Will clocks in space show the same time after slowing down?

[Gib Z]

Imagine someone is traveling in space, very fast past me in a spaceship. We both observe each others clock's running slower than our own, and by the same rate as well, as our relative velocities to each other are equal. So say that spaceship slowed down to zero, and then the clocks were compared. Should they show the same times?

Whoops i spelled dilation wrong in the title :(

 

[ZikZak]

No.

Let's say that you both set your clocks to zero when the ship was right next to you. Then you (being in a nonaccelerating reference frame) simply see the ship slow down to stationary, and since its clocks are slower than yours, they read less than yours do.

If you were the pilot, the situation is more complicated than that, because you accelerated. You must keep track of the clocks that are right next to you on the "highway" you're following. You see, the person on Earth doesn't just use his one clock, he must use a series of clocks along the highway, all at rest with respect to himself, and all synchronized with each other. But in the ship's reference frame, the clocks are NOT all synchronized. Those on the highway in front of him are reading times that are ahead of those behind him, even though they are all ticking at a rate that is too slow.

When he slows down and stops, he agrees that his clock reads less than the clock he stops next to, because that clock was not properly synchronized to zero when he passed you on Earth. When his own clock and the Earth clock read zero, the clock he's comparing to read some large greater than zero time already. So even though it was running slower than his own clock, when he stops, it reads more than his own does. Now that he's stopped with respect to that clock, he must now agree that all the highway clocks are synchronized, and in particular synchronized with the Earth. So everyone agrees that the rocket clock reads less. Again, this is because he accelerated and the Earth did not.

 

[Fredrik]

The first paragraph in ZikZak's post (not counting the "no" as a paragraph) answers your question completely, but you might still want to check out this thread.