Double the result and you have the Earth time to reach your target roughly at rest. You can do the same for ship time using equation 8.
Sometimes a trajectory will be velocity limited. For instance by radiation problems of very fast relativistic craft. In this case, use equation 2 to determine the Earth time to the turnover point, which will also be the point of maximum speed. Plug the result into equation 6 to figure out what the maximum velocity will be. If the velocity is less than the upper limit, then you can do the out and flip trajectory described in the last paragraph. Just double the time to the halfway point.
If the velocity is greater than the limit speed, you will need to calculate a coasting trajectory. The first thing you need to do is figure out the times and distace covered accelerating to your maximum velocity. I observe that there is no equation to do this directly. First lets figure out what the gamma is for our maximum velocity using equation 12. For the next step, we need to rejigger equation 11 a little bit, solving for d.
d = (c2/a)(γ-1)
You can now plug this value of d into equation 2 to figure out the Earth time required to accelerate to vmax and equation 8 to figure out the needed ship time.
Double the times you figured out, those will be our temporary working values t’ and T’. Double the distance, we’ll call that d’
Now we have to figure out the coasting time. For Earth time
tcoast = (d-d’) / vmax
In order to figure out the ship time experienced in the coast phase, we plug the coasting time(Earth frame) that we just figured out and the gamma that we figured out earlier from equation 12 into the following formula:
Tcoast = tcoast / γ
And that’s it. the total Earth time of the trip will be
ttotal = tcoast + t’
and for ship time
Ttotal = Tcoast +T’
That there is about the hardest thing going and I tend to think of it as a bit of an advanced subject, but if your trying to figure out the time for a very long voyage and you don’t want every hydrogen atom hitting your ship like an Oh My God Particle, it might be necessary.
For more on this subject check out The Relativistic Rocket.
Hopefully, this has been helpful.
Comments are always welcome.