In my current worldbuilding project, I decided to put a ban for faster than lightspeed travel, and decided that most space flights be done in slower than lightspeed travels. Assuming efficient engines had been invented in this universe (say, some sort of 80% to 90% energy efficiency), and antimatter are not that hard to farm (just say a new technology (or some) had been invented to increase antimatter productions and storages), relativistic travel is common to travel from stars to stars. So far the setting looks nice, and ready to be filled with stories. As I start writing, I realized that onboard clock and outside observer (assuming it as a reference frame) would experience different passage of time, and that accelerating (and decelerating) to relativistic speed would also contribute to variable time passage felt onboard the ship relative to outside observer.
Common Space Travel Settings
- Accelerate-ballistic-decelerate settings, where ships were accelerated to desired speed, then follows ballistic path, and then decelerates toward rest near its destination. Commonly used, as the ship could be mainly dormant during ballistic path (cruise speed), saves the most power for considerable travel time. Cruise speed varies between common 0.3-0.5 times c to high priority travel between 0.7-0.9 times c.
- Constant acceleration-turnaround-constant deceleration toward rest, very energy consuming, usually used for urgent short-ranged distance travel (for example, interplanetary travel). It differs from the first setting only on turnaround phase, that costs nearly negligible amount of time compared to time elapsed on the whole voyages (while at the first setting most of the voyages were spent on cruise speed).
Mainly the question revolves around relativistic effect felt onboard the ship and an observer at a planet (making a huge assumption that a planet is generally 'static').
In case of first setting
- How to calculate time (both from reference frame of a static observer, and the ship's reference frame) required for the ship to accelerate from rest toward desired speed (or from desired speed toward rest) given x gee of acceleration (ship's reference frame)?
In case of second setting
- How to calculate total time required (both from reference frame of a static observer, and the ship's reference frame) for the ship to cross the distance of n kilometers (or AU) given n gee of acceleration?
Given wide range of possible readers in term of math proficiency, I would encourage easy-to-understand formula, with terms that high-schoolers could understand (high-school education be the baseline for difficulty level). Explanations may follow after the given formula if the answerer desires (and would be appreciated when they do). That said, square root functions, divisions, multiplications (additions and substractions too, obviously), square functions, are allowed.
The purpose of the aforementioned considerations are that everybody (with minimum of high-school education) could read and take advantage of the answers, basically any worldbuilder that requires the formula to calculate similar problem I am facing. And I have to admit that my physics were quite rusty (it had been two years since my high school), and quite lazy to figure it out on the net. I wrote this question mainly to benefit me (let's be honest), but it could also benefits other (mainly) hard sci-fi worldbuilder or any worldbuilder facing similar problem as of me.
Thanks in advance.