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It is my understanding that when moving near the speed of light, time slows down relative to other things not moving so fast. Based on this principle, would it be theoretically possible to travel a thousand light years in a year, with the thousand years only having passed on the place you're moving relative to?

If something is wrong with my premises or question, that knowledge is welcomed, too. (I am still a bit shaky on the whole subject of time-dilation because I don't fully understand how it's possible without a privileged reference frame.) Understanding is the prime goal, here.

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If Alice observes that Bob travels 1000 lightyears in 1000 years, then Bob's speed is c according to Alice and thus, is c in all frames of reference which means that Bob doesn't have a frame of reference at all.

Let's try adjusting the numbers a bit.

Let Alice observe that Bob, moving with a constant speed, travels 999 lightyears in 1000 years. According to Alice, Bob has a speed of 0.999 c.

Bob knows when he has travelled 999 lightyears according to Alice because Alice has helpfully put up a marker for each lightyear (according to her).

How much time elapses on Bob's wristwatch between the event that the 0 lightyear marker flashes by and the event that the 999 lightyear marker flashes by?

According to SR, the elapsed time (in years) according to Bob is

$$\tau = 999\sqrt{1 - (0.999)^2} = 44.7$$

But remember, according to Bob, it is Alice and the lightyear markers that are moving.

And, according to Bob, the distance (in lightyears) between Alice's lightyear markers is

$$d = 1\sqrt{1 - (0.999)^2} = .0447 $$

The point is this: we must be careful in our thinking about distance travelled and elapsed time since these quantities are reference frame dependent.

Now, if we introduce acceleration into the thought experiment, Alice and Bob are no longer are equivalent.

Imagine that Alice and Bob are both on Earth, Bob instantly accelerates to 0.999 c relative to and away from Earth, travels for 44.7 years according to his wristwatch, and then instantly decelerates to zero speed relative to Earth.

Bob and Alice will both agree that he is now 999 lightyears from Earth, that Bob has aged 44.7 years and Alice has aged 1000 years.

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