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Would an observer moving with close to the speed of light ($\sim 99\%$) towards a star system that's $20$ light years away perceive the distance as being less than $20$ light-years? If yes, how much time would elapse in their frame of reference given the aforementioned speed of $.99c$ and $20$ light-years distance?

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marked as duplicate by JMac, Community Nov 13 '18 at 18:15

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In the observer's frame of reference, they observe the star system moving towards them at a speed of $v\approx$, across a contracted distance of $d^\prime={d\over\gamma}$, which takes a time of $\frac{d}{\gamma v}\approx 2.8~\rm yr$. But there is no time dilation from their point of view, since they are moving with a velocity $0$ with respect to themselves.

Meanwhile, from an outsider's point of view, the spacecraft travels over a non-contracted distance $d=20~\rm lyr$, which takes a time of $d\over v$, but with time dilation this is again $d\over \gamma v$ in the spacecraft's rest frame.

In other words, time dilation and distance contraction are just two different ways of looking at the same thing. The spacecraft sees the star arrive in $2.8~\rm yr$ due to length contraction, whereas to an outside observer it takes the spacecraft $20~\rm yr$ of their time, from which they can determine it takes the spacecraft $2.8~\rm yr$ of proper time due to time dilation.

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Yes, you can travel to a star system 20 light years away while aging far less than 20 years. In the star's frame, that's because you're experiencing time dilation. In your frame, it's because the distance to the star is length contracted. The two are, of course, completely equivalent.

Specifically, the length is contracted by a factor of $1/\sqrt{1-v^2/c^2}$, which in your case is $7$. That means you arrive at the star system in about $20/7$ years.

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Would an observer moving with close to the speed of light (~99%) towards a star system that's 20 light years away perceive the distance as being less than 20 light years?

Yes, and you can find the formula for length contraction online in many places, pick the one you like the most.

It should be remembered, however, that only the people on the ship would believe the trip took less time. Observers back on Earth would still see it take ~20 years.

Not that you would care... I'd be happy to get to Alpha Centauri in time for cocktails, I don't mind that's is past bedtime back on Earth!

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