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My advanced human civilisation often has high-acceleration ships travelling at near lightspeed. I know that time dilation would have a significant effect on the time experienced by the crews, but I've no idea how to calculate how much when they're under constant thrust - my usual calculation of averaging speeds won't work here as time dilation isn't linear. How can I calculate the time difference and eventual speed for 1 million G acceleration for 30min real time, or how long it would take to reach 50% lightspeed at 1 million G (for example)?

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    $\begingroup$ 0.5 c = 149,896,229 m/s. 1,000,000 g = 9,806,650 m/s². 149,896,229 / 9,806,650 = 15.29 seconds. $\endgroup$ – AlexP Mar 31 at 20:27
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    $\begingroup$ This is very answerable with some high school physics I believe even though those numbers seem a bit high, however I am very curious: how did you come up with those numbers if you don't know the result? This looks aimless to me. I think a more logical approach would be the opposite: telling us how much time you want to save and we tell you the acceleration required? $\endgroup$ – Raditz_35 Mar 31 at 20:29
  • $\begingroup$ There is a number of relativistic calculators online. However, they all solve it from acceleration and distance, not from acceleration and time. If you like, you can entering different distances until the answer time matches your desired input. $\endgroup$ – Alexander Mar 31 at 20:40
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    $\begingroup$ IMHO, ships (and their occupants!) that can withstand one million G seems somehow less believable than FTL. Any sublight travel will take so long in cruise that a more realistic acceleration level won't change the answer enough to matter. $\endgroup$ – StephenS Mar 31 at 22:58
  • $\begingroup$ Try a web search for "relativistic spaceship calculation" and you should get not only some theory but some online calculators. $\endgroup$ – StephenG Apr 1 at 2:16
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Velocity = acceleration x time, so time = velocity/acceleration. As pointed out in comments, at 1 million Gs, it would take 15.29 seconds to reach half of light speed. It would thus take 30.58 seconds (from the point of view of an outside observer) to reach light speed. Which, of course, you could never actually reach. Obviously, then accelerating for 30 minutes at that acceleration is an absurd premise.

Given the very brief period of time in which they're accelerating to start and decelerating to end the journey to get to and back from that 0.5c cruising velocity, calculating the change in time dilation for that 15 seconds at the start and end is sort of pointless. You simply have to take calculation for the time dilation at your cruising velocity (it's 86.6% by the way; if your trip takes exactly 100 minutes, the crew will experience 86.6 minutes).

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  • $\begingroup$ " It would thus take 30.58 seconds (from the point of view of an outside observer) to reach light speed." You know that only makes sense in a purely Newtonian universe. Accelerating for 30.58 seconds at 1 million G will take a vehicle very, very to lightspeed. The 30 minute acceleration isn't absurd per se. It can be one of two things (1) A ship that begins accelerating at 1 million G & does so for 30 minutes in the rest frame. (2) The ship accelerates for 30 minutes shiptime. Each has quite different time dilation factors. $\endgroup$ – a4android Apr 1 at 7:33
  • $\begingroup$ Accelerations of 1 million G are themselves mind-boggling. A few engineering practicalities spring readily to mind. I am prepared to entertain it as a hypothetical exercise or something in science-fiction. Otherwise lol. $\endgroup$ – a4android Apr 1 at 7:35
  • $\begingroup$ @a4android, in either case, the notion is still absurd because the ship can't keep accelerating: with a non-zero invariant rest mass, it will require energy exceeding the total energy of the universe to maintain its acceleration as it gets closer to lightspeed, which, from an outside observer, it does in under 30 seconds. $\endgroup$ – Keith Morrison Apr 1 at 14:59
  • $\begingroup$ Keith, I am well aware of those slight problems, this is a thought experiment & most thought experiments are full of absurdities. "It is the sign of an educated mind to entertain an idea without accepting it." I merely entertain the idea, as I do its absurdities. $\endgroup$ – a4android Apr 1 at 23:31

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