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So most people want to ask what happens if you go super-close to the speed of light and try to go a bit faster. I want to know what happens if you stop accelerating at close to $.99c$. Not decelerate, just stop accelerating and coast at $.99c$. Would the time dilation effects suddenly cease? Would time pass at a similar rate to those whom the traveler left behind on Earth? Would Doppler shifted light be any different?

I cannot find a similar question asked anywhere. I have my suspicions, but I am woefully ignorant of the math to be able to even start on the problem.

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    $\begingroup$ There is no acceleration term in the Lorentz Transform. $\endgroup$
    – m4r35n357
    Commented Apr 18, 2023 at 8:57

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Your time dilation in some reference frame is based on your velocity in that frame, not your acceleration. So stopping acceleration means that your time dilation will stop changing. It will continue steadily at $\gamma=(1-v^2/c^2)^{-1/2}$.

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