Sorry, there's going to be a lot of preamble here.(adsbygoogle = window.adsbygoogle || []).push({});

I've written a book that involves a lot of space travel, therefore a lot of time dilation (all sub-light travel). I've been calculating the transit times and subjective durations by brute force, i.e. I wrote a program to calculate tau on a second-by-second basis, accumulate time, calculate 'real' acceleration, and accumulate velocity. It works up to a point, but beyond a certain distance, even double-precision variables aren't enough.

It occurs to me that there should be formulae to calculate the results without using an iterative approach. something like

SubjectiveElapsedTime = MagicFunction1(distance,acceleration);

MaximumSpeed= MagicFunction2(distance,acceleration);

ObjectiveElapsedTime = MagicFunction3(distance,acceleration);

So, given a distance that you want to travel, and a given constant acceleration, you could calculate the total time experienced by the occupants, the elapsed time to the universe at large, and the maximum speed of the craft.

Also, of course you accelerate for half the journey and decelerate for the second half, so just consider the first half.

Aaaaaaaaaaaaaanyway, finally coming to the end of my preamble, someone must have derived these formulae at some point, but I can't find them anywhere. Any pointers?

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# A formula would be preferred. :-)

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